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Initiation and development of sand dunes in river channels Venditti, Jeremy George 2003

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INITIATION A N D D E V E L O P M E N T OF S A N D DUNES IN R I V E R C H A N N E L S by Jeremy George Venditti B.Sc. University of Guelph, 1995 M.Sc. University of Southern California, 1997  A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF T H E REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Geography)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A July 2003 © Jeremy George Venditti  UBC  Rare Books and Special Collections - Thesis Authorisation Form  http://www.library.ubc.Cii/spcoll/tlicsaiith.htm  In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e h e a d o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s understood that copying o r p u b l i c a t i o n of t h i s thesis f o r f i n a n c i a l gain s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r , Canada  1 of 1  Columbia  9/22/03 2:! 5 P M  ABSTRACT This study investigated the initiation of bedforms from a flat sand bed and the transition between two-dimensional (2D) and three-dimensional (3D) bedforms in stream channels. Experiments were undertaken in which a narrowly graded, 0.5 mm sand was subjected to a 0.155 m deep, non-varying mean flow ranging from 0.30-0.55 m s" in a 1 m wide flume. 1  The initial flow conditions over the flat beds, prior to bedform development, were examined using laser Doppler anemometry to ensure that the flow agrees with standard models of flow and turbulence over hydraulically rough flat beds. Two types of bedform initiation were observed. The first occurs at lower flow strengths and is characterised by the propagation of defects via flow separation processes and local sediment transport to develop bedform fields. The second type of bedform initiation begins with the imprinting of a cross-hatch pattern on the flat sediment bed under general sediment transport which leads to chevron shaped forms that migrate independently of the initial structure. The chevron shapes are organised by a simple fluid instability that occurs at the sediment transport layer-water interface. Predictions from a Kelvin-Helmholtz instability model are nearly equivalent to the observations of bedform lengths in the experiments. The 2D bedforms initiated by the Kelvin-Helmholtz instability developed into dune features that grew exponentially towards equilibrium dimensions. Dune heights and lengths increased with flow strength while their migration rate decreased. There was no obvious transition from small ripples at the beginning of the runs to dunes when the sandwaves are larger. Morphological estimates of sediment transport associated with the dunes and estimates associated with 'sand sheets,' which are superimposed bedforms on the dunes, were identical, indicating that the material moved over the dunes is controlled by the sheets. Bedform phase diagrams suggest that 2D dunes should be formed under the hydraulic and sedimentary conditions observed in the experiments, but the bedforms became distinctly 3D. Overhead video revealed that, once 2D dunes are established, minor, transient excesses or deficiencies of sand are passed from one crestline to another. The bedform field appears capable of ii  'swallowing' a small number o f such defects but, as the number grows with time, the resulting morphological perturbations produce a transition in bed-state to 3D forms that continue to evolve, but remain pattern-stable. A second set o f experiments was conducted to determine i f the 2D-3D bedform transition could be linked to drag reduction processes. Laboratory measurements o f turbulent fluctuations in clear water over fixed 2D and 3D dune beds with identical lengths and heights were obtained i n a 17 m long, 0.515 m wide flume. The measurements reveal that some 3D bedforms, particularly random arrangements, reduce form drag over dunes. This reduces the applied boundary shear stress and should also reduce or stabilise the sediment transport rate, imparting greater stability to the bed.  in  T A B L E OF CONTENTS Abstract Table of Contents List of Tables List of Figures List of Symbols Dedication Acknowledgements  ii iv vii ix xix xxiii xxiv  Chapter 1: Introduction 1.1 Introduction 1.2 Background 1.2.1 Classical Conception of a Continuum of Bedforms 1.2.2 Bedform Morphology and Terminology 1.3 Theories of the Organisation and Development of Two-Dimensional Bedforms 1.3.1 Initiation of Particle Motion 1.3.2 Perturbation Analysis 1.3.3 Flow over 2D Dunes 1.3.4 Empirical Flow Structure Approach 1.3.5 A n Alternative Theory 1.4 Three-Dimensional Bedform Development 1.5 Problem Statement and Objectives 1.6 Dissertation Layout  1 1 1 3 6 6 7 8 10 12 13 15 16  Chapter 2: Waveforms Developed from a Flat Bed in Medium Sand 2.1 Introduction 2.2 Experimental Procedures 2.2.1 Flow Conditions 2.2.2 Bedload Samples 2.2.3 Echo-Sounders 2.2.4 Video : 2.3 Waveforms 2.4 Dune Morphology and Scaling 2.4.1 Dune Heights, Lengths and Migration Rates 2.4.2 Variability and Measurement Error in Dune Properties 2.4.3 Controls on Dune Growth 2.4.4 Dune Classification and Scaling 2.4.5 Dune Morphology 2.5 Sand Sheet Morphology and Scaling 2.5.1 Sand Sheet Features from Video 2.5.2 Sand Sheet Height, Length and Migration Rates 2.5.3 Variability and Measurement Error in Sand Sheet Properties 2.5.4 Sand Sheet Classification and Scaling 2.5.4 The Origin of Sand sheets 2.6 Sediment Transport Rates 2.6.1 Morphological Estimates of Transport Rates 2.6.2 Dune and Sand sheet Related Transport Rates 2.6.3 Variability and Measurement Error in Transport Rates 2.6.4 Transport Rate Estimate Agreement 2.7 Summary  18 18 20 25 25 26 27 31 31 37 39 42 44 48 50 52 65 66 69 70 71 73 76 79 81  iv  Chapter 3:The Initiation of Bedforms on a Flat Sand Bed 3.1 Introduction 3.2 Experimental Procedure 3.2.1 V i d e o 3.2.2 Echo-sounder Mapping 3.2.3 Laser data  84 85 87  3.3 Initial Flow Structure 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.5.6  88  Velocity Profile Data Analysis Boundary Shear Stress M e a n and Turbulent F l o w Effect o f a B e d Defect Integral scales Summary o f F l o w Conditions  88 90 94 101 101 103  3.4 Bedform Initiation modes  106  3.4.1 Defect initiation 3.4.2 Instantaneous initiation  107 118  3.5 Kelvin-Helmholtz Instability Model 3.5.1 3.5.2 3.5.3 3.5.4  83 83  126  Scenario for K - H M o d e l Testing Depth and Density o f the Active Layer Error Analysis Estimate Agreement  128 131 134 135  3.6 The Development of Bedforms 3.7 Summary  136 138  Chapter 4: The Transition between Two- and Three-Dimensional Bedforms 4.1 Introduction 4.2 Experimental Procedures  140 140  4.2.1 V i d e o 4.2.2 Water Surface and B e d Level Sensors 4.2.3 A r c v i e w Analysis  141 141 142  4.3 A Definition of 3D Bedform Morphology 4.4 Observations of the Transition between 2D and 3D Bedforms 4.4.1 B e d Defect Developed Fields 4.4.2 Instantaneously Developed Fields 4.4.3 Operation o f and Maintenance o f 3D B e d  4.5 Drag Reduction Mechanisms 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5  145 152 152 153 158  161  Calculating Drag Coefficients and Drag Force F l o w Depth and Velocity as a Function o f Time Shear Stress as a Function o f Time Drag Coefficients and Force as a Function o f Time Drag Reduction and Bedforms  163 165 170 174 180  4.6 Summary  190  Chapter 5: Aspects of Turbulent Flow over Two- and Three-Dimensional Dunes 5.1 Introduction 5.2 Experimental Procedure  192 192  5.2.1 Fixed Bedform Design 5.2.2 F l o w Conditions 5.2.3 Measurements and Analysis  193 198 200  5.3 Empirically Derived Structure of Flow over Flat and 2D Dune Beds 5.4 Flow Structure Empiricism and Resolution of the ADV v  205 213  5.5 Mean and Turbulent Flow Fields over 3D Dune Beds 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5  216  Flat bed 2 D Dunes F u l l - W i d t h Lobe ( F W L ) and Saddle ( F W S ) Sinuous Lobe ( S N L ) , Saddle (SNS) and Smooth Saddle (SSS) Regular ( R E G ) and Irregular (IRR) Crests  216 216 218 221 224  5.6 Effect of 3D morphology on Momentum Exchange and Energy Transfers 5.7 Spatially Averaged Flow over 3D Dunes  227 227  5.7.1 5.7.2 5.7.3 5.7.4  Theory Depth Slope Product Measurements Spatially Averaged Velocity Measurements Spatially Averaged Reynolds Stress Measurements  229 231 234 238  5.8 Drag over 3D Dunes 5.9 Summary  242 244  Chapter 6: Conclusions 6.1 The Initiation of Bedforms 6.2 Development of Bedforms 6.3 Transition Between 2D and 3D Dunes 6.4 Drag Reduction Processes  247 248 250 251  References Appendix A Appendix B Appendix C Appendix D Appendix E  253 261 263 265 275 285  vi  LIST O F T A B L E S  2.1  Summary of initial flow parameters.  24  2.2  Bedform dimensions.  34  2.3  Model fitting results  36  2.4  Mean distance from the upstream trough to the slope breaks on the stoss side of dunes. Adjusted values have had extraordinarily large or small values removed from the mean (i.e. \<{B1- TrI) IH > 15 and 1< (B2 - Tr2) IH > 10 were removed from the mean).  49  2.5  Mean sediment transport rates with the standard error as upper and lower boundaries. Subscript i indicates the initial values and e the equilibrium values (mean of estimates/measurements taken after 5000 s). Helley-Smith samples in the dune trough are removed from the averages. No Helley-Smith samples were taken at flow E . The initial Helley-Smith transport values are for a flat bed. For flows A - E , Q is the mean of values in the first 10 min of the experiment. Because the  77  A  si  dunes at flows D and E were generated well upstream of the echo-sounders, no initial dune related transport could be detected. 3.1  Summary of flow parameters.  95  3.2  Values of parameters in Equations 3.10 and 3.11 evaluated from measured profiles.  98  3.3  Integral time and length scales. Values are mean and standard deviation (brackets) of the five 120 s time series extracted from the 600 s time series.  105  3.4  Defect Dimensions.  108  3.5  Initial bedform length scales for instantaneous development runs. L  xv  is measured  122  from one image at the beginning of the run and Z, is measured from a single image taken at time, t . i  3.6  Parameters used in the calculation of the Kelvin-Helmholtz model. Error ranges are ± the standard error for each parameter. Error analysis followed the general rules for the propagation of error when deriving a quantity from multiple measured quantities [Beers, 1957; Parratt, 1961]. Parameters marked with an asterisk (*) are measured quantities.  129  3.7  Depth of the transport layer estimates (in mm). In the Bridge and Dominic [1984] relation q = 0.5 for solitary grains moving over a bed and c = 0 in a bedload  132  transport layer. The subscript cr represents critical values for the entrainment of sediment estimated from the Inman curve in Miller et al. [1977]. 4.1  Timing of critical non-dimensional span values over 2-3D transition.  151  4.2  Model fitted results for crest height and trough height.  167  4.3  Initial values of parameters used to calculate C  4.4  Equilibrium values (after 10000 s) of parameters used to calculate C  4.5  Descriptive statistics for non-dimensional span of 3D bedforms.  182  4.6  Mean 2D and 3D C .  185  D  and F calculated at t = 300 s. D  D  n  vii  and F . D  173 175  4.7  Results o f Spearman Rank Order correlations between A  NDS  test p-values are given for both A  NDS  and C . D  Normality  187  and C , separated by a slash. D  5.1  Summary o f flow parameters. Shear stresses are corrected for side wall effects using the relation supplied by Williams [1970].  197  5.2  Total, skin and form drag coefficient and force. Shear stresses are corrected for side wall effects using the relation supplied by Williams [1970]. Drag coefficients and forces are calculated based on the total shear stress. Values in brackets are for the central segment o f spatially averaged Reynolds stress profiles. Least-squares regression o f inner and outer velocity profiles constructed using profiles taken between Bl and the crest. b is the intercept and b is the slope.  232  Least-squares regression through linear portions o f the Reynolds stress profiles. Values in brackets are or the central segment o f spatially averaged Reynolds stress profiles. b is the intercept and b is the slope.  236  5.3  0  5.4  Q  t  viii  236  l  LIST OF FIGURES 1.1  Typical sequence o f bedforms in alluvial channels. B e d morphologies A - E occur with increasing flow strength [from Simons and Richardson, 1965].  2  1.2  Types o f (a) cross-sectional and (b) planimetric dune morphologies. Panel a is adapted from Bridge [2003] and panel b is adapted from Allen [1968].  5  1.3  Schematic o f the boundary layer structure over a dune [based on McLean,  9  x  R  1990].  is the horizontal distance from the crest to the reattachment point and I B L is an  abbreviation for the internal boundary layer. 2.1  Grain size distributions o f sand used in the experiments. Displayed are the bulk distributions (red) and 109 Helley-Smith bed load samples (black). Bedload size distributions that peak at 0.600 m m (gray) were all from extremely small samples. Most of the catch was likely caused by the introduction o f the sampler to the bed and is therefore regarded as error..  19  2.2  Bedform phase diagram digitised after Southard and Boguchwal [1990]. The  21  mean velocity Uio, flow depth, d , and grain size, D , have been adjusted to lQ  1 0  their 10° C equivalent using the methods in Southard and Boguchwal [1990]. Open circles are the runs discussed here. 2.3  Instrument positions along flume. C Q = Contaq ultra-sonic water level sensors; ES=echo-sounders; 1 and 2 refer to upstream and downstream probes. Flows A - E refer to flow strengths. A plus sign indicates bedforms were developed from a positive defect and a minus sign indicates the bedforms were developed from a negative defect.  23  2.4  Normalised maps of the spatial variation in: (a) the squares that are formed by the grid normalised by the area o f a 0.1 x 0.1 m square (0.01 m ), (b, c) i n 0.10 m line sections that make up the grid normalised by 0.10 m, (d) 0.1 x 0.1 m squares placed at the bed height normalised by 0.01 m and (e) in 0.10 m line sections squares placed at the bed height normalised by 0.1 m. Distance along the flume is x and distance across the flume is y.  28  2.5  Examples o f the waveforms observed in the experiments including (a) long sediment pulses (Flow A - l ) , (b) dunes (Flow A-1-58) and (c) sand sheets (Flow A 2-60). The number in the corners identifies the flow strength (A), followed by the echo-sounder (1 or 2, see Figure 2.3) and bedform number (58 or 60). Circles are the trough heights and triangles are the crest heights. Tr is the dune trough, SFB and SFC are the slip face base and crest, C is the dune crest, BJ and B2 are slope breaks and Tr2 is the upstream dune crest. F l o w over these forms would be right to left.  29  2.6  Dune growth curves for height, H , length, L, and migration rate, R for flows A - D . Line fitting parameters are in Table 2.3. Circles are measurements from echosounder 1 and triangles are measurements from echo-sounder 2 (see Figure 2.3).  32  2.7  Dune growth curves for height, H , length, L, and migration rate, R for Flows E and E . L i n e fitting parameters are in Table 2.3. Circles are measurements from echo-sounder 1 and triangles are measurements from echo-sounder 2 (see figure 2.3).  33  2  2  A  ix  2.8  Bedform height, H , and migration rate, R, plotted as a function o f bedform length, L.  40  2.9  Dune aspect ratio, HIL, plotted as a function o f bedform observed ( H is dune height and L is dune length). The dashed vertical line indicates when the flow strength was changed from flow E(f =17.0 H z ) to flow E ( / =23.5 H z ) .  43  p  A  Circles are measurements from echo-sounder 1 and triangles are measurements from echo-sounder 2 (See figure 2.3). Upper and lower limits for dunes and lower limit for ripples are based on Allen [1968]. 2.10  Normalised dune shape diagrams where bed height, z , is normalised by dune height, H and distance along the dune, x, is normalised by the dune length, L. x IL =0 occurs at the slipface base while z / H =0 occurs at the dune trough. The numbers 1 or 2 indicate the echo-sounder (see Figure 2.3).  45  2.11  Simplified dune crest and stoss types.  46  2.12  Distribution o f crest and stoss types observed in each run.  47  2.13  Example o f the video images grabbed from the overhead video camera. Images taken at t = 42260 s into Run 54 at flow strength B (Image number 546-0461). Lines indicate the features digitised from the image including: dune crest lines (red), sandwave sheet crest lines from the current image (purple), sandwave sheet crest lines from the image taken 10 s prior (blue), sandwave sheet lengths (pink), migration distances over the 10 s image separation (green), distance from the upstream dune crest line to the first sandwave sheet crestline (yellow) and the distance from the first sandwave sheet crestline to the downstream dune crest line (orange).  51  2.14  Sandwave sheet lengths, L ,  53  and migration rates, R ,  min  min  plotted against dune  lengths, L, and migration rates, R as measured from the video. Plots o f L  min  R  min  vs.  and L vs. R are shown for reference purposes. The image averages (open  circles) and averages for each dune (solid triangles) are shown. The circled triangles were removed from the regression between R  min  2.15  and R .  Aggregate sandwave sheet properties measured from the video, including lengths, L , min  migration rates, R ,  54  the distance from the rear bedform crest to the first  min  sandwave crest, x , and the first sandwave crest to the major bedform crest, rear  front '  X  2.16  Sandwave sheet height, H , mjn  plotted against the distance from the downstream  55  SFB normalised by the dune length. 2-17  M e a n sandwave sheet height, Hmm, plotted against run time. Averages are for individual dunes. Equilibrium dune size occurs at ~5000 s during each run (dotted vertical line). The dashed vertical line indicates when the flow strength was changed from flow E (f =17.0 H z ) to flow E (f =23.5 H z ) . p  A  x  p  56"  2.18  Examples o f sandwave sheet sequences including sequences where sandwave sheet height, H „, mi  is increasing (Flow A-1-44), where H  min  (Flow C - l - 2 1 ) and where H „ mi  58  seems to randomly occur  is nearly constant (Flow B-2-22). The number in  the top right corner identifies the flow ( A - E ) , followed by the echo-sounder (1 or 2) and the dune number (e.g. 44). The crest o f the dune is to the left and passes the sensor first. The overall trend is caused by dune passage. The points are successive z  ma  and z  o f sand sheets (i.e. the regular oscillation is exploited by  min  the plotting convention). 2.19  Histograms o f sandwave sheet height, H ,  2.20  Mean sandwave sheet frequency, f ,  2.21  Sandwave sheet length, L ,  min  mill  normalised by dune height, H .  plotted against run time.  60  plotted against the distance from the downstream  min  59  62  SFB normalised by the dune length. 2.22  M e a n sandwave sheet length, Z,»,m, plotted against run time. Averages are for individual dunes. Equilibrium dune size occurs at ~5000 s during each run (dotted vertical line). The dashed vertical line indicates when the flow strength was changed from flow E (f =17.0 H z ) to flow E (f =23.5 H z ) .  63  Histograms o f sandwave sheet length, L ,  64  p  2.23  A  calculated using the relation between  min  R  min  p  and R . The solid bars are for all lengths. The hatched bars are for data that  have been selected to remove lengths where: 1) L  min  H  min  < 3 m m and 3) sheet frequencies, f  min  > 0.50 m, 2) corresponding  > 0.08 H z . The number o f  observations for the unfiltered data ( « , ) and filtered data ( « ) are indicated. 2  2.24  Histograms o f sand sheet aspect ratio (H  is and sheets height and L  min  min  is sand  68  sheet length). The dotted line at 0.1 indicates the typical upper limit o f dunes and the dotted line at 0.05 indicates the usual lower limit o f ripples. 2.25  2.26  Definition diagram for Engel and Lau's [1986] supposition that only that portion of the dune above the reattachment point contributes to the transport rate. The dune shape used i n their formulation (a) and the shape observed in these experiments (b) are displayed.  72  Sediment transport rates, Q , determined from the morphologic characteristics o f  74  s  the dunes (open circles for echo-sounder 1 and open triangles for echo-sounder 2), the morphologic characteristics o f the sandwave sheets (closed squares), and Helley-Smith bedload samples (closed circles). Sand sheet error bars are the standard error o f the estimate. 2.27  Bedform shape factor, / 3 . Circles are measurements from echo-sounder 1 and  75  triangles are measurements from echo-sounder 2 (See figure 2.3). 2.28  Agreement between the dune related transport rate, Q _ , and the sand sheet H  d  80  related transport rate, Q _ . Sand sheet error bars are the standard error o f the s  ss  estimate. 3.1  Calculated boundary shear stress plotted as a function o f velocity, U. Note the different T scale used for T . Error bars are the standard error o f the estimate. S  xi  92  3.2  Measures o f boundary shear stress, T , plotted against the estimate based on the  93  von Karman-Prandtl law o f the wall estimate, T , for both the individual (top) 02  and combined (bottom) profiles. Boundary shear estimates are based on the Reynolds stress profile, x , the Reynolds stress at 5 m m above the bed, T , and B  R  the depth-slope product, r . Error bars are the standard error o f the estimate. s  3.3  Profiles o f mean streamwise velocity, U, streamwise turbulence intensity, I , and u  vertical turbulence intensity, I . w  96  Letters A - E refer to the flow strengths (see Table  5.1). U is normalised by shear velocity, u,, calculated using the von KarmanPrandtl law o f the wall. Height above the bed, z , is normalised by the flow depth, d . Thick lines are Nezu and Nakagawa's [1993] universal turbulence intensity functions (Equations 5.10 and 5.11) plotted using their coefficients. T h i n lines are Nezu and Nakagawa's [1993] functions plotted using coefficients determined from least-squares regressions that are provided in Table 3.2. 3.4  Profiles o f Reynolds shear stress, r , boundary layer correlation coefficient, R , uw  uw  99  and eddy viscosity, e. Letters A - E refer to the flow strengths (see Table 5.1). Height above the bed, z, is normalised by the flow depth, d . Thick lines are functions for the vertical variation in R (Equation 5.13) and e (Equation 5.15) using coefficients provided by Nezu and Nakagawa [1993] and thin lines are the same functions plotted using the coefficients determined from least-squares regressions that are provided in Table 3.2. uw  3.5  Profiles o f mean streamwise velocity, U, over a negative defect at F l o w E . Height above the bed, z , is normalised by the flow depth, d . Open circles are data measured over the defect and lines are profiles measured at the same flow strength without the defect.  102  3.6  Sample autocorrelation functions (act) for 120 s time series drawn from the 600 s velocity measurements. The displayed acf are for the fourth (u4) or fifth (u5) segments o f the 600 s time series at flows A , B , and D .  104  3.7  M a p s o f bed height as a bedform field develops from a positive defect at flow E ( / = 17.0 H z ) . The initial defect dimensions are given in Table 3.3. The maps  110  begins at 8.45 m from the head box. The red dot indicates the location o f the original defect. F l o w is left to right. 3.8  Cross-sections drawn along the centre line o f Figure 3.7. The map begins at 8.45 m from the head box.  111  3.9  Evolution o f bedforms from a negative defect at flow E . Prior to t = 0 s the discharge was being ramped up the desired flow strength. F l o w is left to right.  112  3.10  Bedform L for the first five new bedforms developed from negative and positive defects. The initial defect dimensions are given in Table 5.3. Initial defect length (closed circles); 1 new bedform (upward triangles); 2 new bedform (squares); 3 bedform (diamonds); 4 new bedform (downward triangles).  114  3.11  Evolution o f a positive defect at flow strength E . F l o w is left to right.  115  3.12  Positive and negative defect bedform fields developed at flow strengths D and E . F l o w is left to right.  117  st  rd  n d  th  xii  3.13  Evolution o f a bed through instantaneous bedform initiation process at flow strength A . Prior to t = 0 s the discharge was being ramped up the desired flow strength. F l o w is left to right.  119  3.14  Cross-hatch patterns (a-c), chevrons (d-f), and incipient crestlines (g-i) developed during flows A , B and C . F l o w is left to right.  120  3.15  Histograms o f initial bedform wavelength, L , for each instantaneous initiation  125  t  run. Measurements are from images at t = 60 s (Flow A ) , t = 120 s (Flow B ) , t = 330 s (Flow C ) . Measurements are of all bedforms on the image, from crest to crest, along the streamwise direction only. 3.16  A definition sketch for a Kelvin-Helmholtz instability where fluid 1 has a lower  127  density, p, and a larger velocity, u. Plus and minus signs indicate pressure relative to a mean value at the interface [based on L i u , 1957]. 3-17  Histograms o f surface particle velocity on the bed, U , for each instantaneous  130  initiation run. Measurements were made over the 30 seconds following the onset of widespread sand transport. 4.1  Measurements taken from video images: (a) Crestline length, L ; (b) Linear crest c  length, L  y  ; (c) Bedform area, A  d  143  . Image is Run54-0031 (t - 240 s). F l o w is left  to right. 4.2  Measures o f bedform crestline three-dimensionality.  146  4.3  Figure 4.3: Examples o f non-dimensional span averaged over each image. Clockwise from top left, images are 531-0037 (t = 300 s), 541-0073 (t = 660 s), 571-0106 (f = 1020 s), 541-0223 (t = 2160 s) and 531-0175 (f = 1680). F l o w is left to right.  148  4.4  F i gure 4.4: Non-dimensional span,  150  , during the first hour o f experiments.  A l l observations are plotted in top row o f panels. M i d d l e panels plot crests that exceed a cross-stream extent o f 0.7 m and bottom panels are image means o f data in middle row. 4.5  Transition between instantaneously initiated 2 D and 3 D dunes at flow strength A . F l o w is left to right.  154  4.6  Crest defects developed during R u n 54 at t = 320. A - 1 . 1 . The area in the red oval is defect ' a ' magnified by 200 %. F l o w is left to right.  156  4.7  Progression o f defect ' a ' from Figure 4.6 as it migrates from one crest to the next (t = 320-370 s). The area highlighted red is the parcel o f sand passed from one crest to another. Note the effect o f defect progression on the downstream crest before it cleaves from the upstream crestline. F l o w is left to right.  157  4.8  Schematic o f an advancing crestline lobe as it joins with the downstream crestline and generates bifurcations.  160  4.9  Plan view o f strictly two-dimensional aligned pattern and out-of-phase random pattern o f riblets examined by Sirovich and Karlssen (1997).  162  N D S  xiii  4.10  Change in bed level. Triangles are the heights o f dune crests, z , and circles are  166  c  heights o f dune troughs, z , Tr  measured using the echo-soundings discussed in  Chapter 2. Open symbols are data from echo-sounder 1 and closed symbols are data from echo-sounder 2 (see Figure 2.3). Solid lines are exponential leastsquares regressions through the crests or troughs (coefficients are in Table 4.1). Dashed lines are the change in the mean bed level, z  - (z  bed  c  vertical line in the middle panel at t = 23000 s is where z (t) Tr  + z )/2. Tr  The  was forced to a  constant value. The effects o f this adjustment are shown by the gray lines. 4.11  Water surface level records. B l a c k lines are data from water level sensor 1 and gray lines are data from water level sensor 2 (see Figure 2.3). The datum is the flat water surface before the experiment started.  169  4.12  F l o w depth, d , and mean flow, U, calculated as a function o f time. Vertical lines indicate when bedforms developed at the head box migrated into the measurement section at flows D and E .  171  Measured water surface slope, S, and shear stress calculated from the depth slope  172  4.13  product, z , as a function o f time. Vertical lines indicate when bedforms s  developed at the head box migrated into the measurement section at flows D and E. 4.14  Drag coefficient, C , calculated as a function o f time (solid line). The  176  D  exponential increase in dune height, H , from Figures 2.6 and 2.7 are overlaid for reference (dashed line). Vertical lines indicate when bedforms developed at the head box migrated into the measurement section at flows D and E . 4.15  Drag force, F , calculated as a function o f time (solid line). The exponential  177  D  increase in dune height, H , from Figures 2.6 and 2.7 are overlaid for reference (dashed line). Vertical lines indicate when bedforms developed at the head box migrated into the measurement section at flows D and E . 4.16  Drag coefficient, C , plotted against the areally averaged dune length, L _  4.17  Non-dimensional span, A  D  a  N D S  im  .  179  , time series. Data are image averages o f crests  181  whose cross-stream extent exceeds 0.7 m. 4.18  Drag coefficient, C , and drag force, F , calculated as a function o f time over the D  D  184  first hour o f the experiment. Vertical dashed lines indicate when the transition between two- and three-dimensional bedforms occurred in Figure 4.4. The time when dunes extend across the entire flume is t and t c  max  is when the non-  dimensional span approaches 1.4 for the first time. 4.19  Examples o f when the drag coefficient, C , and the non-dimensional span, D  are out o f phase with one another.  xiv  ,  188  5.1  Dune morphology determined from active transport experiment Distances along the dune and the height o f dune features (slipface base, SFB, slipface crest, SFC, crest, C , stoss slope breaks, BI and B2 and the upstream trough, Tr2) were calculated from the dune dimensions observed at flow strength B (see Chapter 2). Crest data were collated for crest configuration C 3 and stoss data were collated for configuration SI (see Chapter 2 for definitions of C3 and S I ) . These dune dimensions were normalised by the bedform height, H , and length, L, to obtain dimensionless dune morphology. Vertical height above the dune trough, z , and distance along the dune, x, were obtained by multiplying the dimensionless heights and lengths by the desired H (22.5 mm) and L (0.45 m).  194  5.2  Dune morphologies tested. Thin horizontal lines indicate the location o f the dune crest on each plank. Lines down the centre (and along the right lobe o f the sinuous crest) indicate where the profiles were taken.  196  5.3  Example o f bed and water surface profiles over bedforms 5-9 (top) and corresponding flow depths (middle) for the 2 D dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel. See Appendix E for similar diagrams for the other dune configurations. Circles indicate where the probe was mounted in the 0° position. D o w n oriented triangles indicate the probe was in the 45° position. Squares indicate the probe was in the 90° position. U p oriented triangles indicate the probe was in the 0° position but that the lower threshold for data retained after filtering was 40% as opposed to the 70 % retention threshold used for the rest o f the data.  202  5.4  Figure 5.4: M e a n streamwise velocity (z is height above the crest and x is distance along the flume). Note that after profiles were taken between x = 9.630 and 9.855 m over the sinuous lobe, the A D V probe was replaced by another probe which lead to an apparent disruption in the pattern o f U. Only the sinuous lobe measurements are affected. F l o w is left to right.  207  5.5  M e a n vertical velocity ( z is height above the crest and x is distance along the flume). F l o w is left to right.  208  5.6  Streamwise turbulence intensity ( z is height above the crest and x is distance along the flume). F l o w is left to right.  209  5.7  Vertical and streamwise components o f the Reynolds stress ( z is height above the crest and x is distance along the flume). F l o w is left to right.  211  5.8  Correlation between vertical and streamwise velocity components, also known as the boundary layer correlation coefficient ( z is height above the crest and x is distance along the flume). F l o w is left to right.  212  5.9  Turbulent kinetic energy per unit volume ( z is height above the crest and x is distance along the flume). F l o w is left to right.  214  5.10  Select profiles o f mean streamwise velocity, U (solid symbols), and streamwise  217  root-mean- square velocity, U , rm  (open symbols), over the flat bed and the 2 D  dune configuration. In the context o f Figure 5.3 the profiles over the 2 D dunes are the 8 (trough), 18 (stoss) and 2 8 (crest) from the left. Flat bed data was collected during flow strength B (see Chapter 5) with active transport and over the fixed flat bed without transport. The fixed flat bed data are the same in the trough, stoss and crest panels. ,h  th  ,h  xv  5.11  Select profiles o f mean streamwise velocity, U (solid symbols), and streamwise root-mean- square velocity, U  rm  219  (open symbols), over the full-width lobe ( F W L ) ,  full-width saddle ( F W S ) and 2 D dune configurations. In the context o f Figure 5.3 the profiles over the 2 D dunes are the 8 (trough), 18 (stoss) and 2 8 (crest) from the left. Height above the crests, z , is normalised by the dune height, d . th  5.12  th  th  Select profiles o f mean streamwise velocity, U (solid symbols), and streamwise root-mean- square velocity, U  nm  223  (open symbols), over the sinuous lobe ( S N L ) ,  sinuous saddle (SNS) and 2 D dune configurations. In the context o f Figure 5.3 the profiles over the 2 D dunes are the 8 (trough), 18 (stoss) and 2 8 (crest) from the left. Height above the crests, z , is normalised by the dune height, d . th  5.13  th  th  Select profiles o f mean streamwise velocity, U (solid symbols), and streamwise root-mean- square velocity, U (open symbols), over the regular ( R E G ) ,  225  nm  irregular (IRR) and 2 D dune configurations. In the context o f Figure 5.3 the profiles over the 2 D dunes are the 8 (trough), 18 (stoss) and 2 8 (crest) from the left. Height above the crests, z, is normalised by the dune height, d . th  th  lh  5.14  Lateral convergence or divergence o f momentum and turbulent energy over lobe and saddle crestlines.  228  5.15  Spatially averaged streamwise velocity, U, profiles. Circles are averages o f data between B l and the crest (profiles 18-28 in Figure 5.3) and triangles are averages of all data between successive crestlines (profiles 1-28 in Figure 5.3). Regressions are based on the former set o f averages. Inner profile regressions are based on the lower four data points and outer profiles include all points above 0.03 m. Solid symbols denote data points omitted from the regressions in order to improve the fit. Profiles using data from profiles 1-28 in the averages over the sinuous lobe were distorted because o f the probe change and are not plotted.  235  5.16  Spatially averaged Reynolds stress ( T „ ) profiles. Averages include all positive  239  w  T  UW  values observed between two successive crestlines. Circles are averages along  constant heights above a datum that was the crest over the dunes and the bed over the flat bed. Only closed circles were used in the least-squares regressions (thick lines). T w o regressions were calculated for F W S , S N S and SSS profiles. The first uses the closed circles and the other uses the closed circles with centre dots. Al  Time series o f dune crest and trough heights that reveal pulses in sediment observable when multiple consecutive crests (triangles) or troughs (circles) are higher than neighbouring crests. Numbers in top left corner indicates run number (26 - 30) and the echo-sounder (1 or 2) (see Figure 2.3). z is height above the flat bed.  262  CI  Migration rates, R, calculated as the distance the bedform crestline migrated between two video images. See section 4.2.3 for definitions o f the time interval between analysed images. This distance was measured at 10 cm intervals across the flume using the crestlines digitised in Arcview. Black points are all observations, red points are crest averages and blue points are image averages.  266  C2  Areally averaged bedform lengths, L .  267  C3  A l l observations o f non-dimensional span, A .  a  NDS  xvi  268  C4  A l l observations of non-dimensional span, A  NDS  where the cross-stream extent of  269  the crestline exceeded 70 cm. C5  Non-dimensional span, A ^ j , and drag coefficients that represent the total drag,  270  C , plotted as a function of time for Flow A . D  C6  Non-dimensional span, A  N D S  , and drag coefficients that represent the total drag,  271  C , plotted as a function of time for Flow B. D  C7  Non-dimensional span, A  N D S  , and drag coefficients that represent the total drag,  272  C , plotted as a function of time for Flow C. D  C8  Non-dimensional span, A  N D S  , and drag coefficients that represent the total drag,  273  C , plotted as a function of time for Flow D. D  C9  Non-dimensional span, A , NDS  and drag coefficients that represent the total drag,  274  C , plotted as a function of time for Flow E. D  DI  Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the 2D dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  276  D2  Bed and water surface, z , profiles over bedforms 6-9 (top) and corresponding flow depths, d , (middle) for the full-width lobe (FWL) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  277  D3  Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the full-width saddle (FWS) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  278  D4  Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the regular (REG) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  279  D5  Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the irregular (IPvR) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  280  D6  Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the sinuous saddle (SNS) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  281  D7  Bed and water surface, z , profiles over the 8th bedform (top) for the sinuous lobe (SNL) dune configuration. No water surface profile was taken over the sinuous lobe.  282  D8  Bed and water surface, z , profiles over bedforms 6-9 (top) and corresponding flow depths, d , (middle) for the sinuous saddle dune configuration with the smoothed crestline (SSS). Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  283  xvii  D9  B e d and water surface, z , profiles over the flat bed and corresponding flow depths, d , (middle). Profiles were taken at the locations noted in the bottom panel.  284  El  Spatially averaged streamwise velocity, U, profiles. Averages based on all data between successive crestlines (profiles 1-28 in Figure 5.3). Circles are averages calculated at constant heights above the dune crest and triangles are averages calculated at lines equidistant to the boundary, z is height above the crest and z is height above the boundary. Profiles using data from profiles 1-28 in the averages over the sinuous lobe ( S N L ) were distorted because o f the probe change and are not plotted.  286  Spatially averaged streamwise velocity, U, profiles. Averages are based on data between Bl and the dune crest (profiles 18-28 in Figure 5.3). Circles are averages calculated at constant heights above the dune crest and triangles are averages calculated at lines equidistant to the boundary, z is height above the crest and z is height above the boundary.  287  hed  E2  bed  E3  Spatially averaged streamwise velocity, U, profiles. Averages are calculated at 288 constant heights above the dune crest. Circles are averages o f data between Bl and the crest (profiles 18-28 in Figure 5.3) and triangles are averages o f all data between successive crestlines (profiles 1-28 in Figure 5.3). z is height above a datum that was the crest over the dunes and the bed over the flat bed. Profiles using data from profiles 1-28 in the averages over the sinuous lobe ( S N L ) were distorted because o f the probe change and are not plotted.  E4  Spatially averaged streamwise velocity, U, profiles. Averages are calculated at lines equidistant to the boundary. Circles are averages o f data between Bl and the crest (profiles 18-28 in Figure 5.3) and triangles are averages o f all data between successive crestlines (profiles 1-28 in Figure 5.3). z is height above the boundary. Profiles using data from profiles 1-28 in the averages over the sinuous lobe ( S N L ) were distorted because o f the probe change and are not plotted.  289  Spatially averaged Reynolds stress (z ) profiles. Averages include all positive  291  bed  E5  llw  T  IIW  values observed between two successive crestlines. Circles are averages along  constant heights above a datum that was the crest over the dunes and the bed over the flat bed. Lines are averages calculated at lines.  xviii  LIST OF SYMBOLS a , b, c A A  line fitting coefficients bedform cross-sectional area planimetric bedform area  b, b  intercept and slope o f a least-squares regression  b  0  x  Q  C  ~c  drag coefficient, its initial value and its mean equilibrium value  De  C »  drag coefficient based on the form drag component o f shear separated using the Smith and M c L e a n model drag coefficient based on the form drag component o f shear estimated from Reynolds stress profile drag coefficient based on the form drag component o f shear estimated from depth slope product line fitting coefficients  0  C  DR  C  DS  C , ku  CV  coefficient o f variation  d , d  , d  u  A  Ti  flow depth, its initial value, its mean and its equilibrium value  e  A  mean, maximum and minimum flow depth  ' " max » " min  "  d  r  depth o f the transport layer  tl  D, Z )  grain size and its median  50  D  dimensionless grain size parameter  D , D  line fitting coefficients  f  frequency o f sand sheet passage  t  mjn  / ff, ff  ff , ff  ff ,  s  R  F, D  '  02  pump frequency friction factor, an estimate based on the depth slope product, an estimate based the law o f the wall, an estimate based on Reynolds stress profile and an  o  n  estimate based on a measured Reynolds stress near the bed  3  F -, IP De D  F,  ^  r a  § f°  r c e  > i  t s  initial value and its mean equilibrium value  drag force based on the form drag component o f shear separated using the Smith and M c L e a n model drag force based on the form drag component o f shear estimated from Reynolds stress profile drag force based on the form drag component o f shear estimated from depth slope product Froude number  D  F  DR  F  DS  Fr F ,  F ,  tol  Total drag force and its form component and skin component  form  *  F  g I, h u  gravitational acceleration streamwise and vertical turbulence intensity generic bedform height  I  w  H , H, t  fr k k , /t ^ s B  H  dune height, its initial value and its equilibrium value  e  JJ .  s0 2  , k  sand sheet height and its mean value over a dune  sR  ,  time step in autocorrelation equivalent sand roughness, an estimate based on the law o f the wall, an estimate based on Reynolds stress profile and an estimate based on a measured Reynolds stress near the bed xix  K £ I L, L„ L  bedload discharge coefficient characteristic length of a body generic bedform length dune length, its initial value and its equilibrium value  L  areal bedform length  L  distance along bedform crest  e  a  c  L_  Eulerian integral length scale  U  mean initial dune length  LfC-H  Kelvin-Helmholtz instability wavelength  L „ , Lmin  sand sheet length and its mean value over a dune  L;y  oblique striation separation  E  mi  M, n P  M  s  w  mass o f sediment, mass o f water number o f observations porosity o f sand discharge  Q  a , a, . Qse  bedload transport rate, its initial value and its equilibrium value Bedload transport measured using a Helley-Smith sampler  Qs-HS  Qs-cl > Qs-ss  bedload transport estimated from morphology o f dunes and sand sheets Spearman rank order correlation coefficient A D V correlation coefficient  r ADV  r  dune migration rate and its equilibrium value  Re,  Re  R Ri  g  Reynolds number and grain Reynolds number generic bedform migration rate  b  min R(t)  Richardson number sand sheet migration rate boundary layer correlation coefficient autocorrelation function water surface slope, its initial value and its equilibrium value time or experimental time  t 2D  l  3D  time to the development o f 2 D bedforms time to the development o f 3 D bedforms time to when bedforms extend across the channel time o f first bedform observation or time to develop first bedform  lag  time between arrival of bedform S F B at to echo-sounder 1 and 2 time to a local maximum in A NDS-im  pass  '•tot T  time required for a bedform to pass one echo-sounder duration sand sheets are passing one echo-sounder over a dune dimensionless shear stress parameter burst period Eulerian integral time scale  TKE  turbulent kinetic energy  xx  u, Uj, u', U w.,  , u,  u, s  u ,  0 2  ,  iu  mR  t  u, Wj, u  streamwise velocity, its instantaneous value, its fluctuation about the mean and its at-a-point time-averaged mean shear velocity, an estimate based on the depth slope product, an estimate based on the law o f the wall, an estimate based on Reynolds stress profde and an estimate based on a measured Reynolds stress near the bed critical shear velocity for particle entrainment  ~jj JJ  velocity o f upper fluid and velocity of lower fluid in Kelvin-Helmholtz instability mean flow velocity and its equilibrium value  U  velocity at z = 5 m m  2  ;  0  e  5  U  velocity o f a body  U  maximum streamwise velocity  h  U  , U ,  particle velocity and its depth averaged value  [/  ., W  streamwise and vertical at-a-point root-mean-square velocity  U  velocity at z  v , v., V, V  cross-stream velocity, its instantaneous value, its fluctuation about the mean and its at-a-point time-averaged mean volume o f transport layer, volume o f sediment, volume o f water  Vtl,  Vf,  V  w, w , W , w' f  x x  r  vertical velocity, its instantaneous value, its time-averaged mean and its fluctuation about the mean distance along the flume distance from first sand sheet crestline to the downstream dune crestline  front  x .  distance from upstream crestline to the first sand sheet crestline  x  reattachment length  mu  R  x  separation o f water level sensors  wl  y  distance across the flume  y  distance that bedform extended across flume  yM  flume width  h  z, z '  z z, h  , z  '•mm ' d  c  z. Tl  z oB  Z  height about datum, its maximum value and its minimum value max  °  '  change in the mean bed elevation height o f the bedform crest and trough roughness height, an estimate based on the law o f the wall, an estimate based on Reynolds stress profile and an estimate based on a measured Reynolds stress near the bed water surface level and its equilibrium value  z,„p  maximum and minimum water surface level  WS-min  z  P  bedform shape factor  5 £ (p , (j), 7  thickness o f velocity interface eddy viscosity rotation angles  K A  von Karman constant streak spacing  v  X.  streamwise distance lobes o f three-dimensional bedforms extend downstream XXI  X  cross-stream distance between lobes o f three-dimensional bedforms  v  A  measures of bedform three-dimensionality  Aiien-a'  A Allen-b  A ,  non-dimensional span and its image average  NDS  ^  NDS-im  v  kinematic viscosity  6 , 6.  dimensionless shear stress and its critical value for particle entrainment  P  density  CI  p,  p  x  p  density o f upper fluid and density o f lower fluid in Kelvin-Helmholtz instability  2  density o f quartz sand  v  p  density o f the transport layer  tl  p T, T , T  T*  density of water shear stress, an estimate based on the depth slope product, an estimate based on the law o f the wall, an estimate based on Reynolds stress profile and an estimate based on a measured Reynolds stress near the bed. shear stresses separated using Smith and M c L e a n [1977] model  i  critical shear stress for particle entrainment  w  S  T  0 2  ,  T  cr  ^  ;  j&  T,„, , x , sf  initial and equilibrium shear stress based on the depth-slope product i  form  total spatially averaged shear stress, its skin component and its form component  T  time-averaged at a point Reynolds stress  £  average departure of z about its mean  xxii  DEDICATION For Sammi  xxiii  ACKNOWLEDGEMENTS This work would not have been possible without the intellectual and financial support o f M i c h a e l Church. His boundless enthusiasm and encouragement were integral. Sean Bennett o f the National Sedimentation Laboratory, United States Department o f Agriculture ( N S L - U S D A ) in Oxford Mississippi also contributed substantially by facilitating the research. Their combined guidance and patience were invaluable and greatly appreciated. I would also like to thank the other members o f my committee (O. Slaymaker and M . Q u i c k ) for careful and insightful reviews o f the material presented herein. M . Romkens (Director o f N S L - U S D A ) and C . Alonso (Research Leader-Watershed and Channel Processes Unit) kindly provided access to N S L - U S D A facilities. R. M i l l a r , G . Lawrence and M . Quick kindly provided access to the Engineering Hydraulics Laboratory ( E H L ) at U B C . Technical support was provided by J. C o x , J. M i l a m , and D . Wren at N S L - U S D A and by I. L i u and V . Kujala at U B C . The image analysis could not have been accomplished without the help o f several undergraduate assistants, primarily N . Manklow, with some help from C . Christie. Several other tasks including the data collection at E H L - U B C were done with help from N . M a n k l o w , C . Christie, and J. Rempel. A . V i g n a helped prepare many o f the diagrams and D a v i d Printing Services Ltd. provided complimentary printing services. Financial support was provided through a University Graduate Fellowship, a Natural Science and Engineering Research Council o f Canada ( N S E R C ) Post-Graduate Scholarship, and a Research Assistantship provided through a N S E R C operating grant to M . Church. Funds for the experiments were provided by S. Bennett and the Watershed and Channel Processes Unit ( N S L - U S D A experiments) and a N S E R C operating grant to M . Church ( N S L - U S D A and E H L - U B C experiments). The video image frame grabber was purchased using a Ph.D. Research Grant from the American Association o f Geographers - Geomorphology Specialty Group.  xxiv  Chapter 1: Introduction 1.1 Introduction A l l u v i a l river channels are the manifestation o f a suite o f hydraulic and sedimentary processes acting within the channel and watershed. These processes act to modify and adjust the channel system at spatial and temporal scales ranging from those o f individual particle movements to ones o f meander-bend migration and floodplain evolution. Our ability to understand and predict fluvial processes across this range o f scales remains rudimentary because o f the complex nature o f the interaction o f fluids and sediments under the constraint o f varying boundary conditions [McLean et al., 1996]. In sahd-bedded alluvial channels, the bottom boundary consists of a labile bed comprising bedforms o f many different scales and geometries. A deeper knowledge o f how sandy fluvial bedforms interact with the flow field is crucial to understanding sediment transport processes in sandbedded rivers and, ultimately, to the evolution o f alluvial systems. This study seeks to address the processes responsible for the initiation and subsequent growth o f bedforms in alluvial channels. The physics o f these features have fascinated researchers from many disciplines. However, in spite o f nearly a century o f effort, no comprehensive theory capable o f describing the spectrum o f observed shapes and sizes exists [Raudkivi, 1999]. Given this lack o f a complete theory, the research proposed here seeks to determine why a flat sand bed becomes unstable and how this instability leads to bedform development. Before detailing the specific objectives and methodology for this research, a brief background is presented which discusses problems associated with current theory o f bedform development.  