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Velocity structure in gravel rivers Martin, Violeta 1996

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VELOCITY STRUCTURE IN GRAVEL RIVERS by Violeta Martin B.Sc.(Eng.), University of Novi Sad, Yugoslavia, 1989. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1996. © Violeta Martin, 1996 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that the copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia 2324 Main Mall Vancouver, Canada V6T 1Z4 Abstract This thesis investigates velocity profiles in high gradient rivers with rough beds. Experimental work was carried out in the Hydraulics Laboratory of The Department of Civil Engineering at the University of British Columbia. Velocity measurements were taken in the flume with variable slope (S=0.01 to 0.021) and the size of bed material was D5o=5.0 cm. An Acoustic Doppler Velocity (ADV) probe was used and the turbulent flow structure was examined. Velocities in all three directions were measured and Reynolds stresses have been calculated. Existing methods for determination of velocity profiles are based on the theory that was developed for smooth rigid boundaries and low gradient rivers. These theories lead to a logarithmic velocity profile. However, a limited number of previous studies have shown that the velocity profile in mountain rivers is non4ogarithmic and therefore the mean velocity can be under or overestimated. This study shows that the commonly used logarithmic law does not give the best results. Using a different approach to the existing theory, a new velocity profile equation was developed. In addition, the Froude number, flow rate and Manning's roughness coefficient n were calculated. These parameters show that high energy dissipation is produced by large roughness elements and the flow is slightly subcritical. When additional roughness elements were added to the model, the Froude number decreased and Manning's n increased. The equation developed herein has a reasonable fit with the measured data, but additional work is needed to determine the exact range of applicability. Some existing field measurements agree and some disagree with the present theory, therefore further field measurements with the same measuring equipment need to be made to investigate whether natural streams behave in a similar manner. II Table of Contents Abstract H Table of Contents HI List of Tables....... VI List of Figures VH List of Symbols XBH Acknowledgements XV 1. Introduction..... 1 1.1. General.. 1 1.2. Purpose of Present Work VII 1.3. Scope of present work VI1 2. Literature Review 6 2.1. Vertical Velocity Profile and Mean Velocity 6 2.1.1. Boundary Layer Theory 6 2.1.2. Prandtl's Mixing Length Theory 10 2.1.3. VonKarman's Similarity Hypothesis 12 2.1.4. Turbulent Flow Structures 15 2.1.4.1. Two dimensional Turbulent Flow Structures 16 2.1.4.2. Three Dimensional Turbulent Flow Structures 18 2.1.5. Velocity Measurements in Mountain Streams 20 2.1.6. Segmented Velocity Profiles 22 2.2. Shear Stress Distribution 25 2.3. Manning's Roughness Coefficient n 29 2.4. Equipment for Velocity Measurements 36 III 2.4.1. Pitot Tubes 2.4.2. Current Meters 2.4.3. Hot-wire and Hot-film Anemometers... 2.4.4. Laser Doppler Anemometers (LDA).... 2.4.5. Acoustic Doppler Velocimeters (ADV) 3. Theoretical Development 3.1. Mixing Length is Constant 3.2. Mixing Length Decreases 3.3. Mixing Length Increases 3.4. The Modified Logarithmic Law 3.4.1. Assumptions 3.4.2. Derivation of the Formula 3.5. Comparison Between Developed Equations 4. Experimental Investigation 4.1. The First Experimental Set-up 4.2. The Second Experimental Set-up 4.3. Measuring Equipment 5. Experimental Results and Comparison With Theory; First Experimental Set-up: Uniform Roughness Elements 5.1. Flow Rate, Manning's n, Froude Number and Reynolds Number 5.2. Instantaneous Velocities 5.3. Standard Deviations 5.4. Velocity Profiles 5.4.1. Streamwise Velocity Measurements 5.4.1.1. Streamwise Veloc. Meas. for S 5.4.1.2. Streamwise Veloc. Meas. for S 5.4.1.3. Streamwise Veloc. Meas. for S 5.4.1.4. Streamwise Veloc. Meas. for S =0.021, Y=18.5 cm, D=3.5 cm =0.021, Y=20.5 cm, D=3.5 cm =0.01, Y=22.5 cm, D=3.5 cm.. =0.01, Y=32 cm, D=3.5 cm IV 5.4.2. Vertical Velocity Measurements 103 5.4.3. Transverse Velocity Measurements 107 5.5. Reynolds Stresses 110 5:5.1. The Primary Reynolds Stress - pu' v' 112 5.5.2. The Transverse Reynolds Stress -pu'w' 115 5.5.3. The Transverse Reynolds Stress -pv'w' 118 6. Experimental Results for the Second Experimental Set-up and Comparison With the Results of the First Experimental Set-up 121 6.1. Flow Rate, Manning's n, Froude Number and Reynolds Number 123 6.2. Instantaneous Velocities 126 6.3. Standard Deviations 126 6.4. Velocity Profiles....; 130 6.4.1. Streamwise Velocity Measurements 130 6.4.1.1. Streamwise Velocity Measurements for Cross-section 1 131 6.4.1.2. Streamwise Velocity Measurements for Cross-section 2 132 6.4.1.3. Streamwise Velocity Measurements for Cross-section 3 132 6.4.1.4. Streamwise Velocity Measurements for Cross-section 4 133 6.4.1.5. Streamwise Velocity Measurements for Cross-section 5 133 6.4.2. Vertical Velocity Measurements 140 6.4.3. Transverse Velocity Measurements 144 6.4.4. Flow Patterns 148 6.5. Reynolds Stresses 149 6.5.1. The Primary Reynolds Stress - p u V 150 6.5.2. The Transverse Reynolds Stress - pu'w' 153 6.5.3. The Transverse Reynolds Stress - pv'w' 157 7. Discussion and Conclusions 160 Bibliography 164 V List of Tables Table 5-1 Calculated Flow Parameters for the First Experimental Set-up 72 Table 6-1 Calculated Flow Parameters for the Second Experimental Set-up 123 VI List of Figures Figure 1-1 Capilano River at the Suspension Bridge in North Vancouver, B.C 1 Figure 2-1 Development of the boundary layer in an open channel with an ideal entrance condition (Ven Te Chow, 1981) 7 Figure 2-2 Velocity profile near a solid wall (Daugherty et al., 1985) 8 Figure 2-3 Turbulent flow near boundary: (a) Hydraulically smooth; (b)Wholly rough (Daugherty et al., 1985) 9 Figure 2-4 (a) Velocity profile; (b) Prandtl's mixing length 1 (Daugherty et al., 1985) 10 Figure 2-5 Universal velocity distribution law after Prandtl and von Karman (Schlichting, 1955) 15 Figure 2-6 Schematic descriptions of turbulent flow over smooth and rough beds: (a) Smooth bed; (b) Rough bed (Nezu and Nakagawa, 1993) 17 Figure 2-7 Schematic flow patterns of secondary currents: (a) Narrow open channel; (b) Wide river (Nezu and Nakagawa, 1993) 19 Figure 2-8 Velocity profiles, Lake Creek above Twin Lakes Reservoir, Colorado: S=0.029 m/m; intermediate bed material is 247 mm (Jarrett, 1984).... 21 Figure 2-9 Example of a plot of the logarithm of distance above the bed against flow velocity (Bergeron, 1994) 23 Figure 2-10 Velocity profiles and bed micro-topography at the study site (Bergeron, 1994) 24 Figure 2-11 Uniform flow in open channel (Daugherty et al., 1985) 25 Figure 2-12 Schematic diagram for distribution of velocity and shear stress (Kumar, 1989) 27 Figure 2-13 Uniformly spaced cubes in a wide channel (Kumar, 1989) 29 Figure 2-14 The Piezometer and the Pitot Tube (Daugherty et al., 1985) 36 Figure 2-15 Current meters: (a) Cup type; (b) Vane type (Vennard and Street, 1982).... 37 Figure 2-16 Anemometer sensors: (a) Hot-wire sensor and support needles; (b) Hot-film sensor and support needles (Vennard and Street, 1982) 39 VII Figure 2-17 Schematic description of LDA system used in water flume (Nezu and Nakagawa, 1993) 41 Figure 2-18 Schematic of the ADV probe: downward-looking probe head orientation (Kraus, Lohrmann and Cabrera, 1994) 43 Figure 2-19 ADV and LDV comparison (Kraus, Lohrmann and Cabrera, 1994) 44 Figure 3-1 Graphical solution for the log functin 54 Figure 3-2 Graphical solution for the integral 54 Figure 3-3 Comparison between theories for the measurement taken on Bear Creek at Morrison in June 1983. by Jarrett 58 Figure 4-1 Schema of the experimental apparatus 60 Figure 4-2 The first experimental set-up with uniform gravel size, D50=5 cm 61 Figure 4-3 The second experimental set-up with additional roughness elements placed in zigzag pattern 63 Figure 4-4 The downward and upward looking ADV probe heads 65 Figure 4-5 The experimental set-up, S=0.01, Y=26 cm, measurement taken on Aug. 17th 1995. in the big flume of the Hydraulics Laboratory at UBC 67 Figure 4-6 Progressive velocity averaging for case of uniform roughness elements, when S=0.021, Y=18.5 cm, D=3.5 cm 68 Figure 5-1 Streamwise instantaneous velocities: S=0.021, Y=20.5 cm, D=3.5 cm 76 Figure 5-2 Vertical instantaneous velocities: S=0.021, Y=20.5 cm, D=3.5 cm 80 Figure 5-3 Transverse instantaneous velocities: S=0.021, Y=20.5 cm, D=3.5 cm 84 Figure 5-4 Standard deviations for S=0.021, Y=18.5 cm, D=3.5 cm 89 Figure 5-5 Standard deviations for S=0.021, Y=20.5 cm, D=3.5 cm 89 Figure 5-6 Standard deviations for S=0.01, Y=22.5 cm, D=3.5 cm 90 Figure 5-7 Standard deviations for S=0.01, Y=32 cm, D=3.5 cm 90 Figure 5-8 Streamwise velocity profile - theoretical curves vs. measured data (measurements taken with ADV and Ott-meter): S=0.021; Y=18.5 cm; D=3.5 cm 94 VIII Figure 5-9 Streamwise velocity profile - theoretical curves vs. data measured with ADV: S=0.021; Y=20.5 cm; D=3.5 cm 97 Figure 5-10 Streamwise velocity profile - theoretical curves vs. data measured with ADV: S=0.01; Y=22.5 cm; D=3.5 cm 99 Figure 5-11 Streamwise velocity profile - theoretical curves vs. data measured with ADV: S=0.01; Y=32 cm; D=3.5 cm 102 Figure 5-12 Vertical velocity measurements: S=0.021; Y=18.5 cm; D=3.5 cm 105 Figure 5-13 Vertical velocity measurements: S=0.021; Y=20.5 cm; D=3.5 cm 105 Figure 5-14 Vertical velocity measurements: S=0.01; Y=22.5 cm; D=3.5 cm 106 Figure 5-15 Vertical velocity measurements: S=0.01; Y=32 cm; D=3.5 cm 106 Figure 5-16 Transverse velocity measurements: S=0.021; Y=18.5 cm; D=3.5 cm 108 Figure 5-17 Transverse velocity measurements: S=0.021; Y=20.5 cm; D=3.5 cm 108 Figure 5-18 Transverse velocity measurements: S=0.01; Y=22.5 cm; D=3.5 cm 109 Figure 5-19 Transverse velocity measurements: S=0.01; Y=32 cm; D=3.5 cm 109 Figure 5-20 Reynolds stresses: (a) 2D flow; (b) 3D flow 110 