1.2 Background 1.2.1 Classical Conception of a Continuum of Bedforms  Observations in rivers and flumes within the past 50 years have revealed that in flows over sand beds a typical sequence o f bedforms evolves as the velocity increases (Figure 1.1) [e.g., Simons et al,  1  Lower Flow Regime  Upper Flow Regime  . WMCT surface  W a t e r sorfjoe  E PUne bed  A Typical ripple p»«ero  F AnUdune standing w*ve*  G AnUdune breaking **ve  0 Washed-out dunes or transition  H Quite cod pool  Figure 1.1: Typical sequence of bedforms in alluvial channels. Bed morphologies A - E occur with increasing flow strength [from Simons and Richardson, 1965].  2  1961; Guy etal, 1966; Simons and Richardson, 1966]. This continuum o f bedforms is conceived as occurring under a graduated set o f flow regimes. A l l features up to and including dunes (Figure 1.1) are generally termed lower flow regime bedforms and typically are characterised by small bedmaterial transport rates. The lower regime flow sequence is plane bed - ripples - dunes, with increasing resistance to flow. The upper flow regime begins with restoration o f a plane bed on which the lower regime bedforms are 'washed away'. Relatively large bed-material discharges and small flow resistance characterise the upper regime. Following plane bed, antidunes begin to form with upstream breaking waves over the dune crest. This type o f bed is typically followed by the formation of a pool-and-chute morphology [Simons and Richardson, 1966].  1.2.2 Bedform Morphology  and  Terminology  A confusing variety o f terms are used to describe lower regime bedforms (e.g., ripples, dunes, sandwaves, and megaripples) [Ashley, 1990]. Bedforms may be only a few centimetres or several metres in height, up to 1000 m in length, and display an unmistakable regularity, although the characteristics o f any one bedform field can only be quantified statistically [Ashley, 1990; Raudkivi, 1999]. These topographic features are often highly asymmetrical with an upstream (stoss) slope averaging 2° to 6° and a lee slope near the angle o f repose (-30°) [McLean, 1990]. A simple empirical relation, proposed by Flemming [1988] on the basis o f several thousand measurements, relates bedform height, H , and length, L, as # = 0.06771° .  1.1  8 1  Extensive data compilations by Allen [1968] and Flemming [1988] demonstrate that there is a break in the continuum o f observed bedforms defining two sub-populations o f bedforms, which have become almost universally known as 'ripples' and 'dunes'. Ripples are conventionally thought to be restricted to sediments in which D < 0.7 mm [Bridge, 2003]. It is widely acknowledged that equilibrium ripple dimensions scale with the grain size, D, or a grain Reynolds number,  3  Re =U,D/v  (u is the shear velocity and v is the kinematic viscosity). In contrast, dunes, the  g  t  focus o f the work presented here, typically scale with the flow depth [Bridge, 2003]. Despite the efforts o f some authors to generate a process based separation o f ripples and dunes [e.g. Bennett and Best, 1996], there is no consensus on how the terms ripple and dune should be applied. Techniques to divide ripples from dunes vary amongst sources and some authors choose to make no distinction at all. Early work by Allen [1968] indicated that bedforms developed in sand have an aspect ratio, HIL, varying between 0.01 and.0.20, although exceptions can occur, and suggested for ripples 0.05 <HIL< HIL  0.20 while for dunes 0.01 <HIL<  0.10. Bridge [2003] concurred, suggesting  is typically <0.06 for dunes and <0.10 for ripples, but noted that HIL is dependent on a  measure o f the shear stress. From an empirical standpoint, ripples appear to have larger aspect ratios in general, although the data o f Guy et al. [1966] indicate that there is significant overlap in HIL values for ripples and dunes. Ripples have aspect ratios between 0.03 and 0.10 at Re = 7, 0.04 - 0.10 at Re = 10 and 0.08 - O . l O a t Re =\5. g  A t t h e s a m e Re values, dunes have H / L«0A0. g  Based on Guy et al. [1966],  ripples have a lower limit o f 0.05 whereas for dunes, HIL can be much less but only slightly greater. Thus, a separation between ripples and dunes based on the aspect ratio may still be valid. Nonetheless, based on a consensus amongst many researchers, Ashley [1990] suggested dunes are all features with L > 0.60 m while ripples are simply smaller bedforms. In cross-section, bedforms tend to take on one o f three morphologies: symmetric, asymmetric, or humpback (Figure 1,2a). Symmetric bedforms are approximately sinusoidal and are thought to lack flow separation [e.g. Kostaschuk and Villard, 1996; Best and Kostaschuk, 2002]. Asymmetric bedforms are roughly wedge-shaped, with the lee side angle at repose, and exhibit strong separation cells. Humpback dunes have a steep lee side slope but sediment accumulates on their stoss slope whereas asymmetric bedforms usually plane out [Saunderson andLockett, 1983].  4  STRAIGHT  FLOW  SINUOUS  SYMMETRIC  CATERNARY (Out of Phase)  ASYMMETRIC  f T ¥ ¥ ¥ ¥ ¥ l  CUSPATE (In Phase Linguoid)  LUNATE  yyyyyyy yyyyyyy /yyyyyyy yyyyyyy  HUMPBACK  Figure 1.2: Types o f (a) cross-sectional and (b) planimetric dune morphologies. Panel a is adapted from Bridge [2003] and panel b is adapted from^Z/erc [1968].  5  The planimetric morphology of fully developed bedforms is divided into two subclasses: twodimensional (2D) and three-dimensional (3D). Two-dimensional ripples and dunes have fairly regular spacing, heights and lengths. Their crestlines tend to be straight or slightly sinuous and are oriented perpendicular to the mean flow lines. In contrast, 3D features have irregular spacing, heights and lengths with highly sinuous or discontinuous crestlines [Ashley, 1990]. A great number of secondary descriptors have been proposed for the planimetric morphology of bedforms, but Allen's [1968] is the most comprehensive (Figure 1.2b). Two-dimensional morphologies are straight or slightly sinuous. Sinuous crests may be aligned, where the lobes and saddles line up, or out of phase, where lobes line up with saddles. Three-dimensional morphologies are highly sinuous, catenary, linguoid, or lunate. Catenary features have crestlines that open downstream (barchanoid saddles) and have continuous crestlines, while lunate features are the same with discontinuous crestlines. Linguoid features have crestlines that form lobes with discontinuous crests. Highly sinuous, catenary and linguoid morphologies can be aligned or out of phase. Allen [1968] suggested the term 'cuspate' for aligned linguoid features and indicated that aligned lunate features are not common.  1.3 Theories of the Organisation and Development of Two-Dimensional Bedforms 1.3.1 Initiation of Particle Motion Most contemporary ideas on dune development envision the generation of bedforms from a plane bed. At sufficiently small flow velocities, sand particles on a planar bed will resist the fluid forces acting on them and will remain in position. As the flow velocity increases, a threshold is reached where the lift and drag forces due to the flow exceed the submerged weight of the grain and other resisting forces on the bed. The critical stage of initiation of transport is difficult to determine and several methods have been proposed. Early work suggested that the initiation of sediment transport could be described in terms of the ratio of critical bed shear stress and the submerged weight of a single layer of bed materials [e.g., Shields, 1936]. More recent work has reasoned that the 6  entrainment o f sediment can be understood as a statistical interaction process between the distribution of instantaneous bed shear stresses caused by near-bed turbulence and the critical bed shear stresses for individual particles. The overlap o f the distributions define the entrainment level [Grass, 1970]. The seminal work of Kline et al. [1967] and Kim et al. [1971] defined random (or quasi-random) streaks, ejections, and sweeps in the boundary layer generated by bursting processes that are ubiquitous in turbulent flows over smooth boundaries. Sedimentologists drew an analogy between these microturbulent features and the macroturbulent features observed in river channels to provide the instantaneous lift and drag forces necessary to locally entrain sediment from the bed [Best, 1992]. The way in which sediment, once mobilised, is organised into unmistakably regular sandwaves has been the subject o f considerable controversy in the literature. Over the last 50 years, most work on bedform development has focused on one o f two theories.  1.3.2 Perturbation  Analysis  The most pervasive theory for the generation o f bedforms has been perturbation analysis (also referred to as linear stability analysis). Initially proposed by Exner in the 1920s [as described by Leliavsky, 1955], and later developed by Anderson [1953], perturbation analysis involves the linearisation o f the equations o f motion o f both fluid and sediment over an infinitesimally small bed perturbation or defect. Subsequently the response o f the perturbation to flow is used to predict suppression or growth of the perturbation [McLean, 1990]. This approach ascribes the initial instability that generates bedforms to a phase difference between the maximum bedload transport rate and the bed topography. Simply stated, the maximum fluid stress occurs upstream o f the perturbation crest and, assuming sediment transport is some function o f the local stress, deposition occurs at the perturbation crest, producing bedform growth. This stress maximum is thought to be the result o f topographic accelerations over the bedform. In all cases, the shift in transport is upstream o f the shear stress maximum. I f the lag did not exist, incipient bedforms would become unstable and erosion would ultimately clear the defect from the bed. 7  Reasons proposed for this lag include: 1) phase differences between surface and bed waves [Kennedy, 1963]; 2) convective acceleration o f the fluid over the stoss slope [Smith, 1970; see also McLean, 1990; Wiberg and Smith, 1985]; 3) increased sediment suspension by turbulence at the stress maximum, increasing the transport length for grains [Engelund, 1970]; and 4) the effect o f gravity on sediment transported up the dune slope enhancing sediment flux on downward sloping boundaries (near the dune crest) and reducing flux on upward sloping boundaries (near the dune slip-face) [Fredsoe, 1974]. Unfortunately, McLean [1990] notes that none o f these assumptions can be justified in light o f the spatial and temporal variations in turbulence over bedforms.  1.3.3 Flow over 2D Dunes Fluid flows over 2 D laboratory bedforms and negative steps have been studied extensively, and several major components are commonly recognised in the flow field (Figure 1.3). Typically, there is an outer region that is not directly influenced by the detailed geometry of the bedforms insofar as the flow here responds largely to the spatially averaged resistance effect communicated to it from below. The outer region is connected to the mobile sediment bottom v i a several intervening zones whose dynamics differ from classic turbulent boundary layer flows over flat beds. In particular, the presence of a dune leads to significant form drag due to asymmetries in the flow field on the upstream (stoss) and downstream (lee) sections o f the dune. A s the flow moves up the gently sloping stoss side o f the dune, the streamlines converge and the fluid is accelerated toward a maximum depth-averaged velocity near the crest. A t the slip-face crest, the streamlines detach from the surface and flow separation occurs in the lee o f the dune. Provided that the lee side slope is sufficiently steep, a distinct separation cell or recirculation 'bubble' occupies the near-bottom region immediately downstream o f the dune crest. Flow in the separation cell can be quite variable, but in a timeaveraged sense, there exists a counter-rotating eddy with upstream velocity along the bottom. A t the downstream margin o f the cell, the separated flow reattaches to the bottom at a downstream distance that is, on average, 3.5 - 5 H [Engel, 1981; Bennett and Best, 1995; Venditti and Bennett, 2000]. The 8  V Mean Water Surface  OUTER FLOW REGION  --  ? "  a*  ' "  WAKE REGION  Shear Layer 1 < *\  ?  c "™ ™" ^  x  -  ^  -^-^ \ .  <  ~i__f-^"  X  r  ?  :  IBL —  \  ~- *- _-"  SEPARATION^? ZONEIVORTEx\  f  v  )  - POINT OF REATTACHMENT  VE=3  Figure 3: Schematic o f the boundary layer structure over a dune [based on McLean, 1990]. is the horizontal distance from the crest to the reattachment point and I B L is an abbreviation for the internal boundary layer [from Venditti and Bauer, in review].  9  reattached flow accelerates up the stoss side o f the dune, and an accompanying internal boundary layer ( I B L ) grows in thickness from the point o f reattachment toward the dune crest. The character o f the I B L is related to skin friction imparted by grain roughness, although disturbance by eddies from the wake region can be frequent (Nelson et ai, 1993). The near-bottom flow region (i.e., the separation cell and I B L ) is linked to the outer flow region through the intervening wake region and shear layer. The dynamics o f these latter two flow features are relatively poorly understood, although they have generated considerable interest because o f ramifications for sediment transport, the stability o f bedforms, and the existence o f quasi-coherent flow structures [McLean, 1990; Nelson et ai, 1993; McLean et al., 1994; Bennett and Best, 1995; Venditti and Bennett, 2000]. For example, the origin o f kolks and boils in large fluvial systems [e.g., Matthes, 1947], which are typically heavily sediment-laden, has been variously ascribed to the boundary layer bursting process [Jackson, 1976; Yalin, 1992], Kelvin-Helmholtz instabilities on the shear layer [Kostaschuk and Church, 1993; Bennett and Best, 1995; Venditti and Bennett, 2000], shear-layer destabilisation coupled with ejection o f slow-moving fluid from the recirculation bubble [Nezu andNakagawa, 1993] and vortex shedding and amalgamation [Mutter and Gyr, 1986]. M o s t authors now acknowledge some interplay amongst the latter three processes [see discussion in Nezu and Nakagawa, 1993]. These periodic motions can cause orders o f magnitude variations in sediment transport rates at-a-point and are thought to be responsible for much o f the vertical mixing in rivers [Lapointe, 1992, 1996; Kostaschuk and Church, 1993]. Acknowledgement o f the complexity o f the turbulent boundary layer structure over bedforms has led several investigators to examine the generation o f bedforms from a flow structure approach.  1.3.4 Empirical Flow Structure Approach This approach to bedform development can be properly attributed to Raudkivi [1963; 1966] who proposed, from a series o f experiments, that the process o f flow separation over a bed defect and the characteristics o f the separated flow control dune generation. Technological advances in recent years 10  have allowed researchers to detail the characteristics o f the flow field depicted in Figure 1.3. This has resulted in a 'rediscovery' of RaudkivPs ideas in the wake o f the failure o f perturbation analysis approaches to bedform development. Over a flat bed, coherent flow structures thought to be analogous to microturbulent sweeps, ubiquitously observed over smooth surfaces, are envisioned to create flow parallel ridges which flare at their downstream ends creating small accumulations o f sediment or 'defects' [Best, 1992], Reattachment o f the flow over the defects is said to enhance the turbulence intensity, erosion, and sediment transport processes in the lee o f the defect. Movement o f grains is maintained by the turbulent agitation from eddies in the wake. A s the wake diffuses, the shear is not sufficient to maintain transport and material settles into a new bed defect. Once an incipient train o f bedforms is established on the boundary, erosion in the lee o f the defects and deposition at the peaks build the bedforms. A s this process continues, the defects are streamlined to generate ripple forms [Raudkivi, 1963; 1966]. The development o f bedforms from the turbulent flow field described above is only hypothetical; the process has not been interpreted mathematically. Measurement programs have not been initiated to provide experimental evidence that the flow processes described above occur over smaller bedforms and bed defects, although these processes can be observed in the patterns o f sediment movement [Best, 1992]. Also complicating this conceptual model is the fact that flow separation is not always observed over developed bedforms. McLean and Smith [1979] and Kostaschuk and Villard [1996] noted that there was no evidence for flow separation over asymmetrical dunes in the Columbia and Fraser Rivers. Without flow separation, the flow and sediment transport processes described above could not operate. However, it should be acknowledged that initial generation and propagation o f defects is a smaller scale phenomenon than observations in the Columbia and Fraser Rivers. A more substantial problem is that there is a large temporal scale transition between the processes of fluid flow and bedform adjustment that has not been addressed. Turbulent flow fields are highly 11  dynamic and sensitive to small changes in the bulk hydraulics. Similarly, the turbulent flow field is highly disorganised and characterised by the generation o f eddy structures by non-periodic and, possibly, non-linear processes [Venditti and Bennett, 2000; Venditti and Bauer, in review]. In  contrast, changes in the highly organised bed structure are typically slow, since significant volumes o f sediment must be moved. It is not clear how the highly structured, slowly evolving bedforms can result from random turbulent structures unless the turbulence becomes phase locked. Best [1992] noted that burst events were concentrated over fixed, flow parallel ridges in experiments. Y e t the development o f the flow parallel ridges on a sand bed is still dependent on the random turbulent structures.  1.3.5 An Alternative Theory  •  .. ;  •  Interestingly, discussions o f bedform development have largely ignored the possibility that at the earliest stage o f development, bedforms may be simply a manifestation o f pre-existing, wave-like variations in the bed shear stress. However, the literature is not devoid o f reference to this possibility. Gilbert [1914] was first to suggest that bedform initiation is related to 'rhythms in the flow o f water and turbulence'; a view widely held by scientists from the former Soviet U n i o n [Allen, 1968]. Based on Helmholtz's principle o f 'least work', Bucher [1919] held that bedforms are formed to afford a surface o f least friction. Helmholtz's concept was later refined to explain the periodic disruption o f a stratified fluid interface, now known as a Kelvin-Helmholtz wave structure. Unfortunately, Bucher's [1919] ideas have been deemed incorrect because the larger-scale bedforms were envisioned to be inphase with surface waves, a phenomenon now understood to be associated with the upper flow regime and antidunes [Allen, 1968]. Later work by Liu [1957] resurrected the idea o f bedforms generated by a pre-existing fluid flow condition by applying theory for an instability generated at the interface o f a density stratified fluid. The instability was envisioned to occur at the interface o f the sediment laden wall region and the viscous sub-layer region o f a turbulent flow resulting in a shear layer characterised by periodic 12  streamwise variations in velocity along the bed. Unfortunately, difficulties in measuring the velocities and densities o f each layer impeded acquisition o f a mathematical solution by Liu [1957]. So, the concept was extended to develop a criterion for the first appearance o f bedforms based on flow variables (streamwise velocity, kinematic viscosity) and sediment characteristics (mean grainsize, fall velocity). The work is largely discounted because the analysis predicts symmetric bedforms [Allen, 1968]. However, there is no reason to assume that an instability that gives rise to wellorganised variations in sediment transport must continue to dominate the flow once a bedwave is established. This line o f thought deserves revisitation. Nonetheless, moving to this type o f explanation still does not address the fundamental question o f how 3 D bedforms, now understood to be equilibrium forms [Baas et al, 1993; Baas 1994; Baas, 1999], are developed.  1.4 Three-Dimensional Bedform Development The mechanisms responsible for the development o f 3 D bedforms have not been previously addressed in the literature. Several authors have suggested that 2 D bedforms form under 2 D flow conditions while 3 D bedforms occur when the flow is 3 D [e.g., Allen, 1968; Costello and Southard, 1981; Ashley, 1990; Southard and Boguchwal, 1990; Southard, 1992]. Explicit assumptions are  made that 2 D bedforms are in equilibrium with the flow over them and that there is some change in the flow structure at higher velocities triggering change in bedforms. Recent observations have suggested that neither assumption is fulfilled. The flow field over 2 D bedforms appears twodimensional when the time-averaged flow statistics are examined. However, the mean flow structure, characterised in Figure 1.3, is the manifestation o f eddy structures that have been observed to have a strong cross-stream component [Venditti and Bennett, 2000; Venditti and Bauer, in review]. A partial explanation for why both bedform morphologies exist when the flow is 3 D may be that 2 D bedforms are not the equilibrium bedfonn for a given set of flow conditions. The assumption that 2 D bedforms are in equilibrium arises from classical conceptions o f bedform stability and the nature o f the experiments that have defined this stability. Classical concepts o f a 13  continuum o f bedforms assume that transitions amongst lower regime plane beds, ripples, dunes, and upper regime plane beds, are sharp. In many cases, transitions occur within an increase o f a few centimetres per second in flow velocity. The transition between 2 D and 3 D bedforms is envisioned to occur in much the same way, despite the fact that the transition is not sharp [Costello and Southard, 1981]. Most research on small-scale bedforms suggests that, under low flow conditions, bedforms exhibiting a 2 D straight-crested morphology are common and, at higher flow velocities, 3 D forms are established exhibiting linguoid shapes [Southard andBoguchwal,  1990; Southard, 1992]. The time  required to form each morphology is relatively short in both circumstances (tens o f minutes to hours) [Southard, 1992]. The experiments of Baas et al. [1993] suggested that this is not strictly true because although 2 D ripples appear to be in equilibrium, given enough time they w i l l develop into 3 D linguoid ripples. The time required for this transition to occur may be several days in some cases. Given the short time periods commonly used in laboratory experiments, it is easy to understand why the transition had not previously been examined in depth. Furthermore, the frequent changes in mean hydraulics o f rivers and tidal environments may prevent bedforms from reaching equilibrium form. The reason why 3 D bedforms are observed to exist in the upper part o f the stability fields may simply be that it takes less time for a turbulent flow field to move the sediment required for the transition. Further support for this hypothesis can be found in recent experimental fluid mechanics work suggesting that 2 D surfaces offer considerably higher resistance to flow than 3 D surfaces. Experiments by Sirovich andKarlsson  [1997] have convincingly shown that hydraulic drag can be  reduced by up to 20 percent by changing patterns o f perturbations from a strictly 2 D alignment to an out-of-phase (three-dimensional) alignment. This effectively modulates the burst sweep cycle, reducing boundary shear stress. It can be hypothesised that the 2-3D transition reduces the applied stress by disrupting the turbulence structure over a bedform. It is well known that sediment transport rates are dependent on the applied shear stress. A mechanism that reduces (or stabilises) the shear stress should also reduce (or stabilise) the sediment 14  transport rate. Thus, passive drag reduction processes such as those proposed by Sirovich and Karlsson [1997], may protect the bed from erosion by reducing the applied shear stress. This can contribute to the stability o f the bed and the channel by reducing susceptibility to degradation. The physical processes involved in the initiation and development o f equilibrium bedforms are not well understood. Whether the ideas o f Liu [1957] and the numerous investigators interested in the turbulence structure over bedforms can be integrated into a coherent explanation o f bedform initiation has not yet been determined. The mechanics associated with the 2-3D bedform transition have not been elucidated. The contribution o f drag reduction processes to sand bed stability has not been addressed. Therefore, experiments designed to readdress the issue o f bedform initiation and to document the transition between 2 D and 3 D forms are timely.  1.5 Problem Statement and Objectives Following the questions raised in the foregoing review, this research project employed a phenomenological approach to examine the physical processes that transform a flat sand bed into 2 D bedforms and then into a 3 D state. The specific objectives o f the research were as follows:  (1) to obtain a detailed set of observations in a controlled laboratory experiment o f the process that leads to the development o f 2D bedforms on an initially flat bed;  (2) to determine whether the development of small-scale 2 D bedforms can be described accurately as a simple near-bed shear layer instability;  (3) to obtain a detailed set o f observations in a controlled laboratory experiment o f the process that leads to the transition between 2 D and 3 D bedforms;  15  (4) to determine what effects the different bedform morphologies (2D and 3D) have on flow resistance and to determine i f drag reduction plays a role in the 2-3D bedform transition.  The bedforms studied in these experiments are conventionally classified as dunes based on the combination o f grain size and flow conditions selected.  1.6 Dissertation Layout In order to achieve the research objectives, two sets o f experiments were conducted. The first examined the development o f 2 D bedforms on a flat sand bed and the later transition to 3 D bedforms at three different flow strengths. These experiments provide the data for Chapters 2, 3 and 4. The second set o f experiments examined turbulence and flow resistance over fixed 2 D and 3 D dunes and provides the data for Chapter 5. The runs conducted in the first set o f experiments were continuous from the initiation o f bedforms through 2 D and then 3 D morphologies. A s such, it is useful to discuss the type o f waveforms that developed, their scaling and their associated role in sediment transport before addressing the specific research objectives. With this in mind, the next chapter examines the growth of dunes during the experiment and their longitudinal morphology. Included is an examination o f the morphology and dynamics o f ' s a n d sheets,' which are small-scale bedforms superimposed on the dunes. Sand sheets are responsible for sediment flux over the dunes. Chapter 3 addresses research objectives 1 and 2 by discussing the initiation o f bedforms on a flat sand bed. A n extensive analysis o f the initial mean and turbulent flow conditions is presented in order to establish that the bedforms were initiated in flow that does not deviate significantly from conventional models o f uniform flow over flat beds; A series o f micro-scale grain movements and bed deformations are documented which lead to incipient dune crestlines. A n attempt is made to link these bed deformations to some observable flow condition. A simple Kelvin-Helmholtz model o f fluid-solid interface instability is applied in order to predict incipient dune lengths. 16  Chapter 4 addresses research objectives 3 and, to a certain extent, 4 by examining the transition between 2 D and 3 D dunes. First, a definition o f 3 D bedform morphology is presented that does not rely on secondary descriptors. A series o f small-scale grain movements and bed deformations leading to the 2-3D dune transition are documented. A n attempt is made to link the transition and the level o f three-dimensionality to drag reduction processes. Final conclusions concerning drag reduction over 2 D and 3 D morphologies are left to the final new research chapter. Chapter 5 examines turbulent flow over fixed bedforms that had the same length and height, but varying crestline shapes. This provides the best test o f how shifts to 3 D morphologies affect resistance to flow. A l l the chapters containing new research are sufficiently focused so that separate measurements and analysis techniques were used in each chapter. Therefore, each new research chapter has a section called Experimental Procedures in which measurement and analysis techniques are presented.  17  Chapter 2: Waveforms Developed from a Flat Bed in Medium Sand 2.1 Introduction The purpose o f this chapter is to discuss the types o f waveforms observed in the experiments, including their morphology, scaling, and role in sediment transport in the channel. T w o types o f waveforms were identified: dunes and smaller bedforms superimposed on the dunes. The latter features cannot be classified as ripples, dunes or bedload sheets, but have characteristics shared with all these features. Thus, the features are termed 'sand sheets' in recognition o f their geometric similarity to bedload sheets that develop in coarse, heterogeneous sediments. These types o f superimposed wavefonns are known to control the hydraulic surface drag and sediment transport rates [Whiting et ai, 1988; Bennett and Bridge, 1995], with important implications for evolution o f the dunes. L o w relief bedforms (sand sheets) that are in equilibrium with the bedforms upon which they are superimposed have not been examined extensively. Therefore, considerable attention is given to accurately describing sand sheet features and discussing their implications for sediment transport.  2.2 Experimental Procedures The experiments were conducted at the National Sedimentation Laboratory, United States Department o f Agriculture, in Oxford Mississippi using a tilting, recirculating flume 15.2 m long, 1 m wide, and 0.30 m deep. The flume recirculates both sediment and water. The flume was filled with - 2 2 5 0 kg (5000 lbs.) o f washed and sieved white quartz sand from the Ottawa Sand Company with a median grain size diameter, D  5Q  = 0.500 mm. Figure 2.1 shows the grain size distribution o f the bulk  sand mixture. The size o f the flume permits scale issues to be ignored in the experiment. The experimental conditions (velocity, flow depth, and grain-size) for these experiments were reasonable for small sand-bedded channels. It is therefore assumed that the observations in this flume have a 1:1 scaling  18  0  200  400  600  Sieve Size ( L i m ) Figure 2.1: Grain size distributions of sand used in the experiments. Displayed are the bulk distributions (red) and 109 Helly-Smith bed load samples (black). Bedload size distributions that peak at 0.600 mm (gray) were all from extremely small samples. Most of the catch was likely caused by the introduction of the sampler to the bed and are therefore regarded as error.  19  with 'real' river channels. However, discretion w i l l need to be exercised before experimental results are applied to natural channels since many o f the bedform scaling relations may exhibit fractal characteristics. Further, particular effects o f using extremely well sorted sand, not common in natural channels, are not well understood.  2.2.1 Flow  Conditions  Bedform development was observed over five separate flow stages with the pump controller set at frequencies, f  p  = 23.5, 22.5, 21.5, 19.0 and 17.0 H z which corresponded to mean discharges, Q , o f  0.0759, 0.0723, 0.0696, 0.0611, and 0.0546 m s". Runs were - 1 2 hours long. Discharge during run 3  1  30 was increased from 0.0546 m s"' to 0.0759 m s" after 12 hours to observe the effect o f a flow 3  3  1  change. Run 30 was - 2 4 hours in length. These five flow stages were selected to provide a test o f similarity amongst the observations over a significant range o f hydraulic conditions. Further, the hydraulic conditions were selected to cover the two-dimensional (2D) dune portion o f conventional stability diagrams in order to determine whether three-dimensional (3D) dunes would develop, given enough time in a wide flume. Southard and Boguchwal [1990] provide the most extensive bedform phase diagrams and plotting methodology in the literature to date. Figure 2.2 plots the bedform phase boundaries and shows that the hydraulic conditions used here all plot in the 2 D dune field. Several runs were conducted at each /  where different processes were monitored. Labels are  assigned to each flow strength to streamline the text and provide a link to other runs discussed in subsequent chapters. Flow strength A refers to all runs conducted at f  p  22.5 H z , C all runs at f  p  = 21.5 H z , D all runs at f  p  = 23.5 H z , B all runs at /  = 19.0 H z and E all runs at f = p  =  17.0 H z . The  label E is used to denote the high flow portions o f runs where the flow strength was increased from A  f = 17.0 H z to / p  = 23.5 H z . Flow strength E is equivalent to flow strength A . However, the initial A  condition for flows A and E are different. There are no bedforms at the start o f flow A , but A  20  2.0  1  1.0  1  1  I  UPPER  0.9  1  1  1  1 1 1 1 ANTIDUNES 1  P L A N E /  1  1  1  1—1—  11  -  3D  0.8 0.7  DUNES  0.6 0.5  \§  2 0  0.4 RIPPLES  LOWER PLANE  0.3  /  d =0.16-0.25 m  / ,  0.2  .  .  , i  0.1  10  i  i i / i i 1.0  10°C-Equivalent D  1 0  i  i  •  i i  (mm)  Figure 2.2: Bedform phase diagram digitised after Southard and Boguchwal [1990]. The mean velocity U\o, flow depth, d , and grain size, D , have been adjusted to their 10° lQ  ]0  C equivalent using the methods in Southard and Boguchwal [1990]. Open circles are the runs discussed here.  21  bedforms are already present when the flow was increased from E to E and this affects the results. A  Therefore, it is useful to use different labels. A t the beginning o f each experimental run, the sediment bed was artificially flattened using a 1 /4 inch piece o f aluminium angle mounted across the flume, at bed level, on a cart that travelled the length o f the flume. Flattening was done in several cm o f water with the flume pump off. The flume was then carefully filled to 0.152 m. Flow in the flume modified this flow depth, d , and established the water surface slope, S, which were monitored using two ultrasonic water level probes along 2.25 - 2.27 m o f the flume. The probes were located at 8.66 m and 10.91-10.93 m from the head box (Figure 2.3). Prior to the onset o f bedforms, the mean flow velocities (U =Q/ y d w  where y  w  is the flume  width) were - 0 . 3 0 , 0.34, 0.38, 0.41, and 0.44 m s" and the Froude numbers (Fr = U/Jgd 1  ) were  0.29, 0.33, 0.37, 0.39, and 0.41. Table 2.1 further summarises the initial hydraulic conditions for the experiment. The bulk shear stress (T  s  = p gdS, w  where p  w  is the fluid density, and g is  gravitational acceleration) for the experiments ranged between 0.8260 and 1.7811 Pa, shear velocity (w, = TJT Ip S  w  ) ranged between 0.0288 and 0.0422 m s'\ and Reynolds numbers (Re = Ud/v  ,  where v is the kinematic viscosity o f the fluid) were in the fully turbulent range at - 1 0 . M a n y o f the 5  flow parameters listed in Table 2.1 changed as bedforms were initiated and grew. Although it w i l l be discussed in great detail in the next chapter, it is important to note that bedforms seemed to be initiated instantaneously over the entire bed surface (from the head box to tail box) at the flow strengths A , B and C . A t the smaller discharges, sediment was transported but bedforms were not initiated. Bed defects (divots or mounds) needed to be manually generated that could propagate downstream to form the ripple field. During Runs 29 (flow D) and 30 (flow E) the defects were lines in the sand that ran across the flume at - 9 m (30 ft) from the head box. The defects crossed the central 50 cm o f the flume width, were 10 mm deep, and - 2 0 m m wide.  22  LO  I  CO  8 < II  xj\  3  «  C  5|  .  u  xi  . „ ac tfl c  -c *•'  S  <U  b  4) 1/5  u  m  J3  > <-> ? 'Si  — *-  60  10  o  S  u  S  £  ^  C  2  • =  O  „,  "S < x  o o  o ro CD .C  £ o§ 13 ^ o 9 ca*  « o *tt < a>  o SS  E o  O Q .  CD O  CS  c ro  M8|A ospiA  "co  Q  s o  c  ^"°  -o  60 c C  LO  cS  CS  - £  T3  oc£  U5 <D CD £3 T "O 3  .a  CL)  %  ca  ^  c«  c^i  . „  CN  £ C  a o  o o  a u  . o o LO  <  GO  O  o  O  o  LU-  LL  +  a o  a  3 o  + LU  oo UJ  o LL  23  S "os.tt =3  .2 >  .S "°  'Z <L> g .SP  (DO 3  a o  c  1 =  E o o  >  ca  £  3  o o  CJ  23 Q. co  as  o  .  g. ts  u  Table 2.1: Summary of initial flow parameters. Flow Parameter Flow A Flow B /p.  Flow C  FlowD  Flow E  23.5  22.5  21.5  19.0  17.0  0.1516  0.1517  0.1533  0.1530*  0.1534*  U , m s"  0.5009  0.4768  0.4538  0.3993  0.3558  ^ max  0.5929  0.5588  0.5370  0.4556  0.4014  Fr Re  0.4107 75936  0.3908 72331  0.3700 69568  0.3259 61093  0.2900 54580  0.0759  0.0723  0.0696  0.0611  0.0546  12  11  7'  5.5  5.5  Hz  d ,m 1  Q, m s"' 3  SxlO"  4  Determinations based on depth slope product M* , m s"  0.0422  0.0405  0.0324  0.0287  0.0288  T ,Pa  1.7811  1.6337  1.0506  0.8239  0.8260  ffs  0.0569  0.0576  0.0409  0.0414  1  s  s  *Based on average o f two runs. Fr = U /(gd) , 05  Re = p dU/u, w  U =(T / )° . 5  t  s Pw  24  ff = 8T / 0  0.0523 p U, 2  w  2.2.2 Bedload Samples Bedload transport measurements were taken throughout the experiments at each flow strength using a miniaturised Helley-Smith sampler with a 5 0 x 5 0 mm mouth. The body o f the sampler was scaled to the mouth. Bedload samples were collected during runs 53 ( A ) , 54 (B) 57 (C), 55 / 56 (D) and 58/59 (E). A t the beginning o f the experiments, samples o f flat bed transport were taken. A s bedforms developed, samples were taken as single sandwaves travelled into the sampler mouth. Later in the experiments, when the bedforms became too large to sample entirely, samples were taken over the crest, stoss slope and trough o f the bedforms. Most samples over the crest and stoss regions consisted o f minor bedforms that travelled into the sampler mouth as they moved over larger primary bedforms. The sampler destroyed most o f the bedforms on which it was situated, and so the next sample that could be taken was over the next bedform. Thus, samples were infrequent in some sections o f the records. Figure 2.1 shows that the grain size distributions o f 109 bedload samples taken during the flume runs are identical to the bulk grain size distributions. A l l sediment sizes in the flume were in motion at all times. Bedload size distributions that peak at 0.600 mm were all extremely small samples. Most o f the catch was likely caused by the introduction o f the sampler to the bed and is therefore regarded as erroneous.  2.2.3 Echo-Sounders Changes in the bed topography were monitored using two acoustic echo-sounders built by the National Center for Physical Acoustics ( N C P A ) at the University o f Mississippi. The minimum operating depth is about 0.04 m and the reported resolution is about 0.05 mm [personal communication with N C P A ] . The sensors were mounted in the centre o f the channel with a streamwise separation o f 0.133 m at ~10.36 and ~10.50 m from the head box (Figure 2.3). Instrument  25  signals were sampled 60 000 times at 4 H z . This provided 3-4 hour time series o f changes in bed topography. Echo-soundings were collected during runs 26 ( A ) , 27 (B) 28 (C), 29 (D) and 30 (E). There was some minor signal contamination in the echo-sounder records that appears to have been caused by poor quality cables connecting the computer hardware and the probe heads. T o overcome this problem a computer algorithm was developed to first remove any singular spikes or drop-outs in the data. Then whenever the signal variance (calculated over five points) exceeds a set limit, the points are replaced. The algorithm repeats in this fashion until all variances are below the limit. Trial and error indicated that a variance limit o f 5 x 10" m removed the signal contamination 7  2  but retained the bed morphology. Nevertheless, some o f the signal was noticeably altered and deemed unsuitable for analysis.  2.2.4 Video In addition to the echo-sounders, bedform shape, size, and migration rates were recorded using a Super-VHS video camera mounted above the flume and centred at -10.30 m from the head box. Super-VHS video captures video at 60 frames/sec with a resolution o f 600 horizontal lines/frame, both much higher than normal video. The video was focused to capture an area o f the bed 0.8 x 0.9 m. The video was illuminated with four 100 W floodlights mounted to the flume sidewalls and oriented to intersect at the camera focal point. The side lighting produced a glare-free image with light shadows that highlighted millimetre-scale changes in the bed structure. The video records were sub-sampled from the tapes using a frame grabber at intervals that ranged between 1 and 10 sec. This produced series o f images that were then subjected to further analysis. A l l lengths and areas measured from these images were made with reference to a grid with 0.1 m squares, installed at the height of the flume walls in the video view. Since the video camera was focused at the bed, features observed on the bed were actually larger than they appeared relative to the overhung grid. A 0.1 m square was placed at several locations at the bed height to determine the  26  magnitude o f the distortion between the height o f the flume walls and the bed in the video view. Maps in Figure 2.4a show there is ± 5.0 % ( ± 5 mm) spatial variation in the squares that are formed by the grid (i.e. not all the squares are exactly 0.1 x 0.1 m). Maps in Figure 2.4b and 2.4c show there is ± 2 . 5 % ( ± 2 . 5 mm) spatial variation in the lines that make up the grid. This variation does not affect any o f the measurements because measured areal features are digitised in a GIS  software  package that registers JC and y positions on the image based on the corners o f the grid. Figure 2.4d shows that feature areas are 28 % larger on the bed than at the height o f the grid and that feature areas in the centre o f the image w i l l appear to be 4 % larger than at the edges o f the images. Lines on the bed were found to be 15% larger as compared to the installed grid although lines in the centre o f the video view area are ~2.5 % larger than on the outer edge (Figure 2.4e). Lines should be the square root o f the distortion in the area (i.e. Vl .28 =1.13 or 13 % ) . The difference between the observed and expected distortion is caused by the spatial variation in the areas and sides o f the squares that make up the grid. The measured distortion is used for corrections. In order to correct for areas measured with reference to the grid, areas measured from the images are multiplied by 1.28 and all lengths are multiplied by 1.15. The radial distortion in lengths and areas from the centre o f the image is accepted as error in the data set. Only a small portion o f the video data (Run 54 at /  =22.5 H z ) is presented in this chapter. Chapters 3 and 4 contain a more  complete discussion o f this data set.  2.3 Waveforms Examination o f the echo-sounding records revealed three basic waveforms present in the flume experiments: long sediment pulses (Figure 2.5a), dunes (Figure 2.5b) and sand sheets (Figure 2.5c). The long sediment pulses were observed to have a period o f 2-3 hours (although shorter periods are observed) and vertical relief o f 0.08-0.10 m. These waveforms move through the system slowly and are likely bar forms. They are more prevalent in the runs with a larger flow (flows A and B ) and are 27  o  o  o  o  o  o  o  5  C  J  >  o  >  a  5  0  >  F l o w A-1  -100 10  20  30  40  50  T i m e x 10" (s)' . 3  20  1  :)  1  1  c  SFC  B  Flow A-1-58 : B1  -20 -40  t -60  :  t  SFB & Tr  40.25  t  B2  40.50  40.75  41.00  Tr2  :  41.25  41.50  Time x 10 (s) 40 c)  1 3 J H A.  9 11 13 15  6 5  A  kf\rJ\Ky*  i  R  A  17 ^  22  A. 1  27 2*8 30  311  -40 -60 43.0  _i  43.2  43.4  43.6  43.8  44.0  Flow A-2-60 i i i i i i  44.2  44.4  Time x10 (s)  Figure 2.5: Examples o f the waveforms observed in the experiments including (a) long sediment pulses (Flow A-1), (b) dunes (Flow A-1-58) and (c) sand sheets (Flow A-2-60). The number in the corners identify the flow strength (A), followed by the echo-sounder (1 or 2, see Figure 2.3) and bedform number (58 or 60). Circles are the trough heights and triangles are the crest heights. Tr is the dune trough, SFB and SFC are the slip face base and crest, C is the dune crest, Bl and B2 are slope breaks and Tr2 is the upstream dune crest. F l o w over these forms would be right to left.  29  nearly absent in lower flow runs (flows C , D and E), appearing only after several hours. The forms reappeared when the flow strength was increased from E to E . Unfortunately, there are only a few A  observations o f these waveforms per run preventing a more detailed discussion o f their characteristics. Records revealing these waveforms can be found Appendix A . Data are more readily available on the characteristics o f the meso-scale dunes and the micro-scale sand sheets superimposed upon them. It is worth noting that spectral analysis was not employed in this investigation to identify bedform H and L [see Kennedy and Willis, 1977 for methodology). Most spectral techniques are based on sine waves, assume time series stationarity and require the pattern be stable for many cycles. The bedforms observed are not sinuous and dune forms in this investigation grow from a flat bed by gradually increasing both H and L. Bedform dimensions reach a statistical stability only later in the experimental runs. Even after equilibrium is attained, the time series are still characterised by varying H and L, which cause statistically insignificant and erroneous spectral peaks. In order to examine the dune bedforms, the bed height time series were plotted to highlight each bedform as in Figure 2.5b. The time series were decomposed by identifying several bedform features and acquiring a time, t, and bed height, z , for each point. The features included: (1) bedform trough, Tr, the point with the minimum height, z  , in the lee o f the bedform; (2) slipface base, SFB,  mjn  and slipface crest, SFC, which are the lowest and highest points bounding the bedform slipface; (3) bedform crest, C, the local maximum height, z  , on the bedform; (4) a first stoss surface slope  m(lx  break, BI, that commonly defined the downstream extent o f the upstream dune trough, and (5) a second stoss surface slope break, B2, that commonly occurred between the first slope break and the z  min  o f the upstream bedform. Points were often combined on bedforms (i.e. SFB often co-occurred  with Tr and SFC often co-occurred with Q . Stoss surface slope breaks were observed on many bedforms but were absent on others.  30  The sand sheets migrated over the stoss surface o f the primary dune forms at rates much larger than the dune migration rate. The bed height time series were plotted to highlight the dune stoss slope and hence groups o f sand sheets as in Figure 2.5c. The time series were then decomposed to identify t and z o f the sand sheet trough and crest. These smaller bedforms were far too small and numerous to identify other points (SFB, SFC, Bl and B2), although many larger sheets display these features. The morphologic characteristics and sediment transport rate associated with these forms and the dune forms are discussed in more detail below.  2.4 Dune Morphology and Scaling 2.4.1 Dune Heights, Lengths and Migration Rates The development o f dune L and H at flow strengths A - D is shown in Figure 2.6 and flow strength E is displayed in Figure 2.7. Dune height was calculated as H = z  ma  - z  mln  for each echo-  sounder time series, providing two separate H values. Dune wavelength was calculated by: (1) determining a time lag, t,  , between the arrival o f the dune SFB at echo-sounders 1 and 2 (see Figure  2.2), (2) determining the time it took for a bedform to pass one o f the echo-sounders, ? computing a dune migration rate .ft = 0.133 m/V Since each echo-sounder provided a r  to  , (3)  , and (4) computing the wavelength L-R-t  .  value, a common R value produced two estimates o f L for  each bedform. Initial dune heights observed in the time series, H ,  varied between 1.0 and 4.6 mm depending  i  on the run. Initial dune length observed in the time series, L , varied between 0.034 and 0.105 m i  (Table 2.2). There appears to be no relation between H  i  and run U . However, L does seem to i  increase with U with the exception o f flow C . There are greater discrepancies between H measured by echo-sounders 1 and 2 at the lower flow velocities (Table 2.2). Presumably this is because R  31  1  1  ]••••! • •  o m  1  ICQ*  o  -  <  1  o CO  0  «0  •  (  •6 1  PJ?  o CN  4  <  CO  .-  E Q  •  I**  6  X CD  I  <  -  o  1  1  lili  i"""  o:  \  CO  b  :  t  O CO  t>  0  «o  A«  -  o CN  \\ ^<fi  -  X CD  Wfe4  "  1  oo :  •  —1  5:  :  *  c  C3  CO  f „  CO /—*  o o  :  <  o CO  •  <9  o  £  0  LL :  o  a»  O  —  o: oF  il .SP o £ * "O £2  < O ^WOfWj  A.  c3  O  ou_:  oo  "3) c  CC  LD  &:  <  b :  o  • •  cc  o c  " 1  CO  o CN  -  co  SP SI  CO  o X CD  E  C  CO ~~  ^  " Ol  £  - A  \  0 1  o<  CO  I  1  I  .  ,  1  ,  ,  I  i  |  i  i  i  i  |  O !_  j= o  O ° o o  o o<  -  0  o  0  o  •  :  ;  o  %:  v o b (LULU) H  CO  CN (  t  (LU)I  32  s LULU)  i-  y  o CO o CN O  O O  co  ^  CN  >-  s £^ £•§ § C  "c/T  ° :  $cr  o  •  CO CO  co CN " O  5  o  4>  (2  o rf :  o  CO  l-  o LO  i  h  <« §j  co • —  r••rTi  CO  _C0  0 0  O  •  C CN C  CO  o X 0  E  i-  CO  CO  3  "S £2 S  ^  S3 6  ° CN £ 3 6 0  1  Q . CO 0 0  .S  3  CO  1  1 1 1 1  1  1 1 1 1  1 ,  100 o =1=  AA  °A  A  A °  A  10 : 8  0  Flow E (0-14.5 Hr)-  10  20  30  40  Time x 10 (s) 3  50  Flow E (14.5-25 Hr)  Flow E & E (0-25 Hr)  A  60  70  80  Time x 10 (s)  90  100 0  i •  •i •  20  40  1  • • i  1  1  1  60  1  A  i  80  100  T i m e x 10 (s) 3  Figure 2.7: Dune growth curves for height, H , length, L, and migration rate, R for Flows E and E . L i n e fitting parameters are in Table 2.3. Circles are measurements from echo-sounder 1 and triangles are measurements from echo-sounder 2 (see figure 2.3). A  33  Table 2.2: Bedform dimensions. Flow  Run  A  26  B  27  C  28  D  29  E  30a  H mm t  A=2.19 B=2.74 A=1.69 B=1.70 A=1.15 B=2.66 A=3.66 B=1.03 A=4.63 B=3.91 n/a  A  H  lO''  cv  Re  1.172  0.929  0.0650  0.449  0.323  0.860  0.567  0.0371  0.501  35.88  0.319  0.954  0.502  0.0334  0.311  963.41  21.52  0.397  0.383  0.678  0.0172  0.349  3149.42  19.67  0.340  0.300  0.593  0.0097  0.485  n/a  n/a  n/a  n/a  n/a  He  m  s  mm  0.0946J  Instant  47.70  0.277  0.0842|  Instant  41.61  0.1046  Instant  0.0646 0.0336  30b n/a n/a n/a E "f Based on all bedforms observed during first 3 min. A  cv  cv  e  34  m  u  m s"  1  Re  decreases with flow velocity (Figures 2.6 and 2.7) and there is more time for the bedform to evolve between locations 1 and 2. A t flow strengths D and E , H and L are somewhat erroneous statistics as the initial bedforms t  t  were developed at 9.14 m from the head box and migrated 1.22 m downstream to where the probes were located over some time period, t . A t flow D t = 963 s and at flow E r,.= 3149 s. Therefore, t  t  Hj and L are related to this time lag between development and observation. In contrast, at flow i  strengths A and B , the bedforms developed instantaneously and thus H and L should be a fair i  t  representation o f initial bedform values. F l o w C also developed bedforms over the entire bed shortly after the desired flow strength was reached, but the first bedform observation occurred several minutes into the run, which presumably bolstered L . t  The issue o f initial bedform properties is  discussed in greater detail in Chapter 3. Once established, the dune bedforms grow in H and L while R decreases toward equilibrium values. In these experiments, the dunes undergo continuous growth in H and L rather than growth caused by capturing smaller bedforms to form larger bedforms as has been suggested by some authors  [e.g. Costello and Southard, 1971; Leeder, 1980; Raudkivi and Witte, 1990; Coleman and Melville, 1992]. The growth o f the dune bedforms is approximately exponential and can be expressed by the following  H =  a (\-e' "'')  2.1  b  H  2.2 where a , H  a, b L  H  and b are coefficients derived from least-squares regression (Figure 2.6 and L  Figure 2.7). Bedform migration rates decrease exponentially with time and can be expressed using  2.3 where a , b , and c R  R  R  are coefficients derived from least-squares regression (Figure 2.6 and Figure  2.7). Values for a , b, and c are given in Table 2.3. A l l model fits are significant at the 95 % 35  0 0  0  q  =i.  O  00 O  q  q  V  V  0  CN  00  O  O  o  Tf Tf  m  m 00 os — SO 1^1 — Tf Tf m  so  os  —  OS  Tf  CN  10 m  so r-~ m so as vn un C N C N  5S  0  as co  Ti  O  CN — m —  CN T f CN 00 Tf m — cs  as  8  O  00 r-  O CN c i ;  n  0  O  V  3  0  0  0  0  § § § g S q 0  V  0S  0  q  V  0  <=> §  5  V  o  o  —  rn sq  1—  os sq  — 00 CN —>  O  ©  ©  O  oo os r— —  00 os so so Ti 00 O Os m —.' ©' CN  00  C--  O  —  O  m so  10  —  —  Tf Tf  O  CN rn  —  o CN T f —; —: —; o CN  O O O  =1,  ©  V  O O O O  V  O O O  O O O  ©  V  CN T f OO 00 00  Tf  *  o  E  n 3  o o  r~  m  Tf —  —  —  o  CN  OS CN  O  d ©  Ti Ti Tf  .  — '  CN —'• CN —• CNi CN  a  a  o  o o  ©  V  Tf  •^2  o  —  SO SO T f m m Ti O Os cn V")  Tf  Tf  O CN CN CN  CO  so c— 00 Os o CN CN CN CN m  x> o  m  en CN JD  IS  X)  UH  <  D3  U  Q  W tq  E-  36  confidence interval, but more bedforms are observed in the runs with larger flow strengths and consequently the model fit is better (see r  2  values). Equations 2.1 - 2.3 could not be successfully  applied to flow E . When the bed started developing from a flat bed (flow A ) , equilibrium was A  established after only 1.5 - 2 hours. However, 12 hours was not sufficient time for the bed to reach equilibrium when the flow started from a bed that already had bedforms (flow E ) . A  Regardless o f U , H , e  L  e  and R (subscript e indicates equilibrium values) are reached after e  ~1.5 hours, but all are strongly dependent on U , increasing with flow strength. Coefficients o f variation, CV, defined as the ratio of the standard deviation to the mean, are large for H , e  L  e  and  2.4.2 Variability and Measurement Error in Dune Properties There are two sources o f variation in the dune data set. The first is natural variation o f the phenomenon. The second is error associated with the measurements. Neither R nor L were measured quantities but depended on lag and passage times and their calculation from these times imply certain assumptions, primarily that the dune movement is consistent and dune morphology does not vary while being measured. Instrument related measurement error, which nonnally is an additional source o f variance, is minimal because the sampling frequency was large and the vertical instrument resolution ( ± 0 . 0 5 mm) is small compared with the phenomena being observed. However, the practical vertical resolution o f the measurements system is equivalent to D. Nearly all the variation in H is natural and CV  He  ~ 0.28 - 0.40 with a mean for all flow strengths  during the equilibrium phase o f - 0 . 3 3 (Table 2.2). Error associated with the vertical resolution o f the measurement system accounts for 3.7 - 7.5 % (DI oH)  o f the total variation. Villard and Church  [2003] observed similar variation during dune field surveys in the Fraser River where most  CV  H  values were between 0.3 and 0.5. A large degree o f this variation derives from the 3 D nature o f the bed (discussed in greater detail in Chapter 4). 37 •  Villard and Church [2003] observed CV values between 0.4 and 0.7, but some larger values R  occur when the flow conditions had recently changed. Most  CV values are within this range, Re  varying between 0.31 and 0.50 with a mean for all flows o f - 0 . 4 2 (Table 4.2). It is difficult to determine how much of the variation in R is natural and how much is related to the way the measurements were taken. Some of the variability in Potential inaccuracies in t  t  R is related to its calculation from t  lag  .  could derive from the bedform stalling in its downstream progression,  the crestline locally accelerating or decelerating between the echo-sounders, or changes in the bedform morphology before being recognised at the downstream echo-sounder. Variability in the crest movement along its length was most pronounced later in the experimental runs when the bedforms were large. Examination o f the time series suggests that changes in the local migration rate were rare over the 0.133 m echo-sounder separation. SFB was selected to define the arrival o f the dune at a sensor because this location experienced the least change between the two echo-sounders. Quantification o f the error resulting from potential inaccuracies in t  k  cannot reasonably be  accomplished. The dune length also varied naturally but was subject to the greatest potential error as variability in R is combined with error associated with the calculation o f L. A s such CV  U  CV  He  and  is greater than  CV , ranging between 0.42 and 0.68 with a mean for all flows of -0.55 (Table 4.2). This Re  measure o f variability in  L is not strictly comparable to values obtained by Villard and Church  [2003], as they measured dune length directly. Aside from the error inherited from R, variance is derived from the determination o f t  . Bedforms stalled and were subject to acceleration and  deceleration frequently. This is somewhat mitigated by the fact that stalls are likely balanced by greater than average progression. However, the bed was tremendously dynamic as bedforms continuously modified their morphology. Quantification o f the error compounded from the error in determining R and the potential inaccuracies in t  cannot reasonably be accomplished.  38  2.4.3 Controls on Dune Growth Figure 2.8 plots all observations o f H and L at all flows and demonstrates that there is a nonlinear relation between these bedform dimensions. Observed H varied between 1.0 and 96.1 mm and L varied between 0.03 and ~2 m, although there are a few observations o f L > 2 m. A t most flow strengths, bedforms were observed across the entire range o f H and L values displayed. A t flow strength E, observations were restricted to the lower portion o f the range while at flow E , A  observations were restricted to the upper portion. This is not surprising because bedforms at flow E were generally the smallest equilibrium forms observed and there was no period of rapid bedform growth from a flat bed at flow E . A  For H < 40 mm and L< 0.8 m, H and L are coupled as the relation appears linear. Most o f these observations are o f bedforms developing from the flat bed towards the equilibrium. When the bedforms dimensions exceed H ~ 40 mm and L ~ 0.8 m, the linkage between dune H and L is not as well-defined. This suggests that when the dunes are in the equilibrium phase, H and L vary independently within some boundaries. It is not clear what controls equilibrium height and length o f bedforms. There is no consensus on the reasons for bedform growth and what the ultimate equilibrium size of a bedform should be for a given flow, although considerable progress has been made through empirical studies conducted over the last 50 years and through the construction o f bedform phase diagrams. A s mentioned in Chapter 1, perturbation analysis has been applied extensively to the problem o f bedform growth. Unfortunately, the predictions produced by this approach are largely controlled by the conditions imposed upon the equations and thus there is not much physical insight into bedform growth. Recent work by Raudkivi and Witte [1990] and Coleman and Melville [1992] has suggested that bedforms actually grow in length and height by the coalescence o f smaller bedforms, because smaller bedforms have larger migration rates, R, and can overtake the larger features on a bed. A n artificial 39  T  1  1  1I  I I |  1  1  1  1—I  1  I I I |  1  1  1  1I  I  I  0.1  0.01  For L<0.8 m a n d H<0.4 m H=0.043 L , r =0.54, p<0.001  0.001  08 9  _i  < i i 111  i  i  i i iii'i  2  i  i i i 111  i  0.1  10 L(m)  10  -i—i—i—i  i 111  1—i—i—i—i  i 111  1—i—i—i—i  i 11  0<D CD O O  E  0.1  0.01  _i  i i i i i 11  i  i  i i iii'i  0.1  i  i  i i  ' i i '  10 L(m)  Figure 2.8: Bedform height, H , and migration rate, R, plotted as a function of bedform length, L. 40  limit to dune height is imposed to produce equilibrium bedforms. This idea has been referred to as a 'bedform unification model'. Ditchfield and Best [1990] argued against this idea, indicating there is no relation between bedform size and migration rate, R . They also suggested that bedforms may both grow or attenuate without interaction with other bedforms, or they may coalesce as they migrate. Figure 2.8 also plots R against L for all flow stages. For flows A , B and C , the there is a general decrease in R with L while at flows D and E , R with L values are clustered in one region o f the plot. A t flow E there appears to be a positive relation. Thus, the relation is not as simple as A  suggested in unification models. A further problem with applying this theory to the observations here is that, although bedform coalescence was observed during the experiments, it was limited to the sand sheets combining with the dune crestline. This was an equilibrium process that maintained transport over the dune and did not cause dune growth. Widespread bedform coalescence did not occur within the dune population; rather, crest realignment by the growth o f scour induced crest lobes dominated (this observation is described and discussed in greater detail in Chapter 4). M a n y others have ignored the actual dune growth process and sought out an explanation for the limiting height or length o f bedforms. Yalin [1992] has suggested that dune height is generally 0.2 0.25 d and most o f the observations here loosely confirm this scaling. However, H  e  varied over a  wider range (0.13 - 0.32 d ) with only flow strengths B and C being within Yalin's [1992] range. Yalin [1992] attempted to explain the limiting height o f dunes by examining the limiting length and assuming the bedform w i l l maintain a near constant aspect ratio. He attempted to link dune length to a burst period, T , which is the time between large depth-scale turbulent events. Assuming U T /d B  B  is a constant =5, U T ~ 5 d which is similar to observed dune lengths (i.e. 5 - 6 d ). Unfortunately, B  this scaling breaks down when the bedforms become large [see Raudkivi and Witte, 1990]. Others have argued that dune size is limited by a balance between shear stress, sediment transport rate, dune migration rate and form drag. The limiting condition under this balance is flow depth over the crest when no further deposition can occur in the crest region as it begins to act as an upper stage 41  plane when in equilibrium [Bennett and Best, 1996]. This explanation does not address the problem o f why the dunes grow in the first place. Thus, the limiting condition for bedform growth is open to debate.  2.4.4 Dune Classification  and  Scaling  It is necessary to justify the use o f the term dune to describe the large scale features in the experiments; whether these bedforms are classified as ripples or dunes is dependent upon the classification scheme used. U s i n g Ashley's [1990] classification, bedforms where L exceeds 0.6 m are dunes. This artificially divides a data set that does not have individual groupings o f L or H into ripples and dunes. Using a data set like that o f Guy et al. [1966] does not provide a satisfactory classification either. Using u* values determined from velocity profiles rather than values from the depth slope product, the grain Reynolds numbers, Re  g  = u*D/v  for the experiments are 15 ( A ) , 13 (B), 11 (C),  8.5 (D) and 8 (E). Estimates o f u* values determined from velocity profiles are -0.45-0.75 u, . The s  data o f Guy et al. [1966] indicate that ripples have aspect ratios, H IL between 0.08 and 0.10 at Re = 15, 0 . 0 4 - 0 . 1 0 at Re = 10 and 0.03 - 0.10 at Re = 1. g  g  g  Figure 2.9 plots HIL for all the runs as a function o f the bedform number (1 was the first observed). Interestingly, there is no systematic change in HIL is considerable scatter in the data but the average HILflows D and E . There was no change in HIL  with time (bedform number). There  0.05 for flows A , B and C and -0.06 for  when the flow was increased. There is clear overlap  between the ranges from the data of Guy et al. [1966] and the data in Figure 2.8. However, there is no justification for choosing one term over another. In light o f this confusion, the bedforms are classified based on their aspect ratio using the ranges provided by Allen [1968]. Allen [1968] indicated that HIL  varies between 0.20 and 0.01 for  bedforms developed in sand, although exceptions can occur. For ripples 0.05 <H I L< 0.20 while for 42  I  1  1  i •  1  1  o  LO  -a c  pie Q_  CD  C  of dunes  co  unp  t/)  o  ea  o  ppe imit  E  lim  M—  4—  o  CO  CO T3  CD CQ  c  3  X)  low  o  *  XJ «  X)  so  co  c  o  _3  60  5 -c £ cs X")  wer  CD  in  .SP  E  i i  o  >  CO  o o  <  •o o c  6> <0|  0  o  o  CO  O<I  CXI  1 °  o  CM  <  • LU  *  O  <  _ o _  CQ  °I <  A  - — % J -  *  c?  8  << <1j £ 1 A °7  o  0  o  <  LL<1  • o  O  <  < ::0< o x^t  X)  +J •  0  o  CO O CM  fc o H—  XI CD QQ  o  c CO  I* ^ CS  co  CS  <  <  (/}  _CD  ^  Id  o  CL  " cd  <  o< oa  L-  1  oo  TH  o XJ  O  CQ  CD  o o  CD  o o  o  o o  a. CL CO  CN CO •—  3  tsO  IS CO CO  CO  C  K  3  o ~» CO ^ o a  J=,  cd M-l  8-8  co o J2  £  cS  cE  -°  o  2 5i  CS  3  Q so X <3) § O  co  E -22 b -rj-  CO  XJ  4  c  CN  cd  o  CN o cn  CO  CO  <l  00  w  o  o \  43  cd  CO X !  E i  O  CO L-  *%  CM  0  <D  |0  b  £  CO  0  b  CO co  O  •2  o  CM  O  CO  -5 o r-  ^  LO OJ  b  _ccd o  ts •=  CO  "3"  co  CO  e CO  CO  XJ  ° 0 <  O  <  c X!  XJ  J  0  LL  o CO  °<  Q  XI  CO  $  CO  ?  (/I  LO  0 <  <X  o o  1 < |  O o  O < 1  <  _ E _  °  1 1 •; y  c  03 co CO  O S3 3 cd - C  C » co  • in o o  it Xi  cd to  LO  CO  i  CO  CO  cd co co  a.  £ a CO CO  o  ,  CL  cd —  dunes 0.01 <H I L< 0.10. Based on this classification, most of the bedforms are clearly dunes and, since there is no clear grouping o f dune shapes, it is reasonable to conclude that they are all dunes. Reinforcing this classification, the bedforms appear to scale with depth, not grain size, and the bedform phase diagram from Boguchwal and Southard [1990] (Figure 2.2) indicates that the flow strength is too great to develop ripples. That there is no observable change in the bedform steepness with time is significant because it suggests that the bedforms that first develop are not ripples that transform into dunes when they become large. This is emphasised by examining Figure 2.10 which plots the normalised bedform shape where the distance along the bedform, x, is normalised by each dune L and z is normalised by H . The bedforms all have approximately the same shape, regardless o f bedform size, although there is substantial variation on the stoss slope.  2.4.5 Dune Morphology Examination o f the time series revealed different types o f dune crest and different stoss slope types. Bedforms are represented in the literature as having a variety o f streamwise shapes including the typical asymmetric shape that is particularly common in narrow flumes. Even sinusoidal shapes have been used for modelling purposes [e.g. Kennedy, 1963]. However, naturally occurring bedforms tend to have a wider variety o f shapes. Figure 2.1 l a shows the different crest shapes observed in the experiments. T y p e C l displays z  min  downstream o f SFC.  some distance upstream o f SFB and z  max  some distance  Type C 2 is similar but SFC and C coexist. Type C 4 is a relatively smooth  rise from Tr I SFB to SFC IC . Figure 2.12a demonstrates C I is the most common crest type (i.e., the slipface is generally separated from z  min  and z ). max  Types C3 and C 4 are also commonly  observed, although the percentage o f C 4 crests declines with flow velocity. In comparison, type C 2 is rarely observed (i.e., i f SFB is not z , mn  z  mm  and S F C are unlikely to coexist).  44  -0.4 -0.2 0.0  0.2  0.4  0.6  0.8  1.0  x/L  -0.4 -0.2 0.0  0.2  0.4  0.6  0.8  x/L  Figure 2.10: Normalised dune shape diagrams where bed height, z , is normalised by dune height,/^ and distance along the dune, x, is normalised by the dune length,L. x/L =0 occurs at the slipface base while zlH =0 occurs at the dune trough. The numbers 1 or 2 indicate the echo-sounder (see Figure 2.3).  45  1.0  Crest Types SFC J  CI  ,C  SFC/C  C2  'SFB  VSFR TR  TR C  C3  C4  S F C ^ - - - ^ -  TR  SFC/C^.  TR/SFB  Stoss Types c  C  SI  S2  TR2  c.  ^^<TR2  C  \ B 2  S4  S3  Figure 2.11: Simplified dune crest and stoss types.  46  ^-sTR2  Flow A  Flow B  Flow C  Flow D  Flow E  Flow E A  Flow C  Flow D  Flow E  Flow E A  Figure 2.12: Distribution o f crest and stoss types observed in each run.  47  Stoss types observed in the experiments are shown in Figure 2.1 l b . Type SI displays a break in slope some distance upstream o f the dune crest and another some distance downstream o f the upstream z  mjn  (7>2). Type S2 shows a gradual decline from the dune crest to the upstream  z . min  Type S3 shows a slope break that gives the stoss slope a concave shape and S4 has a shape that gives the stoss slope a convex shape. The occurrence o f each stoss slope is somewhat more random than the crest type occurrence. Figure 2.12b indicates type SI occurs most often for Runs 27 and 28 while stoss type S4 is the next most common. For all other runs S2 occurs most often, followed by either SI or S4. Concave (S3) stoss shapes are the least commonly observed. There appears to be a relation between the location o f the lower slope break (B2) and the expected location o f the separation cell. Table 2.4 presents the distance between the upstream bedform trough (Tr2) and both slope breaks (BI and B2) normalised by H . The average distance B2 - Tr2 ranged between 2.5 and 5.1 H . Given that the average measured slipface angles were all ~30° the distance SFC - Tr2 is ~1.75 H . Combined, these distances are equivalent to or somewhat longer than x (3.5 - 5 H ), suggesting the reattachment length controls the location o f B2. BI typically R  occurs a short distance downstream o f x and is formed by accumulating sediment from the dune R  trough. The distance between BI and the crest forms a bedform 'crown' and the distance between the upstream trough and B2 form a bedform 'pan' in the trough. This morphology is somewhat different from the typical asymmetric shape displayed in Figure 1.2, but is common in laboratory settings [cf. van Rijn, 1993] and real rivers [see work by Church, Kostaschuk and collaborators on the Fraser River, personal communication]. Pronounced sand sheets occur over the bedform crowns.  2.5 Sand Sheet Morphology and Scaling The sand sheets migrating over the dune backs were examined using both portions o f the video records and the decomposed echo-sounder time series. The echo-sounder time series were employed  48  Table 2.4: Mean distance from the upstream trough to the slope breaks on the stoss side o f dunes. Adjusted values have had extraordinarily large or small values removed from the mean (i.e. 1 < (Bl to 7>2) I H> 15 and 1 < (B2 to Tr2) IH> 10 were removed from the mean). Flow  ES  ' ' . ' 1 2 1 2 1 2 1 2 1 2 E S = Echo-sounder.  " 21 30 29 24 24 23 6 0 18 17  Unadjusted (Bl to__ Tr2)/H _ " ~ 6.6 14 6.9 26 8.8 17 11.6 21 8.0 18 8.4 15 3.5 8 n/a 3 5.7 13 12.3 9  (B2 to Tr2}/H _ _ ^ ™4.6" "21 3.2 27 6.1 27 5.9 18 5.8 21 4.7 20 2.5 6 3.5 0 5.7 17 3.1 13  49  Adjusted (Bl to Tr2)/H " 6.6 11 6.4 25 7.8 15 8.0 17 6.6 17 6.5 14 3.5 8 n/a 2 6.0 11 4.6 4  (B2 to Tr2)/H 3.8 3.6 5.1 5.1 5.4 3.9 2.5 5.0 3.7 3.7  to determine the sand sheet height, H  min  and frequency, f . min  Unfortunately, the method applied to  determine R and L could not be applied to determine the sand sheet migration rate, R , min  length, L . mln  and  The small bedforms were simply moving too quickly to be identified in one echo-  sounder time series and then identified in the downstream echo-sounder time series (i.e., more than one sheet could pass echo-sounder 1 before being sensed by echo-sounder 2). To overcome this problem, the video records taken at flow strength B were used to determine sheet R  min  and L . min  Additional measurements included the distance from the rear dune crest to the  first sand sheet crest, x , and the distance from the first sand sheet crest to the downstream dune rear  crest, x  from  . N i n e sections o f video taken during flow B were chosen for detailed analysis o f the sand  sheets over five separate dunes at experimental times o f ~2, 3.5, 5.2, 8 and 11.7 hours. Four sections of video were chosen over one dune to determine the consistency o f measurements o f sand sheets on an individual dune, as it evolved. Once sections o f video were chosen, five successive images were drawn from the video, each separated from the next by 10 s. The dune crests and sheet crests were digitised from each image using A r c V i e w . Figure 2.13 defines the measurements taken from the images. Horizontal lines separated by 0.05 m vertically were overlaid on the image and horizontal distances were measured i n c l u d i n g L  min  , x  rmr  and x  .  from  B y overlaying the sheet crests from  successive images, the horizontal distance travelled by the sheet crests could be measured (Figure 2.13). Dividing by the separation time between the images, the migration rate R  min  could be  assessed. Dune length was determined by measuring the areal extent o f the dune and dividing by the cross stream extent o f the dune (i.e. flume width for these dunes).  2.5.1 Sand Sheet Features from Video Sand sheet lengths measured from the video were all approximately the same (Figure 2.13), meaning there was no change in L  min  with distance from the upstream or downstream dune crests.  50  51  Further, L ,  averaged for individual measurement sets, did not display any relation to L. Mean  min  L  min  is unchanged across a range o f R values (Figure 2.14). (For the present purposes, L is the best  index o f bedform size available from the video. Since there is a obvious relation between H and L, L should be a satisfactory scale.) It is safe to conclude that L  min  is invariant with the dune size.  Based on the similarity in the sheet lengths, the measurements can be aggregated (Figure 2.15). Lengths varied between 0.02 and 0.28 m and had a mode o f 0.10 - 0.12 m. The sand sheet migration rate does not appear to increase with distance from the upstream or downstream crests (Figure 2.13), suggesting that the sheets are not accelerating up the stoss slope o f dunes. However, in contrast to L , min  R  accordance with visual observations, R  min  min  does vary with dune size, roughly decreasing with L. In roughly increases with R (Figure 2.14). A strong linear  relation is found by regressing a line through seven o f the nine points. The two points not used in this regression were somewhat anomalous compared to the others. For seven o f the nine measurement IR = 8 - 10 while for these anomalous points this ratio was only 6. It appears the reason for  sets, R  min  this is that the reference dune crestline was accelerating compared to the others surrounding it. Aggregated R that o f L  min  ranged between 0.2 and 5 mm s" (Figure 2.15). The distribution is far broader than 1  min  with significant numbers o f observations at the extreme ends o f the distribution. The  mode lies at ~2 mm s" . 1  2.5.2 Sand Sheet Height, Length and Migration Rates from Echo-Sounders Sand sheet heights, plotted relative to position along the bedform, are displayed in Figure 2.16 and mean heights, averaged over each dune, are displayed in Figure 2.17. The majority o f sand sheet heights varied between 1 and 20 mm (i.e. 2- AD), although some H  min  values were outside this  range over the largest dunes. There certainly is a propensity to find larger sheets near the dune crest (Figure 2.16). However, this does not translate into H  min  52  consistently increasing towards the dune  18  r  16 ; 14 •  E o c  12 ;  'E _l  10 '• 8 : 0.50  r-  0.45 • 0.40  r  'e/3  E o  12  c  'ff  10  :  0.35 • 0.30 • 0.25  :  0.20 :  8 0. 02  0.03  0.04  0.05  40  50  R (cm s")  60  18  70  80  90  L(cm)  1  16 14 12 10 8 0.0  0.1  0.2 0.3 R  min (  C  m  0.4  0.5 0.6  S  Figure 2.14: Sandwave sheet lengths, L ,  and migration rates, R  mjn  min  , plotted against  dune lengths, L, and migration rates, R as measured from the video. Plots o f L  min  R  min  vs.  and L vs. R are shown for reference purposes. The image averages (open circles)  and averages for each dune (solid triangles) are shown. The circled triangles were removed from the regression between R  mm  53  and R .  I  0.5  I  I  I  I  I  I I I  I  I  I  I  I  I  I  I I I  I  I  I  I  I  J  i  i  i  i  0.5+  n=1567  i  i  i  i i  i  i  i  i  i  i  i  i  i i  n=1395  0.4 0.3 0.2 CH  0.1  1  0.0  • I I I I ii i i I i i ii Ii i i l I i r i  i  1,0  i  i  i  I i |  i  I  I  I | I i  I  i | i i i i | I I i  0  F•  1.0  i  i iii ii i ii i i ii  i  i  i  i  i  i i  i  i  n=416 0.8  0.6  0.6  0.4  * 0.4  0.2  0.2 I  i i i i I i i i i I i i i i I i i i i I • i i  0.00  0.05 0.10 0 . 1 5 0.20  i  i  J  i  i  i  L_I ' '  i  i  i  i  i  i  i '  '  i i  n=483  0.8  0 Q  i  i  0.0 0.25 0.00  obs./n  '  '  i '  ' '  ' i ' '  ' ' i '  '  '  '  0 . 0 5 0.10 0 . 1 5 0.20  obs./n  Figure 2.15: Aggregate sandwave sheet properties measured from the video, including lengths, L , migration rates, R , the distance from the rear bedform crest to the first mjn  min  sandwave crest, x , and the first sandwave crest to the major bedform crest, rear  54  x  .  front  0.25  30  i  i  i  [  i  i  i  |  i  Flow A • n=910 :  25  Flow B n=754  20 15 10  -  ""*'*.'•*'' .*  * • '  5 0 30  i  i  i  i  i  FlowC n=920  25  i  Flow D n=424  ; :  20 15 10 5  '• ^^V^>V* i>'•»'.*^••^ •••'" !  t  V-  0 30 Flow  Flow E 25  n=40  n=905  20 15 10 5 0 0.0  0.2  0.4  0.6  0.8  1.00.0  Distance from Slipface Base  0.2  0.4  0.6  0.8  Distance from Slipface Base  Figure 2.16: Sandwave sheet height, H , mjn  plotted against the distance from the  downstream SFB normalised by the dune length.  55  1.0  ]—i—m—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—  T—r—|—i—i—i—i | 1 i i i |—i—i—i—i—|—  Flow B  Flow A 10  ^5  i i i i i i i i i i' i i i i i  i  i i i i i i i i i i  i  • i i i i i i  i i i  i  i  i i' i  Flow C  i i  i i i i i i  i  i  i i i i i i i i  i i  Flow D  10 h  0  8  Q  O  a  i  i i i i  •i i i i i i i  0  10  11 i i • I I  i i i i i i i i i i i i i i i i  20  30  i — n — ' — i — i — r  40  i — i — i — i — i — i -  Flow E  i i i i i i i i i i i i i i  50 0  10  —i  1  20  1  Flow E  30  1  1  1  1  I  I  I  L  i i i i i i  40 1  1  1  50 r  A  10 h  J  L.  i  i  i  I  20  i  i  i  i  I  i  i_  _J  60  40  80  100  T i m e x 10 (s) Figure 2.17: Mean sandwave sheet height, H , plotted against run time. Averages are for individual dunes. Equilibrium dune size occurs at - 5 0 0 0 s during each run (dotted vertical line). The dashed vertical line indicates when the flow strength was changed from flow E(f =17.0 Hz) to flow E (f =23.5 H z ) . mm  p  A  p  56  crests. Although sand sheet sequences could be identified where H  increased towards the crest  min  (e.g. Figure 2.18a), many sequences showed rather random sequences o f H  min  (no pattern) towards  the dune crest (e.g. Figure 2.18b), and many others displayed nearly constant values o f H „ (e.g. mi  Figure 2.18c). Mean sheet height, Hmm, increased during the flume run in the same fashion as H and L, increasing dramatically until an equilibrium was established (Figure 2.17). Thus, unlike sheet length, Hmin increased with bedform size. In fact, the normalised sand sheet height, H /H min  , is centred at  or near 0.1 for all flow strengths with the exception o f flow E (Figure 2.19). Since only 2 bedforms were observed to have sheets during flow E, the distribution is undefined. "Normalised sheet height generally varied between 0.02 and 0.30, although distributions are skewed towards lower values. The frequency o f sand sheet passage over the back o f a dune is the number o f crests observed, n, divided by the period o f time that sheets are passing the probe, t  tot  J mm  hot  expressed as  •  ^••^  Sand sheet frequencies range between 0.01 and 0.15 H z , but, with the exception o f sheets at flows E and E , appear to fall into two groups (Figure 2.20). Over the incipient dunes at the beginning o f the A  runs, many f  min  values are rather large with more than 8 sheets advecting past the echo-sounders per  100 s. The averages varied between ~12 and 16 sheets/100 s. In contrast, over the larger dunes later in the runs, generally fewer than 8 sheets/100 s are observed and, on average, - 3 - 4 sheets/100 s are observed. The larger values o f  all occur over rapidly growing and somewhat diminutive dunes.  A t flow strength E the higher grouping is not apparent; nearly all f  min  values are below 8  sheets/100 s. These dunes had comparatively little sediment moving over the stoss slope and there may have been insufficient amounts o f sand to produce sheets. When the flow is increased to flow strength E the bed already had well developed dunes, and thus differs from other runs in which the A  57  16.8  17.0  17.2  17.4  17.6  17.8  18.0  0 n  18.2 ,  4.85  -10 4. 2.35  4.90  4.95  5.00  5.05  5.10  5.15  5.20  1  1  1  1  1  1  1  1  2.40  2.45  2.50  2.55  2.60  2.65  2.70  2.75  Timex10 (s) 3  Figure 2.18: Examples o f sandwave sheet sequences including sequences where sandwave sheet height, H , min  is increasing (Flow A-1-44), where H  min  randomly occur (Flow C-l-21) and where H  min  seems to  is nearly constant (Flow B-2-  22). The number in the top right comer identifies the flow ( A - E ) , followed by the echo-sounder (1 or 2) and the dune number (e.g. 44). The crest o f the dune is to the left and passes the sensor first. The overall trend is caused by dune passage. The points are successive z  max  and z  min  o f sand sheets (i.e. the regular  oscillation is exploited by the plotting convention).  58  l  0.5+  l  l  l  Flow B n=767  •  0.5  -1  1  0.5+  i  |  r  •  1  •  j* LI  i_  rFlow C 1  —I  1  l  I  l  Flow D  n=957  n=424  0.4 0.3 0.2 0.1 0.0  l—  •  -i—i—i—i—r-  0.5  Flow E  1  1  1  •  1  •  •  0.5+  l  L_  Flow E  n=40  A  n=1003  0.4 0.3 0.2 0.1 0.0 0.00  1 0.05  X  X  0.10  0.15  0.20 0.00  0.05  obs./n  0.10  0.15  0.20  obs./n  Figure 2.19: Histograms of sandwave sheet height, H , mjn  59  normalised by dune height, H  0.25  0.20  0.15  i  i i i  ii  > i  iii  i  i i i i  i  i  i i i i i i  Co\  i  I 1I 1 M i "  F l o w A  i|  r  iii|  i  111  9  | 1 11 1  Flow B -  \  '  1  JD  :o  -  0.13 ^min~  :c? 1-/^=0.16 Hz  Hz  :  cB  0.10  6'  0.05  -O  ° ^ = 0 . 0 3 3 Hz;  Q  U=0-034 H z ;  n  o ;  o P,  i ,  0.00 0.25  i  i i i  i  -  i  i i i i i i i i i  i  i  Jill  i i i i i i i i  1  1  1 1  111  § 1 1 1 1 §  i i i i 111111  1 1 1 1 1 1 1 1  | 1111|  Flow C  1  11 1  Flow D -  0.20  \  O 0.15 h o O  :o  :  I" ^=0-12 Hz  o.  0  g  O  o :  ^=0.033 Hz j  0.00  11 11 i 11 1 1 1 • 1 1 1 1 1 1 1 1  0 0.25  10 1  I  I  I  I 1  20 I  30  11  i  11  40  f „=0.028 H z ;  O  0.05 h  i  i  ,  i  i i  50 0  1  11 1 1 1 11 1 1 1 11 1 1 1 1 1 1  -  - ^ = 0 . 1 2 Hz  :©  69  0.10  -  \  1  m  o  %, °  ;  1 1 1 1 1 1 1 1 1 1  i  i  10  i  •  1 • i • 20 30  40  1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1  | 0.20  50  FlowE  :  1  0.15 h  :  1  :  f ,=0.048 Hz  :  i m  1  0.10 -  Iq.8...  0.05 h  0.00  1  0  °  o  :  1  1  1 1  1  10  1  1  20  Q  ft  Q  1  1 1. i  i  i  1  i  30  i  i  i  i 40 1  i  i  i  M  50  i  i  i 60 1  i  i  i  1  i  70  i  i  i 1i 80  i  i  i  I  i  90  3  mm  60  -  r?°r,  T i m e x 1 0 (s) Figure 2.20: Mean sandwave sheet frequency, f ,  "  °  0  y  i  o o  6  plotted against run time.  i  i  i  100  dunes with large f  values are still quite small in size. It is interesting that when the dunes are  min  most unstable (i.e. at the flow change) f  exceeds 8 sheets/100 s.  min  Using the results obtained from analysis o f the video, the sand sheet lengths were estimated by: (1) determining an approximate value o f R  min  for each dune from the regression between R  min  R (Figure 2.14) and (2) finding the time that each sheet required to pass the sensor (t  and  ). Sand  sheet length was calculated as  T min  =R  7 s  x/ min pass  '"  1  i  J  Sand sheet lengths, plotted relative to position along the bedform, are displayed in Figure 2.21 and mean lengths, averaged over each dune, are displayed in Figure 2.22. Sheet lengths generally varied between a few millimetres and 0.5 m with a few observations in which L  min  exceeded that value.  There was a propensity to observe large sheet lengths near the crest, but, like patterns observed for H , min  there were no consistent patterns in L  min  over a dune.  In accordance with observations from the video, sheet length did not vary with dune size. The initial increase in H'»„•„ is not observed for sheet length. Mean sheet length, Lmin, varied between 0.05 and 0.40 m at the larger flow strengths ( A , B and E ) while at the lower flow strengths (C, D and A  E), Lmin varied between 0.02 and 0.2 m (Figure 2.22). Histograms o f L  min  (Figure 2.23) indicate that the bulk o f sheet lengths were in the range 0.0 -  0.3 m. The distributions tend to be strongly skewed towards smaller lengths with modes that range between 0.02 and 0.08 m. This differed considerably from the lengths observed from the video. Sheets in the video were observed throughout the record and could be seen over the small initial features by eye. However, the sand sheets observed in the video needed to be at least a few mm high before they could be properly identified. This only occurred later in the record, when f  min  8 sheets/100 s. Further, extremely long waves (i.e. > 0.5 m) were generally ignored.  61  exceeded  0.5  I  I  1 . I  I  I  I  I  1  1  I  I  J i  i  |  .i  i  i  i  Flow A .  0.4  0.3  0.2  i  i  i  i  i  i  i  FlowB  n=910  ;  n=754  ;  •  •*'  •;V. \  '*}  'i'i'y. C '  '  ' •"•'• '• •  0.1  0.0 0.5  •  . . I .  i  0.4  i  I  i  i  i  i  ".  i  .  i  Flow C  Flow D  n=920  n=424  Flow E  Flow E^  n=40  n=905  0.3  0.2  0.1  0.0 0.5  0.4 [  0.3  0.2 [  0.1  0.0 0.0  0.2  0.4  0.6  0.8  1.00.0  Distance from Slipface Base  0.2  0.4  0.6  0.8  Distance from Slipface Base  Figure 2.21: Sandwave sheet length, L , mjn  plotted against the distance from the  downstream SFB normalised by the dune length.  62  1.0  -|—i—i • i  -1—m—r-j—r—r-  i—|—i—r—r—i—|—r—i—i—i—[—i—i—i—i—r  Flow B  Flow,  0.1 h  I i II  i—l-.i 1.1,1  1 i i i i—I i i i i I i i—i i I  1 I i—•' • • I • • i • I • t . . I • • . . I  I i i; i i | i i i i | i i i i | i i i i | i  Flow D  0.1 h  i  i i i i i i i  0  J  \~  i  i  i  i  i  i  10  20  J i  i i i  30  i i i i i  40  i Ui i  50  i_  20  40  60  80  100  Time x 10 (s) Figure 2.22: Mean sandwave sheet length, Lmin, plotted against run time. Averages are for individual dunes. Equilibrium dune size occurs at - 5 0 0 0 s during each run (dotted vertical line). The dashed vertical line indicates when the flow strength was changed from flow E (f = 17.0 H z ) to flow E ( f =23.5 Hz). p  A  p  63  i—i—r-i—r-r—i-  -|  1  1  n,=924  ^=767  n =742  n =461 2  NNXNXN  • 0  50  0.05  0.10  r-  1  Flow B  2  0.00  11  1  Flow A  0.15  0.20  0.25  I  0.00  0.05  0.10  i  .  •  I  .  i  0.15  0.20  0.25  0.15  0.20  0.25  t 1^=957  40  n =445 2  30  20  10  I  o  0.00  0.05  0.10  0.15  50 [  40 [  E  30  o 20  10  10  55^  Z l  0 0.0  0 0.1  0.2  0.3  0.4  0.5  0.6  0.00  obs./n  0.05  0.10  obs./n  Figure 2.23: Histograms o f sand sheet length, L , min  calculated using the relation between R  min  and  R . The solid bars are for all lengths. The hatched bars are for data that have been selected to remove lengths where: 1) L „ > 0.50 m, 2) corresponding H mi  min  < 3 m m and 3) sheet frequencies, f  mm  0.08  H z . The number o f observations for the unfiltered data ( « , ) and filtered data ( n ) are indicated. 2  64  In light o f this, the data were filtered to remove sand sheet lengths when: (1) L  min  corresponding H  min  because H  min  < 3 mm, and (3) f  min  > 0.05 m, (2)  > 0.08 H z . I f a sand sheet was removed from the analysis  < 3 mm, its length was added to the upstream sheet. Histograms o f the sand sheet  lengths remaining after this filtering are also shown in Figure 2.23. The histogram for sand sheets measured from the video at flow strength B shows a distribution similar to the one in Figure 2.15. Lengths varied between 0.02 and 0.20 m with a mode o f 0.08-0.10 m, only slightly smaller than the mode from the video. This suggests that differences between the L  min  distribution observed from the  video and the distribution from the echo-sounders is related to the resolution o f the techniques. While the echo-sounders can resolve sand sheets in which H  min  > 0.5 mm, the video can only practically  resolve the larger sheets (> 3 mm). This does not suggest that the L „ distribution produced from mi  the echo-sounders is in error.  2.5.3 Variability and Measurement Error in Sand Sheet Properties from Echo-Sounders There are three substantial sources o f variation in the data set: (1) natural variation o f the phenomenon, (2) error associated with type o f measurements taken and (3) the practical vertical resolution o f the measurements system. Estimating how each source o f error is represented in the individual measurements o f sand sheet properties is difficult. However, variability about the mean values provides some estimate o f the total variability, including the measurement error. The error associated with the type o f measurements taken is likely small for H  min  as height is  measured directly. However, the error associated with the vertical resolution o f the measurement system is quite substantial. For the dunes, D was only 3.7 - 7.5 % o f o~H. For the sand sheets, D can be many times larger than o~H . mjn  On average, the resolution o f the echo-sounder measurements  accounts for 25.1 - 61.6 % o f the total variation. CV  Hmin  65  varied between 0.065 and greater than 1 for  the sand sheets over individual dunes with a mean, calculated for each flow strength, that varied between 0.41 and 0.51. A substantial portion o f the variability in L  min  Error associated with the measurement o f t compounded. CV  Lmin  is related to how the lengths were calculated.  and the regression used to estimate R  min  are  varied between 0.11 to greater than 1 for sand sheets over individual dunes  with a mean, calculated for each flow strength, that varied between 0.43 and 0.58. This variability is equivalent to the variation in t  pass  tptm  a  r  e c a u s e  due to how L  mln  is calculated. Potential inaccuracies in sand sheet  d by the same processes as for the dunes; sheets stalling in their downstream  progression or accelerating and decelerating during the measurement period. Sand sheets moved far more consistently and changed little as they migrated over the dunes. Thus, most o f the variability observed in t  is natural.  The error associated with the regression between R and R  min  L . min  is in addition to the variability in  The standard error associated with the regression between R and R  min  is - 1 6 - 20 % o f R  min  for all flow strengths except E , in which the standard error is ~28 % (recall that there are only two dunes with sheets observed on them at flow strength E). Using rules for combining errors from multiple measured quantities [Beers, 1957], the standard error associated with Lmin varied between 17.6 and 36.6 % o f Lmin for individual bedforms. Mean values, calculated for each flow strength, are ~22 % o f Lmin for all flow strengths but E , for which the standard error is - 3 0 % o f Lmin.  2.5.4 Sand Sheet Classification  and Scaling  Superimposed bedforms are common and have been reported extensively in the literature [see review in Allen 1968 or Ashley, 1990]. The sand sheets described here do not appear to be fundamentally different from previous descriptions o f superimposed bedforms where both scales are in equilibrium at the same time [e.g. Pretious and Blench, 1951; Jackson, 1976]. Superimposed 66  [set Allen and Collinson, 1974].  bedforms are often observed where only one scale is in equilibrium  Those features may be fundamentally different from the sand sheets observed here. Despite their similarity to features previously observed, sand sheets cannot be classified as ripples or dunes, but have characteristics shared with both features. Their relatively small lengths and heights are reasonable for ripples. That the features do not grow as they migrate downstream suggests that they are not growing to scale with a boundary layer depth (i.e. the depth o f the internal boundary layer). Thus, their length and height when they appear on the stoss slope appear to be their equilibrium dimensions, which is typical o f ripple features that scale with grain size. However, the sand sheets tend to have an aspect ratio far smaller than that observed for ripples. Figure 2.24 shows histograms o f the sand sheet aspect ratio (H /L ) mjn  min  at each flow. Based on  the separation between ripples and dunes given by Allen [1968], some o f the sand sheets could be ripple forms. However, most sheet aspect ratios are less than the upper limit for dunes o f 0.1 proposed by Allen [1968]. Further, at flow strengths A , B , C and E , more than 55 % o f the sheets A  have aspect ratios less than the lower limit for ripples (0.05) suggested by Allen [1968] and data from Guy et al. [1966]. Even at flow strength D , 36 % o f the sheets have aspect ratios smaller than 0.05. Only at flow strength E are most sheet aspect ratios greater than 0.05 (Recall that  H  mtn  and  L  min  distributions are not well defined at flow E ) . M o d a l aspect ratios are between 0.02 and 0.05, depending on the run. These aspect ratios suggest sand sheets are not ripples but low relief dunes. In this respect, the sand sheets are similar to bedload sheets, first described by  Whiting et al. [1988], which represent an  additional scale o f sediment transport organisation in poorly sorted sediments. They have been observed over flat gravel beds and superimposed on dunes. Bedload sheets are typically several coarse grain-sizes high, lack a well-defined slipface and have aspect ratios between 0.003 and 0.040 which classify them as low relief dunes  [Bennett and Bridge, 1995]. It is generally agreed that  67  Flow B n=767  0.00 r . 1  0.00 -I  -\ 0.00  . . . . . . . . . . . . . . . . . . . 0.05 1  I  |  1  I  0.10  0.15  1 1  1  I T  J  1  0.20 I  1  0.25  I  0.25  0.00  0.05  0.10  0.15  0.25  Flow E  Flow E  n=40 0.20  ; 0.20  0.15  • 0.15 40.0 %  0.10  "6o!6'% 87.5 % ' 12.5 %  0.05 0.00 0.00  • 0.05  •  1  1  0.10  •  •  1  •  0.15  1  •  1  •  0.20  •  •  0.20  0.25  -i—i—i—|—i—i—i—|—i—i—r  1—I—  |  n=1003  - 0.10  7.7 % '92.3%  0.05  32.4 % 67.6 %  0.00 0.25  A  _i  l  0.00  1  1  •  0.05  obs./n  i—i  i  0.10  T  i  0.15  I  i_  0.20  0.25  obs./n  Figure 2.24: Histograms of sand sheet aspect ratio ( H is and sheets height and L is sand sheet length). The dotted line at 0.1 indicates the typical upper limit o f dunes and the dotted line at 0.05 indicates the usual lower limit o f ripples. min  68  min  bedload sheets are caused by selective entrainment and transport in heterogeneous sediment [Whiting  etal, 1988; Bennett and Best 1995]. Interestingly, since the identification o f bedload sheets by Whiting et al. [1988], descriptions o f them have fallen into two categories, those developed on otherwise flat beds [Wilcock, 1992; Bennett and Bridge, 1995] and those developed on the stoss slope o f bedforms [Livesey et al, 1998]. It is not possible to divide these studies based on sheet aspect ratio. While some bedload sheets over dunes appear to be flatter than dunes, many o f the features could be classified as ripples based on aspect ratio. Livesey et al. [1998] describe bedload sheets with an aspect ratio that would classify them as ripples. Describing the sand sheets over the dunes as bedload sheets seems inappropriate because the origins o f the forms cannot be attributed to selective entrainment and transport. N o sediment sorting can occur along the streamwise axes o f sand sheets because the sand is homogeneous. The height o f the sand sheets is not limited to a few grains. However, the sand sheets do share several morphological characteristics with bedload sheets. A t 0 . 1 / / , H „ mi  is similar to bedload sheet  heights previously observed over dunes as are sand sheet lengths [e.g. Whiting et al, 1988; Livesey et al, 1998]. Most sand sheets did not have a defined slipface and are classified as low relief or incipient dunes based on their aspect ratios.  2.5.4 The Origin of Sand sheets Development o f the sand sheets seems to be related to a minimum 'fetch' length beyond the bedform crest. This is revealed in measurements from the video o f the distance from the rear dune crest to the first sand sheet crest ( x downstream dune crest (x ). fnmt  rear  ) and the distance from the first sand sheet crest to the  Most x  values are between 0.4 and 0.6 m and while most  rear  values are between 0.2 and 0.6 m. While x  rmr  did not vary with L, x  front  x  front  did (Figure 2.13). Thus,  proximity to the dune crest does not appear to be an important factor. However, a minimum x  rear  69  value o f ~0.5 m is necessary for the sheets to form. This appearance is further substantiated by the fact that no sheets appeared in the video until the bedforms reached a wavelength o f -0.5 m. Interestingly, x  R  does not appear to control x . rear  For the first measurement set from the video,  H is -0.025 m and near H for the other measurements from the video. Assuming x  ~5 H , x  = 0.125 m. for the first measurement set and -0.20 m for other measurements sets. x  ~ 3x  e  R  rear  the first measurement set and - 2 x  R  R  R  for  for the other measurements. The reattachment length must play  some role in where the sand sheets begin to form on the dune; echo-soundings indicate sand sheets generally do not form upstream o f x/L =0.8 (see Figure 2.16 and 2.21). The separation bubble would sit at approximately 0.8 < x/L < 1.0 over most o f the bedforms. Given that the length over which the sand sheets develop is somewhat limited, it seems likely that sheets could further develop into either full sized ripples or dunes. These features are similar to those developed at the beginning o f the experimental runs over the flat bed. Therefore it is possible that the same processes that control sandwave initiation and development over a flat bed, discussed in the next chapter, may be operating here.  2.6 Sediment Transport Rates Probably the most compelling reason to study bedforms is to gain a better understanding o f their implications for sediment transport processes. It is therefore logical to end the discussion o f the dunes and sand sheets with estimates o f the transport rates that are associated with each. Although bedload transport measurements were made during the experiments, they were too infrequent to provide a complete picture o f the sand transport associated with the waveforms. Nearly all the bed material was transported as bedload. Sand suspension and sand bypassing bedforms as it travelled downstream were observed to be negligible. Thus, more complete quantification o f the transport can be accomplished by using morphological estimates based on the characteristics o f the waveforms themselves. 70  2.6.1 Morphological Estimates of Transport Rates Assuming all bed material moves as bedload, estimates o f the sediment transport rate can be determined from the morphology o f the dunes and sand sheets separately. Simons et al. [1965] indicate the volumetric dry sediment transport rate of bedforms moving at a migration rate, R , h  per  unit time and unit width is  Q = r3(\-P)R h s  2.6  h  where P is the sediment porosity, h is the bedform height and (3 is the bedform shape factor. Values o f ji can be calculated as  2.7  where A is the cross-sectional area o f the bedform and / is the bedform length. If the bedforms are triangular, ft =0.5. Bedforms in natural channels rarely have a perfectly triangular shape and so a variety o f /3 values has been suggested in the literature [c.f. ten Brinke etal., 1999; van den Berg, 1987]. However, most authors have suggested /3 =0.55-0.60. Engel and Lau [1981] suggested an alternative approach based on the idea that there is a zero net transport point at the flow reattachment point (Figure 2.25a). Upstream o f the reattachment point sediment flux is negative and downstream the flux is positive. Therefore only that portion o f the dune above the zero net transport rate is included in sediment flux (Figure 2.25a). The dune is assumed to be perfectly triangular. In this formulation ji is replaced by K , a coefficient dependent on x  R  ,  H/L , the lee slope angle, and the bedform shape, and H is replaced by <fj , the average departure o f z about z . For a triangular shape £ = H/4  and, using H/L =0.05 (the approximate mean o f the  bedforms under examination here), K is 1.44 or 2.88/3. For a triangular shape, produce estimates that are 72 % o f Q  s  (K,F).  71  Q (ji,H)wi\\ s  Figure 2.25: Definition diagram for Engel and Lau's [1986] supposition that only that portion of the dune above the reattachment point contributes to the transport rate. The dune shape used in their formulation (a) and the shape observed in these experiments (b) are displayed.  72  There are several problems with the Engel and Lau [1981] approach. For example, this approach would seem to give a positively biased estimate o f the transport rate, because logically, the negative sediment flux should be subtracted from the positive flux. Another problem is that the observed dunes differ significantly from the simplified form depicted in Figure 2.25a. Most dunes had a w e l l defined pan that encapsulated the separation cell and so the reattachment point occurred near the base level o f the dunes (Figure 2.25b). Thus, the volume o f sediment below the reattachment point on the dune back is much smaller than envisioned by the Engel and Lau [1981].  2.6.2 Dune and Sand sheet Related Transport Rates  Figure 2.26 plots the sediment transport rates for (1) dunes, Q _ s  Helley-Smith bedload samples, Q _ S  HS  , (2) sand sheets, Q _ , and (3)  d  s  ss  . To estimate the dune related transport rate, h = H and  R = R in Equation 2.6. Values o f H and R are those presented in Figures 2.6 and 2.7. The h  porosity o f 0.5 mm sand is ~0.4 [van Rijn, 1993]. Rather than using an estimated and fixed value o f P , the echo-sounder time series were used to calculate /3 directly from Equation 2.7 with h = H and I - L. In order to determine A , each bedform was plotted using only the features shown in Figure 2.11 (i.e. Tr, SFB, SFC, C, BI and B2). Cross-sectional area is only the portion above a datum line running between Tr and the upstream Tr . Figure 2.27 plots fi against the bedform number. /3 ranged between 0.3 and 0.8 with greater variance later in the runs. Mean /3 varied amongst the runs but is - 0 . 5 6 . T o estimate the sand sheet related transport, h = H'»,/» and R = R b  min  in Equation 2.6. Thus, sheet  heights and migration rates are averaged over a dune. Since the sand sheets have approximately the same morphological scaling as the dunes, a /3 value o f 0.56 is used for all sheets. Both Q _ s  d  and Q _ S  HS  increased in the same fashion as dune L and H , initially rising quickly  and settling into an equilibrium value. Q _ s  ss  was generally larger than Q _ s  73  d  and Q _ S  HS  when the  1000  1000  f  1000  FlowE  5?og  A  •Q  100  10  oa_ liil !  ^ ° A~A  OA  A  ¥  O  A  1 i  V 0.1  i  '  '  • • • • • 10 20 1  1  1  ' ' i • • •' 30 40  i  50  i  i i i i i i 60  i  i i i i i i i 70 80  i  90  i  i  i  i  100  T i m e x 1 0 (s) 3  Figure 2.26: Sediment transport rates, Q , determined from the morphologic characteristics of the dunes (open circles for echo-sounder 1 and open triangles for echosounder 2), the morphologic characteristics of the sandwave sheets (closed squares), and Helly-Smith bedload samples (closed circles). Sand sheet error bars are the standard error of the estimate. 74 s  LO  1  1  * i  1  ' ' * i * * * * i *  Flow A  * * * i  o  0.8  1  1 1  1  ' ' '  1  o A o  *  A  i '  1  °  0.6 CO-  0.4  0.2 0.56 0.0 0  10  1.0  0.8  20  30  40  50  10  20  O  _2  8  A A ,  /DO  A  °  AA  A  A A  0  -O-  oo  O  AA  0  A  50  oo  Cr  O  °*o6 C D °  40  A  A  A  30  -i—i—i—i—i—r  oo  0.6  0.4  0  Flow D  Flow C I- o  co.  70  I i TTi | r i i r | i  • 1111111111111111 i/j^i  T i ^ -Ag  60  A A  A  O  A  0.2 0.57 0.0  1  0  • • '  5  1.0  1  • • ' •  10  1  ' • • •  1  • • • •  15 20  1  • • • '  25  ' • ' '  1  30  1  0.54 —• i i i 1—J • • • ' • • • • '  • ' • '  35 40  10  15  Flow E  Q  1  1  •  20  1  •  25  Flow E o  A  A  0.8 h  O  _OA_  0  ^  ~2~  A 6  Q  1  i l l i | l l l l | l  11111111111111111111111111111111111111  0.6  •  0  K  A  A  A °  CQ.  