Figure 5-21 Reynolds stress - u V for S=0.021; Y=18.5 cm; D=3.5 cm 113 Figure 5-22 Reynolds stress - uV for S=0.021; Y=20.5 cm; D=3.5 cm 113 Figure 5-23 Reynolds stress - uV for S=0.01; Y=22.5 cm; D=3.5 cm 114 Figure 5-24 Reynolds stress - uV for S=0.01; Y=32 cm; D=3.5 cm 114 Figure 5-25 Reynolds stress - \fw' for S=0.021; Y=18.5 cm; D=3.5 cm 116 Figure 5-26 Reynolds stress -u'w' for S=0.021; Y=20.5 cm; D=3.5 cm 116 Figure 5-27 Reynolds stress - u ^ for S=0.01; Y=22.5 cm; D=3.5 cm 117 Figure 5-28 Reynolds stress - u V for S=0.01; Y=32 cm; D=3.5 cm 117 Figure 5-29 Reynolds stress - v V for S=0.021; Y=18.5 cm; D=3.5 cm 119 Figure 5-30 Reynolds stress - v'w' for S=0.021; Y=20.5 cm; D=3.5 cm 119 Figure 5-31 Reynolds stress - V w 5 for S=0.01; Y=22.5 cm; D=3.5 cm 120 Figure 5-32 Reynolds stress - v'w' for S=0.01; Y=32 cm; D=3.5 cm 120 I X Figure 6-1 Positions for additional rocks and fir fifteen verticals where velocities were measured 122 Figure 6-2 Stand, dev. for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67 cm 128 Figure 6-3 Stand, dev. for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17 cm 128 Figure 6-4 Stand, dev. for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 129 Figure 6-5 Stand, dev. for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67 cm 129 Figure 6-6 Stand, dev. for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 130 Figure 6-7 Streamwise velocity profile for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67 cm 135 Figure 6-8 Streamwise velocity profile for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17 cm 136 Figure 6-9 Streamwise velocity profile for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 137 Figure 6-10 Streamwise velocity profile for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67 cm 138 Figure 6-11 Streamwise velocity profile for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 139 Figure 6-12 Vertical velocity measurements for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67cm 142 Figure 6-13 Vertical velocity measurements for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17cm 142 Figure 6-14 Vertical velocity measurements for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 143 Figure 6-15 Vertical velocity measurements for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67cm..; 143 Figure 6-16 Vertical velocity measurements for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 144 Figure 6-17 Transverse velocity measurements for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67cm 146 X Figure 6-18 Transverse velocity measurements for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17cm 146 Figure 6-19 Transverse velocity measurements for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 147 Figure 6-20 Transverse velocity measurements for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67 cm 147 Figure 6-21 Transverse velocity measurements for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 148 Figure 6-22 Flow patterns: (a) Plan view; (b) Side view 149 Figure 6-23 Reynolds stress - u V for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67 cm 151 Figure 6-24 Reynolds stress - uV for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17 cm 151 Figure 6-25 Reynolds stress - uV for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 152 Figure 6-26 Reynolds stress - uV for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67 cm 152 Figure 6-27 Reynolds stress - uV for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 153 Figure 6-28 Reynolds stress -u'w' for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67 cm 154 Figure 6-29 Reynolds stress - u'w' for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17 cm 155 Figure 6-30 Reynolds stress - u'w' for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 155 Figure 6-31 Reynolds stress - u'w' for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67 cm 156 Figure 6-32 Reynolds stress - u'w' for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 156 X I Figure 6-33 Reynolds stress - v'w' for cross-section 1: S=0.021, Y=20.23 cm, Davg=3.67 cm 157 Figure 6-34 Reynolds stress - v'w' for cross-section 2: S=0.021, Y=20.23 cm, Davg=4.17cm 158 Figure 6-35 Reynolds stress - v'w1 for cross-section 3: S=0.021, Y=20.23 cm, Davg=4.33 cm 158 Figure 6-36 Reynolds stress - v'w' for cross-section 4: S=0.021, Y=20.23 cm, Davg=4.67cm 159 Figure 6-37 Reynolds stress - v'w' for cross-section 5: S=0.021, Y=20.23 cm, Davg=4.67 cm 159 XII List of Symbols A cross-sectional area of a stream normal to the velocity [m2] a constant B width of open channel at water surface [m] C Chezy coefficient [dimensionless]; shape factor [dimensionless]; and constant of integration Co drag coefficient of roughness elements [dimensionless] Dc critical grain diameter [m] d median size of the bed material [m] FD drag force [N] Fr Froude number, Fr=u/(gY)05 [dimensionless] / friction factor for pipe flow [dimensionless] g acceleration of gravity [m/s2] k roughness height [m] L length [m] / Prandtl mixing length [cm, mm] n Manning's roughness coefficient [dimensionless]; and the number of measurements in the point P wetted perimeter [m] Q flow rate [m3/s] q flow rate per unit width of rectangular channel [m2/s] Rh hydraulic radius, R=A/P [m] Re Reynolds number, Re=uY/v [dimensionless] S the slope of the experimental channel [m/m] Sf friction slope [m/m] Su, Sv, Sw standard deviation (rms value or turbulence intensity) for u, v, w, velocity [m/s] XIII u, v, w instantaneous velocities in x, y, z, directions [m/s] U, V, W average values calculated from instantaneous velocities in the point [m/s] u', v', w' velocity fluctuations in streamwise, vertical and transverse direction [m/s] u+ dimensionless velocity, u+=u/u* u* shear velocity, u*=(T(Jpf'5 [m/s] Y mean velocity of fluid [m/s] w(y/Y) wake function y the distance measured from the fluid surface [m] y+ dimensionless depth, y+=yu*/v Y the total depth of flow [m] Y average of fifteen depths of flow measured from the water surface to the top of the roughness elements [m] K Von Karman constant, K=0.41 for open channel flow [dimensionless] fj, absolute or dynamic viscosity [Ns/m2] v kinematic viscosity [m2/s] II Coles' dimensionless wake strength parameter p the mass density of the fluid [kg/m3] pu'V primary Reynolds stress [m2/s2] pu'w', pv'w' transverse Reynolds stresses [m2/s2] E summation r shear stress [N/m2] To the total shear stress on the bottom of the channel [N/m2] Si thickness of viscous sublayer in turbulent flow [mm] XIV Acknowledgments The author would like to express her appreciation to those people who have contributed to the preparation of this document. Advisor, Dr. M.C. Quick, has provided valuable assistance and many suggestions, which have significantly improved the quality of the work. His willingness to contribute his time, knowledge and his patience are greatly appreciated. Professor, Dr. S.O. Russell's precious advises added the finishing touch to the overall work. The author is grateful to Mr. Kurt Nielsen for his help in the Hydraulics Laboratory of The University of British Columbia, his expertise and advice. Jason Vine, Jonas Kinfu and Trevor Elliott are graduate students who have contributed to this work with their discussions, support and friendship, and the author is indebted to them. Special thanks to George Barbulovic, a good friend who helped with plotting using AutoCad. At last, but not at least, the author is very grateful to her family for their understanding and encouragement. XV Chapter 1 Introduction 1.1. General In mountain areas, rivers are of high gradient and shallow depths, characterized by very turbulent flow and relatively large bed material. As the terrain gets steeper, the bed material gets coarser and the occurrence of tumbling flow or so called "white water" is more likely (see Fig. 1.1. for illustration). These areas are becoming increasingly important and attractive to people for a diversity of purposes, and a better understanding of hydraulic processes in mountain rivers is an important issue. Some of these purposes are: design of water resource projects in mountain areas; estimation of flood flows in Figure 1.1. Capilano River at The Suspension Bridge, North Vancouver, B.C. (Aug. 1995) 1 steep gravel channels; safe design of steep channels in urban areas; evaluation and estimation of sediment transport; and, all the above factors can influence river use for such purposes as fish spawning, containment of flood without a designed channel, recreational use of rivers, etc. In many earlier studies, hydraulic processes have been investigated in smooth or moderately sloped flumes with deep flows and sandy beds. In many cases, experiments were done in pipes; theories were developed with certain assumptions and then, with some adjustments, applied to open channel flow. Many investigators have shown in the past that these theories give satisfactory results in low gradient rivers. However, in the last twenty years more emphasis was given to measurements in mountainous regions and the results have shown that the hydraulic processes are somewhat different. The key issue is a high value for Manning's roughness coefficient n, and how this relates to the velocity profile. Jarrett (1987) claims much higher n and therefore lower mean velocities. This has a big effect on estimation of discharge Q and is important in many water resource investigations (Jarrett, 1987 and Quick, 1991). The vertical velocity profile was assumed to obey Prandtl's logarithmic law and the mean velocity to occur at 0.6 times the total depth below the surface. Recent measurements showed that the velocity profile is non-logarithmic and that the mean velocity would most likely occur at 0.5 times the total depth (Jarrett, 1987). Various types of current meters, that are still in frequent use, are of large dimensions, so that measurements cannot be made close to the river bed or to the surface. Also, this equipment does not measure velocities in three directions, but rather superimposes the cross-flows to the longitudinal velocity. Propeller meters are not fixed 2 in direction, so they do not necessarily measure the