A O  0.4  A  o  WrT^  O oO  o o Q  A  -QV-  0.2 0.60 0.0  • • • •  0  1  • • • •  2  1  • • • •  4  1  • • • •  6  1  • • • •  8  1  • • • •  1  ' • • •  1  • • • •  1  0.56 i  • ' • •  • • * •  10 12 14 16 18  Bedform  15  1  ' • • •  20  1  • • * •  25  1  • • •  30  1  1  ' • • •  1  • • • •  35 40  1  • • • •  45  1  • • • •  50 55  Bedform  Figure 2.27: Bedform shape factor, P . Circles are measurements from echo-sounder 1 and triangles are measurements from echo-sounder 2 (See figure 2.3). 75  transport rate was increasing towards an equilibrium at flow strengths A , B , C and D . These estimates o f Q _ s  are from sand sheets with a frequency greater than 8 sheets/100 s. It appears that  ss  these sheets were moving more sand than the incipient dune forms and may be a fundamentally different sheet than those observed later in the experiments. This makes sense because in order for the dune to grow, sand sheets must appropriate more sand from the dune back than moves past the dune crest. Table 2.5 provides the initial mean transport rates, Q , and the average equilibrium transport si  rates, Q . se  Transport rate increases with flow strength, which is expected for homogeneous sand.  Interestingly, Q _ se  d  at flow strength E is larger than at flow strength A by ~70 %, suggesting that A  the time required for a run to reach equilibrium when bedforms are already present may be significantly longer than when they are not present. Transport rates differ amongst the different methods used to obtain Q . Sources of variability and error need to be considered before a s  comparison o f Q from the different methods can be accomplished. s  2.6.3 Variability and Measurement Error in Transport Rates Ideally the error associated with each transport measurement or estimate could be determined and error bars given in Figure 2.26. Unfortunately, the error associated with Q _ is derived from the s  d  error associated with H and R . A s noted above, it is not possible to accurately assess the error associated with calculated R values making it impossible to report a total error associated with Q _ . s  Similarly, the error associated with Q _ S  HS  d  cannot be assessed accurately because there is no standard  against which to test the samples. However, several authors have reported the efficiency o f a fullsized Helley-Smith sampler. Both Sterling and Church [2002] and Emmett [1980] indicated that the Helley-Smith sampler has an efficiency o f - 1 0 0 % for 0.5 m m sand. However, these studies  76  Tf m sq CN r n  r-  OS  Tf  <— T f  00  1  d d d d d d  CO  O <U Q. = cn 3  c  c  T3  CO  CN c<-> CN CO sq — CN CN  -a  2 «  CO • = CD  -a e  00  * -  — "EZ  OB  U  1 / 5  ~L  CO •r  •£  x> 3  CO  "O  £ X  3  o  O O  co - 2  e  >, CO  O X)  o  <  <H-  td  O  = -a c  03  cO  -2  a. n. +i 3 P cn CO  2  c CO  ^  CO  -a c 03  to co - 3  cn  3  —  '~  -  .-= *  rn  r-  ~ Tf  OO CN  sq  SO CN  rn rn O  m SD >n '—'  CN  oo in  so  o  Tf  m  os  SO in  co  OS Tf  —' Tf  d I.O) r-^ CN CN  cn  P  >  E  cn  o  <s> £ co 03  E  «4-  so  r-in  CN  —'  m o  o  o  m oo  oo m  m m  m m  CN cn  d  d  d  >Z> <^  SO  m CN rn  5 § ° co  E  \S CO  O  X  o Z  oi  3  CO OA CO  £ CO  - °  E HE 3  P  o O  CO  P  CO  < _  OS  13  cn CN so ^—i i n Tf  CO  •3  Si  IO)  r-~ m  -d co  03 03  CO  I  03  CO  co  .«>  CO  5 OS  SO CN Tf  to u.  so rn  •—  1  CO  c  cu. obi CO  00  Tf  Tf _  o  in  CN  d d_ ~ ^ +' +1 +1 <^> T Zl Z _ O-, Sf O OS  CN  0  CN  0  .  IO)  ~  —' d  CN  so  CO  (3  00  £  CO T 3  CN CN  +1 +1 +1 +1  Q4 3  rn  CO  CO >  c  td  o "5 tt. T3  CO  O)  o  11 r-  o  — CO  c_  O  CN  r~ — ~—  +1 +1 +1  Os  CO  CO  cn  -a  m  Tf  Tf  CO OJ)  CO  j_,  CO  in  rn  Tf  —'  <*> 1  1  G - "o3  cn ~  •—  o  co  co  so  d d +1 +' +1 +1 S - ^ ^  IO)  CO  CO  o  00 00  £ .S  co  rn  Tf  OJ)  3  Tf  1/1 i n in  cO  - 3  CO  -a c  MO so  o  cn  *n  OO  U  =  .. . co  C/l  Tf  rn OS rn  +1 +1 +1 +1 +1 +1 — r - >n CN — rTf sq o r n o —  ^ 3 E »? o  OS — —  —  —  in  CO  (-  03  60 3  « 3  o  < CD, U Q td  examined trapping efficiencies o f coarse heterogeneous sediments. It is not clear i f the results apply to homogeneous sand. In contrast to Q _ S  D  and Q _ S  , the error associated with Q _  HS  S  standard error o f the mean (a/yfn ) can be calculated for H  mjn  associated with the regression used to estimate R  min  [1957]. Thus, the error associated with Q _ S  ranged between 17 and 37 % o f the Q _ S  SS  SS  can be accurately assessed as the  and combined with the error  using rules for combining errors found in Beers  is presented as error bars in Figure 2.26 and generally  SS  estimate with a mean o f ~21 % for each run.  Variability in the transport estimates or measurements can also be examined through estimates o f the mean equilibrium transport rates. Sources o f this variability vary amongst the different methods for estimating Q . SE  In general, CV  varied between 1.16 and 0.17 (Table 2.5). The largest  QSE  variability was observed at flow strengths E ( C V A  Variability in Q  SE  is largest for Q _ SE  HS  Q S E  = 0.87 - 1.16) and A ( CV  = 0.64 - 0.85).  QSE  , intermediate for Q _ SE  D  and least for Q _ SE  SS  (Table 2.5).  The variability in the Helley-Smith bedload samples increases with the flow strength and has several sources. First there is natural variability in the amount o f sediment moving over the bedform, which is related to bedform size and three-dimensional morphology. Another source o f variability in QS-HS '  s w  n  e  r  e  the sample was taken (i.e. over the dune crest, stoss slope or trough). In general  bedload samples taken over the crest and stoss slopes were equivalent in size. Samples over the dune troughs tended to be small or even negligible in size. Error was also incurred when the sampler's position on the bed caused sediment movement into the mouth simply by its presence. A final and substantial source o f error in Q _ S  HS  occurred when the sampler base could not be placed flush with  the bed slope along the streamwise or cross-stream directions. Notes taken during the experiments were used to remove those samples with obvious error. However, it should be noted that no bedload sample was completely free o f these measurement errors.  78  There does not appear to be a consistent pattern in the variability associated with Q _ and se  d  Q_  aside from CV  Q_  are directly related to variability in the bedform heights and migration rates whose variability  se  se  ss  ss  Qse  being largest at flows A and E . The sources o f variability in Q _ and A  is discussed above. The variability in Q _ se  se  d  is larger than for Q _ , because the variability in the  d  se  s  heights and migration rates is greater for the dunes than the sand sheets.  2.6.4 Transport Rate Estimate Agreement Agreement between the morphological estimates o f sand transport and Q _ S  The ratio Q _ IQ _ sj  d  S  HS  is highly variable.  HS  at flow strengths A , B and C are 1.53, 0.62, and 1.43. Q _ S  HS  did not decrease  with flow strength, which contradicts observations made during the experiments and Q _ si  This suggests that there may be more measurement-induced error in Q _ si  between the two methods used to determine Q . The ratio Q _ sl  1.14. A t three o f the flow strengths, Q _ se  se  and Q _  HS  se  HS  HS  'Q .  d  estimates.  than there is variation ranged between 0.62 and  se d  are within 15 % o f each other. Because there  d  are more samples during the equilibrium period o f these flows, the effect o f measurement error is diminished. Figure 2.28 indicates the agreement between Q _ and Q _ s  d  s  estimates are clustered about the 1:1 line and most Q _ s  ss  ss  estimates are within a factor o f 0.5 - 2 x the  Q _ estimate. Agreement is improved somewhat i f only Q _ s  d  s  strengths A , B and C , the ratio Q _ / Q _ se  ss  se  (/  is also variable. Transport  d  and Q _ s  ss  are compared. A t flow  ranged between 0.97 and 1.09 and the error bounds  overlap, indicating that sediment movement over the dune back is essentially equivalent to the volume moved in the dune form. A t flow strengths D , E and E the ratio Q _ A  se  ss  IQ _ se  d  is 1.80, 2.0 and 0.74 respectively. Results  from flow strength E can be dismissed as the transport estimates are based on only 2 o f 17 dunes and 79  1000  ~\—I—I—I I I I  n  1—r  Z  :  / /  OD  'OO  CD  100  O  0  v  10 /  0,  ,  / L  I  I  ' ' ' '  10.  100 Q -d s  1000  (kg hr" ) 1  Figure 2.28: Agreement between the dune related transport rate, Q _ , and the sand sheet k  d  related transport rate, Q _ . Sand sheet error bars are the standard error o f the estimate. s  ss  80  only one o f those points exceeds the range o f Q _ . Similarly, the run at flow strength E can be se  d  dismissed as it did not reach equilibrium and so Q  xe  is interesting that the ratio Q _ IQ _ s  ss  s  A  may not be a relevant statistic for comparison, ft  calculated for individual dunes averages to ~1.0 for all dunes  d  with sand sheets at flow strength E . Thus, there appeared to be some dunes with larger transport A  rates that did not have superimposed sand sheets. However, for Run 29 (flow D), Q _ s  ss  is clearly much larger than Q _ . The ratio s  d  Q . IQ s ss  s  d  calculated for individual dunes averages to ~1.7 for all dunes with sand sheets. The poor agreement is confirmed in Figure 2.26, indicating there is more sediment moving over the dune back as sand sheets than is moving in the dune itself. Error bounds for Q _ and Q _ do not overlap. More s  ss  H  d  sediment appears to be recruited from the dune back than is moving over the dune crest. Thus, the bedforms where sheets were observed may have been growing. However, it is probable that the relation between R and R , min  derived from the run at flow strength B , is not applicable at the lower  flow strengths and that extrapolation o f this relation to flows D and E was too extreme.  2.7 Summary A series o f experiments was conducted in which a narrowly graded 0.5 mm sand bed was subjected to a 0.155 m deep, non-varying mean flow ranging from 0.30 to 0.55 m s" in a 1 m wide 1  flume. Bed waveform morphology was monitored using overhead video and echo-sounders. Three basic waveforms were found in the channel, including long sediment pulses, dunes and sand sheets. Insufficient data are available to examine the long sediment pulses so investigation is limited to the two latter sandwave features. The data presented here are inappropriate to examine initial bedform properties. Equilibrium height and length increase with flow strength. Growth in dune height and length are found to be approximately exponential through time. The dunes that are present during the equilibrium stage are morphologically similar to the initial forms present. Aspect ratios suggest that most o f the forms  81  present are dunes according to the classification scheme of Allen [1968]. There is no obvious transition from small ripples at the beginning o f the runs to dunes when the sandwaves become larger. Several different crest and stoss morphologies are found for the dunes. In the most common crest morphology the slipface is decoupled from the bedform's minimum and maximum bed height. The second most common crest morphology has the dune trough co-existent with the slipface base. The most common types o f stoss slope morphologies exhibit either a smooth, non-linear transition to the upstream dune trough or a pair o f slope breaks defining a pan that holds the separation cell. These slope breaks also define a bedform crown where sand sheets develop. Over equilibrium dunes, ~3 - 4 sand sheets were observed per 100 s. The sheets, many lacking slipfaces, form at -0.5 m from the dune slipface, downstream o f the reattachment point on the dune stoss slope. This distance is invariant with dune size, suggesting there is a necessary 'fetch' length needed for the sheets to begin to grow. The sheets have heights that are typically O . l x the height o f the dune upon which they are superimposed, migrate at 8-1 Ox the dune rate, and have lengths that are nearly constant over the full range o f dune lengths and flow conditions. Aspect ratios are generally < 0.1 with a mode o f -0.025, classifying them as low relief or incipient dunes. Sediment transport rates measured with a Helley-Smith sampler and estimated from the morphology o f both the dunes and the sand sheets are similar in magnitude. The measurement error associated with the Helley-Smith samples is too large to make a meaningful comparison between the samples and the morphological transport estimates. For the three runs with the largest flow strength, the transport rate estimated from the sand sheet morphology was approximately equal to the estimates from the dune morphology. This suggests that the material moving over the stoss slope o f these dunes is equivalent to the material moved in the dune form. This counters arguments that only a portion o f the dune form contributes to the transport rate. A t one o f the lower flow strengths there appears to be more material moving over the dune than is moved in the dune itself. This presents an interesting problem that needs further attention.  82  Chapter 3-The Initiation of Bedforms on a Flat Sand Bed 3.1 Introduction The purpose o f this chapter is to discuss the initiation o f bedforms on an artificially flattened sand bed. The initial flow conditions over the flat bed, prior to bedform development, are investigated in order to establish that the flow agrees with standard models o f flow and turbulence over hydraulically rough flat beds. This establishes that the bedforms are not developed by some aberrant flow condition. Bedform development is examined with and without artificially generated bed defects (small indentations or mounds o f sediment). For both cases, a series o f micro-scale grain movements and bed deformations are documented that lead to incipient dune crestlines. A n attempt is made to link these bed deformations to some observable flow condition. A simple Kelvin-Helmholtz model o f water-sand interface instability is introduced to explain the initiation and organisation o f twodimensional dunes. The model is used to predict incipient dune lengths.  3.2 Experimental Procedure The experiments discussed in this chapter used the same apparatus, experimental design, and flow conditions as in Chapter 2. The runs were conducted at the same pump frequencies, /  = 23.5, 22.5,  21.5, 19.0 and 17.0 H z . The bulk flow hydraulics o f each run can be found in Table 2.1. Several runs were conducted at each /  to accomplish the goals in this chapter, so the same labels are assigned to  each flow strength to streamline the text. Flow strength A refers all runs conducted at / B all runs at f =22.S H z , C all runs at f = p  /  p  21.5 H z , D all runs at f = p  = 23.5 H z ,  19.0 H z and E all runs at  = 17.0 H z . Bedforms were initiated instantaneously during Runs A , B and C and by defect  propagation processes during at flows D and E .  83  3.2.1 Video Runs were video taped using the set up described in Chapter 2 with the Super-VHS video camera mounted above the flume and centred at -10.30 m from the head box. Recall that the side lighting produced a glare-free image with light shadows that highlighted millimetre-scale changes in the bed structure. The video records were sub-sampled from the tapes using a frame grabber at a 10s interval which produced series o f images similar to those in Figure 2.12 which were subjected to further analysis. This video was taken during Runs 53 ( A ) , 54 (B), 55 (E), 56 (E), 57 (C), 58 (D) and 59 (D). In addition to these videotaped runs, another series o f runs was conducted and recorded to determine grain paths and velocities on the bed. The sand surface was seeded with black glass beads (D=0.5 mm). A Plexiglas plate 25.4 mm (1 inch) t h i c k , - 0 . 3 m wide and -0.5 m long was suspended at the water surface with a bevelled edge that faced upstream. The plate surface was illuminated using two 500 W and four 100 W flood lamps such that light penetrated the glass and the view from above was free o f distortions caused by the water surface. The overhead video camera was focused on a portion o f the bed -0.2 x 0.2 m. Video of the particle movements was acquired for each o f the flow strengths as the bed developed from a flat condition. The video was then converted to - 3 0 s long image sequences with a frequency o f 30 H z . This video was taken during Runs 31 ( A ) , 32 (B) and 3 4 ( C ) . Subsequent data analysis revealed that measuring grain paths over time periods greater than a few tenths o f a second was nearly impossible due to the extreme variability in the grain movements and the video resolution. Particles moved along the bed spasmodically, sometimes moving though the entire field o f view and other times moving only a few millimetres. A s particles moved they were frequently buried, while new particles were constantly uncovered and activated. This variability could be quantified, but particles seemed to be extinguished as they moved through the video line boundaries. V i d e o images are typically composed of several hundred horizontal lines that blend together to produce a continuous field when examined by the human eye. However, the lines have boundaries and particles were small enough that they could follow the space between two video lines. 84  In such cases, a particle was hidden during part o f its transport. Therefore it became impossible to determine which particles were coming to rest or being activated by the transport process and which particles were simply obscured by the spaces between video image lines. This occurred despite the fact that the particles were roughly 1.5 - 2 x larger than the width o f the video lines. It would benefit future investigation to use digital images that are not composed o f horizontal lines or to substantially increase the ratio o f the grain size to the video line width or pixel size (i.e. increase to l O x ) . In spite o f this problem, the video could be used to quantify the movement o f a limited number o f the black particles over short time periods. A n image viewer that could display the images grabbed from the video in rapid succession (100 Hz) was used to view 0.167 - 0.667 s sequences immediately after widespread particle transport had begun. Particle movements were then marked directly from the computer monitor onto a sheet o f acetate. The acetate was then calibrated and digitised in a G I S program to determine the particle movement distances over the elapsed sequence time, allowing the calculation o f particle velocity, U . Between 45 and 90 grain movements were identified at each p  flow strength to estimate grain velocity. In order to be recorded as a movement, a particle could not rest for any period o f time during the measurement period and was required to be moving at the beginning and end o f the sequence. The measurement period was varied between 5 and 20 s so as not to bias the mean towards longer movements. Only particles that did not become significantly obscured between video line spaces were accepted. It is important to note that the velocity measurements are for the top o f the transport layer only and U  p  must logically decrease to zero at  some depth.  3.2.2 Echo-sounder Mapping In addition to the video, the bed was mapped using the two acoustic echo-sounders used in the experiments discussed in Chapter 2. The sensors were mounted with a cross-stream separation o f 0.45 m on the motion control system and the bed height was mapped. The motion control system is  85  composed o f an instrument cart that rode on iron roundways installed on each side wall that were levelled to within ± 0.5 mm along the flume. A computer-controlled stepper motor, gear reducer and a sprocket riding on a motorcycle chain on the flume side wall drove the cart. A l s o installed on the cart was a ball-screw axis, driven by stepper-motors, that allowed instruments to be positioned in the cross-stream stream direction. A t the start o f mapping, the sensors were placed at 0.05 and 0.50 m from the side wall, at the upstream edge o f the working section. The motion control system was programmed so that the sensors could be moved: (1) across the flume 0.45 m, (2) downstream 5 mm, (3) across the flume in the opposite direction and (4) downstream 5 mm, all at 10 m s" . This motion was then looped up to 1  600 times. O n either side o f the flume, 25.4 mm (1 in.) wide, 3.66 m (12 ft.) long aluminium stock, extending 0.05 m from the side walls, was hung 5 m m below the sensor height. Each time the sensor reached the sidewalls, the sensor recorded zeros indicating the end o f a cross-stream line o f data. The echo-sounders were sampled at 25 H z so that each sensor would collect - 9 0 data points on each traverse across the flume. The collected motion produced a grid o f 180 x 600 data points that took - 2 8 - 29 minutes to complete. These could then be plotted as contour maps to give detailed snapshots o f the bed topography. Mapping was done before each run and approximately every hour during flume runs. The flume was not stopped during data acquisition so as not to disrupt the topography development. Subsequent analysis of the grids indicated that at the greatest three flow strengths ( A , B and C ) the rates o f bedform development and migration were too fast to provide any useful information (i.e. the distortion o f the bedforms was too great). In contrast, at the lowest two flow strengths, development was sufficiently slow so that distortion was quite minimal and the full developmental process could be monitored.  86  3.2.3 Laser data  Velocity profiles were obtained for all five flow stages (described above) over an initially flat bed using a 300 m W Dantec laser-Doppler anemometer ( L D A ) that measures two-component flow velocities (u and w) and has a reported precision o f ± 0.1 mm s~', a focal length o f 750 mm and a sampling volume o f 20 x 0.2 mm [Dantec Measurement Technology, 1995]. The laser acquired data at 20 H z near the bed, and up to 1 k H z in the upper part o f the profiles when velocity measurements were collected coincidentally. M u c h larger sampling frequencies were obtained when velocity components were measured independently o f each other. Measurements with a sampling frequency less than 20 H z were discarded. The L D A was operated in backscatter mode with burst type detectors so the L D A collected velocity samples whenever there were particles available in the sampling volume. Data were not collected at prescribed time intervals and so the data are irregularly spaced in time. The L D A probe head, connected to the laser v i a a fibre optic cable, was mounted on a 3-axis motion control system that consisted o f three ball-axes, driven by stepper-motors, allowing the probe head to be adjusted quickly and with great accuracy (fractions o f a millimetre). The motion control system was set so that the measuring volume was normal to the flume wall and offset 4 degrees from the vertical plane. This allowed the measuring volume to be placed in the centre o f the flume, 5 m m above the bed, while still measuring velocity data on both components. Rather short 60 s samples were collected at up to 15 points in the vertical plane for each profile. Previous research over dunes in this flume by Venditti and Bennett [2000] indicated that this sampling period is sufficient to acquire reliable estimates o f mean flow and turbulence without bias using an acoustic Doppler velocimeter. Unfortunately, the sampling period could not be extended as the bed developed too quickly. T w o sets o f profiles were taken to ensure consistent results with the short sampling period. Each set consists o f a profile at each flow strength. In profile set 2, an additional  87  velocity profile was taken at the largest flow strength ( / =23.5) as the bed developed rapidly. Profiles were also acquired over artificially made bed defects, mounds or pits in the flat bed, at the two lowest flow strengths. A t the beginning o f the second set o f profiles, a 600 s velocity sample was taken at 5 mm above the flat bed. These long time series were used to examine integral scales o f flow and to provide accurate estimates o f the near-bed flow properties.  3.3 Initial Flow Structure Before discussing the initial bed features developed in the flume experiments, it is necessary to discuss the structure o f the flow that was present as these features were initiated. The data set collected here is o f further interest as it is somewhat unique. Flow over fixed flat sand beds has been examined extensively, as has flow over fixed and mobile sediment beds with bedforms. However, data of flow over a lower regime labile sand bed without bedforms are less common. It is o f interest to determine how the flows used in these experiments compare with conventional models o f flow over sediment beds.  3.3. J Velocity Profile Data Analysis  Analysis o f the velocity profiles began with calculation o f mean and root-mean-square velocities. Time-averaged at-a-point streamwise, U, and vertical, W , velocities were calculated as  U= -±u \W= -±v» X  3.1  X  t  t  ;=1  i=\  where u, and w, are instantaneous velocities and n is the total number o f measurements. The mean streamwise velocity, U, o f a roughly logarithmic velocity profile is U at 0.36 d . Root-mean-square velocities for the streamwise, U , and vertical, W , components, were calculated from rms  rms  0.5  U  mis  = 1=1  -,0.5  • W = ' " rms  3.2  88  The Reynolds shear stress, r , uw  was determined using  u'W = -Y {u,-U)(w -W) i  r  = w  where p  3.3  l  - P w " '  w  '  3.4  is the fluid density.  w  The boundary shear stress was estimated for each profile using four methods. The first method uses the depth-slope product as in Chapter 2 (denoted T ). A second method is based on the von s  Karman-Prandtl law o f the wall. At-a-point U velocity profiles were plotted as a function o f height above the bed, z , and least-squares regression was used to determine the roughness height, z „ , as the z intercept. The shear velocity is calculated from  U,  Z  K  0  is the mean velocity at z and K is the von Karman constant which is assumed to be 0.4.  where U  z  The boundary shear stress based on the law o f the wall (denoted T  0 2  ) is determined from  r = p u> .  3.6  2  w  The law o f the wall is strictly applicable only to the log-layer where the increase in U with z is logarithmic. In fully turbulent open channel flow, this region extends from a few m m above the bed to 0.2d (d is flow depth) [Nezu andNakagawa,  1993]. A s such, equation 3.5 is applied to only the  lower 20 % o f each profile. A third method o f determining boundary shear stress involves plotting x  as a function o f zl d .  im  For uniform flows T  llw  = T {\-zld).  3.7  R  A n estimate o f T  R  can be obtained by using a least-squares regression projected to z / d =0 [see Nezu  and Rodi, 1986; and Lyn, 1991]. A final method is to use a measurement o f  89  T  tlw  at just above the  boundary as the boundary shear stress (denoted x ) [cf. Nelson et ai, 1995]. In these experiments, B  the 600 s u and w velocity measurement was used to determine T  B  u  t  and z  0  using Equation 3.4. Estimates o f  are obtained through the final two methods using Equations 3.5 and 3.6 respectively.  3.3.2 Boundary Shear Stress  Figure 3.1 plots the calculated boundary shear stress values as a function o f velocity. Neither T  s  nor T  S  is based on the profiles so there is only one unique value o f each. Both T  0 2  and T  R  are based  on the profiles so there are two values at each mean velocity (three at flow strength A ) . Error bars in Figure 3.1b and c are the standard error from the least-squares regression. Examination o f Figure 3.1b and c indicate that estimates o f the boundary shear stress are dramatically different for different profiles at the same flow strength, although many estimates are within the error bars associated with the estimate from the other profile set. The most serious deviations in T at flows C and D while for r  R  0 2  between profile sets occur  the greatest deviation is at flow strength B . Perhaps a more significant  problem than these deviations is that T  0 2  and T  R  do not necessarily increase with flow strength when  a profile set (1 or 2) is examined individually. Examination o f velocity profiles showed the mean and root-mean-squared velocities were well represented and little variation was observed between points measured at the same height during different profiles. Velocity covariance ( « T  i w  ) also appeared to be well represented, but significant  differences were observed between the measurement points at the same height but different profiles. This affected all subsequent calculations using the Reynolds stress profiles. There are several reasons why discrepancies in the expected pattern o f T  0 2  and T  R  may occur  while the averaged values seem to be in accordance, including measurement error associated with the L D A . Voulgaris and Trowbridge [1998] have previously noted problems with the L D A resolving turbulence while providing acceptable mean velocities. Another source o f error may be the stage o f  90  bed development. A l l profiles were taken while the bed appeared to be flat however, as w i l l be discussed below, micro-scale bed deformations occur on the flat bed, so it is conceivable that the flow is responding to both grain roughness and form roughness [e.g. Smith and McLean, 1977]. Since the micro-scale deformations are growing with time, this may cause some variability in profiles. Despite these considerations, the most likely cause o f the error is the short sampling time used for the profiles. Given this uncertainty, calculated means, root-mean-sqaures, and covariances were averaged for points measured at the same height during different profiles for each flow stage. When averaging profiles, only points with two (or three) data points were averaged. When averaging U and W (U  and W ) profiles, data at nearly all the heights above the bed could be included.  nm  rms  When averaging T  UW  values, the sampling frequency o f some near bed points was below 20 H z and  there were not two (or three) data points to average. Therefore, the combined T  UW  profiles are  truncated at ~0.2 d . The combined U and x  tm  T  0 2  and T  R  profiles produced more consistent profiles and a consistent increase in  with flow strength (Figure 3.1b and c). One exception was the T  uw  flow B that produced a  profiles taken during  value that was inconsistent with the general pattern even after the  averaging. This point, and the z  B  value taken during profile set 2, are likely biased by some  measurement error and are removed from further consideration. B y examining the increase in T , S  X , combined values o f T B  0 2  and z , R  it is clear that regardless  o f which calculation is used, the boundary shear stress increase with U is non-linear. The boundary shear is nearly the same at flow strengths D and E while at A - C the increase with U is more pronounced. Figure 3.2 displays the boundary shear stress plotted against T  0 2  for both the individual  and combined profiles. Regardless o f whether the individual or combined profiles are used, T  0 2  ~ T ~ T . This is encouraging as it suggests that the flow does not differ significantly from R  B  other documented uniform flows. In contrast, r  s  ~ 2.24T 91  02  . It is significant to note that T  s  values  2.00  1.25  -i—i—i—i—i—i—  a) T , Depth-Slope Product s  1.75 r  1.00  1.50  0.75  1.25  0.50  1.00  0.25 O  0.75 0.30  _i—i—i—.  0.35  O .  i  0.40  0.45  0.50  0.55  0.00 0.30  0.35  0.40  0.45  0.50  0.55  0.35  0.40  0.45  0.50  0.55  1.25  1.25 c) x , Reynolds Stress Profile R  1.00  1.00  5  T  0.75 h  0.75  0.50 r  0.50  i  0.25  i A • •  0.00 0.30  0.35  0.40  0.45  Profile Set 1 Profile Set 2 Combined  0.50  0.55  0.25  0.00 0.30  -1x U (m s  U (m s " ) 1  Figure 3.1: Calculated boundary shear stress plotted as a function o f velocity, U. Note the different x scale used for T . Error bars are the standard error o f the estimate. s  92  \2  ( ) Pa  Figure 3.2: Measures o f boundary shear stress, x , plotted against the estimate based on the von Karman-Prandtl law of the wall estimate, x  0 2  , for both the individual (top) and  combined (bottom) profdes. Boundary shear estimates are based on the Reynolds stress profile, x , the Reynolds stress at 5 mm above the bed, x , and the depth-slope product, / ;  T  s  R  . Error bars are the standard error o f the estimate.  93  represent a different type o f boundary shear than T  0 2  and T which represent local shear, linked only B  to the bed roughness (primarily grain roughness). Shear estimates from the depth-slope product represent the total bed roughness, which may also include roughness induced by the bed topography and the sidewalls. Thus, some difference between the values produced by different methods is expected. However,  is based on the total profile and should be similar to x . The data do not s  reveal a strong correspondence between these values.  3.3.3 Mean and Turbulent Flow Table 3.1 summarises the flow conditions from the experiments based on the combined velocity and stress profiles (including values based on T  s  and T ). B  The roughness height is variable  depending on which method is used to determine the boundary shear stress. However, z  o 0  , ranged  between 0.0064 and 0.0839 mm, an order o f magnitude. This translates into equivalent sand roughness values, k , determined from s  k =30.2z„,  3.8  s  [van Rijn, 1993] that range between 0.19 and .2.53 mm using the T  0 2  estimate. Yalin [1992] suggests  k ~ 2D based on experiments with movable yet stationary grains. This relation yields k ~ 1.0 mm, s  s  which is within the range o f observed k  s  values.  The combined U profiles are presented in the top panels o f Figure 3.3. The velocity profiles are linear through the lower 0.2 d , but there is a kink in the upper portion o f the profiles at -0.5 d . This is not surprising as the profiles are expected to be linear only through the log-layer which is generally accepted to extend from a few mm above the bed to 0.2 d [Nezu and Nakagawa, 1993]. The u and w turbulence intensities, calculated from  j  U rms j »0.2  M  W rms  ^ Q  *02  U  94  Table 3.1: Summary o f flow parameters. Flow Parameter Flow A Flow B /p>Hz  Flow C  Flow D  Flow E  23.5  22.5  21.5  19.0  17.0  d ,m  0.1516  0.1517  0.1533  0.1530  0.1534  J7,ms"'  0.5009  0.4768  0.4538  0.3993  0.3558  U  0.5929  0.5588  0.5370  0.4556  0.4014  Fr  0.4107  0.3908  0.3700  0.3259  0.2900  Re  75936  72331  69568  61093  54580  0.0759  0.0723  0.0696  0.0611  0.0546  max  g.nrV S^oXlO  - 4  12  11  7  5.5  5.5  Determinations based on depth-slope product 0.0422  0.0405  0.0324  0.0287  0.0288  T^Pa  1.7811  1.6337  1.0506  0.8239  0.8260  ffs  0.0569  0.0576  0.0409  0.0414  0.0523  u, ,ms  l  s  Determinations based on Law of the Wall using lower 20% of averaged profde w.o^.ms"  0.0301  0.0255  0.0219  0.0171  0.0156  ,Pa  0.9015  0.6498  0.4807  0.2911  0.2423  ,mm  0.0839  0.0438  0.0225  0.0077  0.0064  £, ,mm  2.5349  1.3235  0.6781  0.2338  0.1926  0.0288  0-0229  0.0187  0.0146  0.0153  T  0 2  z  o 0 2  1  0 2  #0.2  Determinations based on linear portion of the vertical Reynolds stress profde iU ,ms  0.0276  n/a  0.0253  0.0185  0.0187  Tfl,Pa  0.7625  n/a  0.6369  0.3413  0.3496  z ^,mm  0.0509  n/a  0.0526  0.0158  0.0219  A  R  0  k ,mm  1.5374  n/a  1.5899  0.4782  0.6629  ff  0.0244  n/a  0.0248  0.0172  0.0221  sR  R  Determinations based on Reynolds stress measured at 5mm above bed (10 min average) 0.0268  0.0280  0.0217  0.0177  0.0116  T , Pa  0.7176  0.7822  0.4691  0.3114  0.1346  z ,mm  0.0433  0.0744  0.0221  0.0117  0.0005  k ,mm  1.3066  2.2477  0.6682  0.3534  0.0157  ff  0.0229  0.0276  0.0183  0.0157  0.0085  u, ,ms  l  B  B  o B  sB  B  Fr = Ul(gdf\  Re = dU/v,k  = 3 0 . 2 z , ff = 8 T / pU , 2  s  o  95  u,=(T/p f , 5  w  z  oR  ovz  oB  =  (ln e  a IU D  ^  2  o u  PH  >  u-i  "3H  2 ° >  £  p  6  c  03  ao ^  E tt  i  | < rTfi-'i  1  1  ;  ***•< • :  :  •  3  _ : • -  LU  'I  ' ' ' ' I i ' ' i I i  3 3  O  i  i ; i i r r -| i i i i | i i i i | i  1  •  CM  :  • • ••  0 2  LO  CD • '  i I i i L i I I l„i l.J.,1  O O P/z  P/z  96  P/z  are plotted in Figure 3.3. A l s o plotted are the semi-theoretical, universal functions for turbulent intensity provided by Nezu and Nakagawa [1993], calculated as /„=£>„ e x p ( - C ^ z / r f )  3.10  I =D exp(-C z/d)  3.11  w  w  kw  where D , C , u  ku  £> ,,and C M  kw  are empirical constants. These have been previously determined  experimentally for laboratory open channel flows as 2.30, 1.0, 1.27, and 1.0 respectively [Nezu and Nakagawa,  1993]. These values are not universal constants but depend on bed roughness and,  potentially, the presence of bedforms. For example, Sukhodolov et al. [1998] calculated a new set o f constants for flow over a bedform field that were somewhat different from those provided above. In consideration o f this, a new set o f constants was calculated using least-squares regression for the flows in these experiments (Table 3.2). Values o f D , D ,&nd u  by Nezu and Nakagawa [1993], but C  kw  with z/d  w  C  ku  are similar to those calculated  values are significantly different. The distribution o f I  w  is nearly linear for the lower flow strengths and fitting an exponentially decreasing curve  seems unwarranted. The combined T  IM  profiles appear in the top panels o f Figure 3.4 and show a roughly linear  decrease with z , as is expected for uniform flow. A l s o plotted in Figure 3.4 is the boundary layer correlation coefficient, R , calculated as im  R  "  wm  = ^ ^ — U -W rms rmx  3.12  The boundary layer correlation coefficient (-1 < R  uw  < 1) is a normalised covariance that expresses  the degree o f linear correlation between u and w velocity fluctuations. A s such, R  uw  is a ' l o c a l '  statistic that provides insight into the presence or absence o f flow structure at a specific location. In flow over a flat bed, there is little streamwise variation in R  uw  and values o f ~0.5 are typical o f the  near-bed regions, while decreased values o f 0.0 - 0.3 are found in the outer flow region [Nezu and  97  Table 3.2: Values o f parameters in Equations 3.10 and 3.11 evaluated from measured profiles. Flow  n  r  A B C D E Mean  "  2 . 2 8 " " " 0 . 2.49 2.59 2.61 2.61 • 2.52 •  D  C  C  _w  Ww  W» ^  u  WH  8 3 0.84 0.70 0.55 0.39 0.66  T"21 1.44 1.42 1.34 1.49 1.38  98  0.49 0.44 0.23 0.05 0.08 0.26  +C ^kw  1.32 1.28 0.93 0.60 0.47 0.92  o c >^ CD  CO  03  -a c 3 o  CO  o CJ  o CD  2 —;  CL  CD  3 13 CD  L. 3  00  £ I eft  >, -o CD -a c CS  •  ~  .  -  co  in  TZ  CD  0 0  CL>  L-  X)  CD  CJ  • -  c ^3  Co  C  03  _  .23  co  CD >  in  c o 3 C3"  CD  SS  •a  to  3  —  cn  O  CT  Ltf LU  -o >  - o  03  >  CD  • c o «  o  oo cfa • £  CL  'co  3  3  8 _.  C>  • a Os  CD  CD T3 CD  c S3  03 X> r—  -a .b CD  CD  £ S too £ 3C  =  03  o -a  co -t-J  C  5  O  -a t3 -a o CD  o  CO  a>  o  '= I" S  r-  ^  t3 n  CD  3  S§ C  °  CL  CD C  O  CN  lo  ci  CD  £  |  CD «  T3  :•< CD CL  ••  5 3  P/z  O  CO  CD  i-^  d  d  CM  d P/z  99  o  o d P/z  Nakagawa, 1993]. Sukhudolov et al. [1998] provide an expression based on Equations 3.10 and 3.11 for R uw as a function o f z/d  , \-zld  R  "  D^expi-iC^+C^z/d)  w  that is plotted in Figure 3.4 using the constants derived by Nezu and Nakagawa [1993] and those derived here (Table 3.2). The Nezu and Nakagawa [1993] empirical constants provide R  uw  that are too large at all z/d  . In contrast, the empirical constants derived here provide a decent fit to  the data with the exception o f the two lowest flow strengths. Here, the predicted R  values are too  llw  small at all z/d  values  . This is likely because the distribution o f I  w  is not an exponentially decreasing  function with height. A final important characteristic o f turbulent flow is the eddy viscosity e, as defined in the mixing-length concept, which is the strength or magnitude o f the turbulent eddies within the flow and is defined as  £ =  -uw  „ ,  — .  A  3.14  dU/dz  If the entire velocity profile is assumed to be logarithmic £ = KU,z(\-zl  d)  3.15  [Nezu and Nakagawa, 1993]. Bennett et al. [1998] note that, in flat-bed flows, the distribution o f eddy viscosity is generally parabolic, reaching a maximum near 0.5 d . In order to calculate £ , the absolute values o f dU I dz needed to be used because in some places the local slope o f the velocity profile was negative. Plots o f £ for each flow strength are given in Figure 3.4. A l s o plotted are curves that represent the fitted T  u w  curve divided by the fitted U profile curve and equation 3.15. A s  would be expected, the fitted curves match the data nearly identically. These parabolic £ curves reach a maximum between 0.4 and 0.6 d . However, at flow strength D the £ curve reaches a maximum around 0.8 d , suggesting that the zone o f maximum fluid diffusion is shifted upwards in  100  the vertical profile relative to other flows. This seems unlikely given the curves at other flow strengths. It is likely that the curve fits for U and T  im  deviate fortuitously to generate this result.  3.3.4 Effect of a Bed Defect Most authors who have examined the development of sand bedforms have argued that the erosion in the lee o f the initial features, the development o f the scour pit and, ultimately, the propagation o f the initial features is caused by the development o f a flow separation cell [c.f. Raudkivi, 1966; Southard and Dingier, \91\;Best, 1992]. A velocity profile taken at flow strength E just downstream o f a negative defect as it developed is characteristic o f flow separation (Figure 3.5). The defect was ~10 mm deep and had a circumference o f 30 mm at flow E . Profiles taken over the defect bed and over a flat bed match closely in the range z/d  =0.2 - 1. However, below zld  dramatically reduced relative to its flat bed value and U  rms  = 0.2, U is  is increased, providing the turbulent  energy to scour out the bedform lee.  3.3.5 Integral scales Autocorrelations for the velocity time series were derived to determine integral time and length scales in the near bed region using the 600 s time series collected at 5 mm above the bed. A n integral time scale is the time an eddy requires to pass a given point in the flow, and an integral length scale is the characteristic eddy dimension. The Eulerian integral time scale, T , is defined as E  k 3.16  o where R{t) is the autocorrelation function, d£ is the lag distance, and k is the time step at which  R(t)  is no longer significantly different from zero [Tennekes and Lumley, 1972]. With a Taylor [1935] approximation, the Eulerian integral length scale, L , is defined as E  T -U  3.17  F  101  i  i  I  I  V  r  ol  °\ o  T3  75  5  0.1  0.01 0.1  1 0.2  0.3  0.4  0.5 0.02  U (m s" )  0.04  O  0.06  0.08  1  U  (m a" ) 1  r m s  Figure 3 . 5 Profiles o f mean streamwise velocity, U, over a negative defect at F l o w E. Height above the bed, z , is normalised by the flow depth, d . Open circles are data measured over the defect and lines are profiles measured at the same flow strength without the defect.  102  where U is measured at-a-point. Calculation o f autocorrelations required that linear interpolation be used to convert the time series that had irregularly spaced data through time to regularly spaced time series. A sampling frequency of 75 H z was selected for the new time series as that was the minimum observed file averaged sampling frequency for the 600 s time series. Mean and root-mean-square velocities were nearly identical between the regular and irregular time series. Each 600 s time series was divided into five 120 s sections that were detrended. Sample autocorrelations for these 120 s time series are shown in Figure 3.6. In general, R(t) approached zero asymptotically. In those cases when R(t) approached and then oscillated about a zero value, k was determined when R(t) = 0.01. Table 3.3 presents the average streamwise T  E  L  E  . Integral time scales do not vary greatly with flow velocity. Mean T  E  0.271 s while mean L  E  and  varied between 0.225 and  varied between 0.0620 and 0.0758 m. Thus, in the near bed region, the  average or dominant eddy size is -0.07 m.  3.5.6 Summary of Flow Conditions Overall the velocity data suggest that flow over the lower stage plane beds at the beginning o f the experiments is in accordance with conventional models o f uniform flow over flat beds. With the exception o f the depth-slope product calculations, estimates o f the boundary shear stress derived from different methods are similar in magnitude and increase in a similar fashion with U. The roughness heights are consistent with previous observations. The turbulence intensities and R  uw  can be  modelled by the semi-empirical functions provided by Nezu and Nakagawa [1993] with the exception of the vertical intensities at the low flow strengths. Finally, with few exceptions, the momentum exchange (x ) and fluid diffusion ( e ) conform with current formulations for fully turbulent, uw  uniform, open channel flows over flat beds. Bed defects dramatically reduce the streamwise velocity  103  0  2000 i  1  1  4000  1 1 r--i—i  1 1 r-  6000  8000  i  6000  8000  Lags Figure 3.6: Sample autocorrelation functions (act) for 120 s time series drawn from the 600 s velocity measurements. The displayed acf are for the fourth (u4) or fifth (u5) segments o f the 600 s time series at flows A , B , and D .  104  Table 3.3: Integral time and length scales. Values are mean and standard deviation (brackets) o f the five 120 s time series extracted from the 600 s time series. Flow A-l  A-2 B  C D p  U  (cm s" )  T (s)  1  05  E  28.50 (1.29) 29.90 (1.15) 28.00 (0.97) 28.31 (0.91) 26.72 (0.34) 24.73 (0.55)  0.25 (0.03) 0.25 (0.07) 0.27 (0.04) 0.23 (0.04) 0.27 (0.06) 0.25 (0.06)  105  L  E  (cm)  7.04 (0.78) 7.37 (2.14) 7.58 (1.09) 6.36 (1.06) 7.18 (1.59) 6.20 (1.44)  near the bed, increasing turbulence and providing the turbulent energy to scour out the bedform lee. Integral length scales are - d 12 in the near bed region.  3.4 Bedform Initiation modes T w o types o f initiation were observed in the experiments: defect initiation and instantaneous initiation. A t the two lowest flow strengths used in the experiment (D and E ) , a flat bed appears to be stable for long periods o f time, even though the threshold o f motion has clearly been exceeded. M a n y early researchers [e.g. Menard, 1950; Simons and Richardson, 1961; Raudkivi, 1963] have suggested that any sediment motion on a flat bed w i l l lead to bedform development. However, others [Liu, 1957; Bogardi, 1959; Southard and Dingier, 1971] have reported stable flat beds with sediment transport and no bedform development. The flat beds observed here could not be maintained indefinitely. Sediment carried into the head box was ultimately deposited at the entrance to the channel that developed small mounds o f sediment. Eventually, these mounds o f sediment developed into bedform trains that propagated through the flume channel. A t flow strength E the bedform fields took nearly three hours to migrate from the head box to the area where observations were being made (-9.8 m). A t the second lowest flow strength (D) the bedform fields took - 3 5 - 45 m i n to migrate the same distance. A s the bed was flattened, millimetre scale indentations or mounds o f sediment (defects) formed at the sidewalls or occasionally in the channel. A t flow strength D , these defects tended to propagate downstream forming bedform fields. Interestingly, not all defects developed into bedforms. I f sediment entering the flume did not form mounds, it is possible that the stable flat beds could have existed indefinitely. In contrast to this type o f bedform development, at the greater flow strengths bedforms were observed to develop instantaneously over the entire bed. This type o f development occurred within a few tens o f seconds, with the initial appearance o f the pattern that leads to bedform crestlines appearing after only a few seconds o f flow. Hence, this type o f bedform development cannot be  106  linked to defect propagation from the head box or sidewalls. In fact, where defects were observed on the flat bed, they were washed away as the initial bedform pattern was imprinted on the bed. The threshold between the two types o f initiation was just above T threshold for the sand used in these experiments occurs at T  0 2  02  = 0.29 Pa. A practical  ~ 0.30 P a which corresponds to a  dimensionless shear stress  6=  ^  gD(p  s  = 0.0371  3.18  - p J  where D is the grain size (0.5 mm) and p  is the grain density (2650 kg m ). Both types o f initiation 3  v  are examined in greater detail below.  3.4.1 Defect Initiation Following the work o f Southard and Dingier [1971], who examined ripple propagation behind positive defects (mounds) on flat sand beds, the defect type o f bedform development was examined from artificially made defects rather than examining bedform growth from random features on the bed. It seems likely that negative defects (divots) are also o f interest in terms of bedform development. Therefore, both positive and negative bed defects were used in the experiments herein. The defects were generated by either sucking sand into or depositing sand from a large dropper until the desired defect size was attained. The dimensions o f the defects in the runs discussed below are given in Table 3.4. The defects were cone shaped and - 3 0 mm in diameter. Negative defects were 8.2 - 9.4 mm below the mean bed elevation, z and positive defects were 8.2 - 10.4 mm above z . Southard and Dingier [1971] noted that there was some effect o f mound height on the development of ripples in their experiment. However, this conclusion is drawn from a series o f runs using defects that varied in height between 2.5 and 50 mm. The effect o f the variations in defect size here is probably insignificant.  107  Table 3.4: Defect Dimensions. Flow  f  P  (Hz)  D  19  E  17  Run  Defect Circumference (mm)  22 58 21 59 20 56 19 55  28 30 33 26 30 28 30 27  108  •  Defect Depth/Height Relative to z (mm) +8.2 + 10.0 -9.4 -8.2 +8.5 +10.4 -8.2 -8.5  The basic development pattern o f the bedform fields can be observed in a series o f bed maps displayed in Figure 3.7. Although maps were generated for positive and negative defects at flow strengths D and E , they were qualitatively similar at the coarse scale afforded by the bed mapping. Therefore, only one bedform field development sequence is displayed, that developed from a positive defect at flow strength E . Figure 3.8 displays the bed height along the field centre line. The first bedform map was taken without any flow and simply shows the positive defect (Figure 3.7a). The planar nature o f the bed and the defect shape are evident in cross-section (Figure 3.8).  Subsequent  maps (Figure 3.7b-d) display the defect propagating downstream and developing new crests. The initial defect grew in H and L with each bed map while the rest o f the field is composed o f bedforms that decrease in H and L with distance downstream (Figure 3.8). A s the bedform propagation progressed downstream, the bedform field widened at a regular rate until it reached the sidewalls, forming a triangular shape with the initial defect at its pinnacle (Figure 3.7e). In cross-section the bedforms began to take on a uniform H and L (Figure 3.8). The map in Figure 3.7f shows the triangular field continuing to modify the bed, forming larger bedforms. Bedforms developed at the inlet to the flume travelled into the mapped area and merged with the defect bedform field. This is particularly evident in cross-section where the lead bedforms from the head box are much smaller than those in the defect field. In the next bed map (Figure 3.7g) the two bedform fields have merged, but the defect field is still identifiable as larger bedforms. With subsequent bed mappings, the bedform field continues to merge and there is increasingly little observable effect o f the original defect pattern (Figure 3.7g-i and Figure 3.8). The run was extended until equilibrium H and L were reached for this flow strength (Figure 3.7j). A s noted above, mapping o f the bed took too much time to examine the development process in any detail and, for this purpose, the video data are examined. The way defects initially develop bedforms differs slightly between positive and negative defects and thus it is useful to describe them separately. Time-lapsed image sequence animations o f bedform development from defects can be found in Appendix B . Figure 3.9 shows the development o f the first eight bedforms from a negative 109  o  a) Prior to flow  b) t=0-29 min  Vertical Scale 20 mm  c) t=32-60 min  d) t=62-91 min  e) t=92-147 min  f) t=163-215 min  g) t=218-268 min  h) t=270-323 min  0.0  0.5  1.0  1.5  2.0  2.5  3.0  Distance A l o n g the F l u m e (m) Figure 3.8: Cross-sections drawn along the centre line o f Figure 3.7. The map begins at 8.45 m from the head box.  