longitudinal flow component only. Vane type meters respond to cross-flow and vertical flow, therefore they do not give an accurate measure of the longitudinal flow component. These propeller and vane meters do not measure turbulence, thus the three dimensional turbulent flow field cannot be examined. Further development in measuring devices now make such measurements possible. For example, laser-Doppler anemometers (LDA) and acoustic Doppler velocimeters (ADV) produce results that represent the three dimensional turbulent flow structure. The ADV shows great promise for laboratory and field work, because there is little set-up required, as compared with the difficult set-up conditions for LDA. In the present thesis, two approaches have been made: one consists of a theoretical analysis of rough turbulent flow in steep streams, and the second is an experimental measurement of stream velocities. A modified logarithmic equation has been developed, which agrees well with the velocity profiles measured in the flume. But, even this improved equation is an approximation, because it is still based on a two dimensional flow field, whereas the ADV measurements show that the flow is highly three dimensional and the two dimensional assumptions of the shear stress distribution are not valid. The present theoretical work is therefore, a step towards a better equation, but the value of the present work is that it gives a better understanding of the role of the 3D flow in the total behaviour of the channel resistance. A better prediction of the velocity profile and the mean velocity will result in improved estimates of flow discharge and energy dissipation. This is important for flood predictions and flood channel designs, as well as for enhancement in design of structures for energy dissipation in steep spillways. 3 1.2. Purpose of Present Work The purpose of this study is to understand roughness characteristics of steep gravel rivers. An attempt has been made to develop a mathematical equation that will better describe the processes in steep mountain streams and at the same time provide "best fit" to the measured vertical velocity profiles in the flume.. Moreover, shear stress calculations assumed by the conventional equation and by using Reynolds stresses were compared. Also, values of Manning's roughness coefficient were computed using the well known Manning's equation. The author's intention was to develop a usable equation, that will allow practicing engineers to make good estimates of flow in mountain rivers, with only a few on-site measurements. 1.3. Scope of Present Work This study is based on a theoretical analysis of rough turbulent flows and experimental data measured in a laboratory flume, using two basic test conditions. The first condition involved uniform bed material (D50=5.0 cm), which, when compared with the depth of flow, would be considered to be of intermediate scale (1<Y/D84<4) (Bathurst, 1985) and bed slopes that ranged form 0.01 to 0.021. In the second condition additional roughness elements were introduced following a specific pattern and the slope was again 0.021. While in the first condition, the purpose was to investigate the velocity profiles, the purpose of the second was to calculate the increase in flow resistance caused by 3D 4 effects. It is not possible for this thesis to answer all questions regarding hydraulic processes in mountain streams. Instead, we will try to evaluate existing theories, and define a new mathematical equation developed on a modified set of assumptions, which will be used to calculate the velocity profile and the mean velocity. The most important difference between Prandtl's assumptions and the present work is the shear stress distribution. While Prandtl assumed that the shear stress is constant near the wall and over the whole depth, in this thesis a linear shear stress distribution has been presumed, as is predicted by standard open channel theory. Other parameters such as: shear stress, shear velocity, Reynolds stress, Froude number, discharge and Manning's roughness coefficient, will be calculated using conventional formulae. 5 Chapter 2 Literature Review Many researchers in the past have investigated hydraulic processes in pipes and in open channels. The emphasis of this chapter will be on the vertical velocity profile determination and theories that govern the equations for the velocity curve, assuming two dimensional flow. The flow over smooth plane, pipe flow and open channel flow on smooth and rough boundaries will be addressed. This chapter will also outline turbulent flow structures and various studies regarding shear stress distribution, flow resistance, channel roughness and resistance coefficients. Finally, an overview of different equipment types for taking velocity measurements will be presented. 2.1. Vertical Velocity Profile and Mean Velocity 2.1.1. Boundary Layer Theory. - When flow enters a smooth channel, a boundary layer starts to develop. Initially, this boundary layer may be laminar, but will quickly break down to turbulent flow with a laminar sublayer next to the smooth wall. Fig. 2.1. illustrates the velocity distribution variations with the distance from the channel entrance. AB is a zone of laminar boundary layer development, and the velocity profile between the smooth wall and the line AB is approximately parabolic. BC is a zone of turbulent boundary layer development, in which the velocity profile is logarithmic. Inside the region limited by the line ABC and the channel bottom, the velocity profile 6 varies with the distance from the bottom; whereas outside the line ABC, the velocity profile is uniform. At point C, a zone of fully developed turbulent layer is formed, which occupies the total depth of flow and the velocity profile will have a constant pattern further downstream. Figure 2.1. Development of the boundary layer in an open channel with an ideal entrance condition The region inside ABC is called boundary layer and has a thickness S, which is defined as a normal distance from the wall where the velocity reaches 99% of the limiting velocity (Ven Te Chow, 1959). In natural streams, conditions are far from the ideal conditions described above, the flow is turbulent and the zones of laminar and turbulent boundary layer development practically do not exist. In classic boundary layer theory, boundary layer consists of a laminar, transitional and turbulent boundary layers (Fig. 2.2.). Next to the smooth boundary, there is a laminar or viscous sublayer, which is very thin and where the velocity gradient is very steep (Prandtl, 1926). The velocity profile is parabolic, but is difficult to distinguish from a (Ven Te Chow, 1981). 7 straight line. In that region, the shear stress is due only to viscosity, i.e. r=judu/dy. At some distance from the wall, viscous effects become negligible and turbulent shear grows large. Beyond that point, the zone is called turbulent in which the velocity profile is logarithmic. There has to be a gradual transition between the zones (shown by experimental points on Fig. 2.2), a transition zone, where both types of shear exist. thickness of ublayer u Figure 2.2. Velocity profile near a solid wall (vertical scale greatly enlarged) (Daugherty et al., 1985) If the two equations for velocity profiles in laminar and turbulent boundary layers are solved simultaneously, together with some experimental factors, the point of intersection will be at a distance 5i from the wall: 4 = 1 1 . 6 - } = = , (2.1.) 8 where d) is the nominal thickness of viscous sublayer and v is the kinematic viscosity of the fluid. Every surface has irregularities and in reality a mathematically smooth surface does not exist. However, if those irregularities do not pierce through the laminar sublayer, than that surface is called hydraulically smooth and 5i>6e (Fig.2.3a.). For d)<3e, the laminar layer is broken up and the surface is wholly rough (Fig. 2.3b.). In between these values, the surface does not act neither as hydraulically smooth, nor as wholly rough, but in a transitional mode. (a) (b) Figure 2.3. Turbulent flow near boundary, (a) Hydraulically smooth, (b) Wholly rough. (Dougherty et al, 1985) The previous discussion about smooth and rough flow was related to pipe flow. However the process is very similar in open channel flow, except that the symbol e is usually replaced with the roughness height k, which represents the effective height of the irregularities forming the roughness elements. It should be noted that k does not necessarily symbolize the actual or the average roughness height, as two different elements, due to their shape and position, can produce identical roughness effect and will be designated by the same roughness height (Ven Te Chow, 1981). 9 In further considerations the zone of fully developed turbulent boundary layer will be examined only. 2.1.2. Prandtl's Mixing Length Theory. - This theory had been initially developed for flow over a flat plate and was later applied to a pipe flow. It is the most frequently used theory in pipe flow analysis, but has also been widely used in open channel flow for the velocity profile determination. Prandtl (1926.) defined the shear stress (Fig. 2.4a.) in the turbulent flow as: where p is mass density, / is mixing length, du/dy is a velocity gradient at a normal distance y from the solid surface, and the expression - pu'v' is known in modern theory as Reynolds stress (Daugherty et al., 1985). r = p/2 Kdy) = -puv (2.2.) y a u + Au u T du = lu'l dy I Velocity u (a) Figure 2.4. (a)Velocityprofile, (b) Prandtl's mixing length I (Daugherty et al., 1985) 10 In the above equation shear stress is a linear function of distance measured from the wall with the maximum value at the wall T = T Q . Also, Prandtl assumed that in turbulent Prandtl introduced two further assumptions: (1) that the mixing length is proportional to y, and (2) that the shear stress is constant and equal to x0, which is not true but was done to simplify the mathematics. These assumptions need considerable justification, however they have the great benefit of linking the turbulent structure to the mean velocity profile, so that the mean flow can be computed. The mixing length / (Fig. 2.4b.) is defined as the average transverse distance to which fluid masses are carried due to cross fluctuations before losing their identity (Rouse, 1938). where K is the constant of proportionality between / and y determined through experiments by von Karman to be 0.4; and.y is the distance from the wall. Using all these assumptions, Prandtl developed his log-velocity-distribution-law: flow u' is proportional and of same order of magnitude as I v' l = Ky, and (2.3.) du (2.4.) 2 2 T0=PK y Kdy) (2.5.) (2.6.) which after integration gives: 11 u = —In v + C and K (2.7.) u. •0 — (2.8.) where u*0 is the shearing velocity at the wall and C is the constant of integration. This equation is valid only in the vicinity of the wall because of the assumption that T=const. If used for the whole region, i.e. up to y=Y (Y being the total depth of flow in open channel), where the velocity is u=Umax for y=Y, the constant of integration can be determined. This leads to the so called velocity-defect law for the open channel (also found in Schlichting, 1968): or rewritten in terms of the mean velocity: Even though there are weaknesses in Prandtl's assumptions, and the above equation is not applicable very close to the wall, the fact remains that it agrees very well with actual measurements of velocity profiles on flat plates as well as in smooth and rough pipes. It has also been used for smooth and rough open channel flow. The mean velocity can be calculated from the known velocity profile by integration over the depth of flow. 2.1.3. Von Karman's Similarity Hypothesis. - Th. von Karman established a rule for determining the dependence of mixing length on space coordinates. He made an (2. JO.) 12 assumption that turbulent fluctuations differ from point to point by time and length scale factors only, so that they are similar at all points of the field of flow. This is called the similarity rule (Von Karman, 1930). A mixing length / was chosen to be the characteristic linear dimension for fluctuations, while a friction (shearing) velocity u* was chosen to represent the velocity of the turbulent motion. Applying a similarity hypothesis to the stream function at a point with coordinates Xo, y0, von Karman derived the following two equations: {d2uldy% l(du/dy)o B = const, (2.11.) -const, (2.12.) l(du/dy)o where / is the length and B is the velocity scale. Further, he introduced a universal empirical dimensionless constant tc, which has the same value for all turbulent flows; and assumed that the mixing length satisfies the equation: i — du/dy l = K (2.13.) \dluldyl In accordance with the similarity hypothesis, the turbulent shearing stress is as follows: r = pl2 du dy du 2 {du/dy)4 — = px T — - w (2.14.) <^ (d2uldy2) Also, a shearing stress is a linear function of the width of the channel, which is an important condition, but not present in Prandtl's theory. 13 T = T, y (2.15.) where width of the channel is 2h, and coordinate^ is measured from the center-line of the channel. Equating the above equations for the shear stress, integrating two times and applying the condition that u=umax at the center-line (y=0), in order to get the constant of integration, von Karman's velocity-distribution law can be derived: 1 max f In V + - (2.16.) Introducing the shear velocity u,0 = ^T0 I p and rewriting the above velocity equation in dimensionless form, the velocity-defect law is determined: "max - « K . 0 In 1 + . (2.17.) This equation gives an infinitely large velocity at the wall and a kink in the velocity curve near the center-line. The coefficient K is the same as in Prandtl's equation for velocity. Having in mind the difference when determining coordinate y, both equations (Prandtl's and von Karman's) are giving a similar velocity profile, which is very interesting when the shear stress distribution is so different. A comparison of velocity profiles plotted using these two laws is shown on the Fig. 2.5. 14 2.8 2A 2.0 IB-1.2 0.8-OA-11 \\ \ \ \ 1 0.2 OA 0.6 0.8 W Figure 2.5. Universal velocity distribution law after Prandtl (curve 1) and von Karman (curve 2), y being the distance from wall (Schlichting, 1955) 2.1.4. Turbulent Flow Structures. - Significant work on measuring and collecting other researchers' data and results on turbulence in open channel flow, has been done by Japanese authors Nezu and Nakagawa, published in Turbulence in Open-Channel Flow, 1993. Turbulence measurements in water flows started only twenty years ago, but more accurate measurements became achievable with laser-Doppler anemometer (LDA) in 1980's. Velocity is divided into the mean velocity and a turbulent fluctuating component. The flow consists of two regions: a near-wall region, which is controlled by the friction velocity («*) and the kinematic viscosity (v); and an outer region near the free surface, which is controlled by the flow depth (Y) and the maximum mainstream velocity (umcvc). The shear stress varies linearly from its maximum value at the bed to zero at the free surface. 15 The characteristic scale for length is v/u* (viscous length) and for velocity is u*. Several dimensionless numbers will be used in further text: £=y/Y, u+=u/u*, umax+=umw/u*, y+=yu*/v, f=lu*/v. 2.1.4.1. Two Dimensional Turbulent Flow Structures For 2D flows that occur in wide open channels, in the wall region (y/7<0.2) it is assumed that the mixing length distribution is linear. The velocity distribution over a smooth bed in open channel flow is governed by the well known Prandtl's log law formula: u+=-\n(y+) + A, (2.18.) where K is the von Karman constant and A is the constant of integration. In the past, K and A have been adjusted in order to apply the log-law to individual situations. In many other cases values x=0.4 and A=5.5 (determined by Nikuradse for smooth pipe flow) have been adopted without further examination. The von Karman constant and the integral constant have universal values 0.41 and 5.29, respectively, for the wall layers of open channel flows (Nezu and Nakagawa, 1993). They are independent of flow properties or bed roughness. Recent studies have shown that the above formula is valid only in the near wall region. Deviations from the standard log law in the outer region (y/F>0.2) cannot be accounted for by adjusting the values for K and A. Instead, it has been suggested that a wake function w(y/Y) should be added to the log law, which was empirically determined by Coles (1956) and appears to be a useful extension of the log law: 16 u+ =-\n(y+) + A + w(y/Y), K 2TI .[ ny , (2.19.) (2.20.) where II is Coles' wake strength parameter. The velocity defect law than becomes: 1 | V | 211 Jny « v - w = -—In — + cos — (2.21.) Using the above 'log-wake law', the velocity profile (Fig. 2.6a.) will be determined over the whole depth of flow. Ejection Low-Speed Streak Sand Roughness (a) Figure 2.6. Schematic descriptions of turbulent flow over smooth and rough beds, (a) Smooth bed. (b) Rough bed. (Nezu & Nakagawa, 1993) For completely rough flow (Fig. 2.6b.), the roughness elements penetrate into the logarithmic region causing formation of a fluid layer which separates from the grains. Therefore, the roughness coefficient k should be incorporated into the velocity function: 17 where (2.22.) Ar =-ln(k+) + A and k+ = k vl w. Ar decreases as k+ increases, and for a completely rough flow it is constant equal to 8.5. The effects due to roughness elements are classified as follows (Nezu and Nakagawa, 1993): 1. Hydraulically smooth bed (k+<5) 2. Incompletely rough bed (5< k+<70) 3. Completely rough bed (k+>70) The theoretical wall level should be set below the top of the roughness elements (see Fig. 2.6b.) for the value 8 (where 0<S<k). For sand-grain roughness S/k takes values between 0.15-0.3, depending on researcher. 2.1.4.2. Three Dimensional Turbulent Flow Structures Real flows in rivers and open channels are too complex to be represented as 2D flows, as secondary currents have a significant role in the flow structure. River engineers suggested more than 100 years ago, that secondary currents exist due to the fact that the maximum velocity occurred below the free surface in channels that were not very wide Fig. 2.7a.). However, Vanoni (1946) suggested possible existence of secondary currents in wide open channels, as well, as the concentration of suspended sediment varied cyclically in the spanwise direction (Fig. 2.7b.). 18 Figure 2.7. Schematic flow patterns of secondary currents: (a) Narrow open channel, (b) Wide river (Nezu and Nakagawa, 1993) Secondary currents are generated mainly because of anisotropy of v' (vertical) and w' (transverse) velocity fluctuations, which is caused by complex boundary conditions at the solid boundary and the free surface. Primary flow u is affected by secondary currents, and its velocity distortion influences the bed shear stress T0. It was verified by several authors that the bed shear stress increases in regions of downflow of secondary currents and decreases in regions of upflow (Nezu and Nakagawa, 1993). The Reynolds stress 19 -uv is negative in the free surface region, which is consistent with the fact that du/dy is negative in this region (i.e. umax occurs below the free surface). Simple formulae that can describe complicated turbulent 3D processes do not exist, but complex computer models have to be written for this purpose. 2.1.5. Velocity Measurements in Mountain Streams. - Robert D. Jarrett carried out detailed research on eleven mountain streams (S>0.002) in Colorado, USA (1984). He measured velocity profiles in the vertical, and concentrated on determination of Manning's roughness coefficient n and mean velocity V, as well as on the position of the mean velocity in the vertical. He implied that the velocity profile is "S" shaped, but did not describe it mathematically. Two types of current meters were used for velocity measurements: the standard Price type AA meter with open-cup bucket wheel and Price Model PAA meter with solid-cup bucket wheel. Water surface slopes, depths and bed material characteristics were also examined, in order to determine a possible relationship between these and the mean velocity. Data from these investigations have indicated the following (Jarrett, 1989): 1. Flow resistance varies inversely with depth of flow and directly with river gradient (or friction slope). 