Ill  bed defect at the lowest flow strength (E). A s the flow is increased to the desired flow strength a negative defect undergoes the following deformations: l a . Initial divot stretching in the downstream direction to develop a shallow scour pit (Figure 3.9b); 2a. Sediment eroded from the scour pit begins to accumulate at the downstream edge; 3a. The edge is squared off transversely as in Figure 3.9b; 4a. Erosion o f the stoss slope of the new bedform building the crest in height; 5a. A new shallow scour pit develops at the downstream edge o f the new bedform crest; 6a. Steps 2a - 5a are repeated to propagate the feature downstream, forming new bed features (Figure 3.9c-i). Concurrent with this process, the original defect formed a chevron shape rather than developing into a bedform (Figure 3.9d). N e w incipient bed features tended to have crests that were transversely narrow for the first few crests (Figure 3.9c-f). The previously developed crests widened transversely, grew in height, and lengthened in the streamwise direction with time. A s the upstream crests became wider, so did the subsequent forms. Interestingly, once the field grew to include five new crests, the whole field began to migrate downstream. This initial stability o f the bedform field was also observed at the flow strength D , but the migration began after only two new bedforms appeared. Figure 3.10 displays bedform L along the centre line o f the video o f the initial defect and the first five new bedforms as a function of time. The initial negative defect grew in length for a period o f time and reached quasi-equilibrium. A t the higher flow strength (D), L then decreased and the defect eventually disappeared as it was filled in from upstream. Each new bedform had approximately the same L and appeared to grow in a fashion similar to that observed in Chapter 2. Unfortunately, no data on H could be practically derived from the video and the maps were too infrequent to extract detailed information. The development o f a bedform field from a positive defect appears to differ in several respects. Figure 3.11 shows the development o f the first bed feature from a positive bed defect at the lowest 113  20  -i—'—i—i—i—r  i—i—i—i—i—i—r-  Flow E - Negative Defect  15  I 10  i ii  i i  ii•i•iii  • i •  iiii ii  Flow D - Positive Defect  _J—i_j—i—i I i—J i i I  0.0  0.5  1.0  1.5  2.0 2.5  0.0  Time x 1 0 (s)  0.5  1.0  1.5  i_  2.0 2.5  T i m e x 1 0 (s)  3  3  Figure 3.10: Bedform L for the first five new bedforms developed from negative and positive defects. The initial defect dimensions are given in Table 5.3. Initial defect length (closed circles); 1 new bedform (upward triangles); 2 new bedform (squares); 3 bedform (diamonds); 4 new bedform (downward triangles). st  n d  t h  114  r d  115  flow strength (E). In contrast to the process described for a negative defect, as the flow is increased to the desired flow strength, a positive defect undergoes the following deformations: l b . Erosion o f the stoss side o f the initial defect, stretching, and streamlining the feature (Figure 3.1 l b ) ; 2b. Concurrently, the edge is squared off transversely as in Figure 3.1 l b ; 3b. The defect develops arms that stretch downstream forming a barchanoid feature (Figure 3.11c); 4b. Erosion in the lee o f the defect forms a scour pit between the barchan arms; 5b. Once the scour pit is developed, steps 2a-5a are repeated to propagate the feature downstream forming new bed features as in Figure 3.1 l d . Once a few new bedforms have developed, the initial positive defect is planed out to the mean bed elevation. This left only the scour pit that eventually developed into a chevron shape not dissimilar from the feature in Figure 3.9d-i. This also happens to the first new bedform after the initial positive defect is gone. This observation is interesting as it suggests that the positive defect is important only for generating the first scour pit and that it is the scour pits that are propagated downstream rather than the crests. This makes sense because the sediment that forms the ridges must be picked up from the pits. Thus, scour pits seem to be more stable features than the mounds that are generated at their downstream edge. Initial growth o f the bedform field from this first scour pit is similar to growth from the initial negative defect. Regardless o f the defect form, bedform fields developed from negative (Figure 3.12a) and positive (Figure 3.12b) defects at flow strength E have approximately the same form, when the fields are composed of five or more bedforms. The observed processes were qualitatively similar at different flow strengths, but the development was much quicker at the larger flow strength (D). Bedform fields developed from negative (Figure 3.12c) and positive (Figure 3.12d) defects at flow strength D enveloped the field o f view in only 20 min compared to 45 min at the lower flow strength. A t both flow strengths, the positive field was slightly more developed than the negative field although 116  117  3  the effect was more pronounced for the larger flow. There was more sediment initially available in the positive case as a negative defect must excavate the sediments from the scour pit initially while the positive defect offers an ample supply o f sediment above the mean bed elevation in the defect itself. The positive defect bed began developing with less energy expended and is more efficient in generating the first bedform. The positive beds developed more quickly because o f this advantage. Artificially made defects placed on the bed at the larger flow strengths ( A , B and C ) failed to persist. The defects were simply washed out and little or no remnant was observed only a few seconds after the flow was started.  3.4.2 Instantaneous Initiation A t flow strengths A , B and C , bedform initiation occurred spontaneously over the entire bed surface. The development o f the bed from flat to two-dimensional dunes is documented by a series o f video images in Figure 3.13. Initially, the bed was covered with lineated striations, oriented along the flow, with spacing approximately equivalent to the expected streak spacing X = l O O v / w , [see review s  in Best, 1992] (Figure 3.14a). These linear streaks did not appear to play any significant role in further development o f the bed. Instead, the bed undergoes the following deformations: 1. A cross-hatch pattern is imprinted on the bed (Figure 3.13b); 2. Chevron-shaped scallops develop at the nodes o f the cross-hatch (Figure 3.13c); 3. Chevrons begin to migrate and organise into incipient crestlines (Figure 3.13d); 4. Crestlines straighten into two-dimensional features (Figure 3.13e-h); 5. Bedforms grow in height and length. The time required to move through these developmental stages decreased significantly with increasing flow strength. The cross-hatch pattern is composed o f striations at oblique angles to the flume centre line (Figure 3.14a-c). Angles between the oblique striations and the flume centreline were measured from  118  the images in Figure 3.14a-c. The angles ranged between 10° and 70°, with a mean between 35 and 40° for all three instantaneous initiation runs. The oblique striations were ~1 - 2 D in height. Williams and Kemp [1971] have indicated that flow separation occurs when  Re = u,H/v>4.5.  3.19  Even when H = D, Re > 4.5. Therefore it is likely that some flow separation occurred over the striations. It is difficult to measure the oblique striation spacing as the pattern on grabbed images is somewhat weak in spots (Figure 3.14). This is not an indication that the cross-hatch was not present in those areas. Lack o f observations in some areas represents a limitation o f the observation techniques. (The features are best observed by eye, on the video, or in time lapsed image sequences. Time lapsed image sequence animations for each run can be found in Appendix B.) Where the oblique striations were well represented on the grabbed images, their separation, L , xv  was ~0.045 -  0.048 m (Table 3.5). Assuming the cross-hatch angles were ~35 - 40°, the streamwise separation o f the nodes is ~0.064 - 0.067 m. Both separations are  »X . S  Once the cross-hatch pattern was developed it began to migrate downstream, and in doing so, formed chevron shaped defects at the nodes o f the cross-hatch (Figure 3.14d-f). These chevron shapes had a developing crestline that flared upstream. Similar features have been identified by Gyr and Schmid [1989] when the flow strength was increased quickly. The chevron shapes are also remarkably similar to rhomboid ripple marks [see discussion in Allen, 1986]. These features are commonly found on steep seaward-facing beaches or on bar faces where flow is shallow and temporary (transient). Allen [1986] notes rhomboid ripples are transitional to transverse ripples. Gyr and Schmid [1989] attempted to link their observations to turbulent sweep events that are ubiquitous in turbulent flows and are characterised by +u' and - W velocities. They noted that, in order for turbulent events to deform the entire bed, forming the chevron shapes, the events would need to be phase locked. 121  Table 3.5: Initial bedform length scales for instantaneous development runs. L  xv  is measured from one image at the beginning o f the run and L is t  measured from a single image taken at time, t . t  Flow A (Run 53) 4.51  FlowB (Run 54) 4.84  Flow C  60  120  330  Li, cm 0~ L , cm  8.27  8.56  9.08  1.63  1.45  1.47  Li-inax  13.78  12.08  12.53  Lj-miti  4.27  5.64  5.18  Parameter  L„, cm  t  122  4.69  The chevron features seem to differ from those described by Williams and Kemp [1971] and Best [1992] that are characterised by flow parallel ridges, flared at their downstream edges, with small accumulations o f sediment at the mouth. Best [1992] also argued that these features were generated by coherent turbulent structures. Grass [1970] laid the foundation for this idea by noting that initial sediment transport is a product o f 'sweep' impacts on flat sediment beds. The high-speed sweep structures alternate with low-speed streaks in the cross-stream direction, giving rise to hairpin vortices. Noting a significant difference between the size o f the sweep structures and the size o f initial bed defects, Best [1992] suggested that defects are formed by multiple hairpin vortices (composed o f multiple sweeps), whose size is commensurate with the size o f initial defects. Best [1992] also noted that burst events were concentrated over flow parallel ridges in experiments. Regardless o f the specific defect feature, it is widely accepted in the literature that sweep events generate the chevron shaped defects observed in the flume. While there is an increasing acknowledgement amongst the scientific community that the coherent structures examined by Grass [1970] and Best [1992] are organised, they are still considered to be rather random in space and time. In fact, data presented by Best [1992] seems to demonstrate a rather random distribution o f grouped sweep structures without ridges installed in the flume. It is therefore difficult to imagine how the flow parallel ridges are established by random events in Best's [1992] experiments and how this might lead to the chevron patterns observed herein. It is not clear how random events could give rise to the regular cross-hatch pattern observed. Significant flow parallel ridges were not observed in the instantaneous development experiments. Unfortunately, without spatially detailed flow measurements the causal mechanisms o f the cross-hatch pattern cannot be defined. Once chevrons were migrating, they quickly began to organise transversely, forming flow perpendicular ridges at preferred along-stream spacing (Figure 3.14g-i). These incipient bedforms grew in size, with H increasing more quickly than L. In fact, L remained nearly constant until the crestlines were fairly well developed and two-dimensional (Figure 3.13). In the present experiments,  123  the initial bedform length, L , varied between 0.04 and 0.14 m and had a mode that varied between t  0.08 and 0.10 m (Figure 3.15). Mean L, decreased with increasing flow strength, as did the time required to develop the two-dimensional forms, t (Table 3.5). It should be noted that the specific t  value o f Li w i l l vary slightly depending on when the measurement was taken. While the origin o f the chevron pattern is still open to debate, the organisation o f the chevron pattern into crestlines has not before been addressed in any detail. The process o f instantaneous bedform development and, in particular, the organisation o f the chevrons appears similar to the expression o f classic hydrodynamic instabilities [see Lawrence et al, 1991]. I f it is assumed that an interfacial instability is generated along the top o f the sediment transport layer, the sheared density interface occurs between the water flow and the pseudo-fluid flow o f the transport layer whose density is composed o f both solid and fluid components. A variety o f hydrodynamic instabilities is defined in the Richardson number range -3 < Ri < 1 [Lawrence et al, 1991]. The Richardson number is defined as  */ =p^ -Au4  3.20  2  where A p and Au are the density and velocity differential, respectively, p  2  is the density o f the  lower, more dense fluid (the transport layer) and S is the thickness o f the velocity interface. When Ri < 0.07 the Kelvin-Helmholtz instability dominates. In the flow system observed here Ri < 0.02 (using the definitions below). Therefore, it is o f interest to test the theory o f a Kelvin-Helmholtz instability to determine i f the flow system envisioned above conforms to a classic hydrodynamic instability.  124  Figure 3.15: Histograms of initial bedform wavelength, Z , , for each instantaneous initiation run. Measurements are from images at t - 60 s (Flow A), t= 120 s (Flow B), r = 330 s (FlowC). Measurements are of all bedforms on the image, from crest to crest, along the streamwise direction only.  0.00 16  0.05  0.10  0.15  0.20  0.25  0.00  0.05  0.10  0.15  0.20  0.25  0.00  0.05  0.10  0.15  0.20  0.25  obs./n  125  3.5 Kelvin-Helmholtz Instability Model Liu [1957], drawing on presentations of theory presented in Lamb [1932], Prandtl [1952] and Rouse [1947], notes that an interface between two fluid of densities p, and p  2  moving at velocities  u and u (Figure 3.16) will be stable if: x  2  _ f gLKdL.P2 -P* In  .  2  (Ui  U2 <  pp ]  3.  2 1  2  If the squared velocity difference exceeds the right-hand side of Equation 3.21, the interface is unstable and begins to undulate, forming a Kelvin-Helmholtz wave with a defined wavelength, L_ K  . An along-stream (streamwise) variation in velocity is generated as the waveform causes the  H  streamlines to converge (increased velocity) and diverge (decreased velocity). According to the Bernoulli principle, a decrease in velocity is counteracted by an increase in pressure and an increase in velocity is counteracted by a decrease in pressure. The pressure variations are accentuated with time, which results in the concentration of vorticity at the crests of the waveforms and the generation of discrete vortices (Figure 3.16). In the relation represented by Equation 3.21, it is assumed that: (1) flow is two-dimensional, (2) both fluids are moving in the same direction, (3) fluids are inviscid, (4) turbulence is not present or can be ignored, (5) both fluids have infinite depth and (6) only gravitational forces are acting on the fluids. The wavelength at which the interface becomes unstable can be predicted by rearranging Equation 3.21 such that  K-H  L  =("l  ~  U  2  7>  S  P  2  7-  3  2  2  -Pi"  Liu [1957] was first to suggest that this is a viable mechanistic explanation for the establishment of bedforms. He considered the less dense, faster moving layer to be flow in the viscous sub-layer and the denser, slower moving fluid to be the sediment bed. Unfortunately, difficulties in measuring the velocities and densities of each layer impeded acquisition of an arithmetic solution for Equation 3.22 126  Figure 3.16: A definition sketch for a Kelvin-Helmholtz instability where fluid 1 has a lower density, p , and a larger velocity, u. Plus and minus signs indicate pressure relative to a mean value at the interface [Based on Liu, 1957].  127  Therefore, the seminal work of Liu [1957] relied on applying an instability index that contained information related to the sediment bed and the near bed flow, which was simply a Reynolds number calculated using u, and D.  The measurements taken during the present experiments allow the  arithmetic solution to Equation 3.22 that eluded Liu [1957].  3.5.1 Scenario for K-H Model Testing The interfacial instability scenario envisioned here is slightly different from that considered by Liu [1957]. In this conceptual model the upper fluid is considered to be the water column and is not restricted to the viscous sub-layer. The velocity of this upper layer is taken as the velocity measured at 5 mm above the bed, U , and the density is equal to p 0  5  w  (Table3.6). U s i n g u =U t  would be an  oversimplification of the problem but it may be a valid approximation. The best way to approximate w, would be to incorporate the velocity profile through the log-layer ( z < 0.2 d ) or even the entire d . This would greatly complicate Equation 3.22, and is not necessary for an initial test of the Kelvin-Helmholtz model. The lower fluid depicted in Figure 3.16 is considered to be the active sediment transport layer. The velocity o f particles at the surface of the transport layer (U  ) is measured from the video and  generally varied between 0.010 and 0.075 m s" , depending on the run, and has a mode that varied 1  between 0.015 and 0.035 m s" (Figure 3.17). A s flow strength increased, mean particle velocity, 1  Up, increased. If we assume U particle velocity is U  pd  decreases linearly to zero at some depth, the depth averaged  = U J2 (Table 3.6). Measurements made by Paulos [1998] suggest that the P  profile is in fact linear and that this is a valid approximation of U  U  pd  . A s with the upper layer it  would be useful to incorporate the grain velocity profile through the transport layer into an estimate o f u , but this would also complicate Equation 3.22. Determining an appropriate value for p 2  more difficult than for p, as the lower layer is composed of solid and liquid components. 128  2  is far  Table 3.6: Parameters used in the calculation o f the Kelvin-Helmholtz model. Error ranges are ± the standard error for each parameter. Error analysis followed the general rules for the propagation o f error when deriving a quantity from multiple measured quantities [Beers, 1957; Parratt, 1961]. Parameters marked with an asterisk (*) are measured quantities. Parameter  Flow A  Flow B  Flow C  •Surface Particle Velocity, ( / , c m s ' '  3.66 ± 0 . 2 2  3.40 ± 0 . 2 5  2.67 ±0.21  Depth-averaged U , U  1.83 ± 0.11  1.70 + 0.13  1.33 ± 0.11  30.92 ±0.03  29.52 ±0.03  28.96 ± 0.02  3.01 ±0.07  2.55 ± 0.13  2.19 ± 0.11  0.99 ±0.013  0.90 ±0.028 '  0.82 ±0.026  7.96 ±0.51  5.76 ± 0 . 3 6  2.85 ±0.41  7.96 ±0.51  5.76 ± 0 . 3 6  2.85 ± 0 . 4 1  15.19+1.14  13.08 ± 1.24  9.27 ± 0 . 9 4  1.81 ±0.11  1.53 + 0.12  1.09 ± 0 . 0 9  0.29 ± 0 . 0 2  0.22 ± 0 . 0 1  0.17 ± 0 . 0 2  10"  1.52 ± 0.11  1.31 ± 0 . 1 2  0.93 ± 0 . 0 9  , Kg m"  1265+ 104  1235±130  1249 ± 142  10.04 ±2.03  10.27 ± 2 . 6 5  9.85 ± 2 . 7 4  8.27 ±0.18  8.56 ± 0.15  9.08 ± 0.16  p  , cm s"  1  t  *Flow velocity at z = 5 mm, U ,cm  s"  1  05  *Shear Velocity, u, ,  cm s"  1  02  Depth of Trans. Layer, d ,  mm  tl  * Initial Dry-Mass Transport Rate, Q ,  kg s" x 10" 1  sj  3  Mass of Sand in Trans. Layer, M ,  kg x 10"  3  s  Mass of Water in Trans. Layer, M , s  kg x 10"  Volume of Active Layer, V , m x 10 3  3  5  tl  Volume of Sand, V , m x 10" 3  5  s  Volume of Water, V ,m x i  5  w  Combined Density, p  3  u  Predicted Wavelength L _ K  H  , cm  •Observed Initial Wavelength, L , cm i  129  Figure 3.17: Histograms of surface particle velocity on the bed, U , for each instantaneous initiation runs. Measurements were made over the 30 seconds following the onset of widespread sand transport.  0.00  0.05  0.10  0.15  0.20  0.25  0.00  0.05  0.10  0.15  0.20  0.25  0.00  0.05  0.10  0.15  0.20  0.25  obs./n 130  3.5.2 Depth and Density of the Active Layer The density o f the active sediment transport layer is calculated as the sum o f the solid and liquid component masses per unit volume. A volume o f the active layer, V , can be determined as tl  3.23 where y  w  is the flume width, t is some period o f time (1 s in all calculations here), and d„ is the  depth o f the transport layer. N o suitable, systematic measurements o f d  tl  could be obtained during  the experiments. However, limited observations indicated d - 2 - 3 D and that d tl  tl  increased with  flow strength. Depth o f the active transport layer is an issue that has been debated in the literature, resulting in several relations to predict d  [see review in Bridge and Bennett, 1992]. Einstein [1950] suggested  tl  d ~ 2 - 3 D, in his seminal treatise on bedload transport. Other relations are based on the tl  assumption that d  tl  varies with T (or u,). The most frequently referenced relations are those o f  Bridge and Dominic [1984], Bagnold [1973], and van Rijn [1984a], all o f which are composed o f an empirical fit between d  tl  and some measure of the shear stress above its critical value for entrainment  (Table 3.7). The van Rijn [1984a] relation also includes some measure o f the sediment's submerged weight and the fluid viscosity. Calculated values o f d  tl  in Table 3.7 indicate most values are within the range suggested by  Einstein [1950] and the range observed during the experiments. The Bridge and Dominic [1984] relation provides a rather small depth estimate, which is not surprising as the empirical fit is based solely on the data of Abbott and Francis [1977]. Their study examined the motion o f solitary particles (D =6.4-  8.2 mm) over a fixed bed composed o f rounded pea-gravel D =4.8 - 9.6 mm.  Extension o f the empirical relation to sand sized particles represents a radical extrapolation, and evidently produces erroneous results, van Rijn [1984a] and Bagnold [1973] back-calculated d  tl  131  Table 3.7: Depth o f the transport layer estimates (in mm). In the Bridge and Dominic [1984] relation q = 0.5 for solitary grains moving over a bed and q = 0 in a bedload transport layer. The subscript cr represents critical values for the entrainment o f sediment estimated from the Inman curve in Miller etal. [1977]. S o u r c e E s t i m a t e F l o wA F l o w B F l o wC d„ = 2-3D  Einstein ( 1 9 5 2 )  Bridge and Dominic ( 1 9 8 4 ) d„  = (2.53(0-ecrf+q)D 5  1-1.5  1-15 .  1-1.5  0.35  0.27  0.20  Bagnold ( 1 9 7 3 )  d„ = 1.4(w./w,)°-Z)  0.99  0.90  0.82  van Rijn ( 1 9 8 4 a )  d„ = 0.3 D, r D  13 .0  10 .0  0.73  6  r c  07  05  J  i D,^D(^{  K  v  P S  /  P W  -\)}  J  T  ; T  0.2  ~ cr X  132  values from sand transport rates measured by Williams [1970] in which median D = 1.35 mm. Although either the Bagnold [1973] or van Rijn [1984a] relations can reasonably be used, the former is used in the calculation o f V here. tl  The mass o f the sediment in V , M , can be determined from the initial dry-mass transport rate, tl  s  M =Q t. s  3.24  sl  There are two candidate Q values for each run; that measured by the Helley-Smith sampler, Q _ si  S  HS  and that determined from the nascent waveforms using the morphological method, Q _ . s  d  Unfortunately, there appears to be a larger amount o f error associated with Q _ - (see discussion in S  HS  Chapter 2). A value o f Q _ is calculated using all bedforms observed during the first 10 m i n o f the s  d  echo soundings. During this period H was generally less than 5 mm, which is fortuitous as the transport rate increased significantly when H exceeded 5 mm. These values o f Q _ are similar to s  QS-HS '  D  u  tw  '  t r i  a  d  reasonable increase in the transport with flow strength.  The mass o f the water in V , M , can be determined as tl  M  w  = P {V„-V )  W  W  3.24  S  where V is the volume of sediment grains in the transport layer calculated as s  M V=—i.  3.25  Ps  Using M  and M , the density o f the active transport layer can be calculated as  s  w  p = ' M  +  M  H  » .  3.26  V„  Table 3.6 provides estimates o f Q _ ,M , M s  d  S  w  and p  n  bedform initiation was observed.  133  for all three runs in which instantaneous  ,  3.5.3 Error Analysis  .  ,  Before proceeding with a discussion o f agreement between predictions and observations, it is useful to review the error associated with the measured quantities and its propagation through the calculation o f L _ K  H  . Error analysis followed the general rules for the propagation o f error when  deriving a quantity from multiple measured quantities [Beers, 1957; Parratt, 1961]. Standard errors o f the estimate (cr/-Jn ) are given in Table 3.6 with each parameter. The error range associated with a parameter is 2x (or ± ) the standard error. A l l measured quantities used in the calculation o f L _ K  associated with d  tl  have some associated error. The error  H  is derived from the error associated with the least-squares regression used to  determ ine w»o2 • The standard error o f the estimate o f w«o ranges between 0.69 and 1.64 mm s and 2  *o.2-e;r/ *o.2  M  M  =  5 - 11 % (the subscript err indicates the error range). However, the error associated  with the d estimate from Bagnold's [1973] equation is not great, d _ ld tl  tl  because d,, is determined from ( « « / « , . )  M  C)  err  tl  is only 2.7 - 6.4 %  and u* = [x I p , ) . These power functions effectively 0 5  M  reduce the effect o f the error derived from u, .' 0 2  The error associated with U  05  is excessively small because such large samples are used to  determine mean velocity at z = 5 mm. This is not the case for the error range associated with U  p  which is 4.21 - 5.05 mm s"'. This translates into large errors in U  where U _ ./U  p d  pil  en  pi/  = 12 - 16 %.  This is caused by relatively small sample size relative to the magnitude o f variation in U . The error range associated with p  is rather large (p _ ,.,./ p 1  tl  compounds errors associated with V , M  /;  tl  s  e  tl  = 1 6 - 2 3 %) as this calculation  and M , which have large associated errors. The bulk o f w  the error associated with V is derived from the measurement o f U tl  pd  , as only errors associated with  it and d are combined in the calculation o f V . The error associated with M d  tl  s  is also large and  entirely derived from the error in Q . The number o f samples used to determine Q was 21 for flow si  si  134  A , 17 for flow B and 5 for flow C , resulting in an error range for Q  si  and 1.02x 10" kg s" . Thus, Q _ ./Q 3  si  V  s  («: M ), s  = 1 2 - 2 9 %. Since M  1  en  si  that varied between 0.72 x 10  is calculated as a function o f V  w  and  tl  its associated error is also large.  A p p l y i n g Equation 3.22 compounds these measurement-related errors resulting in a relatively large degree o f error in L _ K  where the error range is 40.6 mm for flow A , 53.1 mm for flow B and  H  54.9 m m for flow C . Ultimately, L _ _ ./L _ K  error associated with Q  si  H  en  K  = 4 0 - 56 %. Nearly all the error is derived from  H  and U . The error associated with Q  si  increasing the number o f samples. The error in U  p  could be reduced by significantly  could also be reduced by increasing the number  of observations, but an improved technique is required to do so. In contrast to the error associated with the prediction, the error range associated with the observed bedform length is 3.56 mm for flow A , 3.10 mm for flow B and 3.22 mm for flow C . So, L _ ./Li t  3.5-4.1 %. The error in the predicted L _ K  H  is much larger than the error in the observed  =  en  Lt.  3.5.4 Estimate Agreement Predictions o f L _ K  H  using Equation 3.22 are provided in Table 3.6. The predicted threshold for  an unstable Kelvin-Helmholtz wave for this flow system ranges between 0.0985 and 0.1027 m, depending on the flow strength. Observed Lt is slightly smaller, 0.0827 - 0.0908 m, and so is median L , 0.08 - 0.10 m. Both mean and median lengths are within the error ranges associated with L _ f  K  Predictions o f L _ K  L _ ILi K  H  H  H  .  are extremely sensitive to the estimate o f sediment transport rate. The ratio o f  would be unity i f Q  si  was increased only 1 0 - 2 0 %.  It is interesting to note that when the bedform length becomes equivalent to L _ K  H  , the bedforms  crestlines begin to breakdown and become three-dimensional. This suggests that the K e l v i n Helmholtz instability dominated the system through the two-dimensional stage.  135  3.6 The Development of Bedforms T w o types o f bedform initiation have been described above: defect and instantaneous initiation. Initiation o f a bedform field from a bed defect has been described previously in the literature [e.g.  Raudkivi, 1966; Southard and Dingier, 1971; Williams and Kemp, 1971; Leeder, 1980; Ses/, 1992]. Negative defects on the bed initially grow downstream due to scour over the upstream lip o f the defect. This piles sand grains scoured from the pit at its downstream edge, generating further downstream flow separation, which in turn generates a new scour pit and mound o f sand grains at its downstream end. Through this process, negative defects grow downstream. After several pits are developed, the accumulation o f sediment at the downstream end o f the pits begins to square off and resemble bedforms. Each new scour pit is wider than the previous and, as the pits are propagated, the bedform field grows laterally. Through this process the initial defect is seen to generate a bedform field that covers the entire bed. Positive defects differ in their response to the flow in that they are initially planed off. Concurrently, a scour pit develops in the lee o f the defect. Propagation o f the scour pits continues in the fashion o f the negative defect. It appears that these scour pits are more stable features than positive defects which quickly decay if there is no scour pit upstream. Regardless o f the defect form, bedform fields have approximately the same form when the fields are composed o f five or more bedforms. Positive defect fields tend to develop more rapidly than negative fields, because there is more sediment initially available and exposed to flow in the mound. The observed processes were geometrically similar at different flow strengths, but the development was much quicker at the greater flow strength. Beginning with Raudkivi [1966] most investigators have generally accepted that the driving mechanism behind defect propagation is flow separation. In fact, velocity profiles taken over the defects in these experiments demonstrate the characteristics o f flow separation. It is therefore interesting that a separate initiation mode was observed in these experiments that was unrelated to defect initiation processes. 136  A t the greater flow strengths, bedform initiation seemed to be instantaneous and defects are actually washed out. A cross-hatch pattern is first imprinted into the flat bed surface, which breaks up into individual chevrons that migrate independently. It has been suggested that the origin o f these chevrons is linked to coherent turbulent flow structures that are ubiquitous in anisotropic shear flow [see Gyr and Schmid, 1989; Best, 1992]. This seems unlikely as one must argue that spatially and temporally random events must lock in place to generate the cross-hatch pattern. Best [1992] suggested that this may be the case i f flow parallel ridges are developed because hairpin vortices tend to stabilise over this kind o f topography. Lineated striations are observed over the bed, but these are not coincident with the scaling o f the cross-hatch or chevrons. Unfortunately, the origin o f the crosshatch cannot be determined from the data herein, but its similarity to rhomboid ripple marks suggests its development may be linked to the sudden (transient) onset of flow and transport. The organisation o f the chevrons can be explained as a hydrodynamic instability that occurs at the shear interface between the sediment transport layer and the near-bed flow. Once sediment transport is initiated, the active sediment transport layer begins to act as a pseudo-fluid with a density that is composed o f solid and fluid components. A n instability develops at the interface o f the fluid flow and the transport layer causing an oscillating along-stream pressure gradient. The pressure variations are accentuated with time and the waveform eventually breaks, generating localised sediment erosion and organisation o f migrating features. When the waveform breaks, there is a concentration o f vorticity at the crests o f the waveform and the generation o f discrete vortices [Lawrence et al., 1991]. The integral scale, which is a measure o f the average sized eddy in the flow, is within the error boundaries o f L _ K  H  for two o f the instantaneous initiation runs. Thus, the integral length scale is  coincident with the length at which vorticity is concentrated in the Kelvin-Helmholtz instability.  Predictions from the theory o f a Kelvin-Helmholtz instability are encouraging because L , and median L are within the error range o f L _ i  K  H  . This result has not been attained in the literature  previously. Ultimately, it appears that the instantaneous initiation o f bedforms is controlled by a  137  simple hydrodynamic instability. This is in direct contrast to theories o f bedform development based purely on the motions o f coherent turbulent eddy structures, but it statistically complements the observed regularity o f the initial bedforms. A more advanced solution than Equation 3.21 is necessary to take into account the velocity profiles in the near-bed region and through the sediment bed. Improved estimates o f the initial transport rate, using more advanced technologies, are critical. There is also a need to examine the active layer through measurements in order to better characterise its depth and density. Finally, further research is needed to define the grain sizes, grain size distributions and transport intensities over which the two types o f initiation processes described here dominate.  3.7 Summary Bedform initiation was examined on a flat bed, composed o f a homogeneous 0.5 m m sand. The bed was subjected to a 0.155 m deep, non-varying mean flow ranging from 0.30 to 0.55 m/s in a 1 m wide flume. Bed deformation was monitored using overhead video and echo-sounders. In order to confirm that the flows were typical steady, uniform flows, velocity profiles were taken using laser Doppler anemometry. Overall the velocity data suggest that flow over the lower stage plane beds at the beginning o f the experiments was in accordance with conventional models o f uniform flow over flat beds. With the exception o f the depth-slope product calculations, estimates o f the boundary shear stress derived from different methods are similar in magnitude, and increase in a similar fashion with U . The roughness heights are consistent with previous observations. The turbulence intensities and R  uw  can be modelled by the semi-empirical functions provided by Nezu and Nakagawa [1993], with  the exception o f the vertical intensities at the low flow strengths. With few exceptions, the momentum exchange ( T ) and fluid diffusion ( £ ) are typical o f fully turbulent, uniform, open i w  channel flows. Bed defects dramatically reduce the streamwise velocity near the bed, increasing turbulence and providing the turbulent energy to scour out the bedform lee.  138  T w o types o f bedform initiation are observed in the experiments. The first occurs at lower flow strengths and is characterised by the propagation o f defects via flow separation processes to develop bedform fields. This type o f bedform development has received some attention in the literature and widespread support in the earth sciences. The second mode o f bedform initiation has received less attention. This form o f bedform initiation begins with the imprinting o f a cross-hatch pattern on the flat sediment bed, which leads to chevron shaped forms that migrate independently o f the initial structure. The chevron shapes are organised by a simple fluid instability that occurs at the sediment transport layer-water interface. Predictions from a Kelvin-Helmholtz instability model are nearly equivalent to the observations o f bedform lengths in the experiments. It is likely that the instability model holds only for intense transport conditions when the bed constitutes a fluid layer, and is less applicable to situations where the active transport layer is discontinuous. Further research is needed to define the grain sizes, grain size distributions, and transport intensities over which the two types o f initiation processes described here dominate.  139  Chapter 4: The Transition between Two- and Three-Dimensional Bedforms 4.1 Introduction The purpose o f this chapter is to examine the transition between two-dimensional (2D) and threedimensional (3D) dunes. A definition o f what constitutes a 3 D bedform is provided using a statistic called the non-dimensional span or sinuosity o f dune crests. A critical value o f the non-dimensional span is proposed to divide bedforms with straight or slightly sinuous 2 D crests from 3 D bedforms. Video records are examined for patterns in the breakdown o f dune crestlines that give insight to the physical mechanisms o f the transition. The problem o f whether 3 D bedforms develop only at greater flow strengths w i l l be addressed. Finally, a possible explanation is provided for why the transition occurs. It is well known that sediment transport rates are dependent on the applied shear stress. A mechanism that reduces (or stabilises) the shear stress should also reduce (or stabilise) the sediment transport rate. Drag reduction is a mechanism that may protect the bed from erosion by reducing the applied shear stress. This can contribute to the stability o f the bed and the channel by reducing susceptibility to degradation. In light o f this idea, the shift from 2 D to 3 D morphology is examined as a mechanism that reduces drag. Drag coefficients and forces are examined as bedforms develop from 2 D transverse ribs through to equilibrium 3 D dunes.  4.2 Experimental Procedures The experiments discussed in this chapter used the same apparatus, experimental design, and flow conditions detailed in Chapter 2. The runs were conducted at the same pump frequencies, /  =  23.5,22.5,21.5, 19.0 and 17.0 H z . The bulk flow hydraulics o f each run can be found in Table 2.1. Several runs were conducted at each /  to accomplish the goals in this chapter, so labels are again  assigned to each flow strength to streamline the text. A s before, flow strength A refers to all runs  140  conduced at f  p  = 23.5 H z , B all runs at /  19.0 H z and E all runs at /  = 22.5 H z , C all runs at /  = 21.5 H z , D all runs at /  =  = 17.0 H z . A s discussed in Chapter 3, bedforms were initiated  instantaneously during Runs A , B and C , and by negative defect propagation processes during flows D and E . A short note on scaling is necessary before any discussion o f 3 D bedforms can be accomplished. If a dune is 1 m long in a 0.1 m wide flume, the sinuosity o f the crest may be low. I f that same dune is formed in a 2 m wide flume, the crest sinuosity is free to develop to larger values. The level o f three-dimensionality o f a bedform is certainly a function o f L/y  w  (L is bedform length and y  w  is  width o f the flume). In the 1 m wide flume used for these experiments, only the longest bedforms had L > y , w  so this is probably not a problem. However, all observations discussed in this chapter  need to be considered in this light.  4.2.1 Video Experimental runs were recorded using the Super-VHS set up discussed in Chapters 2 and 3. Videotape records were sub-sampled with a frame grabber at an interval o f 10 sec. This produced a series o f images suitable for further analysis. The video was taken during Runs 53 ( A ) , 54 ( B ) , 55 (E), 57 (C) and 59 (D).  4.2.2 Water Surface and Bed Level Sensors During the videotaped runs, changes in water surface level were monitored using two ultrasonic water level sensors built by Contaq Technologies Corporation. The water level sensors emit an ultrasonic signal toward a boundary (liquid or solid) and the signal return time is sensed and converted to a distance using hardware and software provided by the manufacturer. The minimum operational distance to a boundary is —0.15 m and the reported resolution is 0.172 m m . The sensors  141  were mounted in the centre o f the channel with a streamwise separation x  wl  - 2.26 m at 8.66 and  10.92 m from the head box (see Figure 2.3). The video was taken towards the downstream end o f this span. Instrument signals were sampled at -2.5 H z for 30 min periods through out the experiment. The signals appeared to be relatively free o f signal contamination, but the high sampling frequency o f the sensors produced far more detailed information than was necessary for the purposes o f this investigation. A s such, a 250 s running mean was calculated for the time series at 25 s intervals. This provided nearly continuous measurements o f the water surface level and slope, S, determined as  £ _ ws 1  ~~ ws 2  z  z  wl  X  where z  wsx  and z  are the water levels at sensors 1 and 2 (Figure 2.3).  WS2  Echo-sounder bed height measurements are used in this chapter to determine the change in bed level surface with time. Records used are those discussed in Chapter 2 taken during runs 26 ( A ) , 27 (B), 28 (C), 29 (D) and 30 (E). The sensors were deployed in the centre o f the channel with a streamwise separation o f 0.13 m at 10.36 and 10.50 m from the head box (Figure 2.3).  4.2.3 Arcview Analysis In order to examine the morphology o f the bed the video image series were analysed using a G I S software package. Figure 4.1 details the two sets o f measurements made. In order to calculate an areally averaged bedform length, L - A jy a  h  h  , the area between two crestlines, A , was digitised and b  the distance that the bedform extended across the flume, y , was measured along the cross-stream h  axis o f the flume. A could only be digitised when both crestlines were visible in the video view and b  within ~1.2 m o f each other, placing a practical limit on L . The second set o f measurements a  consisted o f a measurement o f the length along the bedform crest, L , and a measurement o f the c  142  Figure 4.1: Measurements taken from video images: (a) Crestline length, L ; (b) Linear crest length, L ; (c) Bedform area, A . Image is Run54-0031 (r = 240 s). Flow is left to right. c  y  d  143  linear distance between the end points o f the crestline, L  which is measured according to the  v  orientation o f the bedform crestline. In nearly all cases, y ^ h  L. v  Digitising data from video images is enormously time consuming so different measurement periods were used based on the rate o f bedform change, which is a function o f flow strength and, ultimately, the sediment transport rate. Measurement period also varied based on data requirements to answer the research questions. It was necessary to have extensive data at the beginning o f the higher flow runs, when bedforms were initiated instantaneously, because these data were needed to define when the transition between 2 D and 3 D bedforms had occurred. The rest o f the records were needed to examine the variability o f measured quantities through time and so the data density was lower in all but one run. For Run 54 (B), measurements o f A , y , b  b  L and L c  v  were made at 60 s intervals throughout the  12 hour experiment, providing a complete data set o f the variation in measured quantities. F o r Run 53 ( A ) , at a slightly greater flow strength than 54, images were digitised at 60 s intervals for the first 4 hours. For Run 57 (C), at a slightly lesser flow strength than 54, measurements were made at 60 s intervals for the first 1 hour and at 120 s intervals when t = 2 - 4 hours. Images were digitised at 5 min intervals after 4 hours for Runs 53 and 57 to ensure the patterns were consistent with those observed during Run 54. Bedforms developed from negative (or positive) defects provide a fairly complex morphology before they extend across the whole channel (see Chapter 5). Therefore images were digitised after t = 4 hours. The rate o f change for these bedforms was not great, so images were digitised at 5 min intervals. A l l measurements were made with respect to the overhung grid in the video view. Due to distortion caused by the distance between the grid height and the bed height, all areas measured from the images are corrected by multiplying by 1.28 and all lengths are corrected by multiplying by 1.15. See Chapter 2 for details and justification o f these corrections.  144  4.3 A Definition of 3 D Bedform Morphology Numerous researchers have examined the character o f 3D dunes [e.g. Allen, 1968; Ashley, 1990], yet few o f these researchers have attempted to define the three-dimensionality o f a bedform numerically. Most researchers have instead relied on descriptive terms that characterise the crestlines o f bedforms such as straight, sinuous, catenary, linguoid, cuspate and lunate  [Allen, 1968].  These terms are typically applied to the whole bedform field. This approach to documenting bedform shape provides a necessary element o f simplification in order to characterise bed structures in the rock record or general patterns in active rivers. However, the assignment o f a particular term is somewhat subjective since a bedform field may be composed o f bedforms with different degrees o f three-dimensionality. Terms also vary widely in the literature, meaning that a single bedform field may legitimately have several terms that describe its morphology. In light of this,  Ashley [1990]  reviewed the literature and recommended that the terms 2 D and 3 D be used as primary descriptors o f the bed morphology in lieu o f the terms listed above. While this makes defining the areal morphology o f a bedforms more universal, the final designation o f descriptor is still somewhat subjective. It is therefore beneficial to define a criterion that uses actual bedform characteristics to define the level o f three-dimensionality. Allen [1968] was first to suggest a numerical measure o f areal bedform morphology, defining the degree o f 'waviness' o f a crestline as  where A is the cross-stream distance between lobes o f a sinuous, catenary, linguoid, cuspate or v  lunate crestline and X  x  is the streamwise distance that the lobes extend downstream (Figure 4.2). In  a later writing, Allen [1969] suggested bedform three-dimensionality could be characterised by the ratio  145  ALLEN [1968; 1969]  NON-DIMENSIONAL SPAN  Figure 4.2: Measures o f bedform crestline three-dimensionality.  146  A  Allen-b  = L/X  4.3  y  (Figure 4.2). In fact, Allen [1969] attempted to functionally link A _ Allen  b  to a Froude number.  Unfortunately, both measures suggested by Allen [1968; 1969] require rather simple plan geometries of bedforms. Tf a bedform crest is composed o f multiple lobes, it becomes inordinately difficult to define X , v  X and L. In this study, in particular, the streamwise extent o f lobes is not uniform along x  a single crestline and it is difficult to break the bedforms into individuals. It may even be misleading to break a single crestline into several paired lobes. A n alternative to  Allen's [1968; 1969] measures o f crestline morphology is the non-dimensional  span (or sinuosity) defined as A  NDS  4.4  ~  (Figure 4.2). This measure is ideal when the crests o f bedforms can be linked into a continuous crestline. Even if the bedforms are isolated individuals on the bed,  A  NDS  measured from the ends of the crestline. Figure 4.3 provides examples o f over the image,  A _ NDS  im  . When A _  straight crested 2 D dunes. A t  NDS  A _ NDS  lm  lm  jm  NDS  v  values averaged  = 1.10, the crestlines remain largely 2 D but converge at A _ NDS  highly sinuous and linguoid [in the terminology of  NDS  A  L is  = 1.02, the minimum observed, the bed is composed o f  several locations and the crestlines bend. When  A _  can be defined as  im  = 1.21, the bed is composed o f a mixture o f  Allen, 1968] bedforms. A t the larger values o f  in Figure 4.3 (1.29 and 1.39), the bed is largely composed o f linguiod bedforms with some  crestlines that are highly sinuous. A s w i l l be discussed below, instantaneously developed bedforms provide a clear transition between 2 D and 3 D bedforms. Bedforms that develop from bed defects eventually become 3 D , but lag effects complicate the transition process. Therefore only the greater flow strength runs are used to define a numerical criterion that divides 2 D dune fields from 3 D fields.  147  148  Figure 4.4 plots A  as a function o f time for the first hour o f the experiments. There is  NDS  considerable scatter in A  which is reduced by excluding values where L does not exceed 0.7 m  NDS  v  (as is done in Figure 4.3) which inordinately bias the image averages. A general trend can be observed in  values, but plots o f A _ ms  clarify the pattern. Initially, the majority o f  im  values are in the range 1.1 - 1.3 with A _ NDS  A  NDS  = 1.1 - 1.2 when the bed is composed o f organising  jm  chevron shapes. A s the crestlines straighten and become 2 D , A _ NDS  jm  begins to drop approaching its  definitional minimum o f 1.0. A t flow strength B , crestlines that are continuous across the channel form at t = 120 s, while c  minimum values o f A  NDS  occur at t  2D  = 420 s, ~5 min later (Table 4.1). Similar trends can be  observed at flow strengths A and C where t  - t ~ 300 s and thus appear not to depend on flow  2D  strength. A t t  c  there are several observations o f A  2D  ~ 1.00, but A _  NDS  flows A , B and C , respectively. That A _ NDS  NDS  im  = 1.02, 1.08 and 1.07 at  is much lower at flow strength A may be significant;  jm  at this flow the velocity gradient that leads to Kelvin-Helmholtz instabilities is the greatest and thus the concentration o f vorticity may be greatest. However, this observation could be fortuitous as lower values o f A _ NDS  jm  may occur in between the sampled images at flows B and C .  Roughly 20 - 30 min beyond the minimum, A _ NDS  increased to values that were as large as  jm  1.38-1.39 (Table 4.1). There was a nearly linear increase in A _ NDS  after which A _ NDS  variability in A  jm  NDS  beyond the first hour is discussed later. The same near linear increase can be  NDS  = 1.07 - 1.08 to 1.31 - 1.29, after which  jm  approaches 1.4, defining a local maximum A _ NDS  im  at t  max  A s crestlines begin to break up and become 3 D , A  NDS  im  (Table 4.1).  0.70 m are above a value o f 1.2. The scatter about A _ NDS  149  A _  rises until nearly all observations with  NDS  v  from 1.02 - 1.39 at flow A ,  varied between 1.2 and 1.4 as the bedforms grew towards an equilibrium. The  observed at Flows B and C , but from A _  L>  jm  jm  also increases substantially  0.0  0.9  1.8  2.7  3.6  0.0  0.9  1.8  2.7  3.6 0.0  0.9  1.8  2.7  Time 1 0 (s) 3  Figure 4.4: Non-dimensional span,  , during the first hour of experiments. A l l  observations are plotted in top row of panels. M i d d l e panels plot crests that exceed a cross-stream extent of 0.7 m and bottom panels are image means of data in middle row.  150  3.6  Table 4.1: T i m i n g o1'critical non-dimensional span values over 2-3D transition. Flow  Run  A B C  53 54 57  t  c  0)  60 120 360  NDS  A  1.1332 1.1895 1.1256  () 300 420 660  2D  {  A. NDS  HD ( )  1.0192 1.0764 1.0730  1080 900 1020  S  151  S  NDS  *max ( )  1.2195 1.2169 1.2141  1920 2160 2040  A  S  NDS  A  1.3873 1.3909 1.3795  compared to when A  NDS  was less than 1.2 because the bed is composed o f bedforms with varying  degrees o f three-dimensionality. Given these observations and the description associated with A _ NDS  jm  = 1.21 in Figure 4.3, it seems logical to assert that when A  is 2 D , and when  NDS  is less than 1.20 a crestline  exceeds 1.20 a crestline is 3 D .  