2. Velocity head coefficients vary inversely with relative roughness (Rf/D84), and directly with friction slope and Manning's n. 3. Flow is predominantly subcritical. 20 4. Vertical velocity profiles are S-shaped and not logarithmic (Fig. 2-8); velocities are slower near the streambed than for a logarithmically distributed profile and are faster near the water surface. 5. Price AA current meter may consistently over-register velocity in turbulent mountain-river flows. Figure 2.8. represents a typical velocity profile resulting from Jarrett's study, with a logarithmic velocity profile superimposed. At v/V=1.0, the point velocity equals the mean velocity. In reviewing Jarrett's measurements, I have plotted the measured velocity profiles and generally velocity profiles were S-shaped for higher gradient rivers (S«0.02) and large bed material, whereas the backward looking curve near the surface was observed for lower gradient rivers (S«0.002). Bathurst (1987) also observed increased velocities near the surface and S-shape profiles for relatively high gradient rivers. RATIO OF POINT VELOCITY TO MEAN VELOCITY IN THE VERTICAL (v/V) Figure 2.8. Velocity profiles, Lake Creek above Twin Lakes Reservoir, Colorado. S=0.029 m/m; intermediate bed material is 247 mm. (Jarrett, 1984). 21 Jarrett developed the following equation for Manning's n determination: n = 0.325° 3 8 i T ° 1 6 , (2.23.) where S is the friction slope in m/m and R is the hydraulic radius in m. The above formula is defined for S=0.002 to S=0.052 m/m and R=0.15 to R=2.2 m. From Manning's equation, the mean flow velocity is than: V = 3 .17R M 3 S 0 U . (2.24.) Both equations are applicable for relatively clear water flow in stable channels with minimal bank vegetation, few obstructions and regular banks. This velocity equation is very interesting, as it implies that the mean velocity does not vary considerably with changes in slope. It can be justified physically, knowing that as the slope increases more energy is dissipated in turbulent tumbling flow and hydraulic jumps. The velocity does not increase indefinitely, but stays within certain limits. Jarrett also observed that the mean velocity will most likely occur at 0.5 of the total depth. In comparison, standard open channel theory for the logarithmic velocity distribution predicts the mean velocity occurrence at 0.6 of the total depth, measured from the water surface. 2.1.6. Segmented Velocity Profiles. Normand E. Bergeron obtained velocity profiles using a Marsh McBirney electromagnetic current meter model 523 for nine gravel bed streams located in Quebec, Canada and New York, Vermont and New Hampshire, USA. These velocity profiles are similar in shape to Jarrett's data. He carried out statistical tests of the linearity of flow velocity profiles (1994). His findings indicated that 22 the velocity profile is often not semilogarithmic, but segmented (Fig. 2.9.), which means that it can be divided in two or more semilog-linear segments. N C n o 0 20 40 60 80 100 120 Flow velocity, u (cm/s) Figure 2.9. Example of a plot of the logarithm of distance above the bed (In z) against flow velocity (u). (Bergeron, 1994) A spline modeling technique was used to model the segmented velocity profiles. Splines are defined as segmented polynomials of degree n whose function values agree at points where they join (Smith, 1979). The following linear regression equation is used to describe a spline model of degree n with j=n-l knots: u = a + bx (In z) + b2 (In z - In knotx )A+.. .+bn (In z - In knotj )A, (2.25.) where the function "A" is defined as follows: Inspecting the spline models, Bergeron concluded that the knots tend to occur in the near surface and near bottom region of the flow. Free surface effects will be reflected lnz-Inknot. - \nz-\nknot. In z -In knot. = 0 if lnz >lnknot, if lnz < In knot, J J 23 in near surface knots. Streams with large bed material will have near bottom knots higher than streams with small bed material. Bergeron also examined the dependence of velocity profiles on stream bed topography (Fig. 2.10.). 0 20 ' 40 60 ' 80 1 00 ' 120 U O ' 160 160 ' 200 ' 220 ' 240 ' 260 ' 260 ' 300 ' 320 ' 340 ' 360 Horizontal distance, cm The boltom mark indicales the posilion ol Ihe velocity profile on Ihe stream bed protile. 5^  The top mark indicales Ihe location ol the value u = 0 cm/s for each velocity profile. N O-J 1 i For velocity profiles 5. 6 .7. and 15. velocities below the horizontal bar are negatives. 0 60 u (cm/s) Fig. 2-10. Velocity profiles and bed micro-topography at the study site (Bergeron, 1994) It can be observed that the flow separates after larger roughness elements, creating a zone of recirculating flow in the lee of the obstacle, where the velocities can be even negative. Upstream from the obstacle and downstream, where the wake effects have dissipated completely, the velocity profile is semilogarithmic over the entire depth of flow. He concluded that the velocity profile segment located below the near bottom knot corresponds to the portion of the flow affected by a flow separation. Above this knot, the velocity gradient is steeper because the flow is influenced by grain roughness and form drag due to large bed obstacles. 24 2.2. Shear Stress Distribution In the flow of any fluid, work is done to overcome resistance caused by fluid viscosity, and therefore energy is dissipated. "The basic resistance mechanism is the shear stress by which a slow-moving layer of fluid exerts a retarding force on an adjacent layer of faster moving fluid." (Henderson, 1966, pp. 13) For uniform steady flow, standard open channel theory predicts a linear shear stress distribution, which has a minimum value of zero at the free water surface and a maximum value at the channel bottom, which can be calculated using the following equation: T0=rjSf=jRhSf, (2.26.) where y is a specific weight of fluid, A is the cross-sectional area, P is the wetted perimeter, Rh is the hydraulic radius, and Sf is the friction or energy slope. Figure 2.11. Uniform flow in open channel (Dougherty et al., 1985). 25 In most open channels, the bed slope is small. For steady uniform flow the bed slope is equal to the water slope and to the friction slope, i.e. So=Sy^Sf (Fig. 2.11.). The shear velocity can be calculated from the shear stress: = = ^ g R " S f ' ( 2 ' 2 7 ' } Many texts state that the shear velocity does not represent a physically real velocity, yet it has dimensions of velocity and is a useful velocity scale with which to normalize mean velocity and turbulence. In next paragraphs, it will be argued that shear velocity does have a real physical significance. Another definition of a shear stress in a turbulent flow moving over a solid surface has been given by Prandtl (1926), where the equation 2.5. can be written as: , = ^ y , (2.5a) where p is the mass density, / is the mixing length, and du/dy is the velocity gradient at a normal distance y from the solid surface. The shear stress can be expressed as function of velocity fluctuations: x = -ptiv" (2.28.) The shear velocity (equation 2.27.) becomes then equal to: u=y[u7V, (2.29.) where u' and v' are velocity fluctuations in longitudinal and vertical direction of flow, respectively, and -u'v' is one of the Reynolds turbulent stresses. The minus sign is necessary, as the product u'v' is negative on the average. It can be seen here that, despite the above mentioned statement that the shear velocity does not represent any physically 26 real velocity, when using Prandtl's boundary layer assumptions, u becomes identical with the root mean square of turbulent velocity fluctuations u' and v', and therefore gives an approximate representation of the turbulence intensity. This is interesting because, knowing the mean properties of the flow (S, Rh), one can estimate the turbulent fluctuations. For semi-rough conduits, Kumar and Robertson (1980) proposed a somewhat different shear stress distribution in the zone of roughness elements (Fig. 2-12). They suggested that the shear stress distribution is dependent on the resistance of the roughness elements, and viscous resistance of the smooth boundary. This type of distribution would affect the velocity distribution, which would consist of two logarithmic profiles. Above the roughness elements, a conventional logarithmic velocity distribution is assumed to apply. Below the top of elements, the velocity profile is logarithmic again, but the roughness elements produce an additional resistance to the flow, which should be accounted for. j Velocity Distribution Shear Stress Distribution Figure 2.12. Schematic diagram for distribution of velocity and shear stress (Kumar, 1989). From Fig. 2.12., it can be seen that the total shear stress at the boundary, r0, is a combination of: rs which is the viscous shear caused by the smooth boundary between the 27 roughness elements, and zr which is the shear stress due to the form drag of roughness elements (Kumar, 1989). rs + TR=T0, or ^ + ^ = 1 , (2.30.) i FD=-CDApPU2 (2.32.) In the above equations the following symbols were used: FD is the drag of all the roughness elements; Co is the drag coefficient of roughness elements; A P is the projected area of all elements on the boundary on a plane normal to the flow; U is the approach or mean velocity in the channel; Uk/2 is the velocity of flow at the height of k/2 above the smooth surface; k is the average height of roughness elements; B is the width of given boundary area; and L is the length of given boundary area. Also, "in the region above the viscous sublayer and up to the average height of the roughness elements, Prandtl's mixing length equation (2.5. a) has been shown to apply" (Robertson and Chen, 1970) and the equation 2.6. becomes: du u IT — = — - (2.6a) dy xy y T0 It was also observed that the relative viscous shear stress, T/TO, decreases rapidly with increase in concentration of roughness elements, A (Fig. 2.13.). X is denned as a plan area of elements divided by boundary area over which they are resting. With increase in roughness density, A, the smooth boundary shear stress TS becomes negligible, which in turn, affects the shear stress and velocity distribution, as seen in Fig. 2.13. 28 * - 0.25 She^f S c r e e s V e l o c i t y D i n t n but i o n D L a c r l b u C l o n Figure 2.13. Uniformly spaced cubes in a wide channel (Kumar, 1989). In the present thesis, the above approach has not been adopted, because the roughness elements, i.e. the rocks on the bottom of the channel, were placed next to each other, so that a smooth boundary did not exist. 2.3. Manning's Roughness Coefficient n Manning presented his formula for the first time in 1889, and was later modified to a simpler form that is very well known worldwide, and is frequently used for velocity calculations in open channel flow. The equation was first developed in meter-second units and later modified for feet-second units, by introducing the conversion coefficient 1.49. The same n values can be used in both unit systems and the equation is simple to use. In SI and Imperial system of units, respectively, the formula is as follows: V = -R2h"SV2, or (2.33.) n 29 V = ^R™Sm, (2.33a) where I7 is the mean velocity, n is the roughness coefficient, Rh is the hydraulic radius, and S is the slope of the energy line. The Manning n is not dimensionless, having dimensions TL" 1 / 3. Although the Manning equation looks very simple, the Manning n itself is a complex factor which involves many aspects of the flow and is therefore not easy to define, as will be discussed below. There is no exact method of selecting the coefficient n, although many attempts have been made to identify the factors influencing the n value. The following factors (Ven Te Chow, 1981) have the greatest influence on the coefficient n in both - artificial and natural channels: Surface roughness - represented by the size and shape of grains forming the wetted perimeter. Fine grains have small value of n and are relatively unaffected by change in flow stage. Larger grains, such as gravel and boulders, cause higher values of n particularly at low flow stages. Vegetation - a kind of roughness that retards the flow and reduces the capacity, depending on the type of vegetation, height, density and distribution. It can change the value of n considerably during the year. Channel irregularity - consists of irregularities in wetted perimeter and variations in cross-section, shape and size along the channel length. Such irregularities are: sand bars, sand waves, ripples, holes and depressions, humps. Some other factors are: channel alignment, silting and scouring, obstructions such as bridge piers, size and shape of the channel, stage and discharge, seasonal change, 30 suspended material and bed load. For mountain rivers, n also depends on turbulence intensities, sediment transport and water jumps (Ugarte and Madrid, 1994). Ven Te Chow (1981) gives four general approaches on how to determine the value of n. •0- To understand the factors that influence the value of n and therefore narrow the wide range of guesswork. To consult a table of typical n values for channels of various types. <• To examine and become familiar with the appearance of typical channels whose roughness coefficient is known. To determine n analytically from a known velocity distribution in the channel cross-section and on the data of either velocity or roughness measurement (Ven Te Chow, 1981). In reality, n is determined from experimental measurements of the mean properties of the flow (velocity, hydraulic radius and slope). It is still not well defined from a theoretical basis. Manning's formula is suitable for all fully rough flows, but not for transitional flows, unless n is recognized as dependent on Reynolds number. Many authors in past and present days have investigated the roughness coefficient n. Here, only some of the theories will be presented: 1. The relationship with the well known Chezy Formula: V = CjRji and 149 (2.34.) 31 where Vis the mean velocity in ft/s, and C is Chezy's resistance factor. 2. The relationship with the Darcy-Weisbach friction factor/: >l/6 I f Imperial units: « = 1 . 4 9 ^ / % | r - (2.35.) >l/6 / f SI units: n = R];\\^ (2.35a) 3. Strickler's empirical equation (1923), based on experiments carried out on gravel-bed streams: » = 0.034</,/6, (2.36.) where d is the median size of the bed material in ft. Therefore, Manning's n varies with one-sixth power of the bed material diameter (or in other words - the roughness height), which leads to the conclusion that Manning's n varies only slightly with roughness height. Strickler's equation (2.36.) can be presented in another form: n= ^ , (2.36a) where C is the shape factor and Dc is the critical diameter. Madrid (1992) recommended following values: C=10 for D 5 0 and C=3.4 for D g 4 in the range of relative submergence 1.0<Rh/D84<12.5. These values for C will minimize the difference between the estimated and measured roughness. 4. Limerinos (1970) proposed the following formula for calculation of n: 32 5. Jarrett (1984) carried out research on steep mountain rivers in Colorado (S>0.02) and proposed an empirical formula for n (equation 2.23.): n = Q32Sf%R^6. It can be seen that in this equation, Manning's roughness coefficient does not depend on the river bed material. Jarrett's equation is discussed in more detail in Section 2.1.5. of this thesis. 6. Higginson and Jonston (1988) proposed a method for Manning's n estimation, using a Stickler's formula modified to account for channel properties: £2 .922^3 .978 n = ns + O.OlLog ^ 0 6 3 ^ 0 2 5 , (2.38.) where ns is an estimate of n using Stickler's formula, S0 is the bed slope (in cm/m), Rh is the hydraulic radius (in m), D35 grain size (in mm)for which 35% of the material is finer, V is the mean velocity (in m/s). The method was extended for flows that are less than bank-full, and for the 50% capacity flow, the result is: £2 .654^3 .786 n50 =ns+ O.OlLog . (2.39.) It has to be noted here that the estimation of n50 for a flow which is half of the channel capacity, uses values for Fand Rh that are for the baiik-full flow. 33 The above power laws are very precise, calculated up to three decimals, which is probably far too precise and not justified, because the whole hydraulic model is not precise itself, as it is based on the assumption of a two dimensional flow, whereas the flow in mountain rivers with rough beds is highly three dimensional. 7. Ugarte and Madrid (1994) examined data for rivers in United States (Jarrett, 1984), United Kingdom (Bathrust, 1985) and Chile and developed following empirical formulae: n = nb + An, (lADcy 0.183(7J84) 1/6 21 An 4i (A,) 1/6 f{Sf,RJDM,Fr) (2.40.) where nt is the base value of roughness and the expression is based on Strickler's equation, and has been determined by multiple regression using 62 observations of Chilean rivers; An is an increase in the roughness coefficient and is a function of the friction slope Sf, relative submergence Rt/D84, and Froude number Fr. For the large scale roughness (Rj/Ds^l.O), the following equation is proposed: n •• 0.183 +In 1.74621S,0:1581>i V --0.2631 D, 1/6 84 4i ' (2.41.) where D84 is considered in meters. Using this equation to estimate the mean velocity, the mean error is about 5.7%. For the small and intermediate scale of roughness (l<Rh/D84<12.5), the following empirical equations are proposed: 34 n and n -0.183 +In ruoi4so/7S5(Rh/DM)oo2U^ 7 0.2054 D. 1/6 84 0.219 +In ^1.32595;0932(JR,/Z)50)00260A F 0.2054 D, 1/6 50 4i ' where .D^ and D50 are in meters. The mean error is 2.2% when estimating the mean velocity. Again, the power laws are incredibly precise, but the accuracy of the entire calculation does not justify such power laws. The values should be rounded of, and the equations tested again. Ugarte and Madrid have tested some of the empirical formulae for calculating the roughness coefficient on data collected in Chile. Strickler's equation gave an average error of approximately 45% if C=1.0 and 14.8% if the appropriate shape factor, C, was used. Limerinos' formula lead to an average error of 14.5%. Jarrett's equation gave an average error of 19%. All these formulae appear to be highly specific for the particular rivers studied. They should be used with caution and checked against measured values, such as Barnes (1967), or Hicks and Mason (1991). 35 2.4. Equipment for Velocity Measurements The most frequently used devices for taking velocity measurements in fluid flow are discussed in this Section. The basic principle of operation is described, so are the disadvantages to the use of the particular device. Finally, the equipment used in experimental work for this thesis, the ADV, will be discussed. 2.4.1. Pitot Tubes. - In 1730, scientist Henri Pitot introduced a bent glass tube into an open flow, which had an open end facing upstream. He found that the water level in this tube rose above the free water surface to the value of the velocity head, u2/2g, where u is the local velocity of the flow (Fig. 2.14.). Using Pitot tubes, one can therefore measure the total head of the flow (pressure head + energy head) and can calculate the velocities. Pitot tubes are still in use. However, it had been shown by many in the past that there are several disadvantages that have to be considered: a b Figure 2.14. The piezometer and the pitot tube (Daugherty et al, 1985). 36 4- For velocity profile measurements taken in shear flows, there is an error due to the asymmetry of the flow near the tip of the tube. This error increases toward the solid boundary, where the velocity gradient gets higher. <• Pitot tubes are not convenient for velocity fluctuation measurements, because the measurement system tends to average the pressure fluctuations as well. If the flow is very turbulent and the water surface fluctuates widely, than it is very difficult to determine the value of the velocity head, which leads to additional experimental errors. 