4.4 Observations o f the T r a n s i t i o n between 2 D a n d 3 D B e d f o r m s Before proceeding with a discussion o f the processes involved in the 2-3D transition, it is useful to simply note that the ultimate bedform morphology was 3 D . This answers an important question raised in Chapter 1: do 3 D dunes develop at larger flow strengths than 2 D dunes? A s noted in Chapter 2, Baas and collaborators [Baas et ai, 1993; Baas, 1994; Baas, 1999] have demonstrated that, given enough time, ripples w i l l always transform from a 2 D to a 3 D morphology. This also appears to be true for dunes as 2 D dunes were observed to be temporary and transitional features. Southard and Boguchwal [1990] provide the most extensive bedform phase diagrams and plotting methodology in the literature to date. A l l the observations made during the present experiments plot in the 2 D dune fields (Figure 2.2). The idea that 2 D features develop at lower flow strengths than 3 D dunes [Costello and Southard, 1981; Southard and Boguchwal, 1990] does not seem valid, at least for the grain size employed in the experiment. The 2-3D dune transition was observed to occur shortly after the features developed for all experimental runs. The transition is far more pronounced when the 2 D bed is developed instantaneously, rather than from a bed defect, and so it is useful to discuss the two sequences separately.  4.4.1 Bed Defect Developed  Fields  It is difficult to define distinct 2 D and 3 D phases for the bedform field developed from defects at flows D and E. The bedform field developed from defects is initially 2 D and then undergoes a 152  transition to 3 D morphology with time. A l s o , newly developed crests can have a 2 D form that undergoes a later transition to 3 D . Figure 3.9a-e demonstrates that the features developed from a defect are straight crested. However, a disturbance in the crestline appears in Figure 3.9f and is amplified with time in Figure 3.9g-i. The defect bedform field that appears in Figure 3.9h has some features that would be considered 2 D and others that would be considered 3 D . In Figure 3.9g-i the third and fourth crests and the top portion o f crests 5 - 8 crestlines are approximately 2 D while the rest o f the field is approximately 3 D . This kind o f mixed morphology can also be observed throughout the sequence o f maps presented in Figure 3.7. Bedforms that migrate from the head box are generally 2 D but become 3 D as they interact with the defect field. Once the defect-developed field is overtaken by bedforms developed at the head box, the entire field becomes 3 D (see Figure 3.7g). However, since sediment transport rates are low for the defect- developed bed, the 2-3D transition may lag behind at newly developed portions o f the bed. Essentially, the transition between 2 D and 3 D bedforms occurs at different times and at different locations in the bedforms field. Nevertheless, once the defect pattern and head box developed bedforms have interacted and the dunes are fully developed, their morphology is 3 D as in Figure 3.7h-j. Given the complicated lag in the 2-3D transition at flows D and E , it is best to examine the processes in which the transition occurs quickly and entirely.  4.4.2 Instantaneously Developed Fields Figure 4.5 depicts a series o f morphological changes that the 2 D bed undergoes during the transition to 3 D when the bed is developed instantaneously. Time-lapsed image animations o f the 2 D - 3 D dune transition can be found in Appendix B . Once organised by the Kelvin-Helmholtz instability, the bed is composed o f straight or slightly sinuous crested dune features (Figure 4.5a). The crestlines begin to bend along their cross-stream length and become convex downstream during the greatest flow strength examined ( A ) (Figure 4.5b). 153  A t flow B bending is less perceptible, and does not occur at flow C , the lowest instantaneous initiation flow strength. Presumably this phenomenon is caused by side wall drag, where the wall fluid is moving slightly slower compared to fluid in the centre of the flume. The cross-stream gradient is strongest at the greatest flow strengths. A s the 2 D bed (and 2 D bed with bending crests) persists, crest defects sporadically form, which are minor, transient excesses or deficiencies o f sand in the crestline. Figure 4.6 shows several crest defects developed in the 2 D field that appear similar to dune blowouts observed along coastal sand dunes formed by sea breezes [Hesp and Hyde, 1996]. Crest defect development can probably be linked to turbulent busting over the 2 D bed, but the measurements taken herein are inappropriate to test this hypothesis. Crest defect features are passed from crest to crest and can be seen to migrate though the dune field (Figure 4.7). When the defect is formed a volume o f sediment is generated downstream causing a convex downstream bulge in the crestline in-line with the defect. The parcel o f sand that composes the defect is elongated downstream, flattening it. The dune crestline acts as a barrier to flow. Once it is removed in the form o f a crest defect, the downstream crestline is exposed to a greater flow velocity than the rest o f the downstream crest and another blowout is formed in line with the previous defect. The defect sand parcel is thus cleaved from the crestline and passed to the downstream crest. Crest defect features move through the bedform field at a velocity ~ 2 x the dune migration rate. The defects in Figure 4.6 moved at an average rate o f 2.9 mm s" while the bedforms were moving at 1  only 1.5 mm s" . The larger defects, such as A and F, moved through the dune field more quickly 1  than the others, with velocities o f 3.6 and 3.5 mm s"' respectively. Smaller defects, such as D and B , tend to move slowly; both moved at a velocity o f 2.5 mm s" . 1  Crest defects are o f little consequence when the bed is composed o f straight or slightly sinuous dunes. Defects are successfully passed from one crestline to the next without altering the 155  156  Figure 4.7: Progression of defect 'a' from Figure 4.6 as it migrates from one crest to the next (t = 320-370 s). The area highlighted red is the parcel of sand passed from one crest to another. Note the effect of defect progression on the downstream crest before it cleaves from the upstream crestline. Flow is left to right.  157  morphological characteristics o f the crest (Figure 4.7). Crest defects are generally passed through a few crestlines and then disappear when the bed is 2 D and so the field appears able to ' s w a l l o w ' a limited number o f defects. However, the number o f crest defects increases with time and the 2 D field is eventually overwhelmed (Figure 4.5c-d). When this occurs, crest defects slow their downstream march and eventually stop, causing permanent changes in the crestline. The defects interact and the bed falls into a cycle where the 3 D bed is maintained, indefinitely.  4.4.3 Operation and Maintenance  of the 3D  Bed  Allen [1973] provided a description o f the mechanisms associated with a 3 D bed that Baas [1994; 1999] aptly called ripple 'birth-and-death processes'. According to Allen [1973], death can occur by a number o f different processes. Rapid deceleration o f a ripple or crest can occur i f the crestline becomes strongly concave downstream which leads to a concentration o f the lee side vortex and a deep scour pit. The increased erosion can accelerate the next crest's (crests') downstream migration, starving downstream crests o f material and allowing the accelerating ripple to overtake the downstream ripples. Increasing the local height o f a ripple may also lead to downstream features being planed off, and one or more crestlines being destroyed by the efficiency o f the separation vortex. Allen [1973] indicated that new ripples may be developed by a crestline splitting along its crossstream length at spurs, which are ridges o f sand on dune backs perpendicular to crestlines. Alternatively, a new ripple o f limited height may be generated on an existing ripple back that grows in height and splits the original ripple length. Allen [1973] noted that all these processes are at work on the bed at any one time. Only in exceptional cases did these processes lead to the generation or extinction o f a ripple. M o s t of Allen's, [1973] description focused on what was happening to produce or destroy bedforms along a streamwise transect cut into the bed (a 2 D plane). This is appropriate because 158  Allen's [1973] primary interest was in explaining certain types o f cross-stratification that are typically examined in a 2 D plane o f the rock record. A l l the processes that Allen [1973] described are occurring; in a 2 D plane however, these processes are secondary to a dominant processes observed to maintain the 3 D bed. This process is the growth o f crestline lobes that extend downstream (Figure 4.8). Lobe extension is caused by localised increases in bedform height across the flume and erosion o f a scour pit in the lee o f the bedform. A s the increased available sediment extends the lobe, it w i l l sometimes stop and be planed out. Frequently the lobe w i l l begin to locally starve the downstream crestline o f sediment while the rest o f that crestline continues to move (Figure 4.8b). Eventually, the upstream and downstream crestlines w i l l meet and two fragments w i l l join (Figure 4.8c) forming a new crestline and two bifurcations (Figure 4.8d). Concurrent with this process, local crestline heights are increased in downstream dunes forcing new crest lobe growth downstream. A s long as H and L vary across the channel, this process of lobe extension w i l l continue, inevitably maintaining the 3 D bed.  Once the bed becomes 3 D it cannot become 2 D again without some external driving force, such  as a change in flow rate. During flow E , when the flow rate was increased (see Chapter 2), 2 D dunes briefly re-established themselves, but became 3 D a short time later. Given these observations, it is necessary to ask i f there really is a birth-and-death process that occurs on the 3 D bed. A simple answer is ' n o ' because new bedforms are rarely created or destroyed when the bed is examined in plan. There is a constant rearranging o f crestlines that gives the appearance o f the birth and death process in the 2 D plane. This observation may have eluded previous investigators as their research was conducted in flumes where the ratio L/y  w  was too small,  inducing scale effects on planimetric dune morphology. A s discussed in Chapter 2, Raudkivi and Witte [1990] have suggested that bedforms actually grow in length through the birth and death cycle by the coalescence o f smaller bedforms, because smaller bedforms have larger migration rates, R , and can overtake the larger features on a bed. 159  A  B  Crest starved no advance  -1 Crest will advance  — —  N  N  Flow / /  -  ^  ' , ~ >. ^ /  Crest will advance \  N \ \  \  \ \ \ \ — —^w \ \  Advancing Lobe  Sediment delivery  f§l|§l|> Localized High  Localized High  D  Old crest  Localized High planed off  Figure 4.8: Schematic of an advancing crestline lobe as it joins with the downstream crestline and generates bifurcations.  160  Ditchfield and Best [1990] argued against this idea, indicating that there is no relation between bedform size and the migration rate, R . They also suggested that bedforms may both grow or attenuate without interaction with other bedforms, or alternatively, they may coalesce as they migrate. In contrast to Ditchfield and Best [1990], a strong relation was observed between L and R for some runs (see Figure 2.8). Bedform coalescence was observed during the experiments, but it was limited to the sand sheets combining with the dune crestline. This was equilibrium process that maintained transport over the dune and did not cause dune growth. Widespread bedform coalescence did not occur amongst the dune population; rather, crest realignment by the growth o f scour induced crest lobes dominated. It appears that the bedform unification models advocated by Raudkivi and Witte [1990] and by Coleman and Melville [1992] need to be re-examined.  4.5 Drag Reduction Mechanisms Recent advances in the aerodynamics literature have suggested that surface drag can be significantly altered by the arrangement o f perturbations on otherwise flat surfaces. Sirovich and Karlsson [1997] examined flow over 'riblets' (basically the opposite o f dimples in a golf ball) in an effort to determine what sort o f riblet patterns would be effective in reducing drag on airplane surfaces. This research demonstrated that a strictly 2 D aligned pattern o f riblets (Figure 4.9a) produced a larger drag than a smooth surface while an out-of-phase random pattern (Figure 4.9b) produced a lower drag than a smooth surface. Hydraulic drag can be reduced by up to 20 % by changing the arrangement o f perturbations. The researchers demonstrated that random orientations of riblets effectively modulate the burst-sweep cycle, reducing boundary shear stress. The work o f Sirovich and Karlsson [1997] may have profound implications for sandy bedform evolution in river channels. The aligned pattern o f riblets can be considered analogous to 2 D bedforms while the random orientation o f riblets is analogous to the random 3 D linguoid 161  AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA  AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA b) Random Pattern  a) Aligned Pattern  Figure 4.9: Plan view o f strictly two-dimensional aligned pattern and out-of-phase random pattern o f riblets examined by Sirovich and Karlssen (1997).  162  morphology, identified as the equilibrium bed morphology here and in other studies [see references above]. Thus, a sediment bed composed o f randomly arranged bedforms, as opposed to straight crested 2 D bed features, may reduce or stabilise the shear stress. Transport and erosion rates are strongly dependent on the shear stress. Therefore decreasing or stabilising shear stress can contribute to the stability o f the bed by reducing the likelihood o f degradation. In order to test these ideas, it is necessary to examine the behavior o f drag as changes in bed morphology occur, which requires the calculation o f a time dependant drag coefficient, C , and drag force, F . D  D  4.5.1 Calculating Drag Coefficients and Drag Force The standard form o f the drag coefficient is 4.5 where u* is the shear velocity and U is the mean channel velocity. The drag force is calculated from  F  where p  w  D  = ~ C  D  P  W  U  4.6  A  H  is the density o f water and A is the effective cross-sectional area o f a body [Roberson h  and Crowe, 1993]. For dunes, A =H h  • L , where H is dune height. v  It is important to note that F , and the associated C , represent the total drag. F o l l o w i n g D  D  Einstein and Barbarossa [1952] the total drag force F  is the sum o f the skin drag, F , which is  tot  sf  related to the resistance to flow offered by the sediment grains, and form drag, F , fnrm  which is  resistance to flow offered by the bedform morphology. It is widely thought that the transport rate over a dune stoss slope is related only to F . r  Based on an approach to divide these quantities by Smith and McLean [1977], Nelson et al. [1993] present equations that may be used to calculate F  sf  163  and F  form  separately. Unfortunately, the  calculations require detailed velocity profiles downstream o f the reattachment point, for each bedform, which are not available, but this problem can be circumvented. One grain size was used in all experiments, so the grain related roughness remained approximately the same for all the bedforms observed, whether 2 D or 3 D . Some variation in F  sf  may be caused by the intensity o f sediment  transport over different bedforms but that is, arguably, another component o f F  m  [Wiberg and  Nelson, 1992]. Since direct calculation o f F  and F  sf  form  is not possible, this investigation is primarily  concerned with the variation o f F  mt  (and the associated C ). Under the assumption that F  constant, most o f the change in F  m  is related to change in F  D  sf  form  is near  which is thought to be unrelated to  the transport rate over a dune stoss slope. This complicates the idea that drag reduction can decrease or stabilise transport rates. However, this should not be a problem because when a bedform field is considered as a whole, the transport rate increases with the total applied shear force  (F ). tot  In order to calculate C (t) and F (t), u* and U must be calculated as a function o f time. The D  D  shear velocity can be calculated from its definition  4.7  where  T (t) = p gS(t)d(t) s  and p  w  4.8  w  is the density o f water, g is acceleration due to gravity, d is flow depth. Because S has  been measured continuously, only d need be estimated from measurements, which as a function o f time is  d(t) = d -z (t) + z (t) st  becl  4.9  ws  164  where d  is its initial value at the beginning o f the experiment, z  sl  is the change in the mean bed  bed  level and z  ws  is the change in the water surface. The mean velocity can be adjusted as a function o f  time using d(t) such that U(t) = Q/{d(t)-y )  4.10  w  where Q is the discharge and y  w  is the channel width.  4.5.2 Flow Depth and Velocity as a Function of Time The first step in calculating C (t) and F (t) is to determine z D  D  bed  . Unfortunately, bed level  could not be monitored while the video and water level sensors were being operated - the echosounders disturbed the water surface. However, bed level erosion as a function o f time can be estimated from echo-soundings collected during other experimental runs. Thus, the echo soundings discussed in Chapter 2 are used to determine best-fit curves through crest heights, z , and trough c  heights, z . and the mean bed level as a function o f time is Tl  The height o f dune crests increases exponentially to an asymptote and can be expressed by z (t) c  = c +a (\-e-»c') c  4.12  c  while the height o f dune troughs decrease exponentially to an asymptote and can be expressed by z .(t) = c +a e~ '  4.13  bT !  Tl  Tr  Tr  where a , b and c are coefficients determined from least-squares regression (Figure 4.10 and Table 4.2). A s with the exponential fits to the growth in L and H in Chapter 2, r values varied but were 1  generally lower for the lesser flow strength runs where there are fewer observations. A l l but one o f the regression fits are significant at the 95 % confidence interval. Only the z  c  model fit at flow E is  not significant, but there is an apparent visual fit o f the curve to the data (Figure 4.10). A t flow C , 165  50  I 1 1 1 I  0  o  1  cf CTB A  OO  cf  1  -  Ill  1111.111111111111111111  I  9  &  •  : Flow A I  I 1 I  1  1 1 1 1 1 1  —  E E  ^  0  -100 25  • :  •  -50  N  :  ^ A A  A  • -100 50  i  A  -50  A •  A A  4  4  8  : Flow B  o : -  1111  iii  Flow E E E  AA  A  N  -25 0  10  20  30  40  50  i3  Time 10° (s) Figure 4.10: Change in bed level. Triangles are the heights o f dune crests, r circles are heights o f dune troughs, z ,  (  , and  measured using the echo-soundings discussed in  Tr  Chapter 2. Open symbols are data from echo-sounder l and closed symbols are data from echo-sounder 2 (see Figure 2.3). Solid lines are exponential least-squares regressions through the crests or troughs (coefficients are in Table 4.1). Dashed lines are the change in the mean bed level, z  heil  =-(: + Z .)I2 . The vertical line in the middle r  Ti  panel at / = 23000 s is where z .(l) was forced to a constant value. The effects o f this n  adjustment are shown by the grav lines. 166  Table 4.2: Model fitted results for crest height and trough height. Flow A B C D E  Run 26 27 28 29 30a  c  c  mm -2.123 -0.650 -0.255 0.174 -1.485  a  c  mm 14.205 24.971 12.994 12.421 6.498  r  2  b  c  xlO"  4  2.488 0.506 1.500 0.338 2.450  P~ c z  c  z  0.18 0.85 0.58 0.30 0.04  <0.001 <0.001 <0.001 0.001 0.567  167  c.  a  b  mm  mm  xlO"  -37.240 -31.072 -35.778 -13.348 -13.873  36.463 28.313 32.469 25.069 2315.1  Tl  Tr  r  p~z  0.68 0.68 0.77 0.18 0.30  <0.001 <0.001 <0.001 O.001 0.015  2  Tr  Tr  4  2.117 1.322 0.603 8.396 16.780  z (t) does not reach an asymptote, which becomes problematic later in the analysis. Therefore, Tr  after 23000 s z (t)  is forced to a constant value.  Tr  The next step in calculating C (t) and F (t) is to determine z (i). D  D  ws  The rise in the water  surface was measured at - 1 0 . 9 m from the head box (sensor 2 in Figure 2.3) which is just downstream o f where the measurements that went into estimates o f z  hed  were taken. Initially  z  ws  dropped several millimetres when flow was initiated. However, within a few minutes bedforms developed that increased resistance to flow, causing a rise in the water surface that varied amongst runs, but generally decreased with flow strength (Figure 4.11). With growing bedform size, z  ws  continued to increase. It is expected that z (t)  would increase asymptotically and reach an  ws  equilibrium value, z  , after some time (subscript e indicates an equilibrium value). Beyond this  WSe  point, no significant increases or decreases in z  ws  would be expected. Slope o f the z (t) ws  curves all  tended toward zero after some time period. While some records achieved what appears to be equilibrium, others displayed a gradual increase over the remaining hours o f the experiment. It is likely that this gradually increasing portion o f the records represents an equilibrium state. The water level rise in these records was < 2 m m . Slight changes in the volume o f water in the flume, changes the volume o f sediment available for transport along the channel, or potentially, a large water temperature change could cause a gradual increase in the trend o f z . ws  of z  WSe  Keeping this in mind, values  increase with flow strength.  The variability in z  ws  about z  WSe  also increased with flow strength. A t flow strength A ,  minimum and maximum running averages o f z , after z ws  WSe  at flow E, this separation is < 1 m m . N o t all the variation in  is reached, are separated by 8 mm while is associated with the increased  resistance to flow; some o f it is related to the progression o f the dunes. According to Bernoulli's principle, the water surface w i l l fall over the dune crest due to accelerated flow and rise over the  168  N  E E N  E  E CO  N  N  E E NJ IN  12 10 8 6 4 2 0 12 10 8 6 4 2 0 12 10 8 6 4 2 0 12 10 8 6 4 2 0  | i i i i i i i i  Flow A  I  I  Ill 1  t i r i i i i [ i i  i  i i i i i i |  i  Flow B  I i i i i i i  i  i  i  | i  i  Flow C  Flow D  12 i Flow 10 8 6 4 2 0 '•  i I n  I  II  1'''  i i i i i  '1  E  i i i  ;  :  '-'•"•*" "  \  /  \ •  10  20  T i m e x10  30  40  50  (s)  Figure 4.11: Water surface level records. Black lines are data from water level sensor 1 and gray lines are data from water level sensor 2 (see Figure 2.3). The datum is the flat water surface before the experiment started.  169  dune trough due to decelerated flow. When the dunes are large, as at flow A , the rise and fall o f the water level over the dune crest and trough is several mm about the mean for that bedform. A s equilibrium bedform size decreases with flow strength (see Chapter 2), this variability is reduced. Figure 4.12 displays d(t) , calculated using Equation 4.9. F l o w depth increases asymptotically towards a maximum for all runs and U(t) responds by decreasing asymptotically. Note that had the original z .(t) function at flow C been used, d(t) would have been a continuously increasing Tl  function and U(t) would have been a continuously decreasing function, which would have been counter to observations during the run. Interestingly, in flows D and E , both d and U only begin to change significantly when bedforms migrate into the field o f view from the head box. Obviously, these cross-channel bedforms have a dramatically different effect on resistance to flow than the bed defect developed fields.  4.5.3 Shear Stress as a Function of Time Both S(t) and r (t) s  are displayed in Figure 4.13. Generally, S underwent an initial period  when it increased after the flow was established and then varied about what appears to an equilibrium value after equilibrium 3 D bedforms were established. A t flow E , S does not appear to reach an equilibrium value, which is not surprising given the increasing trends in the water surface level sensor records at this flow. Initial water surface slope, 5,, is o f the order 10" and increases 3  with flow strength (Table 4.3). Once equilibrium 3 D bedforms have developed, mean slope, S , e  increases substantially. A t  flow strengths A , B , and C the increase is 30.6 - 45.7 % . A t the lower flow strengths, S  e  is up to  109.4 - 154.5 % S (Tables 4.3 and 4.4). Variability in equilibrium values o f parameters is f  quantified using the coefficient o f variation, CV, which is the standard deviation divided by the mean. The variability in S  e  is relatively large compared with the early portions o f the record 170  | iiii iiiii [ i iiii ii ii |i iiii iiii |i iiiiiiii |iiiiiii i i  Flow A .  Flow A ] 1  i  I i i  i ii  i i I  1  '  iIi i  i  1  i  i  i I  iii I  i i  ii i i  i  i  ii  i i I  i  i i  ii i ii  i i i  0.43 0.47  i i i  i  i  i>  ' • '  | I 1 I 1 1 1 I I I | 1 I I ! 1 1 I I I | I I 1 I I I I I I | I I 1 I 1 1 I I I | I I I I 1 I I 1 I  Flow B  0.46  r  'co  0.44  ~° 16.0 15.5  0.43  Flow B • ' '• • • • • • • • • • 1  17.5  0.45  1 1 1  '  i  1  i ii i  i i i i i  i  i i  0.42 0.45 rr  i i  r|~m-rn i"i~i~n~'"r"f "f n~rpr  17.0  _ co  1=)  •''<  ''  Flow C  \  0.43 0.42  ] i i i i v r t T f p  r i ! i i i i i | r r i i i Ti i i | t Ti Ti i i i l | i i ITTI'TI  Flow D  0.40 CO  '  rn-rj'T t i i i i ? i i | i i i i i i i i i | r i i J I TI I I  0.44  0.41 0.41  i  0.39 0.38 0.37 0.36 0.37  » I I 1 I 1 1 | I 1 i 1 i i i i I | i  Flow E :  0.36 V)  0.35 0.34  1=)  10  20  30  40  50  T i m e x1Cr (s)  0.33 0.32  50 T i m e x1(T (s)  Figure 4.12: F l o w depth, d , and mean flow, U, calculated as a function o f time. Vertical lines indicate when bedforms developed at the head box migrated into the measurement section at flows D and E .  171  03 CL co  i  1  1 1 1 1 1 1 1 1 1  1  Flow B 03 Q_  i  i I I I ! I I I  I  j  | I I I I I I I I I | I I I I I I I I I | I I I I I I I I I | I I I I I I I I I  ri  i i i i i i i  i  | i ri  II  i t  rrp  i i i i i i  r  i  i > i i i i i i i  rT] i i  FlowC  Flow C 03 CL co  1  I  1  I  •  1  '  1  r i r i i i i i | i i i i i i i i  i i i i  i i  i  I I, I,I I I I I • I I  | i i i i i i i i i | i i r i i i i i i | i i i i i i i i i  T  i  i  1111111111  I | i i i i i i i i | i i i i i i i i i | r i i i i i i i i | i i i i i i i i i | i i i i i i i i i  Flow D  I  Flow D  03 CL  1  I  1  • •  I I I I I I I I I I I I I I  1  I  I I I I  I I T j T m ~ r r v r i i i i i i i i | i i i i i i i i i | i i i i i i i Ti | i i i i i i i i i  ' |  5  Flow E • 03 CL  i  '1 1  i  'i  1  i  Flow E  3  1 0 10  20  30  40  50  Time x10 (s)  l.r  '.J  1  1  1 1 1  1  I  1  10  I I I I  20  30  40  Time x10 (s)  Figure 4.13: Measured water surface slope, S, and shear stress calculated from the depth slope product, x , as a function o f time. Vertical lines indicate when bedforms developed s  at the head box migrated into the measurement section at flows D and E .  172  50  Table 4.3: Initial values o f parameters used to calculate C  D  and F  D  Flow  Run  d (m)  Ui (m s"')  Sj x 10~  z, (Pa)  A B C D E  53 54 57 59 55  0.1593 0.1593 0.1580 0.1574 0.1561  0.4763 0.4540 0.4404 0.3883 0.3497  12.18 9.60 9.39 4.26 4.49  1.9043 1.4992 1.4562 0.6569 0.6875  i  4  173  calculated at t = 300 s.  F  Di  0.0168 0.0145 0.0150 0.0087 0.0112  (N)  3.24 4.55 2.09 1.04 0.90  (CV  Se  = 1 3 . 2 - 3 8 . 1 %) and is largest at the greatest flow strengths. Some o f this variability may be  related to changes in bedform morphology, but a large portion is related to the rise and fall o f the water surface with position over the dune, which is considerably enhanced or damped in the calculation o f slope. The slope is enhanced when sensor A is over a dune trough and sensor B is over a dune crest. Slope is damped when the opposite positioning occurs. Since the water level deviation over a dune increases with bedform size and dune size increases with flow strength, variability in S increases with flow strength (Table 4.4). Bulk shear stress calculated using Equation 4.8, is strongly dependent on S and follows the same pattern, increasing from an initial value and then varying about an equilibrium value. Initial shear stress values, z , varied between 1.90 and 0.66 Pa, decreased with flow strength and were in si  accordance with values reported in Chapter 2. Note that z  Si  at flows D and E are for defect  developed fields and not full width fields that later encroached upon the measurement section from the head box. Mean equilibrium shear stress, T & , is 40.1 - 52.3 % larger for flows A , B , and C while at flows D and E , ise is 163.2 - 120.9 % larger. Variation in T  Se  equilibrium slope; CVz  is within ± 1 % o f CV .  Se  calculation o f C  D  and F  Se  D  is similar to that observed for  The only fluctuating properties in the  are d , S and U, which are already strongly interrelated. Therefore, the  variability in equilibrium C  D  and F , w i l l be the same as for x . D  Se  4.5.4 Drag Coefficients and Force as a Function of Time Figures 4.14 and 4.15 display C (t) and F (t), D  is useful to make a brief comment related to C  D  D  respectively, with H overlain for reference. It  before proceeding with a discussion o f its variability  or any interpretation o f patterns. The drag coefficient is often treated as a constant for a given body, but C  D  tends to increase with body size [Halliday et ai, 1993]. In fluid mechanics, C  D  body is widely accepted to decline with increasing Reynolds Number, Re = £U /v h  174  for a given  (£ is a  Table 4.4: Equilibrium values (after 10000 s) of parameters used to calculate C  D  Flow A B C D E  Run 53 54 57 59 55  de  U  (m)  (m s- )  0.1711 0.1665 0.1655 0.1626 0.1645  0.4437 0.4343 0.4206 0.3758 0.3319  aS  Se  e 1  ox  e  lO" 15.91 13.99 13.22 10.84 9.40 4  e  10" 6.07 4.27 3.09 1.43 2.33 4  (Pa)  (Pa)  2.6685 2.2836 2.1466 1.7291 1.5187  1.0148 0.6921 0.5033 0.2264 0.3815  175  and F  FDe  °c  De  0.0271 0.0242 0.0243 0.0245 0.0276  L  0.0103 0.0072 0.0057 0.0032 0.0071  (N)  120.45 110.65 86.69 37.99 30.80  OF*  (N)  44.93 33.19 23.15 5.19 8.19  E E  E E  E E  E E  0.04 D  CJ  i  0.03 0.02  10 1  x  0.01 0.1  0.00 10  20  30  40  50  Time x1Cr (s) Figure 4.14: Drag coefficient, C , calculated as a function o f time (solid line). The D  exponential increase in dune height, H , from Figures 2.6 and 2.7 are overlaid for reference (dashed line). Vertical lines indicate when bedforms developed at the head box migrated into the measurement section at flows D and E .  176  250 I E E  E E  E E  E E  E E  10  20  30  40  Time x 1 0 ( s ) 3  Figure 4.15: Drag force, F , calculated as a function o f time (solid line). The D  exponential increase in dune height, H, from Figures 2.6 and 2.7 are overlaid for reference (dashed line). Vertical lines indicate when bedforms developed at the head box migrated into the measurement section at flows D and E. 177  characteristic length o f the body, U  h  is the relative velocity o f the body to the fluid and v is the  kinematic viscosity o f the fluid) [Roberson and Crowe, 1993]. In sediment transport studies, it has been accepted for some time that sediment particle C  D  decreases towards an equilibrium value,  regardless o f shape or density, with increasing Re [Rouse, 1946]. The reason for this change in C  D  has been conclusively linked to changes in the boundary layer structure over particles at higher flows [Stringham et al., 1969]. It is significant to note that this experimentally determined relation holds only for particle sizes less than a few centimeters because C  w i l l increase with the body size at a  D  constant flow. A larger body, such as a bedform, w i l l generate a greater level o f turbulence and hence a larger value o f u, than a smaller body. The increase in turbulence w i l l decrease U, but this occurs disproportionately because boundary layers develop. In separated flows, as over dunes, a dramatic increase in u, with bedform size occurs with relatively modest decreases in U (as is demonstrated above). In summary, C  should increase with dune size at a constant flow.  D  The drag coefficient experiences a general increase with bedform size (represented by H ) until an equilibrium H is reached, after which C  D  varied about an equilibrium value, CDC. Values o f  Coe are similar for all runs, varying between 0.024 and 0.027 and, unlike S  e  (or T & ), CDC shows  no variation with flow strength (Table 4.4). Initial values o f the drag coefficient, C , Di  and C are similar (0.014 - 0.017) and C e is 61.3 - 66.9 % greater than C D  is smaller (0.009 -0.011) and C This is because C  DI  D  e  » C  Dl  Di  at flows A , B  . A t flows D and E , C  (146.4 - 181.6 %) when compared to flows A , B and C.  at the lesser flow strengths is computed for the bed defect fields where part o f  the bed remains smooth. C  D  increases when a defect field is overtaken by bedforms developed at the  head box. Figure 4.16 plots C  D  L  a  Di  against the areal bedform length (L ) a  measured from the video (plots o f  as a function o f time can be found in Appendix C). The relation has a strict linear upper limit,  178  1 7 9  but there is considerable scatter for some o f the large bedforms where C  D  is often less than  expected. Some o f this scatter is probably related to drag reduction. However, S and ultimately C  D  are most strongly affected by fluctuations in z  associated with the bedforms when the dunes are  ws  large. Overall, C  does appear to increase with dune size as expected.  D  The drag force is calculated using Equation 4.6 where H is estimated from the least-squares regressions calculated in Chapter 2 (see Figures 2.6 and 2.7 as well as Table 2.3). It is assumed that L =y. v  w  The drag force experiences a strong increase with bedform size. When H increases  exponentially, all other variation in F  D  is damped. This compromises the use o f F  D  o f drag reduction as the bedform size increases. However, F  D  as an indicator  can be used when equilibrium H has  been reached. Initial values o f the drag force, F , varied between 0.90 and 4.55 N and equilibrium Di  drag force, F De, varied between 30.80 and 120.45 N . Both show a decrease with flow strength (Tables 4.3 and 4.4). There is a dramatic 23 - 3 6 X increase in FDe over F  Dj  during the experiments.  4.5.5 Drag Reduction and Bedforms Image-mean non-dimensional span is plotted as a function o f time in Figure 4.17 (plots o f raw A  NDS  and A  NDS  > 0.70 m data are in Appendix C). Descriptive statistics for A  NDS  presented in Table 4.5. For the 3 D stage, A _ NDS  = 1.34 - 1.43 and CV  im  instantaneous development runs, minimum A _  Ams  NDS  im  < 1.2 and A _ NDS  [m  after r  ~ 1.2 for the lower flow  NDS  NDS  generally larger ( A observed where A  N D S  NDS  im  are  = 7 - 1 5 % . F o r the  strength runs during the 3 D stage. Individual crests were observed where minimum A < (Table 4.5). M a x i m u m A _  m a x  1.05  > 1.6 for all runs, but the greater flow strength maximums are  = 1.8 - 2.3) than the lesser flow strength maximums. Individual crests were was as high as 10, but the maximum for most runs was in the range 2 - 5  (Table 4.5). 180  2.5  Figure 4.17: Nondimensional span,  Flow A  Q  1.5  1.0 2.5 Flow B 2.0 Q  <  1.5 h  1.0 2.5 FlowC  :  •  2.0 h CO Q  <  -  1.5 *  •. -  v  1.0 2.5  Li  .  •  i  i  '  —!  1  •  1  1  1  -  i 1  •  s* *"  i  i  i  . i  •  '  i  i  '  i  i  i  i  '.  1i — • — i — i — i — i — • — i — i — i — i — . — p — i — • — i — • —  F l o w D '• 2.0 co  *  Q  <f  1.5  •  ;•  1.0 2.5 Flow E 2.0 co Q Z  <J  NDS  time series. Data are image averages of crests whose cross-stream extent exceeds 0.7 m.  2.0 co  <  A ,  1.5  ^'r ••.••'.V  1.0 0  10  20  30  40  T i m e 1 0 (s) 3  181  50  Table 4.5: Descriptive statistics for non-dimensional span o f 3D bedforms. Flow  Ima£;e Mean Max  Run A NDS-im  A B C  D E  53 54 57 59 55  1.3422 1.3971 1.3355 1.4256 1.3986  G ^  NDS-im  0.1187 0.2088 0.1318 0.1274 0.0925  Raw Data Min  MaxA  NDS  A NDS-im  1.8204 1.2617 1.9810 1.7608 1.6510  182  A NDS-im  1.1277 1.0650 1.0617 1.2033 • 1.1986  3.2395 4.9570 10.1481 3.3540 4.4790  1.0457 1.0623 1.0413 1.0390 1.0470  The most detailed  record was generated for F l o w B , where there is clear evidence o f at  least two dramatic increases in the three-dimensionality o f the bedforms in the video view, separated by ~4.5 hours, at 5.5 and 10 hours. There is no relevant time scale associated with the flow or the flume that might cause this. However, low relief bars are known to develop in flumes. Indeed, the long sediment pulses described in Chapter 2 seem to be evidence o f them. It is easy to envision a scenario that might lead to changes in the flow velocity and depth, as a bar form passes, that could account for this increase in  A . NDS  In order to determine i f drag reduction is occurring during the 2-3D dune transition, it is necessary to compare the A  NDS  C  D  and F  to C  D  and F  D  time series. Figure 4.18 plots the first hour o f the  time series with a line indicating 2 D and 3 D stages detennined when A <  D  NDS  1.2 respectively. A s noted above, the F  1.2 and >  time series is strongly dependent on H , which obscures  D  any other patterns when H is increasing exponentially. There is no dramatic reduction in C  D  the dunes have become 3 D ; there is actually a 15-27 % increase (see Table 4.6). The C  D  after  time series  also occurs against a background o f increasing bedform size, so it would be expected that it would increase like F , but this is not the case. D  There is a minor increase in C  D  and C (no C  after bedforms are initiated, between t = 0 s and t for Flows B c  data are available before t at Flow A ) . Following this period, C  D  c  D  - 0 . 0 1 5 , and then begins increasing, closer to where r has an important consequence. When C  D  m a x  occurs (when A  NDS  remains stable at  increases to ~1.4). This  should be increasing, it is not, suggesting that indeed the  change in morphology is reducing drag. Based on the dune H data from the echo-sounders presented in Chapter 2, there is an increase o f 1129 (Flow A ) , 567 (Flow B ) and 439 % (Flow C ) in H between t and r c  m a x  . Image averages o f areal bedform length increase 413 (Flow A ) , 244 (Flow  B ) and 186 % (Flow C ) over the same time period. This occurs with relatively little change in C . D  183  Figure 4.18: Drag coefficient, C , D  and drag force, F , calculated as a function o f time D  over the first hour o f the experiment. Vertical dashed lines indicate when the transition between two- and three-dimensional bedforms occurred in Figure 4.4. The time when dunes extend across the entire flume is t and t c  approaches 1.4 for the first time.  184  mm  is when the non-dimensional span  Table 4.6: Mean 2D and 3D C, Flow  Run  2D t (s)  A B C  53 54 57  0-1080 0-900 0-1020  2D Co 0.0150 0.0162 0.0144  3D t (s) 1080-3600 900-3600 1020-3600  185  3D CD 0.0172 0.0191 0.0183  2D/3D (%) 15 18 27  Another way o f examining whether drag reduction occurs with increases in A  NDS  A  and C  NDS  records to determine i f they are out o f phase. If an increase in A  D  decrease in C  NDS  0  and C  D  is to compare causes a  the two are dynamically linked. In order to further examine this possibility,  A  NDS  time series were correlated. It is important to note that there are increasing and equilibrium  portions o f some o f the time series. These portions need to be separated. The increasing portions o f the A  NDS  and C  time series w i l l inevitably produce positive correlations which betray the true  D  nature o f the phase relation. Therefore, at flows A , B and C , time series were split into stages when H was increasing (prior to 5000 s) and when H was in equilibrium (after 10000 s). A t flows D and E, A  NDS  data are already restricted to the equilibrium portions o f the record.  The first step in the correlation analysis was to determine i f A ^ and C  D  are normally  distributed using a Kolmogorov-Smirnov Normality Test. I f the normality test p value exceeds 0.05, at the 95 % confidence level, the distribution is normal. Table 4.7 demonstrates that few o f the selected time series portions are normally distributed, so Pearson Product Moment Correlation is inappropriate. Spearman's Rank Order Correlation does not assume the data are normally distributed and, therefore, was used for the analysis. For the equilibrium portions o f runs at flows A , B and D , the correlation coefficients, r , are negative, as expected, but none are statistically significant (Table 4.7). A t flow strength C , r is positive but the result is not significant at the 95 % confidence interval. A t flow strength E , r is positive and the result is significant at the 95 % confidence interval. These results are surprising as visual inspection o f the time series suggests that there are portions where A  NDS  and C  D  are clearly  out o f phase (Figure 4.19; full records are in the Appendix C). Most other portions show no relation or are poorly in phase. O f the two dramatic increases in A  NDS  run at Flow B , C  D  at roughly 5.5 and 10 hours during the  is significantly lower than the mean for one and higher than the mean for the  other. 186  Table 4.7: Results o f Spearman Rank Order correlations between A  NDS  and C . D  Normality test p-values are given for both  and  C , separated by a slash. D  F l o w R u n  Normality ^ T e s t  n  r  0 . 0 0 4 / 1 0 3 0 . 0 2 0 0.001/ 54 ' B 5 4 9 0 . 4 . 1 8 * 0 . 3 8 8 * / C 5 7 9 1 0 . 0 9 3 * 0.001/ D 5 5 9 0 O . 0 0 1 0 . 3 8 8 * / E 8 5 5 9 0 . 0 9 3 * "Statistically significant at the 95% c o n f i d e n c e interval A  5 3  187  0 . 1 3 2  0 . 1 8 2  0 . 0 3 6  0 . 3 9 7  0 . 0 7 2  0 . 4 9 4  0 . 0 6 0  0 . 5 7 5  0 . 2 5 8  0 . 0 1 7 *  1.0 9000 1.8  0.01 9500 T  1  10000 1  1  1  10500 1  1  11000  11500  12000  1—I-  • Flow E  0.024  °  O H 0.015  0.010 11000  11200 11400 11600 11800 12000  12200  12400  Time x 1Cr Figure 4.19: Examples of when the drag coefficient, C , and the non-dimensional span, D  A , NDS  are out of phase with one another.  188  There may be a simple explanation for this outcome. Nearly all the variation in the C  D  time  series is derived from the water surface slope and level. It is assumed in the derivation o f Equation 4.8 that S is equivalent to the energy grade. Given that a portion o f z  and 5 is dependent on the  ws  passage o f bedforms, it is not clear that this, assumption is satisfied. This is particularly true during the equilibrium phase o f the experiments when bedform position can cause several millimetre variations in z  m  . Unfortunately, it would be nearly impossible to remove this effect from the  current data set. Another significant impediment to finding a strong out-of-phase relation between A  NDS  and C  D  is that A  is estimated for only a 1 m long section o f the flume. Water surface  NDS  slope is estimated over 2.26 m o f the flume, so there is a large portion o f the bed that may not have a similar  A . NDS  It would appear from C , calculated over the first hour o f the experiments, that drag reduction D  does play an important role in the 2-3D transition. The drag coefficient fails to increase against a background o f increasing bedform size. Increasing the level o f three-dimensionality may also serve to reduce drag once 3 D bedforms have been established. Portions o f A  NDS  and C  D  time series are  out o f phase. However, the nature o f the data set, in particular whether the slope can be linked to the energy grade, precludes any confident statements in this regard. In light o f this another experiment was designed to examine the turbulence structure and the form drag over fixed 2 D and 3 D dunes. The dunes, modelled after those observed in the experiments discussed here, were 0.45 m in length and 25 mm in height. Only the shape o f the crestline was varied between runs, providing a data set where C  D  could be accurately known for 2 D and 3 D  bedforms. These data are described in the next chapter and are more suitable to determine i f the 23 D dune transition is linked directly to drag reduction processes.  189  4.6 Summary The transition between 2 D and 3 D dunes in a homogeneous 0.5 mm sand was examined. A flat sand bed was subjected to a 0.155 m deep, non-varying mean flow ranging from 0.30 to 0.55 m/s in a 1 m wide flume. Changes in the planimetric configuration o f the bed were monitored using highresolution video. A video capture card was used to yield a series o f 10 s time lapsed digital images. The images reveal that, once 2 D dunes are established, minor, transient excesses or deficiencies o f sand are passed from one crestline to another. The bedform field appears capable o f ' s w a l l o w i n g ' a small number o f such defects but, as the number grows with time, the resulting morphological perturbations produce a transition in bed state to 3 D forms that continue to evolve, but are patternstable. A l l the observations made during the experiments plot as 2 D dune fields on conventional bedform phase diagrams, despite the observation that the ultimate dune morphology is 3 D . Stable 2 D dunes were not observed in this calibre sand, which calls into question the reliability o f bedform phase diagrams using crestline shape as a discriminator. The idea that 3 D arrangements o f bed roughness elements passively reduce drag, when compared to smooth or 2 D arrangements, was examined as a possible explanation for why the transition occurs. A defined increase in the drag coefficient is lacking during the early stages o f the experiment and during the 2-3 D dune transition, suggesting that passive drag reduction processes may be at work. Further, an out o f phase relation between the drag coefficient and the non-dimensional span suggests these quantities may be dynamically linked at certain times. However, correlation analysis fails to reveal a relation between the drag coefficient and the non-dimensional span. Changes in the drag are somewhat obscured by other processes - primarily dune growth - as the transition occurs. A l s o problematic is that the measured water surface slope may not be representative o f the energy gradient. Unfortunately, these problems cannot be resolved here. Thus, another experiment was conducted using fixed bedforms with varying crestline shapes. Dunes were scaled to the live bed  190  experiments discussed here, had the same size and were subjected to the same flow conditions. These experiments are discussed in Chapter 5.  191  Chapter 5-Aspects of Turbulent F l o w over T w o - and Three-Dimensional Dunes 5.1 Introduction The purpose o f this chapter is to discuss a set o f experiments designed to examine the turbulent flow over fixed dunes with different crest shapes but constant wavelengths and heights. F l o w is examined over a fixed flat bed and six dune morphologies including (1) straight-crested, twodimensional (2D), (2) full-width saddle (crestline bowed upstream), (3) full-width lobe (crestline bowed downstream), (4) sinuous crest, (5) regular staggered crest and (6) irregular staggered crest. The flow structure is discussed in the context o f recent experiments that have elucidated the flow structure over 2 D dune morphologies. The effects o f three-dimensional (3D) morphology on momentum transfer, mixing and energy exchanges from the mean flow to turbulent frequencies are discussed. Spatially averaged flow characteristics are presented for each morphology and drag reduction over 3 D dunes is examined.  5.2 Experimental Procedure The experiments were conducted in the C i v i l Engineering Hydraulics Laboratory at the University o f British Columbia using a tilting flume channel, 17 m long, 0.515 m wide and 1 m deep. The head box exit was fitted with a honeycomb o f 0.025 m (1 in.) P V C pipe that was 0.3 m (1 ft.) long to ensure quasi-uniform flow out o f the head box. A Styrofoam float damped water surface waves. A n adjustable sluice gate at the rear o f the flume controlled the width o f exiting flow and, ultimately, flow depth. Flume slope was adjusted by a pair o f hydraulic jacks at the rear o f the flume. In order to generate flow in the flume channel, water was pumped to the head box tank (1.5 x 2 x 3 m) from an underground tank via an axial pump driven by a constant speed electric motor. Flow rate was controlled by a screw valve installed on the inflow pipe that dumped water into the head box from above. F l o w rate was measured with an acoustic pipe flow meter, mounted upstream o f the control valve. Discharge was maintained by the valve to within ± 3 . 3 x 10" m s"'. 4  192  3  5.2.1 Fixed Bedform Design  Bedforms were constructed from 14 pressure treated 16 ft 2 x 4 planks. A predetermined bedform shape was carved into each 2 x 4 plank eleven times and the boards were bolted together to form bedforms that spanned the flume channel. The accuracy of the carved bedform heights was approximately ± 1 mm. The template used for bedform shape was based on the earlier experiments a t N S L - U S D A (discussed in Chapters 2-4). The along-stream and vertical locations of bedform features, identified in Figure 2.11 for crest type C3 and stoss type SI, were determined at flow strength B. These dimensions were normalised by the bedform height, H , and length, L, to derive dimensionless bedform morphology for a dune at flow B. Dimensionless morphology is averaged to produce a dimensionless scale model. By selecting a desired H or L, a corresponding L or H value can be calculated from the average bedform aspect ratio, HIL,  for flow B. The dimensionless scale model  is converted to actual dimensions by multiplying through by H and L. It was decided that, since the transition between 2D and 3D dunes occurred during the exponential increase in bedform size (i.e. in H and L), the model bedform dimensions should be less than equilibrium size for the flow strength. Ideally, bedforms should be the precise size they were when the 2D-3D transition occurred. Unfortunately, at the transition H ~ 10-15 mm and L ~ 0.2 - 0.3 m. The bedforms would be too small to take measurements over using the available instrumentation (an acoustic Doppler velocimeter or A D V ) . In light of this, L = 0.45 m was selected as a desired length. This provided that at least 10 bedforms could be carved into the 16 ft planks. With L = 0.45 m and average HIL  = 0.05 for flow B, the corresponding H = 22.5 mm. A smaller  H would have been too difficult to carve into the planks and the A D V sampling volume, ~0.9 cm , 3  would have been roughly the same size as the step height. The resulting morphology appears in Figure 5.1. The variability in H across the 11 bedforms was < 5 %.  193  r  E o in  o o  cn  0>  o in co  O C  co  C  U  EL  .cn  cn „ •— Q cn  1—J  o o  CJ r  CD  cS  a.  .  fN  t o  cn  J=  2  CJ CJ  "5  w  cn cd  X)  E  -o  ej — O cn CJ  O CN  3 -C  .2 C+.  •a O in  o  x:  D . cj O  $  in o  1 1 1 1  in o in  N  in o  CN CN r -  in  CM  o  -  ir> o  (%) H / Z (LULU)  T -  194  Z  CJ  CJ  1  CJO  c  .60 _o LL.  CS  CJ  "O x:  -*-»  CJ  •a » 60 c  .2 o_o  -g  X)  -a E 5  (S3 XI  3  cr °  5  -o c  .2 ° t5  X)  cn tC  'cn C£ CJ ~-  C O -w  o cu  ca  £ 2  cn  CJ  C T3 O C  cj  ^  i- x :  S  F  cS CJ  o  c cs  C  CS  2 2  4-i CJ  cn cn  C 3  C O  cj  C  ja  CJ  c2 .2? CN  E JS  c  -a T3  60 o^~ sr. 6o "5  xT  u, S CJ ri -e  "  00  "O  co o  3  .2 2P  y. 5 CJ •53 CJ  cj  o o  .M cS  <u n 3 T3 3 T3 P  m  ie £uP o  cn CJ cn c O 3 +•» x; -5  "  ~I  eq  CS  rt  cn ^3  —  o in  _ S3  o  ^ o .EE  O O  CN  cj  .  O .60 c/T  Ja  o-  E  H  °3  CN CN  § £  cn  cj  cj cj c <+-  E  o G_  —  "cn  c  O  O  o cS  O O  2  .2  o a. CQ  C  O-cS cn 03  CO x:  „ a . <o  CN  p 60 o  o cn o  4)  c cS  X)  § 3 £C CO  CL)  o m  o  43  -O  CJ  -3 <u  CJ  CJ  J =  cS  -S  .60 .E  '53 xt> "cS cj CJ  2  3  £  ^  •  XI  w  B y staggering the alignment o f the boards, various dune crestline configurations could be generated. S i x different crestline configurations were designed (Figure 5.2). A straight crested 2 D dune (2D) was created by lining up the crests carved in the wooden planks. With the understanding that Sirovich andKarlsen [1997] examined flow over regular and irregular shapes (see Chapter 4), a set o f regular and irregular crested 3 D bedforms was designed. The regular crested dune morphology had the boards staggered at 92 mm relative to one another (the dune trough lined up with its neighbouring board's crest). This is an extreme example o f a regular dune crest. Practically, it is unlikely that a dune with lobes and saddles so tightly spaced could occur naturally. Staggering o f the boards for the irregular dune morphology was essentially random. The irregular crest could form naturally and similar morphologies were observed during the active transport runs discussed in Chapter 4. The regular and irregular features had A  NDS  A  NDS  = L IL c  v  where L  c  is the crestline and L  v  =2.4 (Table 5.1). Recall from Chapter 4,  is the linear distance between the crest endpoints.  More conventional 3 D dunes were designed with crestlines that bowed downstream over the whole flume width; these are referred to as full-width lobes ( F W L ) (Figure 5.2). Another set was designed with crestlines that bowed upstream; these are referred to as full-width saddles ( F W S ) (Figure 5.2). For both configurations, the two centreboards were staggered at 35 mm relative to the next and the other boards are staggered at 45 mm relative to the next. The full-width bedforms had a non-dimensional span, A = NDS  1.43.  A sinuous crested bedform was composed o f two lobes, one saddle in the centre o f the flume and two half saddles at the side walls with approximately the same A  NDS  as the full-width bedforms  (Table 5.1). For the sinuous crest, the boards were staggered by 35 mm from the next (Figure 5.2). Measurements were taken over a sinuous lobe ( S N L ) and a sinuous saddle (SNS). Once measurements over the sinuous crested bedform were complete, the crestline was smoothed with wood filler to remove the jagged edge, producing a new bed morphology, a sinuous and smooth  195  2 D Straight  Regular Crest  Full Width S a d d l e  Irregular C r e s t  Full Width L o b e  Sinuous Crest  Figure 5.2: Dune morphologies tested. Thin horizontal lines indicate the location o f the dune crest on each plank. Lines down the centre (and along the right lobe o f the sinuous crest) indicate where the profiles were taken. F l o w was from bottom to top.  196  Table 5.1: Summary o f flow parameters. Shear stresses are corrected for side wall effects using the relation supplied by Williams [1970]. Flow Sinuou SinuousFlow B Flat 2D FWL FWS REG IRR Parameter s Smooth --  --  1.0  1.43  1.43  2.59  2.31  1.33  1.33  0.0723  0.0376  0.0376  0.0376  0.0376  0.0376  0.0376  0.0376  0.0376  --  --  0.1394  0.1412  0.1413  0.1442  0.1458  0.1438  0.1448  max>  --  --  0.1640  0.1632  0.1623  0.1660  0.1696  0.1657  0.1683  d ,m  0.1517  0.1535  0.1507  0.1520  0.1533  0.1561  0.1569  0.1549  0.1562  V, m s"  0.4766  0.4756  0.4846  0.4803  0.4763  0.4679  0.4653  0.4713  0.4674  Fr  0.39 72300  0.39 73010  0.40 73010  0.39 73010  0.39 73010  0.38 73010  0.38 73010  0.38 73010  0.38 73010  1.06  0.767  1.43  1.41  1.59  1.49  1.33  1.27  1.16  A NDS Q, m s3  min >  d  d  1  1 1 1  m  1  Re S X1 c r  3  Determinations based on depth-slope product Ut ,  m s"  0.0405  0.0323  0.0437  0.0436  0.0465  0.0454  0.0430  0.0417  0.0400  T ,Pa  1.6337  1.0447  1.9072  1.9052  2.1610  2.0571  1.8507  1.7422  1.6028  ffs  0.0560  0.0370  0.0651  0.0662  0.0763  0.0753  0.0685  0.0628  0.0588  S  i s  197  saddle (SSS). This provided an opportunity to examine the effect o f the jagged dune face on shear stress estimates. Upstream and downstream of the dune configured bed, a flat bed section was installed at the level of the first dune trough. The flat bed extended ~6.5 m downstream from the head box and ~3 m downstream o f the dune field. The use o f fixed bedforms is problematic. A s discussed below, an active bedload layer has been shown to increase turbulence [Best et al. 1997]. Fixed beds eliminate the migration o f the bedform that theoretically increases the resistance to flow. A l s o , the solid boundary offered by wood bedforms eliminates the exchange o f fluid between the interstitial space within the bedform and the fluid flow. However, recent work by Venditti and Bauer [in review] suggests that these conditions may affect the turbulence structure over a dune only nominally. A more easily corrected problem is that a smooth bedform does not have the same grain roughness as a natural bedform. To reduce this problem, the flat and dune field portions o f the bed were painted with contact cement to which a layer o f 0.5 mm sand was adhered.  5.2.2 Flow Conditions Since the dune morphology was based on dunes observed during flow strength B , the experiments were run at the same discharge, Q =0.0723 m s" , corrected for the smaller width o f the flume, 3  1  0.0376 m s" . The initial flow depth, d , at flow strength B was 0.152 m and the initial slope, S, was 3  1  1.06 x 10" . A n attempt was made to set d and S to these values by adjusting the sluice gate and 3  the flume tilt to provide constant depths over the upstream flat portion o f the bed. The flow depth established was 0.153 m. Unfortunately, the flume tilt was found to be slightly different than 5* over the flat bed portion o f the bed (i.e. at a flume tilt o f zero, S ^ 0). Thus, S was not perfectly matched between the model and the prototype. Bed slope over the flat portion was - 7 2 % o f S at flow B and shear stress, T = p gdS, s  w  was - 6 4 % o f r  s  at flow strength B (p  198  w  is water density and g is  gravitational acceleration). In spite o f this, the velocities, U = QJ(d- y ), were nearly identical w  (Table 5.1) and the flow was uniform in both cases. Best et al. [1997] suggest that a transport layer modulates turbulence by increasing roughness heights and near-bed turbulence intensities while reducing mixing lengths. It is likely that this is part o f the reason why the fixed flat bed data yield a lower value o f z  s  than flow B , but the same d and  U . However, turbulence modulation is not sufficient to account for the entire deviation. The fixed flat bed data w i l l be used to examine the vertical structure o f the boundary layer. Thus, the difference in T , between the active transport and fixed bed conditions, w i l l not affect the results greatly. S  However, care is needed when using  in some o f the spatially averaged flow calculations below.  Over the 2 D dune bed, the sluice gate opening and S were readjusted so that d over the crests was similar to d over the crests o f dunes with H =22.5 mm at flow B . Quasi-equilibrium flow was achieved by adjusting the slope to obtain the same depth (±1 mm) over five successive bedform crests. Over subsequently tested dune configurations, the sluice gate opening was maintained so that only the flume tilt was adjusted to attain quasi-equilibrium flow. A s such, d and U are a function o f flume tilt (ultimately S) and crestline configuration. Table 5.1 summarises the bulk hydraulic conditions for each crest configuration. Only one set o f hydraulic conditions is presented for the sinuous crests (SNS and S N L ) . The mean flow depth, d , was calculated as  d =d  min  where d  min  +H{\-P)+  -""  Zws  lx  ~ ~ Zws  5.1  min  is the minimum depth (over the crest), j3 is the bedform shape factor, z _ ws  maximum water surface over the dune length and z _ ws  min  max  is the  is the minimum water surface over the  dune length. Recall from Chapter 2, fi = Aj'(HL) where A is the cross-sectional area o f the bedform.  199  The minimum depth was calculated over the same 4-5 dunes that were used to establish quasiequilibrium flow. Mean flow depth over the dunes varied between 0.151 and 0.157 m depending on the bed configuration. Mean flow velocity, calculated using d , varied accordingly, between 0.484 and 0.465 m s" . Over the 3 D beds, d is larger (U smaller) than over the 2 D bed, but d was largest (U 1  smallest) when A  was greatest ( I R R and R E G ) and when the crestline was complicated (sinuous  NDS  crests). Depths and velocities were nearly identical over the 2 D bed and simple 3 D forms ( F W L and FWS). Froude numbers, Fr = Xjj-^gd , were ~0.38-0.40 and the Reynolds number, Re-Ud/v  was  73010 (v is the kinematic viscosity) demonstrating that the flow is both sub-critical and fully turbulent. Bed slope varied between 1.16 x 10" and 1.59 x 10" , was greatest over the full-width 3  3  saddle and regular crestlines, intermediate over the 2 D and full-width lobe crestlines and lowest over the I R R and sinuous crests. Bulk shear stress was corrected for side wall effects using the relation proposed by Williams [1970] and varied between 1.6 and 2.1 Pa. Values o f r  s  patterns as S. Corresponding shear velocities, u  = ^jz / p  t$  s  w  follow the same  , varied between 0.040 was 0.046 m s"  and corresponding friction factors, ff = 8 T / p J J , varied between 0.059 and 0.076. 2  5  5.2.3 Measurements and Analysis Velocity measurements were obtained using an A D V that measured three-component flow velocities (streamwise, u, cross-stream, v , and vertical, w) at 50 H z . The A D V has a reported precision o f ±0.1 mm s" and a focal length o f -0.05 m. A D V signals are affected by Doppler noise, 1  or white noise, associated with the measurement process [Lohrmann et ai, 1994]. The presence o f this noise at high frequencies may create an aliasing effect in frequencies greater than the Nyquist frequency (herein f =25 Hz). To remove possible aliasing effects, a Gaussian low-pass filter with a n  200  half-power frequency o f 25 H z was applied to the velocity time series, removing all variance at frequencies above / „ [Biron et al., 1995; Lane et ai, 1998]. The A D V manufacturer provides signal quality information in the form o f a correlation coefficient, r  ADV  . The manufacturer suggests that when r  ADV  does not exceed 0.7 the signal is  dominated by acoustic noise and, as a rule o f thumb, that at-a-point measurements should be discarded when r  ADV  does not exceed 0.7 for more than 70 % o f the record [Sontek, 1997; Lane et al.,  1998]. However, recent work by Martin [2002] has suggested that signal quality is reduced in highly turbulent regions o f flow and suggested that the threshold r  may be < 0.7. Martin [2002]  ADV  conducted sensitivity analyses that revealed accurate mean velocities could be obtained when r  ADV  exceeded 0.4 and accurate Reynolds stresses could be determined when r  ADV  The measurements presented here also indicated that low r  ADV  exceeded 0.7.  values occurred in highly  turbulent regions o f flow. Near the bed, along the lee slope and in the separation zone, r  ADV  frequently less than 0.7 for < 70 % of the record. L o w r  ADV  was  values appear to be distributed randomly  in these records (i.e. low correlations do not appear to be related to a specific turbulent motion). In recognition o f these observations, a filter was designed to remove data points in the time series when r  ADV  which r  ADV  did not exceed 0.7. Then at-a-point measurements were removed from the data set in did not exceed 0.7 for > 70 % o f the record. When the measurement was near the bed,  along the lee slope or in the separation zone, at-a-point measurements were accepted when r  ADV  exceeded 0.7 for > 40 % o f the record. A t a sampling rate o f 50 H z , a record in which only 40 % is retained is still sampled at a nominal rate o f 20 H z . In this respect, the data can be viewed as having been collected in a 'burst sampling' mode rather than a 'continuous' time-dependent sampling mode. A test section was defined over the eighth dune, 9.5-10.0 m downstream o f the flume entrance. A total o f 35-37 profiles o f velocity were taken along the flume centre line and spaced at 0.014 to 0.018 m apart. Each profile consisted o f 10-15 vertical measurement locations sampled for 90 s. Figure 5.3 201  150  M  i  i  i  i  i  i i  -|—i—i—i—i—|—i  i  i  i |  i—i—i—i—f  2D  100 J=  i  Flow  50  N  0 -50  -I  9.2 170  T  1  1  1  1  1  I  9.4 1  1  1"  t""I  L—l  I  1  9.6  I  1  I  1_  -I  9.8  1  1  1  10.0  L_l  L _ l I  10.2  -i—|—i—i—i—i—|—i  11  I  I  I  I  I  I  10.4 i  i—i—|—r  I  I  I  1  I  I  10.6  '  I  '  I  I  10.8  I  1_  11.0  -  160  E E  150  T3  140 130  1  9.2 150 125 100  E N  75 50 25  n  1  9.4  r-  8 8  -25  I  L _ l I  10.0  • • • • •  •  I  I  I  6 0 l l oi ° 2o o oo o o o o o o o oo o o V VV v o o V V  V V  8  V  V  V V  V  V  °  I  I  I  I  10.4  i  I  i  10.6  \  '  I  '  i  '  10.8  •  8  8  8  8  8  8  V  v V  *  Hi  HI _J  9.7  •  8-S 8 8 8 a o 8  8  9.8  •  9.9  1  1  10.0  •  '  I—  10.1  Distance A l o n g F l u m e (m) Figure 5.3: Example o f bed and water surface profiles over bedforms 5-9 (top) and corresponding flow depths (middle) for the 2 D dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel. See Appendix E for similar diagrams for the other dune configurations. Circles indicate where the probe was mounted in the 0° position: Down oriented triangles indicate the probe was in the 45° position. Squares indicate the probe was in the 90° position. U p oriented triangles indicate the probe was in the 0° position but that the lower threshold for data retained after filtering was 40% as opposed to the 70 % retention threshold used for the rest o f the data.  202  I  '  11.0  •  V  V  -50 9.6  I  I  .10.2  • • • • • • • • • • • • • • • • • • • • • •  88  v vv R vv  0  9.8  — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I —  • • • • •  o o  9.6  1  shows the location o f the velocity measurements over the 2 D dunes. Locations were similar over the other dune configurations (see Appendix D for velocity measurement location diagrams for all configurations). Over the sinuous crestline, the centre line profiles were over a saddle (SNS). A n additional set o f profiles was taken nearer to the side wall over a lobe ( S N L ) (Figure 5.2). Only one set o f hydraulic measurements was taken over the sinuous (ragged crest) configuration. In addition to the profiles over the dunes, 6 profiles were taken over the flat bed at 0.10 m intervals between 5.0 and 5.5 m. For most measurements, the A D V probe head was oriented towards the bed at 0°. The lowest point in each velocity profile was at a height o f - 0 . 0 0 5 m above the bed and highest point was at -0.08-0.11 m, depending on the position over the dune. Poor data quality caused by acoustic feedback was observed at several heights above the bed. This is a common feature o f A D V measurements [see Lane et al, 1998]. In order to obtain data at these heights, the A D V probe head was rotated 45° in the cross-stream plane to reduce the feedback. To augment the velocity profiles, the A D V probe head was also rotated 90° in the cross-stream plane to obtain a measurement that was typically 0.11-0.12 m above the dune crests. Data collected when the probe head was at 45° and 90° needed to be rotated into the vertical plane. The streamwise velocity has the same magnitude and direction, regardless o f whether the probe was in the 0°, 45°, or 90° positions. Thus, the data need to be rotated only in the v-w  plane.  The v and w velocity components were rotated according to the following convention V  ,  T  =v cos<p + w sin<p m  m  where subscripts m and r refer to the measured and rotated velocity frames, respectively, and q> is the angle o f the probe head (0°, 45° or 90°). The subscript p indicates the rotation is necessary for realignment when the probe is in the 45° or 90° position. Care was taken in orienting the probe head so it was aligned with the maximum streamwise  203  velocity in the 0°, 45°, or 90° positions. However, small misalignments were still common, which presented the possibility o f having slightly different planes o f reference for the measurements in different configurations. This is a potentially serious problem when different probe head alignments are used. Therefore, rotations are necessary in the u-v  and u-w  plane. A u and v rotation follows  the convention  u =u cosy ra  m  + v s\ny  ^  rp  m = - « „ , sin y + v,. c o s y  v  p  where y is a misalignment angle in the u-v  plane and the subscript a distinguishes this rotation  from the realignment necessary when the probe is in the 45° or 90° position. A u and w rotation follows the convention ra2 = «  U  r a  C O S 0 + W ,sin0 r ;  w =-u ra  5.4  sin (j) + w cos (j)  m  rp  where <p is a misalignment angle in the u-w  plane and the subscript 2 indicates that this is the  second rotation o f the u component. In order to estimate the rotation angles y and 0 , calibration files were taken each time the probe was readjusted at 0.06 and 0.09 m above the flat bed portion o f the flume at 5.0 m from the head box, where v and w components o f velocity could be expected to average to zero. B y assuming the mean v  ra  velocity is zero, Equations 5.3 can be rearranged such that  7  =  t  an-'-^  5.5  (capital u, v and w represent mean at-a-point velocities). In order to estimate 0 , the data need to be rotated by applying Equations 5.3. Then, assuming mean w  m  velocity is zero, Equations 5.4 can  be rearranged such that  <i) = tan"  W 5.6  1  U,.„  204  Calibration files were used to estimate 7 and 0 , which were averaged between the two heights. Values o f 7 and tp varied between -4° and 3° but were typically smaller. Data collected over the dunes are then rotated by 7 and <j> using Equations 5.3 and 5.4, placing the entire data set in the same plane o f reference. B y approaching the rotations in this manner, deviations in the mean vertical and cross-stream velocities over the dunes are accepted as real.  5.3 Empirically Derived Structure of Flow over Flat and 2D Dune Beds The mean and turbulent flow structures over fixed 2 D dunes have been described previously  [Nelson etal, 1993; McLean et al, 1994; Bennett and Best, 1995; Venditti and Bennett, 2000; Best and Kostaschuk, 2002]. Figure 1.3 shows the main characteristics o f the flow are (1) convergent, accelerating flow over the dune stoss, (2) flow separation at the dune crest, (3) flow reattachment at ~3.5 - 5 H [Engel, 1981; Bennett and Best, 1995; Venditti and Bennett, 2000], (4) a turbulent wake and shear layer originating at the crest, extending and expanding downstream, (5) an internal boundary layer (IBL) that grows from the reattachment point downstream beneath the wake towards the crest, and (6) an outer, overlying wake region. O f particular interest here is the shear layer and wake region and their turbulence characteristics. The wake region resembles flow behind a cylinder [McLean, 1990], and three-dimensional rollers, kolks, and internal boils occur along the shear layer, dominating the macroturbulent flow structure (See chapter 1 for a more complete description o f flow over 2 D dunes). The flow field over the 3 D dunes is examined empirically v i a several simple yet informative turbulence and velocity relations. Contour maps for all flow and turbulence parameters were constructed, but only selected results are presented here. Calculation methods for the mean and turbulence parameters are presented below, followed by a description o f how each has been observed to vary over flat beds and 2 D dunes in previous research. The descriptions o f flow over 2 D dunes are based on work by Nelson et al. [1993], McLean et al. [1994], Bennett and Best, [1995] and Venditti  205  and Bennett [2000], unless otherwise indicated. The focus of each of these studies is different, but descriptions of the flow field are consistent. Figures 5.4 and 5.5 show contour maps of time-averaged streamwise, U, and vertical, W , flow velocities defined as n  i  i  £/ = —TV  «  and W = — Sw  5.7  i  where u] and w are instantaneous velocities and n is the total number of measurements. Over a flat t  bed, U and W should not show any along flow variation, unless the flow is non-uniform. Flow over a 2D dune is highly non-uniform, accelerating over the dune stoss slope due to convergence and decelerating over the dune trough due to expansion. The vertical velocity responds with flow directed towards the bed downstream of the lee slope and water surface directed flow up the stoss slope from reattachment towards the crest. Mean cross-stream flow, V, is negligible across 2D crestlines but this does not indicate that eddies do not have a significant cross-stream extent. Over the 3D dunes considered, V will probably be negligible as profiles are taken over symmetric crestlines, with the exception of the irregular bed. Here, V will not necessarily average to zero along the dune profile. Streamwise root-mean-square velocity, U , rm  was calculated from  -|0.5  U  2  =  rms  5.8  and the streamwise turbulence intensity, displayed in contour maps for each bed in Figure 5.6, was calculated as /  5.9  u Ut  s  The pattern of I  u  a flat bed, I  u  over a dune is a reflection of the pattern in U  rm  . Recall from Chapter 3 that, over  decreases exponentially towards the bed following Nezu and Nakagawa's [1993] semi-  theoretical universal functions. Previous research suggests that, over 2D dunes, maximum 206  U  nm  z (m) O  O  O  z (m) O  O  O  z (m) O  O  O  O  z (m) O  O  z (m)  O  O  CT!  O  O  z (m) O  O  O  O  z (m) p  o  z (m)  o  O  CJ1  O  O  z (m) O  O  O  O  values occur just downstream o f reattachment and local highs occur within and just downstream o f the separation cell. Elevated U  values occur in the highly turbulent wake region.  NM  The Reynolds shear stress, z , is plotted in contour maps displayed in Figure 5.7 and was tlw  determined using  U'W = -Y (U,-UXW -W)  5.10  T„„,=-P w w'.  5.11  J  I  ,  i v  Recall from Chapter 3 that, for flow over a flat sand bed, T „  w  should increase linearly from near zero  at the water surface to a maximum at the bed with no along-stream variation. M a x i m u m values o f T,  w  over 2 D dunes occur at, and just downstream, o f the reattachment zone and along the shear layer.  Large T  uw  values extend downstream o f the dune crest, defining the wake and the I B L below.  T  uw  should tend towards zero (or even be slightly negative) moving up in the water column. Some authors have observed large Reynolds stresses in the I B L upstream of the dune crest [e.g. Smith, 1970; Nelson et al., 1993]. However, others have failed to observe this pattern [Bennett and Best, 1995; and Venditti and Bennett, 2000], probably because o f insufficient measurements near the bed. It is unlikely this phenomenon could be observed here using the A D V technology. The boundary layer correlation coefficient, R , uw  R„ = w  "  • U  nm  is plotted in Figure 5.8 and was calculated as  5.12  -W  nm  Recall from Chapter 3 that the boundary layer correlation coefficient (-1 < R  uw  < 1) is a normalised  covariance that expresses the degree o f linear correlation between u and w velocity fluctuations. A s such, R  uw  is a ' l o c a l ' statistic that provides insight into the presence or absence o f flow structure at a  specific location. In flow over a flat bed, there is little streamwise variation in R  uw  and values o f  ~0.5 are typical o f the near-bed regions, while decreased values o f 0.0 - 0.3 are found in the outer  210  z (m) O  ro  O  O  z (m) O  O  O  z (m) O  O  O  z (m)  to  z (m)  z (m)  flow region [Nezu and Nakagawa, 1993]. Over a 2 D dune, .R  tlw  is largest in the separation cell (-0.7)  and elevated along the wake where values are typically > 0.5. L o w R  uw  values (< 0.4) are observed  in the I B L and outer layer. The turbulent kinetic energy per unit volume is plotted in contour maps in Figure 5.9 and is calculated as  5.13  where v'=v -V t  and v. is an instantaneous velocity. TKE represents the energy extracted from the  mean flow by the motion o f turbulent eddies {Kline et ai, 1967; Bradshaw, 1977). TKE production involves interactions o f the Reynolds stresses with mean velocity gradients and, ultimately, TKE dissipation occurs v i a viscous forces after being passed through the inertial subrange o f the turbulence spectrum [Tennekes and Lum\ey, 1972]. Since most turbulence production occurs at the boundary [Kline et al., 1967], TKE can be expected to be largest near the bed for the flat bed case and decrease towards the water surface. Over a 2 D dune Venditti and Bennett [2000] indicated that TKE defines the wake structure and reaches a maximum at reattachment. Elevated levels are common in the separation cell.  5.4 Flow Structure Empiricism and Resolution of the A D V Before examining the results o f the experiments it is useful to consider the turbulent scales that can usefully be resolved using the A D V technology. The A D V is a somewhat coarse instrument to examine fine scale turbulence structures. The use o f the A D V probe in this experiment is predicated on several assumptions. The first is that only the larger scale eddies are o f interest. A t a sampling rate that varied between 20 and 50 H z , it is not reasonable to expect eddy frequencies in the viscous subrange to be resolved. Further, a large degree o f averaging occurs within the sampling volume o f the A D V which is ~0.9 c m . The probe can only practically resolve eddies that are larger than the 3  sampling volume. Both these constraints (sampling frequency and sampling volume) suggest that 213  z  to  ( ) m  z (m)  2  (m)  only those eddies in the productive subrange (coherent structures) can reasonably be resolved. This should not preclude comparison o f the results with measurements made using technologies that do not suffer from these limitations (i.e. hot wire (or film) and laser Doppler anemometry). However, some care must be taken before absolute comparisons can be made. In light o f this, it should be noted that some o f the turbulence structure cannot practically be resolved over the low relief dunes observed here, because the bedforms are only 2.5 x the height o f the A D V sampling volume and the instrument cannot be operated within 0.005 m o f the boundary. I f the separation cell is small or weakly defined as a result o f the 3 D morphology o f the dune, it may not be observable at all. A l s o , because the A D V cannot be operated within 5 mm o f the boundary, a component of the flow must extend vertically beyond this height in order to be observed. Components o f the flow such as the separation cell, reattachment point and the I B L stress maximum may not be observable over all dunes. Given these limitations, the character o f the turbulence structure w i l l be considered in a relative sense. For example, i f the separation cell is observable over one dune morphology and not over another, it must be small or weakly defined over the latter. Its absence in the contour plots should not be interpreted as an indication that flow separation is not occurring over the dune. A s such, most o f the conclusions drawn about how the turbulence structure interacts with various 3 D morphologies w i l l be focused on readily observable components o f the flow field (i.e. wake, I B L , separation cell and outer layer). More advanced technologies need to be applied to flow over the dunes in order to resolve the fine scale turbulent flow properties. Fortunately, such resolution is not central to the present purpose.  215  5.5 Mean and Turbulent Flow Fields over 3 D Dune Beds 5.5.1 Flat bed  Flow over the flat bed does not vary significantly from the empirical model reviewed above. Figure 5.10 displays U and U  profiles over the flat bed. Flow over the fixed flat bed is similar to  rm  flow over the active transport layer bed at flow B . The mean velocity profiles are log-linear through the lower 0.2 d and U  nm  profiles increase exponentially towards the bed conforming to the general  form ofNezu and Nakagawa's  [1993] semi-theoretical universal functions. The difference between  the profiles is largely a result o f the larger shear stress at flow strength B with the active transport layer. The mean velocity profile is slightly steeper and U  is larger, which is in accordance with  rms  Best et a/.'s [1997] observations o f how a transport layer affects the flow. Contour maps o f U, W and the turbulence parameters over the flat bed show no along-stream variation, indicating the flow is indeed uniform. Near the bed (up to -0.33 d ), T  UW  ~ 0.2-0.4 Pa,  R  uw  exceeds 0.4 and TKE ranges between 0.5 and 1.5 Pa closest to the bed. In the middle flow region, between -0.33 and 0.66 c/ , X  U W  and R  uw  decrease towards zero and eventually become negative  above - 0. 66 d , indicating the decoupling o f u and w motions and the absence o f momentum exchange.  TKE tends towards zero above -0.33 d indicating there is little or no energy transfer  from the mean flow (turbulence production) away from the bed.  5.5.2 2D Dunes  Flow over the 2 D dunes displays most o f the features described in previous research. Figure 5.10 displays U and U  n  m  profiles over the trough, stoss slope and crest o f the 2 D dunes and over the flat  bed for comparison. A shear layer is observable in velocity profiles (Figure 5.10). F l o w is generally decelerated over the dune trough and accelerated over the dune crest when compared to the fixed flat bed profiles. Over the dune trough, U is decelerated and U . rm  is increased in the separation cell.  Contour maps indicate there is an upstream directed flow near the slipface and that the shear layer 216  -i—i—i—|—i—i—i—|—i—i—i—[-  I I 1—I—I—I— CD O  05 O  •  -I  o I  I—I  1  1  l — ;  1  1  | 1 1 1 [ 1  I"  i  i  i  i  i i  * :  CO  i i  <J>  o  CD  R p  o  E,  O  CO  E  o  X>  E  CO  CN O  £ : oo  i  o i  i  c  CN T3  c 3  cn cn  cS H» O cn  c  o  5C  CO  x;  to  CO  > o  E  cn  CO  o  o o  tc o o u.  o  Q.  x:  OO  '  o co  o  -4-*  > xf 60 o o 3O -o cn CS s« ^ ' x: co CO  05  CD  left. xed fla cn  CO  o o  i  e  IO  HC Ml  CO Q. cS x : co O x: *00 CO CO cn IH CO co x: cS CS m CO ts IC3 -o cn 00 "O CO c CO CO X) cn ts Q CN o noj  i  HH  CO Q cs o cn o t T3 .£ -o o C CS 60 "oo  -  oo  CO  i  cS CO  dsu  CD O  O O  t_  E  2D  1I I l  i i 1  atb da  1  i  pen s ym bo  1  i i i I  CO  i °.  c  <c CO AO  i  CO  x:  o  -4H  i  •  CO  x:  O  . l l l l — I — i  o o  CO O  same  <  er th nd res panels.  O  Z>  PH  ss) an etr  CN O  E  E  J2 H  4H  dun  co  LL < CN  CO  aver th uring 'he fixed  o  CD O CD  itext of  <D • £  .Hud  CO  <C  T3 t—  cn  « S t;  o JD  CD  o o  CO  E  CO CO ••  o _o "  >r> CO  o o o  P/z  P/Z  217  n. o 5 co  CO In  3  60  CO  >  CO '-  CS 3  CO  x:  rn in  co  cn  I>  ~  O3 «> • £  T3 £ 3 T3 T3 L.  00 CO CO X) X )  extends from SFB to B2 (see Figure 5.1 for definitions). Mean streamwise flow over the stoss slope is nearly identical to that over the flat bed with the exception that the turbulence intensities are larger, which is the case over the whole 2 D dune profile. Mean w contour maps show plunging vertical flow over the dune lee slope and water surface directed flow between B2 and BI. Near the crest, mean vertical flow is negative (towards the bed). There is a well-defined wake structure observable in the I  u  - 1.25 and TKE - 2 Pa isolines that  extend from the lee slope through the trough region and up over the downstream crest. A n I B L seems to develop at BJ and extend towards the crest. Reynolds stress reaches a maximum ( T , = 1.0 - 1.2 h m  Pa ) in the separation cell and is elevated through the trough region and in the wake structure. Large positive R  uw  occurs in the separation zone, but is large over most o f the lower portion o f the flow. A s  was observed over the flat bed, R  uw  decreases towards negative values in the upper 0.33 d o f the  flow. The reattachment point and I B L maximum stress are not observed, but this is likely due to limited resolution of the A D V .  5.5.3 Full-Width Lobe (FWL) and Saddle (FWS)  Mean u profiles over the full-width lobe (Figure 5.11) indicate that flow is retarded over the entire bedform when compared to the 2 D dune, presumably due to increased flow resistance. There is no apparent shear layer observable in the dune trough profile; instead a smooth increase in U with height above the bed is observed. The contour map o f U suggests there is a larger decelerated flow zone downstream o f the dune crest over the trough to BL This decelerated zone extends well above the dune crestline and confirms that no distinct shear layer is present. Compared to the 2 D dune, there is an extensive zone where U < 0 occurs, abutting the slipface. Vertical flow plotted in the W map shows a much weaker plunging flow in the lee o f the dune but a strong and extensive vertical current above B2-B1. A weak vertical flow component persists downstream o f this zone and even over the dune crest. 218  ~i  1  fTT-r—i—i—i  1  —1—1—1 1 i i i 1 _J CO . Q CN L0_ LL  1—  CD  d •<cr  o  tf) E  CM O  CD  o  •  <  i  i  1  i  i  | i  i  i  | i cn o  i  o  O  ,  i  ,  CO  -  O O  ro  i  i  i  1  > i  1  i  ~i—I—I—I—I—r-  I  1  1  I  1  9  1  72 CD  CD sC CD O ^ CD .=  'tf)  8 * CD > CD  "2 £ 5 M  c  o ~  ^ -5  o  g  •£  jT~  o o  i  CD  3  o  o  1  CD  CD Q. O  m  C O  O  CiJ OO  y= —  ro O  iiiii—i—r  -  11 i i i—i—i—i  -i—i—— | i—r~  1—q  I  I  1  1  1  I i  I  1  1  CD  ci  oo  _  O O  C3 CD  O  f= <+.  C OD  CD  tf)  o  E  CD CD  CO  Z>  o  d o d  O  CD  u. c  _CD  CM  CD ^—v 5— 11  C/3  Q. O  _1 CO  d  P/z  CD  d  I  I  I  I  d P/z  219  I  I  I  I  I  CM d  I  I  I L.  o d  o o  "O  =5 Q S rP _c 7 _£  CC!  CD  -S  oo  CD  o co  CD  £ —  cS  S  C  j  ,  c/l co CD u  ~ CD  o  "5 3 ^ -—  -  u o  >•  O  d d  CD  -5  a v Q  •  CD  </)  _c J  .S  CD  s=  13 op g> ^: °- • > CD c£ _n CD  Turbulence appears to be enhanced over most o f the full-width lobe bedform when compared to the 2 D dune. Profiles o f U  rm  (Figure 5.11) reveal larger turbulence intensity in the upper portions o f  the flow but similar values near the base, although U . rm  is larger at the trough profile, owing to more  vigorous mixing in the separation cell. Contour maps o f I  and T  u  reveal low values in the upper  uw  and lower portions o f flow with a strong core o f elevated values that occupy -0.33 d . The wake structure extends from the upstream lee slope downstream over the next dune crest, as defined by the I  u  - 1.25 and TKE - 2.5 Pa isolines, and is somewhat stronger than in the lee o f the 2 D dune. In  fact, it is likely that the core o f more turbulent flow is composed o f stacked wakes from upstream bedforms as suggested by Nelson and Smith [1986]. L o w I , u  r , uw  R  and TKE values indicate  uw  there is an I B L beneath the wake on the stoss slope that is thicker than over the 2 D dune. The separation cell is well defined and appears as an elongated zone where x  uw  most o f the dune, R , = 0.6-0.8, except in the I B L where R m  abutting the lee slope, where R  tlw  uw  exceeds 0.6 Pa. Over  is near zero, and in the separation zone  is negative.  Mean u profiles over the full-width saddle indicate larger velocities over the dune profile compared to a 2 D dune (Figure 5.11), presumably due to decreased flow resistance. There is some evidence o f a shear layer in the U profile over the trough, but it is not as pronounced as over the 2 D dune. The U contour map suggests a low velocity zone where U < 0 occurs near the lee slope. However, this zone is not extensive and the shear layer is diminished at one step height downstream of SFC. The W contour map indicates a strong plunging flow over the dune lee slope compared to the 2 D dune bed and a responding surface directed flow between B2 and Bl.  Vertical flow over the  crest is negligible. Turbulence appears to be suppressed over most o f the full-width saddle bedform. Profiles o f U .  (Figure 5.11) reveal diminished turbulence intensity in the centre portion o f the flow. Values o f  U .  are similar over the 2 D dune and the full-width saddle at the base and surface o f the profiles.  rm  rm  220  This observation is confirmed by the contour maps o f I  u  which is < 0.5 over most o f the flow field.  / „ is larger in the lower 0.5 d but is less than over the 2 D dune. Interestingly, there is no apparent wake structure and consequently no observable I B L in any o f the contour maps. The separation cell is small, absent, or only weakly defined. I , u  T, ,R w  uw  and TKE are somewhat larger in the trough  area bounded by SFC upstream and Bl downstream, suggesting some turbulence is generated over the dune but not on the scale o f the 2 D dune.  5.5.4 Sinuous Lobe (SNL), Saddle (SNS) and Smooth Saddle (SSS) Mean u over the sinuous crest follows essentially the same pattern as over the full-width lobe and full-width saddle (Figure 5.12). F l o w velocity (U) is retarded over the sinuous lobe and enhanced over the sinuous saddle when compared to flow over a 2 D dune. However, the magnitude of the retardation or enhancement is smaller over the sinuous lobe and saddle when compared to the full-width bedforms. Although this pattern persists over the whole bedform, the U velocity differential between the lobe or saddle and the 2 D dune is less pronounced over the stoss slope and crest compared to the trough profiles. There is no apparent shear layer developed in either the lobe or saddle profiles, which agrees with observations over the full-width bedforms (Figure 5.12). Over the sinuous lobe and saddle trough, there is a smooth decrease in U towards the bed. A contour map o f U indicates there is a large decelerated zone over the sinuous lobe that is similar to the full-width lobe, but not as extensive. Negative U velocities are observed near the lee slope. Over the sinuous saddle, the decelerated zone is quite limited and there are no negative U velocities observed. A short shear layer diminishes ~ 1 H from the upstream  SFC.  Contour maps o f W indicate that plunging flow occurs with a surface directed return flow at Bl over both the sinuous lobe and saddle. In agreement with observations over the full-width lobe and saddle, the magnitudes o f the plunging and return flows are greater over the lobe. The magnitude o f 221  these flows is similar to those over the full-width bedforms, indicating that this feature is unaffected by the width o f the saddle or lobe. Over the sinuous lobe crest, vertical flow is negligible while, over the sinuous saddle crest, flow is directed towards the bed. The characteristics o f the turbulent flow over the sinuous bedforms also bear strong similarity to the full-width bedforms. Over the sinuous lobe, the upper portion o f the U  nm  profde is larger than  over the 2 D dune (Figure 5.12). Over the sinuous saddle, turbulence is suppressed in the central core o f the flow but similar U  r m s  values are observed at the top and base o f the profiles (Figure 5.12).  Over the sinuous lobe, contour maps reveal a central core region that has larger I  u  and  z  uw  values compared to flow above and below. This region is less pronounced over the sinuous lobe when compared to the full-width lobe. The wake structure is well defined by the / „ - 1.25 and TKE - 2.0 Pa isolines and is better defined over the sinuous lobe than over the full-width lobe. This suggests that the width o f the lobe tends to concentrate the strength o f the wake over lobes. The central core is not as turbulent as over the full-width lobe; this may be attributed to the neighbouring saddle, which has no observable wake, interfering with the maintenance o f mixing and the development o f stacked wakes higher up in the flow. L o w values o f / „ , z , im  R  uw  and TKE indicate  that there is an I B L beneath the wake on the stoss slope o f the sinuous lobe that is similar in depth to the I B L over the full-width lobe. The separation cell is quite pronounced over the sinuous lobe trough and is o f similar magnitude compared to the full-width lobe. In the elongated separation cell, T , = 1.4-1.6 Pa. ira  Since there is no wake developed over the stoss slope o f the sinuous saddle there is no apparent I B L . Further, the separation cell is weakly defined or non-existent. z  im  reaches a maximum o f only  ~0.6 Pa. Interestingly, the flow structure over the smoothed sinuous saddle is nearly identical to that over the ragged edged sinuous saddle, suggesting that the effect o f the ragged edge is relatively minor.  222  Recall from Figure 5.2 that the sinuous crest is composed o f two lobes, one saddle in the centre o f the flume and two half saddles at the side walls. I f a single estimate o f the level o f turbulence (based on any o f  , T , uw  R  uw  and TKE) over the sinuous dunes was made, it would be logical to average  the results over the lobe and the saddle. Since the sinuous lobe appears to have a slightly higher turbulence level than the 2 D dune and the sinuous saddle a significantly lower turbulence level than the 2 D dune, it is likely that an average would produce an equal or lower turbulence level when compared to the 2 D dune.  5.5.5 Regular (REG) and Irregular (IRR) Crests Before discussing the effect o f the regular morphology on the flow structure, it is necessary to note that the regular crest morphology is not realistic. Recall from Figure 5.2 that the regular measurements were made over a saddle where the trough was recessed to the neighbouring crestline. It is unlikely that a crestline could develop with a lobe and saddle within < 0.1 L. However, this morphology was included as an end member test o f the theory o f Sirovich and Karlsen [1997] (see Chapter 4). In light o f this, the flow structure associated with the regular morphology w i l l be discussed briefly. The irregular bed deserves more attention because it is perfectly likely that this bed morphology could develop. In fact, much of the equilibrium morphology in the active transport experiments resembled the irregular form with asymmetric lobes and saddles as a constantly varying component o f the bed morphology. Mean u profiles over the 2 D and regular dune morphologies are identical above the dune height and U decreases substantially below crest height (Figure 5.13). The combination o f the dune step and the hollow created by neighbouring crests generates a strong shear layer. There is no significant difference in U profiles over the stoss and crest o f the regular and 2 D dune morphologies, suggesting that the effect o f the 3 D form is localised to the trough region. Vertical flow inferred from the  224  I I I I  I I  I  OJ O  CD «>  a  CD  o  CO  E  1 *  OJ (V !C •£ C w CO u  -73  -1  s< —  CD  I fi "o  CO O  6 0  c CD >^ x  .ii  O O  X!  .  ro  CD  ^  CD  Q x  CD  9 §  3 60 E  1 1—1 1 1  T l  11 1 1—1—1—1 1  1  -|—r-i—i—|—i—i—i—T  1—-  -i—i—i—i—r  CD  ^  O  d 1  d  -3  _ £  eg  o <r>  ^ S X CDc/> o B b -4^  CD O  CO  10  c  CD  E  a.  c •—'  o Oo •=oo CD C N  -  CN  (/)  ;  - d  o  '  1  1  1  o d  CS t-  : co  1  1  i i i—i—i  1  1  1  n—i—i—|—i—i—i—|—r-  r-  1  _o  , o ' o  o d I  1  1  1  I  1  CD >  OJ  c/ct  o  CD  d  _CD  e CS 3  <C cr  o <o  CM  • • • • L_J  L  o CO  =>  oO CD  ro  o d  o o  2  I  3  03  P/z  CD  d  d  P/Z  225  CN  d  o d  O CD CD 3 "o  C „ -a 00 ~  CD  X  60 3 O  C  ^ X CD  —  Q"oo CS  CD C/3  o  d  3 60 IC  CD  "g £ o  E  d  00  '  -g -a ^ c » ^ 2 o 2n x!  CO  =>  '—I  C  *  co —  Di  CD  M  Q  2  iri  .SP £ UL.  CD  , , b [V  tfl  3  6 0 C N CD £ CD CD  E X  X  W contour map is negative in this trough region. Weak plunging flow occurs at the streamwise extent o f the neighbouring lee slopes and a return flow occurs at Bl. There is no significant difference between the U  n  morphologies except there is no increase in U  rm  m  profiles over the regular and 2 D dune  in the hollow at the lee slope in the regular dune  trough profile. This is probably because there is no significant circulation in this region. A contour map o f I  tl  indicates that a weak recirculation cell develops at the position o f the neighbouring lee  slopes. There is no strongly defined wake over the regular morphology but elevated values o f / „ , T  uw  and TKE occur between the upstream SFB and the upstream crest. Thus, the wake does not  appear to have a lower boundary. L o w x  im  and R  uw  over the stoss slopes may indicate an I B L , but  this is difficult to assess. In other respects the patterns o f / „ , T  and TKE do not vary significantly  uw  over the regular and 2 D dune morphologies. Recall from Figure 5.2 that profiles over the irregular dune were taken where the neighbouring crests were staggered upstream and downstream. The crestline in the centre o f the flume formed an asymmetric lobe. Profiles o f U over irregular and 2 D dunes are identical above the dune height. Below this point U is larger than over the 2 D dune morphology (Figure 5.13). There is no shear layer observable in the trough profile and this is confirmed by examining the U contour map for the irregular morphology. In fact, the reduction in U is quite minimal in the trough zone and no upstream-directed flow is observed. The pattern o f W velocities is similar over the irregular dune and the 2 D dune. Plunging flow occurs over the dune crest with an upward directed return flow at Bl.  A  significant difference in the mean flow quantities over the irregular and other morphologies is that there is a noticeable mean cross-stream velocity over the lobe, especially in the dune trough where V exceeds 0.08 m s" at the bed. 1  There is no observable wake, I B L or separation cell developed over the irregular dunes. However, there is a zone o f more turbulent flow, observable in x  uw  226  and TKE contour maps, in the  bedform trough that means some separation induced mixing must be occurring. Turbulent intensities are moderately larger in the outer layer over the irregular dune than over the 2 D dune configuration but, near the bed, I  u  is much less.  5.6 Effect of 3D morphology on Momentum Exchange and Energy Transfers Over a flat bed, momentum exchange and mixing, as indicated by the distribution o f  T , uw  decreases with distance from the bed. Energy transfer from the mean flow, as indicated by the distribution o f TKE, is confined to the lower 0.33 d . The presence o f a 2 D dune dramatically alters these distributions. The separation cell dominates momentum exchange but mixing is strong along the wake layer. A s over a flat bed, little mixing occurs between 0.66 d and the surface, because momentum exchanges at the bed have little impact. Similarly, the wake structure and separation cell dominates energy transfer from the mean flow. Turbulent energy is negligible above 0.66 J o f flow. Introduction o f three-dimensionality to the 2 D crestline changes these distributions by rearranging the flow structure. Over a lobe, energy transfer from the mean flow (TKE) is dominated by the wake structure. When compared to a 2 D dune, a greater amount o f energy is transferred from the mean flow to turbulent flow, resulting in a slower mean flow velocity. Momentum is diffused laterally over the topographic high (Figure 5.14) and upwards into the flow column and a larger level of mixing is maintained over the whole dune field (Figure 5.7). Over a saddle, momentum is concentrated in the hollow (Figure 5.14) formed by the crestline, which results in less energy extraction from the mean flow and less mixing in the lee o f the bedform (Figure 5.7). Therefore, flow velocity is enhanced as less energy is being extracted for mixing.  5.7 Spatially Averaged Flow over 3D Dunes In order to make a final assessment o f Sirovich and Karlsen's [1997] theory it is necessary to calculate a drag coefficient, C  = 2{u*/U) , 2  D  and a drag force, F 227  = 0.5C plf A 2  D  D  h  associated with  DIVERGENCE  Figure 5.14: Lateral convergence or divergence o f momentum and turbulent energy over lobe and saddle crestlines.  228  the dunes on each bed ( A  b  is the effective cross-sectional area o f a body). For dunes A is the b  product o f the dune width and H . Calculation o f these quantities require the calculation o f u for t  each bed morphology. Individual measurements o f Reynolds stress or individual profile based estimates o f shear stress are relatively meaningless over a dune as u, varies tremendously across a dune surface. Therefore, spatially averaged u, values must be estimated. These data can also be used to determine how 3 D morphology affects the skin and form stress components o f the total applied shear stress.  5.7.7  Theory  Over dunes the total spatially averaged shear stress, x  m  component, T - , and a form drag component, x ., ?/  T  fnrm  , is composed o f a skin friction  so  / M ~ form . s / T  ^.14  + T  [Einstein and Barbarossa,  1952; Smith and McLean,  1977; McLean  et, al, 1999]. The skin friction  component is related to the resistance to flow offered by the sediment grains and the form drag component is resistance to flow offered by the bedform morphology. A n additional component, the friction associated with the transport layer, can be added to x  tol  when there is sediment transport  [Wiberg and Nelson, 1992].  Evaluating the shear stress components can be a difficult enterprise and theoretical models have varying degrees o f success when applied to real data. Thus, estimates o f the shear stress components w i l l be made using measurements. There are two general approaches to estimating components o f x . The first method involves tot  the calculation of x  tol  over a flat bed with the same grain roughness as over the dunes formed at the  same flow strength. Over a flat bed, x  m  ~ T - because there is no form drag (assuming channel v/  shape effects can be ignored), van Rijn [1984b] and Yalin [1992] have both proposed models in 229  which x  oc f(U', p- , C _ )  sf  w  D  skjn  and C _ D  the dune stoss slope and the flat bed, x  sf  the flat bed ( T  / l a t  ) . However, x  sf  x skin  f(d, k ). s  Thus, assuming U and d are similar over  over the dune stoss slope is roughly equivalent to x  sf  over a dune is not equivalent to X  over  because skin friction is only  m  relevant between the reattachment point and the slipface crest. Thus, x  sf  over a dune is related to the  total stress over a flat bed by  if ~ flat  T  L-x  R  }  5.15  X  where x  L  is the reattachment length. Equation 5.15 assumes that the width o f flow over the flat bed  R  and the dune bed is the same. The form drag can be estimated from Equation 5.14 (i.e.  t form  =  ^"tot ~ if )• T  Total shear stress can be determined by any number o f methods. Assuming the water surface slope can be accurately determined, x  tot  can be determined from the depth-slope product  ^s=P gdS.  5.16  w  Alternatively, x  tot  can be determined by plotting x  uw  as a function o f z/d  . Over 2 D dunes there is  typically a region o f approximately linear decrease with height above the bed. Through this region, x ,=x (l-z/d). im  5.17  R  A n estimate o f x  R  can be obtained by using a least-squares regression, through the linear decrease  region, projected to the dune height [Lyn, 1993; Nelson etal, o f X should reflect x R  un  1993;McLean  etal,  1999]. This value  [McLean etal., 1999].  Another method o f estimating the stress components in Equation 5.14 is based on the idea that spatially averaged velocity profiles over dunes are concave-down. Smith and McLean  [1977]  proposed that these velocity profiles can be divided into two log-linear segments, where the upper profile represents the total drag and the lower profile represents the skin friction component. Shear  230  velocities for the upper and lower portions o f the velocity profile can be estimated from the von Karman-Prandtl law o f the wall  U ' l , z  —  Ut  =- l n — K  5.18  Z  0  where U is the velocity at some z and K is the von Karman constant which is assumed to be 0.4. z  The roughness height, z , is the intercept o f the velocity profile plotted as a function o f height above 0  the bed. Shear stress for each profile segment can be determined from x =  pu. 2  w  t  McLean et al. [1999] recently criticised this method for estimating total boundary shear stress, noting topographic effects are felt throughout the water column unless the flow is much deeper than the bedform height. Errors in x  lut  dune crest w i l l underestimate x  tot  can be up to ± 3 0 % . They note that local velocity profiles near the and those in the dune trough w i l l overestimate x  tot  . They further  suggested that a profile over the stoss slope between the two points may be as accurate as a spatial average. Nelson et al. [1993] and McLean et al. [1999] also note that lower velocity profiles should encompass measurements in only the I B L . I f a spatial average uses measurements near the upstream dune crest, portions o f the wake and the separation cell w i l l be included, leading to erroneous estimates o f X . The message appears to be that spatial averages should be constructed in sf  consideration o f these problems and regarded with caution.  5.7.2 Depth Slope Product Measurements Table 5.2 shows side wall corrected r _ s  tot  determined from the depth slope product (Equation  5.16) using d and S values presented in Table 5.1. Over the flat bed, x _ s  A p p l y i n g Equation 5.15 to determine x  sf  lot  = 1.0 Pa.  over the dunes is complicated because x  R  could not be  measured from flow maps as in other investigations [e.g. Bennett and Best, 1995; Venditti and Bennett, 2000]. Assuming x  R  = 3 . 