2.4.2. Current Meters. - Current meters are mechanical devices consisting of a rotating element, whose speed of rotation depends on the velocity of the flow, as determined by calibration. The same type of device is used for velocity measurements in air, but in that case, the device is called an anemometer. The rotating element can contain a set of cups or a propeller, and therefore the device is called a cup type or a propeller type current meter, respectively (Fig. 2.15.). (a) (b) Figure 2.15. Current meters: (a) Cup type; (b) Vane type. (Vennard and Street, 1982) 37 These devices are frequently used for fluid and air flow velocity measurements. However, some drawbacks to the use of this equipment are recognized below: ^ Cup type current meters are not suitable if there are eddies or other irregularities in the flow, as the cups always rotate in the same direction and at the same rate regardless of the direction of the flow. This device will not detect whether the velocity is positive or negative, or if the velocity is at right angles to its plane of rotation. <~ Vane or propeller type current meters that rotate about an axis parallel to the flow, will register the velocity component along their axis and will rotate in the opposite direction for negative velocities. Propellers used to have rather large dimensions (~14 cm), and therefore velocities near the solid boundary or the surface could not be measured. Lately, a propeller meter of small dimensions (3 mm) has been developed in Japan, but is mainly used for flume measurements. The maximum response frequency of this type of device is about 10 Hz and it cannot be used to measure the fine structure of turbulence. 2.4.3. Hot-wire and Hot-film Anemometers. - Hot-wire anemometers were developed in early 1920's and are used for velocity measurements in gases. Hot-film anemometers were developed later, at the end of 1960's, and they are capable of measuring the mean and the instantaneous velocities in liquid flows. So far, these devices are the most frequently used devices for measuring the turbulent velocity fluctuations. The sensing element consists of a thin wire, or a metal film laid over a glass rod and 38 coated to protect the film (Fig. 2.16.). The sensors can be either a constant temperature or a constant current circuit, the constant temperature one being more accurate. (a) (b) Figure 2.16. Anemometer sensors: (a) Hot-wire sensor and support needles; (b) Hot-film sensor and support needles (Vennard and Street, 1982). The operation of a hot-wire anemometer with constant temperature sensor is based on simple electric properties of the wire used. The higher is the velocity of the flow past the wire, the higher is the heat transfer rate to the surrounding fluid. In order to keep the wire at constant temperature the electric current through the wire needs to be increased in order to compensate the elevated heat loss. This is achieved by a balancing device which is increasing the voltage. The variability of the voltage correlates to the changes of the velocity of fluid. The signal sent from this device has to be analyzed with the aid of an analog to a digital converter. Hot-wire anemometers can measure accurately gas-flow velocities from 0.30 to 150 m/s. Velocity fluctuations with frequencies up to 500,000 Hz can be measured. If the wire is mounted normal to the mean flow direction, it will measure fluctuations in that direction. Other probe configurations can measure fluctuations for other velocity components. 39 Hot-film anemometer is more suitable for velocity measurements in liquid flow. This device operates by the same principle as the hot-wire anemometer. However, several factors can attribute to inaccuracies in taking measurements. These are: Changes in water temperature; •$* The effect of impurities in water flow; The effect of air bubbles attached to the sensor; and The insufficient insulation between the probe connector and the water (Nezu and Nakagawa, 1993). Hot-film anemometers should be calibrated for each set of measurements, and the calibration coefficient should be determined from a test before and after the measurements are taken. This procedure requires additional equipment that can pull a probe at a constant speed through still water. 2.4.4. Laser Doppler Anemometers (LDA). - Intensive development of this type of device started in the 1980's. LDA is an instrument of first choice for taking turbulence measurements, as it does not require experimental calibration and no probe is introduced into the flow, just a laser beam. The relation between the flow velocity and the Doppler frequency can be determined from theory (theoretical calibration). Fig. 2.17. shows the basic principle of operation for LDA that is commonly used for turbulence measurements in open channel flow. The beam splitter splits the laser beam into two parallel beams of equal intensity. A front lens focuses the two beams onto the measuring volume (point P) in the flume, which is usually a small ellipsoid (0.3 mm in diameter and 2 mm long). 40 Glass Photo-Mult ipl ier Shif ter Figure 2.17. Schematic description of LDA system used in water flume (Nezu and Nakagawa, 1993.) The light emitted from point P by very small scattering tracers is gathered by a lens and focused on the photo multiplier. The photo multiplier sends burst output signals of frequency /&, which are produced by means of frequency shifter of frequency fs. These signals are than analyzed with the aid of an analog to digital converter. The Doppler change frequency /D, is represented by the difference fb-fs. The flow velocity is u=A/(2sin0)*fD, where: X is the wavelength of the laser light, and ^ is the half angle of beam intersection. From the described principle of operation, it is obvious that the presence of small particles in the fluid, which always move at the fluid velocity, is necessary as they will scatter the light. LDA is capable of measuring reverse velocities and turbulence in flows with zero mean velocity. Advanced LDA equipment with five-beam method can measure all three 41 velocity components simultaneously. Several problems with LDA equipment are as follows: The slope of the flume has to be set very accurately, as it is difficult to rotate the laser-beam plane precisely, so that the beam plane is truly tangent to the flume bed. 4- A forward-scatter system should be adopted, because the intensity of the back-scatter light is much lower. However, if the opposite wall of the flume is opaque, than the back-scatter system is the only choice. For most open channel flows, additional seeding for scattering particles is not necessary. Overseeding causes incoherent signals from many particles. Conventional LDA systems are impossible to use for velocity measurements in large basins simulating estuary and lake flows, in channels with large roughness elements that can block the laser beam, in channels with opaque side walls, etc. The back scatter LDA should be used in this cases. 2.4.5. Acoustic Doppler Velocimeters (ADV). - This is a relatively new device developed in the early 1990's. ADV measures the point velocity fluctuations in all three directions, based on the Doppler-shift velocity-measurement principle. It has small dimensions and does not disturb the flow, does not need frequent calibration, can measure velocities in a range from 0 to 2.5 m/s with resolution 0.1 mm/s, the accuracy of the device is better than ±0.25% or ±0.25 cm/s - whichever is greater (Kraus, Lohrmann and Cabrera, 1994), and is therefore very convenient for water flow measurements. The 3D ADV consists of a probe attached to a stem, which is attached to the signal conditioning module. This module is connected to the signal processing module installed inside a 42 computer. The probe itself consists of a transmit transducer and three receivers placed at 120° azimuth angles. The angle between the transmitter, sampling volume (which is 5 cm below the transmitter) and each receiver is 30° (Fig. 2.18.). The probe can be oriented in three different ways, so that its transmit transducer emits an acoustic signal downward, upward or sideways, which will be referred in later text as a downward, upward or sidewise-looking probe. This is very advantageous, as the measurements can be taken as close as 5 mm from the solid boundary or to the free surface. Large roughness elements on the bottom of the channel do not represent any problem in taking measurements, as the device can recognize the distance from the boundary. However, the wavy water surface can cause some inconveniences if one wants to take measurements too close to it. 5.6 cm Figure 2.18. Schematic of the ADV probe: downward-looking probe head orientation (Kraus, Lohrmann and Cabrera, 1994). The principle of operation is as follows: the transmit transducer produces and emits periodic short acoustic pulses, which are picked up by scatterers (micro-bubbles or 43 impurities in the fluid; seeding material) in the sampling volume and echoed back to the receive transducers. It is assumed that the scatterers are traveling with the velocity of the fluid flow. The frequency of the echo is Doppler displaced according to the relative motion of the scatterers.. The total velocity vector can be computed from the known geometry of the beams and three measured velocity projections with the aid of a transformation matrix. This matrix is determined empirically in the process of calibration and will not change as long as the geometry and the physical dimensions of the probe do not change. The measure of the scattering strength and a central parameter of an ADV in operation is a signal-to-noise ratio (SNR), which represents the strength of the echo relative to the electronic noise level. Air bubbles are an excellent natural seeding for achieving a high SNR. c <v c a E i ° 1 N c o 1 I -30 A D V : Solid line L D V : Dash-dotted line 8 10 12 14 16 18 20 Time (s) Figure 2.19. ADV and LDV comparison (Kraus, Lohrmann and Cabrera, 1994). The ADV has been extensively tested in different settings in the laboratory. When tested in a tow channel, the results have shown that the rotation of the probe around its 44 vertical axis minimally changed the measurements (less than 0.5%). The measurements taken with ADV correspond very well with the measurements taken with LDA (Fig. 2.19.). There are not too many disadvantages to ADV known to