5 - 5 / / yields a separation length o f 0.08 - 0.11 m, which may be  231  Table 5.2: Total, skin and form drag coefficient and force. Shear stresses are corrected for side wall effects using the relation supplied by Williams [19701. Drag coefficients and forces are calculated based on the total shear stress. Values in brackets are for the central segment o f spatially averaged Reynolds stress profiles. Parameter *s-„„  ^  ,,,„„,„  „_  •ws  IRR RFG Determinations based on depth-slope product  „  , ,.  ,  SNS  SNL  SSS  1.0447  1.9072  1.9052  2.1610  2.0571  1.8507  1.7422  n/a  1.6028  P  a  1.0447  0.6540  0.6540  0.6540  0.6540  0.6540  0.6540  n/a  0.6540  0.0000  1.2532  1.2512  1.5070  1.4031  1.1968  1.0883  n/a  0.9489  0.0092  0.0163  0.0165  0.0191  0.0188  0.0171  0.0157  n/a  0.0147  0.0120  0.0219  0.0219  0.0248  0.0236  0.0212  0.0200  n/a  0.0184  P  a  c us  L  ^ I-'WL  a  .V-Ion,, ,  T  2D  P  <  r.v-,/- -  Flat  Determinations based on spati; illy a vera:»ed velocity profiles ;  T._„„, Pa  0.5944t  1.1097  4.4905  0.0246  0.8915  0.0941  0.0859  0.1764  2.1953  '-M  0.59441-  0.3577  0.1490  0.71 15  0.4836  0.2464  0.8008  0.23 11  0.4833  0.0000  0.7520  4.3415  -0.6869  0.4078  -0.1522  -0.7149  -0.0547  1.7120  0.0052  0.0100  0.0393  0.0002  0.0080  0.0009  0.0008  0.0016  0.0213  0.0068  0.0122  0.0515  0.0003  0.0102  0.0011  0.0010  0.0020  0.0252  1.0601  0.1642 (0.3751)  -  T  *-/on,l  T  P  a  '  P  a  cv  . T  X  «-.v/  .  R form '  ^ DR  P  P  Determinations based on spatially averaged Reynolds stress profiles 0.0630 0.0835 0.3679 0.9555 1.4066 0.6242 0.4226 (0.2880) (0.2047)  a  '  0.3679  1  P  a  0.2303  0.2303  0.0000  0.7252  1.1763  0.0033  0.0081  0.0122  0.0042  0.01 10  0.0161  0.2303 -0.1673 (0.0577) 0.0006 (0.0025) 0.0007 (0.0033)  0.2303  0.2303  0.2303  0.3939  0.1923  -0.1468 (-0.0256)  0.8298  0.0057  0.0039  0.0018  0.0095  0.0072  0.0048  0.0023  0.0122  tBased on lower 0.2 d o f velocity profile (see Chapter 3 for explanation).  232  0.2303  0.2303 -0.0661 (0.1448) 0.0015 (0.0034) 0.0019 (0.0043)  an accurate estimate over the 2 D dune, but over the 3 D morphologies, contour maps o f the flow structure suggest x  may vary. Based on the size o f the separation cell, it appears that the  R  reattachment length would be longer over the lobes and shorter over saddles when compared to a 2 D dune. Without measurements, the reattachment length used in Equation 5.15 cannot be adjusted for different dune morphologies. However, it is likely that the skin friction contribution to the total shear stress is not great upstream o f B2 because o f the dune topography. Therefore, x  R  w i l l be assumed to  be the distance from the upstream dune SFB to B2 or 0.13 m. Based on this assumption, T  S /  = 0.65  to(  ). F o r a  Pa over the dunes. Over the 2 D dune,  r _ = 1.91 Pa (z _ = 0.34T,_,„, ) and z _ s  lol  s  sf  s  /orm  = 1.25 Pa ( 0 . 6 6 r , _  2 D dune with an aspect ratio o f 0.05, Smith and McLean [1977] suggest the ratio o f skin to total shear stress should be - 0 . 4 , which is near the value observed over the 2 D dune here. Despite the fact that I  and %  u  regular bed,  T _ S  ( 0  ,  ~ 2 Pa (T _ s  sf  uvl  ~ 0.30 - 0 . 3 2 T _ , ). Over the irregular dune, where turbulence was s  greatly suppressed, T _ , and T _ s  true o f T _ s  lot  to  s  and T _ s  are substantially reduced over the full-width saddle and the  form  w  are nearly equivalent to values for the 2 D bed. The same is  over the full-width lobe where / „ and T„ , are substantially enhanced.  furm  h  Only over the sinuous beds is T _, reduced below its value over the 2 D dune. Here t _ s  Pa (T _ s  sf  ~ 0.38 - 0.41 x _ ). s  ot  s  tot  = 1.6 - 1.7  A s expected, the sinuous dunes with the smoothed crest produced a  tot  lower total boundary shear stress. When this array o f T _ s  lol  values is used to calculate u* and ultimately C  D  and F , the same D  groupings appear. For the 2 D , full-width lobe, irregular and sinuous lobe morphologies C  D  ~ 0.016  and F ~ 0.022 N . For the full-width saddle and the regular dunes the drag is moderately larger ( C D  = 0.019 and F ~ 0.024 N ) and for the smoothed sinuous crest, drag is reduced (C = 0.015 and F D  D  = 0.018N. 233  D  D  The most substantial problem with these estimates o f the shear and the drag is that they do not reflect the flow structure. The depth-slope product may simply be too coarse a measure. Spatial averaging is used below in an attempt to alleviate this problem.  5.7.3 Spatially Averaged Velocity Measurements Spatial averages o f velocity were calculated along constant heights above a datum. Over the flat bed the datum is the bed and over the dunes the datum was the dune crest. In order to avoid biasing the inner velocity profiles by velocities collected in the separation zone or the wake near the dune crest, averages o f U were constructed over the dune back, between BI and the dune crest. The vertical rise in between these two points on the dune is minimal (~3 mm), so whether the inner profiles are constructed at constant heights above the bed, or constant heights above a datum, does not affect the profiles (see Appendix E ) . It can be argued that the total shear stress should be based on a spatially averaged profile over the whole dune and not just over the portion where the skin stress contributes. Thus, two sets o f profiles were constructed. The first set were the upper portion o f the inner profiles over BI and the dune crest. The second set was averaged over the whole dune length. Whether the outer profiles are constructed at constant heights above the bed or, constant heights above a datum, does not affect the profiles significantly (see Appendix E). There was no significant difference between the outer profiles averaged over the whole dune and those averaged over the stoss slope (see Appendix E or Figure 5.15). The outer profiles discussed below are averaged over the stoss slope o f the dunes. Constructing the inner and outer profiles in this fashion should mitigate some o f the concerns expressed by McLean et al. [1999]. However, the results w i l l be considered in light o f the fact that values o f x  lol  could vary by ± 3 0 % about their true value.  Figure 5.15 presents the spatially averaged velocity profiles over the flat, 2 D dune and 3 D dune beds. Tables 5.3 and 5.4 provides least-squares regression results that were used to estimate the slope  234  1 i i i—i—i—i—i  : ° : :  pr—i—i—i—i—i  CD O  i  LO O  1  x:  Z)  -a  an  :  — x ; .ti =  S  r —  CD C) LO  o  :  -  i . . . . .  J  •>a; o  CQ cS c CD cn XI CD 2 CD ? "5 3 jcnX> x :  _ "3O •  - CO : CO :W  .  —i—i  '  CO O  cs  r  ' -ts o  CD O LO  CD 00 CS  CO  CD tCS cn JD O  CO 2  1  r  Um  o  -  b cn JD  CD  <G  o  LO O  ci  CO i  CD  9 1 i i i — i i i i  :  -I ci LO O  t -  :  OcpcPo  CD O  o  -I o  LU  LO  1  o  :  1 i i i—i—i—i  : LL 1 ' » ' l •  •  '  (LUO)  i  Z  CO  o  1 •  o  : co  Z  -i  CO  LO O  o  o  : Dd  icr: :  •  L_i i i i i  i *  (LUO)  235  Z  CO O  53  JD  CN  3  CD Q . - 0 v. — % CD O ^CD C M XJ c cs . J S' cn (U c n ^3 0 0 O — cS u  x;  — IC O "O fe Q- CD  p t! fe O  ^~ cs  CL o  xi Cs o—.cS  O CS g cn c+-<  -s  i-J J3cn CcsDg)  ^ CD CD > CS E _ o CS CD Um ~ Um CS cn • O cS CL, CD - 2 "o ,ti -g S  C  .2  C O CD CL.  O .£ x : CD _ . CD CD T3 cn — ^ cS C CS cC 2 CD CS XI o r-. '—^ CD fe E « ^ S &• _  CD CD CD > CS  (u ~ "  <+  © 3 CS  n ti o o  g> S.2 o ^ 2 § E ? ? cn oo [V c  CD T3 CD  c  o  T3 CD CD Q. cn cn SJ cs CD > g ^ Is (CDDOX)CD i i CD T 3 C O  'C ^ CS D . oo ty^  -8  - n  CD  o  8  o 00 Um o_ -a o. <u. 2 cn ^ cn  CO  o  CD  5.2 o  IS c u. ~ _ p cn cn o CD CD C > CS  d  | T i—i—i—i  cS cS  o C/D  LL  co  c > O O C L cn  IC ^  -  CD  op  O CN i s «  cn  Table 5.3: Least-squares regression o f inner and outer velocity profiles constructed using profiles taken between BI and the crest. b  0  imeter  is the intercept and b is the slope. x  Flat  2D  FWL  FWS REG Inner Profile  IRR  SNS  SNL  SSS  ^0  -2.62  -4.01  -4.24  -3.27  -3.25  -5.42  -3.00  -5.08  -4.13  6,  6.77  8.71  13.27  6.15  7.42  10.45  5.55  10.61  7.77  ->  1.00  0.99  0.92  0.98 0.96 Outer Profile  0.97  0.98  0.84  0.98  -2.59  -0.78  -19.99  -2.89  -9.04  -11.25  -6.06  -1.94  ~  5.90  3.72  34.11  6.63  17.79  18.50  13.52  31.81  -  0.99  0.98  0.27  0.87  0.22  0.88  0.97  0.75  r~ b  0  r  2  Table 5.4: Least-squares regression through linear portions o f the Reynolds stress profiles. Values in brackets are for the central segment o f spatially averaged Reynolds stress profiles. b  0  is the intercept  and b is the slope. x  Parameter  r  2  Flat  2D  7.49  8.65  16.71  -20.36  -9.06  -11.88  0.99  0.98  1.00  "  FWL  m  FWS 9.05 (5.35)  REG . '  (-18.56) 0.87 (0.98)  -14 03  1 4 3  6 4  236  8  7 6  y  y  IRR , . -  SNS~ 10.58 (6.11)  -33 39  " (-29.86) 0.90 (0.99)  1 4  U  u , y z  1 2 6  8 3  SNL " S S S , .^ 8.50 (5.48) c  1 4  6 5  -13 82  " (-14.62) 0.94 (0.99) 5 L 7 4  and intercept o f the upper and lower velocity profile segments. Over a flat bed, profiles are similar to those presented in Chapter 3 with a linear increase in U with log z .  T„  T O (  = 0.6T _ s  (the subscript *  i o (  indicates a value derived from a spatially averaged velocity profile), which should not be surprising. A similar trend was observed in Chapter 3 and attributed to the footprint o f the lower portion o f the velocity profile. The depth-slope product is dependent on side wall resistance in addition to bed resistance. Profiles over the 2 D dunes have two log-linear segments and upper segments have a lower slope (i.e. higher shear rate) (Table 5.3). Thus, s-t<>t  T  >  *tot  T  a n  d "sf ~ 0-32 T , T  T O  ,,  T» , > TO  near the  T  tsf  r, and x sf  /r*  tfomi  to/  > 0 (Table 5.2). Over the 2 D dune,  value (0.4) suggested by Smith and  McLean  [1977]. Values o f T , magnitude as  I ; /  T„ , T O  produced from the lower segment o f the velocity profiles are o f the same order o f over the flat bed, but varied between 0.15 and 0.80 P a over the different dune  morphologies. There is no relation between T ,  S /  and U . In the absence o f more information, it is  difficult to assess whether this is a problem in the method used to split the skin and form components of the total stress, or real variation. Values o f T , ,  0 /  also show confusing variation. Over the full-width lobe dunes, there are two log-  linear segments and the upper segment has a lower slope than the inner segment, as was observed over the 2 D dune (Table 5.3). However, x,  lot  at 4.5 P a ( T »  4/  =0.033 T „  TOR  is much greater than its value over the 2 D bed  (x _ ) lot  2D  ) . This observation is in agreement with expectations based on the  turbulent flow structure over the full-width lobe dune. Over the full-width saddle and the sinuous saddle (smoothed and ragged) crestlines there are two log-linear segments, but the upper segment has a lower slope (Table 5.3), indicating a lesser shear stress. For these profiles,  T,  tot  < T,  sf  and  T  ,  tform  by difference, is less than 0 (Table 5.2). Profiles over the irregular, regular and sinuous lobe crestlines have weakly defined slope breaks, where the upper and lower profiles have approximately 237  the same slope (Table 5.3). Over these dunes, T,, , < r 0  below, it is not unreasonable to have i  tform  t r  and x,  forin  ~ 0 (Table 5.2). A s is discussed  ~ 0 when there is no evidence o f strong separation or a  wake structure. However, it is not clear how x,  < 0 could occur under any circumstance. The  form  suggestion by McLean et al. [1999], that the outer log-linear portions o f the spatially averaged velocity profile may not accurately reflect x  m  , appears to be particular relevant over 3 D bedforms.  5.7.4 Spatially Averaged Reynolds Stress Measurements  Figure 5.16 presents the spatially averaged x  uw  profiles. Reynolds stress was averaged along  constant planes above a datum. Over the flat bed the datum was the bed and over the dunes, the datum was the crest. Averages include all positive x  im  crestlines. Profiles w i l l differ i f x  uw  values observed between two successive  is averaged along equidistant lines above the boundary  (Appendix E). However, the spatially averaged x  uw  profile should only be linear well above the  bedform where values are not directly affected by flow separation. Thus, the more meaningful profile is from averages at constant planes above the dune crest [see Nelson et al, 1993 and McLean et al, 1999]. Over the flat bed, profiles are similar to the profiles observed in Chapter 3 over the flat beds with active transport, except at the boundary. There is a linear increase in x  uw  with decreasing z , but near  the bed ( z < 0.02 m or z/d < 0.15), x , decreases slightly. This is not surprising as the flat beds m  discussed in Chapter 5 had active sediment transport which would have increased x  m  in this region  [see Best et al., 1999]. Over the 2 D dune, a substantial portion o f the spatially averaged x  uw  profile is linear and similar  to those presented by Wiberg and Nelson [1992], Nelson etal. [1993], Bennett and Best [1995] McLean et al. [1999] and Venditti and Bennett [2000]. A t a height o f - 0 . 0 4 cm ( z / d ~ 0.25), there is  238  CD  T3  CD 60 CS CD > CS  CD >  o -a  CD CD cn  3  CD  CS CD 60 CS CD >  <  T3  co  CS  O  C  CJ  I  f -  CD  >  c+-  o ^ CD  t-  CS  a* C/3  cn cn cS  CD  O I J  1 1  1  T  i  i  i  i  i  1 i i1 |i i  i  i  i  i i  i  i  i  iyi  | i  e  1 1  \  '• Q : <M  d  jr r  ;  o  O  c  o d  "  >^  CD Di  o  -o  CD 00 cS u* CD > CS  CM O  \ '  i  J<  i  • • '  1  • • > •  1  • • • •  O d  5  e  OO  CD  JCD  O  r-  -g * =S g z .=  o 3  O  CM  -  :  4-  O  3  cn  J  cn  60 CD  >  ^  CD  X  C  CD  o  -  L! H  CD  CD  a-  C  to cfl  e CD CD  cS  CD  X  T3 CD  CS CD >  _CD  o CJ x -a  X x> >, O o  * °O  =  es  \C  cS  a.  c/>  CS  .SP  cS  X  _3  o  "8 s  CU  cn  cn CD  cn  ^  a  tCD cn  >•  2 T3  'CD XT'  > S o cS  C  c  O  . 'cn  cS CD  CD  o -o oo o CD t-  3  00  CM  o  d  d  00  o d  CD o d  •>cf  o d  CM o d  o o d  (w)z CM  o  d  d  00  o d  CD o d  (LU)Z 239  o d  CN o d  o o d  00,5  c o  CD  I-  T3  p  cn  r—  13  £ .2  an inflection in the curve and x  decreases in both vertical directions. The spatially averaged profile  uw  over the regular dune morphology has essentially the same form, which is not surprising. Apart from the fact that the regular dune lacked a well-defined wake, the distribution o f / „ and T„„, over the 2 D It  7  tin'  and regular dune morphologies were substantially similar. Over the full-width lobe, there is an inflection in the x  uw  0.53) and, as over the 2 D dune, x  uw  profile that occurs at -0.08 m ( z / d =  decreases in both vertical directions. Profiles o f spatially  averaged x , over the sinuous lobe and irregular beds are composed o f three segments (recall that im  profiles over the irregular bed were taken along the flume centre line over a minor lobe). A s over the full-width lobe, an inflection occurs in the centre core o f the flow at -0.09 m (z/d =0.60) over the sinuous lobe and -0.08 m (z/d =0.53) over the irregular lobe. Spatially averaged Reynolds stress decreases in both vertical directions about the inflection. Another inflection occurs at the base o f the profile, -0.04 cm (z/d =0.25), where x  uw  x  uw  increases in both vertical directions. The magnitude o f  at the inflections is largest over the full-width lobe, intermediate over the sinuous lobe and lowest  over the irregular lobe. Spatially averaged r  tlw  segments show decreasing  profiles over saddles are also composed o f three segments, but all three T  llw  with increasing z . There are breaks in the slope o f the curve at -0.03  and 0.05 m (z/d =0.20 and 0.33). Top and bottom portions o f the profiles have similar slopes that differ in magnitude over the various saddles. It is a relatively simple task to identify the linear portions o f the spatially averaged x  uw  flat, 2 D , regular and full-width lobe morphologies, allowing an estimate o f x  R  Appropriately, x  m  over the  using Equation 5.17.  (x ) increases through flat - 2 D dune - full-width lobe morphologies. R  Unfortunately, it is not obvious which portion o f the spatially averaged x  uw  reasonable estimate o f x  m  profiles provide a  over the other bed morphologies. It is assumed that the uppermost  240  segment o f the three-segment profiles represent the total boundary shear stress when extended to the bed. Results o f least-squares regression through these portions o f the profiles are presented for all bed morphologies in Table 5.3 and estimates o f  can be found in Table 5.2.  The side wall corrected, spatially averaged, boundary shear stress over the flat bed, z _ , R  0.37 Pa or -0.33 z _ . Again, this pattern where s  s  above for depth slope product estimates o f shear stress, z _^ is not equivalent to z _ R  R  Equation 5.15 needs to be applied using the assumption that x  R  R  t0  R  = 0.24z _  sf  R  lol  j]at  and  can be approximated as the distance  from the upstream dune SFB to B2. Based on this assumption, x _  R  was  < T was observed in Chapter 3. A s noted  Mt  Over the 2 D dune, z _ , = 0.96 Pa (z _  m  = 0.23 Pa over the dunes.  sf  ) and z _ R  form  = 0.73 Pa ( 0 . 7 6 ) . The  ratio o f skin to total shear stress is less than the value (0.4) suggested by Smith and McLean [1977]. Over the full-width lobe, z _ R  tot  =1.41 Pa (z _ = 0.16z _ ) which is greater than its value over the R  sf  2 D bed ( T , _ ) . Over the sinuous lobe, z _ 0 /  2 0  R  morphologies, z _ < R  m  z _ . lol  2D  R  >z _ .  m  m  Irregular bed z _ R  lol  1D  tot  Over the regular and irregular dune  is less than regular bed z _ . The ratio o f skin R  tot  to total shear stress varied between 0.16 and 0.54 over these dunes. Spatially averaged z  profiles over saddles are more complex and pose a substantial impediment  uw  to determining stress values. Regressions extended to the boundary through the upper segments o f these profiles indicate z _ R  m  < z _ . Regressions though the lowest segment produce similar results. R  sf  Regressions through the steeper central profile provide greater z _ R  tot  values o f 0.29 Pa ( F W S ) , 0.20  Pa ( S N S ) and 0.38 Pa (SSS), which are either greater than or roughly equivalent to z _ R  sf  . One o f two  possibilities can explain these results. Either the model upon which 5.17 is based upon breaks down over saddle shaped features due to momentum convergence, or these bed morphologies are so effective at reducing form drag that this component o f the stress is not felt throughout the profile and skimming flow is generated. Both possibilities are interrelated.  241  The basic assumption that underlies Equation 5.17 is that all turbulence production occurs at the boundary. Over a flat bed, discrete, coherent structures are generated that break down and cascade turbulent energy to smaller scales. With distance from an obstacle or rougher patch o f the boundary, fluctuations in velocity and hence the turbulence, are reduced. Naturally, this also occurs with distance above a boundary, resulting in a reduction in the turbulent stresses with increasing height. Over dunes, most turbulence production occurs not at the boundary, but in the separation zone, and this turbulence is dissipated along the flow and vertically [Nelson et ai, 1993; Venditti and Bennett, 2000]. A p p l y i n g a spatial average assumes that the turbulence is dissipated along-stream and that an average w i l l remove the spatial non-uniformity over one bedform length [McLean et ai, 1999]. Over saddles, separation occurs but wake generation is inhibited. Thus, there is no strong alongstream dissipation o f energy, and so the saddle shaped dune beds are acting like flat beds in that the lower portion o f Reynolds stress profiles respond to skin friction over the stoss slope and crest. M i n i m a l contributions to T _ R  m  are made by 1 _ R  .  f o r m  This idea deserves further examination.  5.8 Drag over 3D Dunes The effect o f the 3 D dune morphology on drag can be examined using drag estimates based on spatially averaged Reynolds stress profiles. Overall these results are consistent with the observations of / „ and x  uw  and F  DR  over the dune. Values o f C  D  and F  D  indicate that over the 2 D dunes, C  DR  = 0.008  = 0.011 N . Lobes generally enhance drag while saddles cause significant drag reduction.  The change in drag (enhancement or reduction) is measured as  CDR-2D  where the subscript 2D indicates a measurement over the 2 D dune and + values indicate enhancement while - values indicate reduction.  242  The degree o f drag enhancement is dependent on the cross-stream extent o f the lobe. Compared with the drag over a 2 D dune, sinuous lobes that occupy half the flume width increase total drag by - 1 7 % while full-width lobes increase total drag by - 5 0 %. The apparent reason for the enhancement is an increase in the form drag caused by the lateral and vertical divergence o f momentum and turbulent energy over lobes. The degree o f drag reduction over saddle forms does not depend on their cross-stream extent, as form drag does not contribute significantly to the total stress. This appears to be caused by the lateral and vertical convergence o f momentum towards a small area in the lee o f the bedform. A s discussed in Chapter 4, Sirovich and Karlsen [1997] theorise that drag is reduced by random or irregular distributions of roughness elements. A final test of whether this theory can be applied to bedforms can be made by examining the drag generated by the 2 D , regular and irregular bedforms. Compared with the drag over a 2 D dune, the regular dune morphology reduces total drag by - 3 0 % by reducing the form drag component. A s noted above, the regular dune is an end member case for examination of the Sirovich and Karlsen [1997] theory. It is useful to examine the more probable regular 3 D dune morphology (i.e. the sinuous case). Recall from Figure 5.2 that the sinuous crest is composed o f two lobes, one saddle in the centre o f the flume and two half saddles at the side walls. If a single estimate o f C  DR  logical to average the results over the lobe and the saddle which gives C  DR  is made, it would be ~ 0.006. Thus, total drag  is reduced by - 3 0 % over the sinuous crested dune. The irregular bed morphology reduced total drag by - 5 2 % , even though measurements were taken over a lobe. Lobes appear to increase drag when they are full-width forms or are in a regular morphology, but i f in an irregular morphology, drag is reduced over a lobe. The apparent reason is that the wake is prevented from developing and there can be no significant divergence o f momentum. Thus, the form drag component is reduced.  243  The original idea behind the investigation o f drag reduction was to determine i f a shift 2 D to 3 D bed morphology reduced or stabilised the shear stress, and in doing so, reduced the transport rate, imparting greater stability the bed (see Chapters 1 and 4). The reduction in the total stress observed is caused by a reduction in the form component. It is often assumed that sediment transport rates over dune stoss slopes are related to the skin friction component. Thus, it is not clear how a reduction in the form drag w i l l affect the transport rate at the individual bedform scale. However, at the scale o f the bedform field, the reduction in the total drag should reduce transport rates leading to greater stability o f the bed.  5.9 Summary and Conclusions A set o f experiments was designed to examine the turbulent flow over fixed dunes with different crest shapes but constant wavelengths and heights. Laboratory measurements o f turbulent fluctuations in clear water over fixed flat, 2 D and 3 D dune beds were obtained in a 17 m long, 0.515 m wide flume. Flow over nine bedform morphologies is examined: (1) flat bed, (2) straight-crested 2 D dunes, (3) full-width saddle (crestline bowed upstream), (4) full-width lobes (crestline bowed downstream), (5) sinuous crest (saddle portion), (6) sinuous crest (lobe portion with a ragged crest), (7) sinuous crest (lobe portion with a smooth crest), (8) regular staggered crest and (9) irregular staggered crest. Measurements o f velocity were made at a sampling rate o f 50 H z using an acoustic Doppler velocimeter at 350-500 points over a dune in each morphology. The time averaged turbulence structure over the fixed flat bed is dynamically similar to flow over a flat bed with active sediment transport. The observed flow field over the 2 D dune agrees well with an empirically derived model o f flow over mobile and fixed bedforms. Three-dimensional bedforms appear to modify the flow field over a dune significantly. Lobe shaped dune crestlines appear to enhance the level o f turbulence producing a better defined wake structure and more vigorous mixing in the separation cell than observed over 2 D dunes. This causes lateral and vertical divergence o f momentum and turbulent energy. Whether the lobe exists in a 244  regular configuration (i.e. sinuous crestline) or an irregular configuration seems to be relevant. Irregular lobes seem to affect the flow in the same fashion as saddle shaped dune crestlines where the separation cell is only weakly defined and the wake does not appear to be a significant component of the flow field. This causes the lateral and vertical convergence of momentum and turbulent energy to a small area in the lee of the dune. Spatially averaged flow over the bedforms was examined in order to estimate the total, form and skin drag over the dune morphologies. Estimates of the shear and the drag from the depth-slope product do not reflect the turbulence structure. Spatially averaged velocity profiles over 2 D dune beds show the characteristic outer and inner log-linear profiles that can be successfully linked to the total and skin related shear. Flow over the 3D dunes differs from flow over the 2 D dune significantly enough that inner and outer profile segments cannot be consistently defined. The suggestion by McLean et al. [1999] that the outer log-linear portions of the spatially averaged velocity profile may not accurately reflect z  [ol  appears to be particular relevant over 3 D bedforms. U s i n g the inner and  outer log-linear portion of spatially averaged velocity profiles to estimate total, skin and form drag over 3 D dunes can produce erroneous results. Spatially averaged Reynolds shear stress profiles appear to be the most accurate way to estimate the total boundary shear. Skin friction can be estimated from shear stress measurements over the flat bed allowing calculation of the form drag. Compared to flow over a 2D dune, lobe shaped crestlines enhance form drag while saddle shaped forms reduce form drag. Total shear stress over saddle shaped crestlines is nearly equivalent to the skin friction suggesting that, when there is no significant separation cell and wake, the form drag is rather insignificant. Irregular crestline morphologies passively reduce drag more effectively than regular configurations. A regularly sinuous crested bedform reduced drag by - 3 0 % . In contrast, an irregular crest morphology that resembled the 3 D beds discussed in Chapter 4 reduced drag by - 5 2 %. These results suggest that drag reduction does occur during the transition between 2 D and 3D dunes. The  245  ideas of Sirovich and Karlsen [1997] concerning drag reduction over aligned and randomly oriented surface roughness elements are relevant for dunes. Further work is required to determine what implications this finding has on sediment transport processes in sand-bedded river channels.  246  Chapter 6: Conclusions 6.1 The Initiation of Bedforms The first aim o f this study was to examine the processes that lead to the development o f twodimensional (2D) dunes in river channels. A specific goal was to examine a conceptual model termed herein the flow structure approach. The model indicates that bedforms are generated from random turbulent events that generate bed defects. These defects are propagated downstream by flow separation processes and eventually develop into bedforms. This model has received some attention in the literature and widespread support in the earth sciences yet there are two major problems with it. The first is that there is a large discrepancy between the time scales o f bed development and turbulence. The second problem is that it is not clear how random turbulent events can produce one of the most regular forms observed in nature. In order to examine this problem a series o f experiments was designed to examine bedform development in homogeneous 0.5 mm sand in a 1 m wide flume. The sand bed was flattened to remove all variation greater than 1 mm and subjected to a 0.155 m deep, non-varying mean flow ranging from 0.30 to 0.55 m s"'. The initial flow conditions over the flat beds, prior to bedform development, were examined using laser Doppler anemometry to establish that the flow agrees with standard models o f flow and turbulence over hydraulically rough flat beds. With the exception o f the depth-slope product calculations, estimates o f the boundary shear stress derived from different methods are similar in magnitude and increase in a similar fashion with mean flow strength. The roughness heights are consistent with previous observations. The turbulence intensities and boundary layer correlation can be modelled by the semi-empirical functions provided by Nezu and Nakagawa [1993], with the exception of the vertical intensities at the low flow strengths. With few exceptions, the momentum exchange and fluid diffusion are typical o f fully turbulent, uniform, open channel flows. This establishes that the bedforms are not developed by some aberrant flow condition.  247  T w o types o f bedform initiation are observed in the experiments. The first occurs at lower flow strengths and is characterised by the propagation o f defects v i a flow separation processes to develop bedform fields. This type o f bedform development is in accordance with the flow structure approach described above. However, the defects examined here were artificially made because the flat bed seemed capable o f remaining flat in their absence. Thus, a perturbation was introduced to the system and not internally generated by the flow structure (or random turbulence). The second type o f bedform initiation observed differs quite substantially from the flow structure approach to bedform development and has received less attention. This mode o f bedform initiation begins with the imprinting o f a cross-hatch pattern on the flat sediment bed which leads to chevron shaped forms that migrate independently o f the initial structure. The chevron shapes are organised by a simple fluid instability that occurs at the sediment transport layer-water interface. Predictions from a Kelvin-Helmholtz instability model are nearly equivalent to the observations o f bedform lengths in the experiments. It is likely that the instability model holds only for fully mobile bed conditions because the bed constitutes a fluid layer that would be less applicable to situations where the active transport layer is discontinuous. Thus, neither type o f bedform development observed here support the conceptual model based on turbulent flow structures. The observed defect propagation confirms that flow separation is a viable mechanism for bedform development. However, it is not clear that turbulent events can generate the initial defects. The fluid instability model o f bedform development operates without external perturbation o f the system and is capable o f producing the regularity observed in bedform fields.  6.2 Development of Bedforms The experiments in which bedform initiation was examined were extended to determine how the initial forms evolved. It was also o f interest to determine what the equilibrium bedform dimensions and dynamics are at these flow stages. Three basic waveforms were found in the channel, including  248  long sediment pulses, dunes and sand sheets. Insufficient data was available to examine the long sediment pulses so investigation was limited to the dune and sand sheet waveforms. The initial 2 D bedforms developed by the Kelvin-Helmholtz instability developed into dune features that grew exponentially towards equilibrium dimensions. The instability scaling is not preserved because the presence o f bedforms leads to a further mutual adjustment o f the flow and the bed geometry. Dune heights and lengths increased with flow strength while their migration rate decreased. The dunes that are present during the equilibrium stage are morphologically similar to the initial forms present. Aspect ratios suggest that all the forms present are dunes according to the classification scheme of Allen [1968]. There is no obvious transition from small ripples at the beginning o f the runs to dunes when the sandwaves are larger. Smaller bedforms, termed sand sheets, developed over the stoss slope o f the dunes. Interestingly, both the dunes and the sand sheets were observed to be in equilibrium at the same time, meaning both bedforms were active sediment transport agents. The sheets, many lacking slipfaces, form at approximately 0.5 m from the dune slipface, downstream o f the reattachment point. This distance is invariant with dune size, suggesting there is a necessary 'fetch' length needed for the sheets to begin to grow. Approximately 3-4 sand sheets were observed per 100 s. The sheets have heights that are typically O . l x the height o f the dune upon which they are superimposed, migrate at 8 - l O x the dune rate, and have lengths that are nearly constant over the full range o f dune lengths and flow conditions. Aspect ratios are generally < 0.1 with a mode o f -0.025, classifying them as low relief or incipient dunes. Sediment transport rates, measured with a Helley-Smith sampler, and estimated from the morphology o f both the dunes and the sand sheets, are similar in magnitude. Unfortunately, the measurement error associated with the Helley-Smith samples is too large to make a meaningful comparison between the samples and the morphological transport estimates. However, for the larger flow strengths the material moving in sand sheets over the stoss slope o f the dunes is equivalent to the material moved in the dune form. 249  6.3 Transition Between 2D and 3D Dunes One o f the primary characteristics o f the dunes observed during these experiments was that they were distinctly three-dimensional (3D) after an initial 2 D stage that ended when the bedform lengths extended beyond the Kelvin-Helmholtz wavelength. Thus, a second aim o f the study was to examine the 2-3D dune transition in river channels. The primary goals in this portion o f the research were to determine what constitutes a 3 D bedform, how they develop and why they develop. A definition o f what constitutes a 3 D bedform was provided using a statistic called the nondimensional span or sinuosity o f dune crests, which is the distance along the bedform crestline divided by the distance across the channel between crest endpoints. A critical value o f 1.2 is proposed to divide bedforms with straight or slightly sinuous 2 D crests from highly 3 D bedforms. Overhead video records were examined for patterns in the breakdown o f 2 D dune crestlines that gave insight to the physical mechanisms o f the transition. The video revealed that, once 2 D dunes are established, minor, transient excesses or deficiencies o f sand are passed from one crestline to another. The bedform field appears capable o f 'swallowing' a small number o f such defects but, as the number grows with time, the resulting morphological perturbations produce a transition in bed state to 3 D forms that continue to evolve, but are pattern-stable. A possible explanation for why the transition occurs is provided by examining recent advances in the aerodynamics literature that have suggested that surface drag can be significantly altered by the arrangement o f perturbations on otherwise flat surfaces. Aligned (2D) arrangements o f roughness elements were found to increase drag while random (3D) arrangements o f roughness elements greatly reduced drag. Thus, the shift from 2 D to 3 D dune morphology was examined as a mechanism that reduces drag. Time dependent drag coefficients were calculated using continuous records o f water surface slope and bed elevation. A defined increase in the drag coefficient is lacking during the early stages o f the experiment and during the 2-3D dune transition, suggesting that passive drag reduction processes may 250  be at work. However, changes in the drag are somewhat obscured by other processes, primarily dune growth, as the transition occurs. An out of phase relation between the drag coefficient and the nondimensional span suggests that these quantities may be dynamically linked at certain times. However, correlation analysis fails to reveal a relation between the drag coefficient and the nondimensional span because the measured water surface slope may not be representative of the energy gradient. Unfortunately, these problems could not be resolved through these data. Thus, another experiment was conducted using fixed bedforms with varying crestline shapes.  6.4 Drag Reduction Processes A series of experiments was designed to examine turbulent flow over fixed dunes with different crest shapes, but constant wavelengths and heights. Laboratory measurements of turbulent fluctuations in clear water over fixed flat, 2D and 3D dune beds were obtained in a 17 m long, 0.515 m wide flume. The bedform dimensions and the corresponding flow conditions were identical to one flow stage in the experiments described above. Flow over nine bedform morphologies was examined: (1) flat bed, (2) straight-crested 2D dunes, (3) full-width saddle (crestline bowed upstream), (4) full-width lobes (crestline bowed downstream), (5) sinuous crest (saddle portion), (6) sinuous crest (lobe portion with a ragged crest), (7) sinuous crest (lobe portion with a smooth crest), (8) regular staggered crest and (9) irregular staggered crest. Measurements of velocity were made at a sampling rate of 50 Hz using an acoustic Doppler velocimeter at 350-500 points over a dune in each morphology. The time averaged turbulence structure over the fixed flat bed is dynamically similar to flow over a flat bed with active sediment transport. The observed flow field over the 2D dune agrees well with an empirically derived model of flow over mobile and fixed bedforms. Three-dimensional bedforms appear to modify the flow field over a dune significantly. Lobe shaped dune crestlines appear to enhance the level of turbulence producing a better defined wake structure and more vigorous mixing in the separation cell than observed over 2D dunes. This causes 251  lateral and vertical divergence o f momentum and turbulent energy. Whether the lobe exists in a regular configuration (i.e. sinuous crestline) or an irregular configuration seems to be relevant. Irregular lobes seem to affect the How in the same lash ion as saddle shaped dune crestlines where the separation cell is only weakly defined and the wake does not appear to be a significant component o f the How field. This causes lateral and vertical convergence of momentum and turbulent energy towards a small area in the lee of saddle shaped dunes. Spatially averaged Reynolds shear stress profiles appear to be the most accurate way to estimate the total boundary shear. Skin friction can be estimated from shear stress measurements over the flat bed allowing calculation o f the form drag. Compared to flow over a 2 D dune, lobe shaped crestlines enhance total drag by increasing form drag and saddle shaped crestlines reduce total drag by decreasing form drag. Total shear stress over saddle shaped crestlines is nearly equivalent to the skin friction suggesting that, when there is no significant separation cell and wake, the form drag is rather insignificant. Irregular crestline morphologies passively reduce drag more effectively than regular configurations. A regularly sinuous crested bedform reduced drag by - 3 0 % . In contrast, an irregular crest morphology reduced drag by - 5 2 %. The ideas of Sirovich and Karlsen [1997] concerning drag reduction over aligned and randomly oriented surface roughness elements are relevant for dunes. These results have important implications for the stability o f a sand bed. Most o f the bedforms observed during the active sediment transport experiments were 3 D and similar in shape to the irregular bed configuration. 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Animations are in Graphic Interchange Format (GIF) and can be viewed by opening the file in most non-Microsoft image viewers (e.g. A C D S e e j or web browsers (e.g. Netscape).  V i d e o clips: 1) F l o w A - Instantaneous Bedform Development (Run 53). 2) F l o w B - Instantaneous Bedform Development (Run 54). 3) F l o w C - Instantaneous Bedform Development (Run 57). 4) F l o w D - Bedform Development from a Negative Defect (Run 55). 5) F l o w D - Bedform Development from a Positive Defect (Run 56). 6) Flow E - Bedform Development from a Negative Defect (Run 59). 7) F l o w E - Bedform Development from a Positive Defect (Run 58). 8) F l o w A - 2 D - 3 D Bedform Transition (Run 53). 9) F l o w B - 2 D - 3 D Bedform Transition (Run 54). 10) F l o w C - 2 D - 3 D Bedform Transition (Run 57).  263  Appendix C Miscellaneous data extracted from the video image sequences in Arcview.  265  1.0  T—i—i—i—i—r  0.8  • Flow A  -i—i—i—i—i  -i—i—i-  i  i  Measurement Crest Average Image Average  0.6 CO  E tt  0.4  •  •  i  0.2  !  0.0 -0.2  J  I  L.  0.6  T  1  1  c/>  1  1  1  1  1  1  1  1  1  1  1  1  r  I  I  J~  0.2  E  cn  1  1_  I  • Flow B  0.4 T  _1  ^^^^^^^W  o.o  ^^^r^M^^^^P  «^n^^^^^^^  *^^^^^B^^^rW  -0.2 •  -0.4 0.3  -i—i—i—i—i—i—i-  0.2  FlowC  •Pco  0.1  E  0.0  cn  -o.i  i  -i  • •••  •  i  i  1  r~  •*•  i—i  •  *  •  •  • i_  —]  1  1  i  i  i_  i  .  -0.2 -0.3  1  1  1  1  j  i_  10000  20000  30000  i  40000  50000  Time (s) Figure C1: Migration rates, R, calculated as the distance the bedform crestline migrated between two video images. See section 4.2.3 for definitions of the time interval between analysed images. This distance was measured at 10 cm intervals across the flume using the crestlines digitised in Arcview. Black points are all observations, red points are crest averages and blue points are image averages.  266  o o  A - A l l Obs.  T—i—r—r—r—i— A  A  A *  ^  &  AA AA  £  A A  A  % A A  A - A l l Obs. Image Ave.  C - All Data Image Ave.  EJ - AlrData Image Ave.  I —f'—'—'——I—i—'—'—' I 1 -  1  9  <  • o  o R  :  o  °  *  o  8  ° ' $  o  °  '  8  j  o  °0  o  A - F W Obs.  B - F W Obs.  FW Obs.: '. .i .  I I I I 1111  1111  I I  1111  I I I I  1111  1111  1111  £ A  A  A-FW Obs.] Image Ave.  i •20• •30 • 40 '50  10  Time x 10 (s) 3  1111  B - F W Obs. Image Ave. 0  10  20  30  40  50 0  C - F W Obs. Image Ave. 10  Time x 10 (s)  267  30  Time x 10 (s)  3  Figure C 2 : Areally averaged bedform lengths,  20  3  L. a  40  50  Figure C3: A l l observations of non-dimensional span, A .  Flow A  W D 5  CO Q  <  2  Flow B  CO Q  <  2  Flow C  CO Q  <  2  *...  1 4  Flow D 3 CO Q  Flow E  CO Q  <  2  1 10  20  Time 10  30 3  40  (s) 268  50  2.5  Figure C4: All observations of non-dimensional span, A where the cross-stream extent of the crestline exceeded 70 cm.  Flow A  NDS  2.0  1.5 t;  1.0 2.5  Flow B 2.0  1.5  1.0 2.5  Flow C 2.0  1.5  '*.***—V—  1.0 2.5  Flow D 2.0  1.5 ••'  •• X  1.0 2.5  Flow E 2.0  1.5  1.0  0  10  20  30  40  Time 1 0 (s) 3  269  50  270  271  272  273  274  Appendix D Bed and water surface profiles over dunes and corresponding flow depths for each dune profile. Profile and at-a-point velocity measurement locations are also displayed in each figure. In the lower panel o f each diagram, circles indicate where the probe was mounted in the 0° position. D o w n oriented triangles indicate that the probe was in the 45° position. Squares indicate that the probe was in the 90° position. U p oriented triangles indicate that the probe was in the 0° position but that the lower threshold for data retained after filtering was > 40% as opposed to the > 70% retention threshold used for the rest o f the data.  275  n—i—|—I—I l i—r—i—i i i |  150 f  —\—i—i—i—i—|—r i—i—i—|—i—i—i—i [ ii -  2D  100 \  Flow  E  50 \ N  •  0 '  :  -50 ^ 170 160 'E  rr1  _l L—l 9.4  9.2 1  i  1  1  1  1  1  i  1  1  1  I  I I  I I I I  9.6  1  i  1  1  9.8  1  i * *  1  I • I 10.0• I10.2  1  I . . . .  10.4  I  i  1  1  1  10.6  10.8  I .1  11.0  -|—i—i—i—i—|—r—i—•—i—|—i—i—i—i | i i i—i |  :  -  150 \ 140 j 130 Li 150 r  T  125 !• 100 !• , „ 75 jE  N  50 \ 25 \ 0f -25 j-50 t  9.2 1  1  9.6  9.4 1  1  • • • • •  1  •  1  1  1  9.8 1  1  1  10.0 1  1  1  10.2 1  1  1  J  10.4 1  1  1  1  10.6 1  1  1  • • 11.0 •• IL  10.8 1  1  1  • • • • • • • • • • • • • • • • • • • • • • • • • • •  1  •  1  r  •  O ° O 8 8888 o888° o eVvVVVVVVVV oVo oo Voo ooVoo V V;8 8I gv 8v8vV8V8VV8vVv8V8v8•7VVgVV8 VV8VgVo 8o o VVVV fi a a 8 v VV v O 0  0  J7  ^4.  _J  9.6  L_  _J  3  I ' I I9.8  9.7  9.9  10.0-J  I  I  t  I  10.1  v  l_  Distance Along Flume (m) Figure D l : Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the 2D dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  276  ~l  150  -|  1  1  r-  '  ~1  FWL  100  E j=.  r-  r-  Flow  50  N  10.0 150  ~J  1  1  1  T— -|  125  • • • • •  •  10.5  1  1  1  1  1  1  r-  • • • • • • • • • • • • • • • • • • • • • • • • • • •  •  •  100  E E  50  N  25  7  5  \ t-  0 E-25 \ -50 9.8  _)  9.9  10.0  i  10.1  i_  _i  10.2  i  i  i_  _l  I  L_  10.3  Distance A l o n g F l u m e (m) Figure D2: Bed and water surface, z, profiles over bedforms 6-9 (top) and corresponding flow depths, d , (middle) for the full-width lobe (FWL) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  277  9.4  9.5  9.6  9.7  9.8  9.9  Distance A l o n g F l u m e (m) Figure D3: Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the full-width saddle (FWS) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  278  ->  150  r  1  -i  REG  100  1  r-  Flow  E  JL  50  N  0 -50 8.5  —J  •  1  9.0  170  ~>  '  _i  L_  I  9.5  1— -i  1 1  i  10.0 r  i  i —i  •_  1  10.5  r-  160  150  140  8.5  125  10.0 • • • • • -i—r—i •  -50  9.5  •  9.6  •  • • •  r-  •  •  10.5  • • • • • • • • • • •  •  9.7  9.8  9.9  10.0  Distance A l o n g F l u m e (m) Figure D4: Bed and water surface, z, profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the regular (REG) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  279  150  "i  1  1 1  1  r  IRR  100  Flow  If 50 N  -50  -J I  '  1  9.0  L_9.5 —J • • • 10.0 1  180  •  •  l  l_  -l 1 1 [ l 1 1  10.5  I  170 h  140 125 100 75 50  9.0  •  V  V  v  V  V  V  8 e  V v v  •T  88 8 8 8 ^a I& & a 1 V  V  V  "  ?  ?  A  A  A  9.7  r—i  o o o o  V  A  A  V  V  v v v  888 a &* a *  1—  £ ] • • •  O Oo Oo  V  A  9 6  •  00008  o ° o o  V  8 8 § 8 8  •  10.5 •  o o o o Oo Oo o Oo Oo  -25 -50  10.0  • • • • •  25 0  9.5  V v  V  v  f j  A  9.8  -1 I 9.9  _i  10.0  i :  10.1  Distance Along Flume (m) Figure D5: Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the irregular (IRR) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  280  Distance A l o n g F l u m e (m) Figure D 6 : Bed and water surface, z , profiles over bedforms 5-9 (top) and corresponding flow depths, d , (middle) for the sinuous saddle ( S N S ) dune configuration. Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  281  "1 1 1 1 | 1 1 I I | 1  150  Flow  SNL  100  If 50 N  0 -50  9.6  9.7  9.8  9.9  10.0  10.1  10.2  Distance A l o n g F l u m e (m) Figure D 7 : B e d and water surface, z, profiles over the 8th bedform (top) for the sinuous lobe ( S N L ) dune configuration. N o water surface profile was taken over the sinuous lobe.  282  T  1111 11111 11111 11i11 1111 11111 11111 1 r 1  _50 I— —'—•—•—i— — —•—i——i—" " i i 1  9.0  180  1  1  9.2  9.4  1  9.6  • • • • i i—J i i i i i i i i i 9.8  10.0  1  1  10.2  10.4  I11111 1111 11111 1111 1111 11111 i 1111 1 r 1  Lj  9.5  11111 11111 111•I••••'•ii 9.6  9.7  9.8  9.9  i  I  10.0  i  •  •  •  '  10.1  Distance A l o n g F l u m e (m) Figure D8: Bed and water surface, z , profiles over bedforms 6-9 (top) and corresponding flow depths, d , (middle) for the sinuous saddle dune configuration with the smoothed crestline (SSS). Profiles were taken over the 8th downstream dune at the locations noted in the bottom panel.  283  154.0 -i  1  1  1  _j  i  i  L.  1  i  r——i  r-  _i  1  153.5 153.0  _ E E^  152.5  "D  152.0 151.5 151.0  1i  i  i  1  •  i  i  i  I  •  •  i  i  I  E, N  5.0  5.1  5.2  5.3  5.4  5.5  Distance Along Flume (m) Figure D 9 : Bed and water surface, z , profiles over the flat bed and corresponding flow depths, d , (middle). Profiles were taken at the locations noted in the bottom panel.  284  •  Appendix E Conventions for plotting spatially averaged profiles o f mean streamwise velocity, U, and Reynolds shear stress, r  tm  differ amongst sources. Thus, it is useful to compare the different ways  the profiles may be constructed. Spatial averages can be calculated using either all the data between two successive crestlines or only data collected over the stoss slope o f the dunes. Spatial averages may also be constructed at equal heights above a datum, z , such as the dune crest, or at lines equidistant from the boundary, z  hed  .  Figure E l plots averages based on all data between two successive crestlines relative to z and bai  •  z  to z  hed  Velocity measurements in the separation zone are included when profiles are calculated relative and reduce the averages in the lower portion o f the profile. This does not occur i f the profiles  are calculated relative to z . The upper portions o f the profiles are similar. Figure E2 plots averages o f data collected on the stoss slope o f the dunes only, relative to z and z  hed  . Data from the separation zone is not included in the averages. The vertical rise over the stoss  slope is only 3 mm, so the same data points are used to form averages and the profiles are identical. Figures E3 and E4 both plot averages based on all data between two successive crestlines and averages o f only these data collected over the stoss slope o f the dunes. In Figure E 3 , both sets o f averages are plotted relative to z . The profiles are nearly identical. In Figure E4, both sets o f averages are plotted relative to z  hed  . The lower portions o f the profiles include data from the  separation zone and average velocity is reduced. The upper portions o f the profiles are nearly identical. These plots indicate that the inner (lower) profile is affected by the plotting convention while the outer profile is not. This is particularly true i f averages are plotted relative to z  bed  . Nelson et al.  [1993] and McLean et al. [1999] indicate that including data from the separation zone in spatial averages w i l l produce biased inner profiles and that averages should include only data over the dune  285  I ii i i i  1  r  A  I  -  99 I 1 1 1—I—I I I I  8o  o  o  in o  w  co co co  CO  CO  CD  1  1  ' '  I  I  I  '  «8 <*  '  CD O  in  :<5©  o  O o o '  •  <  »o  CO  1  4  <  °  CM  I i i i—i—i—i  1  r-  H  ci  1  1  I  I• •  •  o o  T— o  (LU)  P 8 q  Z JOZ  1 11  1  1  1  I  CM » HO CD O  I _J  tea  e CD  >  2  1  O'  >  _CD  C  "cD 60  r-  CO  o  CD O LO O  o  •«fr O CO  cc cc  o 1  O  O O  O (LU)  286  P 8 q  2  Z JOZ  •• • i i  =1  CNj O  O  •  o cs .2 CS t8 is T3 - CD C  CS  "5 ^c CS CS o -a  C c« O  CD  CD  W  _e CD -a o ~ 60 8 CD > 2 cS c XI 3 x> ° cn ^ es CD CD CD+ J X ) bp x:  o  60 fe CD  x:  «> xj  cn *3 CD cn !G c60 3 2 '5S CS n. -c c CD _CS CD cn X ) CD CD CD CJ  >  ci  CN I i i i i ' HO  9§ o  ci  J-  cs IG £  «>  X>  CD  —I  I i i i—i—i—i  cS  -o  oD £ C o  o  <•  CN  •  1  LU  ci  • • I I  1  u  < 1 CO 1  I  ° o o < < <  ci  o  - 1  LO  o  m o  d  | I I—I—I—I 1 1  o  x:  CO  CD O  <<jM<k]  LO  o w o CQ "a Q .  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However, since the outer profile is not affected by the plotting convention, it is acceptable to calculate total shear stress from the outer profile over the stoss slope. Spatially averaged r  tlw  profiles are meant to represent the total shear stress. Averages over the  stoss slope will not represent the total stress and are excluded from consideration here. Figure E5 plots averages based on all data between two successive crestlines relative to z and z  beJ  . Profiles  have similar shapes but the difference in the datum shifts the profiles down, which will affect the estimated spatially averaged boundary shear stress. The spatially averaged z  uw  profile should be  linear only well above the bedform, where values are not directly affected by flow separation. 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