the author of this text. Those can be summarized as follows: The device is fairly new and more tests have to be done to gain better knowledge about its capability and accuracy; <• The device is expensive, but cheap compared with the LDA. 45 Chapter 3 Theoretical Development Several studies of mountain streams have shown that the velocity profile does not follow Prandtl's logarithmic law. In mountain streams a logarithmic profile does not develop, because of shallow depths of flow and increased drag from the bed material, which usually consists of coarse gravel and larger rocks. This observation is very important, because the widely used assumption that the velocity profile is logarithmic leads to several possible errors in estimating the flow. Namely, the mean velocity and the discharge are usually overestimated, while the Manning's roughness coefficient is underestimated. Accurate river discharge and other reliable hydraulic data are important in different areas, such as: sediment transport estimation, hydrological predictions, geochemical processes, and effects of climate changes. The aforementioned problems call for a better understanding of hydraulic processes in high gradient rivers, and for the development of equations which will describe those processes better than the existing ones. In the present work an attempt has been made to develop an equation for the velocity profile. Several approaches have been tried with different sets of assumptions concerning the mixing length variations. These will be discussed in Sections 3.1., 3.2., and 3.3. However, it will be shown that these variations did not develop a useful equation for velocity distribution. Then, another approach has been tried, in which all Prandtl's assumptions were valid except for shear stress, which 46 was assumed to have a linear distribution. This will be discussed in Section 3.4. and a graphical comparison between all developed equations will be given in Section 3.5. In addition, it has been assumed that the large roughness elements will produce much higher resistance to the flow, than the smooth surface of the glass walls on the channel sides, and consequently the wide channel approximation has been assumed to be valid in the further discussion . 3.1. Mixing Length is Constant The assumptions are as follows: Prandtl's equation for shear stress applies and the shear stress is a linear function of the depth of flow, y, which is measured from the bottom of the channel: r = pl2 — (3.1.) \dyJ and T=T0{^j (3.2.) The large roughness elements on the bottom of the channel produce considerable turbulence and mixing. Therefore, the mixing length, /, is assumed to have an initial value dependent on the size of the rocks, D50, and stays constant over the depth of flow: / = aD50 - const. (3.3.) 47 If now the two equations for shear stress are equated (i.e. equation 3.1. is equated with equation 3.2.), and the substitution for mixing length is made, the equation for velocity gradient becomes: dy = aDjpi Y " ^ After integration, the integration constant can be solved from boundary conditions, i.e. i v = « m a x for y=Y, giving C equal to umax. Then, the velocity distribution, would be the 3/2 power low, as follows: 2 1 \ Tn t \3/2 or 2 W* / „ \3/2 w - « -3oD 5 0V7 if the substitution ^jr01 p = w» is made in equation 3.5. Using the usual steady flow shear stress relationship, r0=yRS (3.7.) or r0=gpYS (3.8.) for a wide channel approximation (R=Y). If the substitution for To is made, the velocity distribution (equation 3.5.) becomes: U = U--3aD, (Y-yf2 (3.9.) 50 48 The assumption about a mixing length having some initial value that depends on the size of the roughness elements seemed reasonable, because of the nature of flow in mountain areas. However, a unique value for constant a could not be found either for flume data or the real data measured in Colorado rivers (Jarrett, 1984). The equation (3.9.) has also been tested, but using Dg4 size instead of using the mean roughness size D50. This was done because it was speculated that the larger roughness elements will govern the turbulence in the flow, rather than the mean size. The measured velocity profile (Jarrett's data, 1984) was plotted and an attempt was made to fit that profile with the theoretical one by varying the value for constant a, because everything else in the equation was determined in the field. But, in order to force a fit to the real data, a was nowhere near a constant and its value ranged from 0.06 to 40.0, in extreme cases, although the most usual range was from 0.1 to 0.5. This wide range for a, which changed from case to case, indicated that this approach was not correct, and a universal solution could not be found. 3.2. Mixing Length Decreases Continuing on from the above analysis, the following modified assumptions were tested: as before, Prandtl's equation for shear stress was assumed, and also the shear stress was taken to be a linear function of the depth of flow, y, (measured from the bottom of the channel), but the mixing length, /, was given an initial value at the bottom, which then decayed towards the surface. The maximum value of the mixing length, equal 49 to KY, is at the bottom, where the turbulence is the biggest, and then decays to zero as y reaches Y. If the two equations for shear stress are equated (equations 3.1. and 3.2.), the relationship is as follows: Y-y\ Jdi?2 ( 2 1 0 ) where l = K(Y-y) (3.11.) After integration and substitution for the total shear stress r0 = ^ jpgYS, the following 1/2 power law equation for the velocity was obtained: This equation gives the opposite curvature to the velocity profile than is observed in natural streams or laboratory flumes. Therefore, this approach was abandoned. 3.3. Mixing Length Increases One more approach was tried in which Prandtl's mixing length theory was used, including his assumption of constant shear stress. The difference from Prandtl's work was that the mixing length / was given an initial value equal to KD&4 at the bottom of the channel and then increased towards the surface. Measuring y from the bottom of the channel, the following relationships apply for / and v. 50 1 = KD, 84 (3.13.) and T = p .dy) (3.14.) These assumptions led to the following equation for the velocity distribution: max ~ 84 y-Y\ l - l n £ (3.15.) where the constant of integration was determined for boundary condition y=Y, where u=umax, and is equal to: c = « m a x - - ^ A ^ ( i + i n r ) KDm \ p (3.16.) When equation 3.15. was applied, negative values for velocity were calculated for values of y/YzO-0.6. It became obvious that this equation did not lead to a realistic velocity profile, and therefore, it was also abandoned. 3.4. The Modified Logarithmic Law 3.4.1. Assumptions. - The basis for a theoretical development makes use of Prandtl's mixing length theory. It has been assumed that PrandtPs definition of the shear stress in turbulent flow is valid (equation 3.1.), i.e.: 51 r = pl2 Kdy) ' where p is mass density, / is Prandtl's mixing length, and du/dy is a velocity gradient. The minor difference with respect to Prandtl's work is that here the distance y is measured from the water surface, which lead to the following expression for /: where K is the von Karman's constant, 7 is the total depth of flow, and / is a linear function of the distance from the boundary. The value of the mixing length on the bottom most important difference from Prandtl's work is that while Prandtl assumed a constant shear stress equal to the shear stress on the bottom of the channel, To, here a linear shear stress distribution has been assumed, which is predicted in standard open channel flow theory. The maximum value of the shear stress is on the bottom and is equal to T0, while the minimum is at the water surface and is equal to zero. If y is the distance measured from the water surface, then the shear stress can be expressed as: 3.4.2. Derivation of the Formula. - Two equations for the shear stress (equations 3.1. and 3.18.) can be equated, and together with the substitution for / (3.17.), the following expression can be written: (3.17.) is equal to zero and on the free surface it has its maximum value equal to KY. The (3.18.) 52 L0 2 yy = p* r \ 2 f y\ ^ Y) rdu^ (3.19.) Recognizing that du/dy is negative if y is measured from the surface, du can be expressed as: du= 1 K\p(X-ylY) dy (3.20.) In order to integrate equation 3.20., the following substitutions have been made: y/Y=sin2t, and consequently l-y/Y^o^t. The velocity u, at some distance y from the surface, is than: u = — K \ p - I n .yl^yTYJ + const. (3.21.) The constant of integration can be solved from known boundary conditions, i.e. «=M m a x : when y=0. The above equation becomes: ( i + J w F n l (3.22.) This equation is not applicable in the zone of the solid boundary, i.e. when _y=7, u tends to minus infinity. The total discharge Q can be determined from the above equation by integration over the cross-sectional area of the rectangular channel: Q = \udA = juBdy = B\ 2 frn "max + - , — K V p In l + JyJY' .fi^yTY) \dy (3.23.) and 53 ^ „ \Y IB T n ^ max ^ ^ 2 3 V 7 {in o v J The In function to be integrated has no analytical solution, because as y—>Y, the function ln\ , becomes infinite (Figure 3.1.). However, a numerical solution for ^l-y/Y, the integral jln| l + Jyir ^l-y/r) dy tends to the value 7, as represented in Figure 3.2. Figure 3.1. Graphical solution for the log function Figure 3.2. Graphical solution for the integral 54 Making substitutions w» = ^ Jr01 p and \ ln| dy-Y in equation 3.24., the discharge, Q, becomes: Q = BY\umax-2 u, 3~K (3.25.) The mean velocity can be calculated if the discharge, Q, is divided by the cross-sectional area (A=BY): 2 ut - Q or - 2 M , (3.26.) (3.27.) Equation 3.22. for the velocity distribution can be rewritten in terms of the mean velocity: - 2 « , 2ut u = u + + 3 K K (3.28.) The equation for the velocity profile expressed in terms of the mean velocity (equation 3.28.) will be evaluated in Chapter 5. However, comparison between the equations developed in Sections 3.1. to 3.4. is given in the Section 3.5. 55 3.5. Comparison Between Developed Equations In the following graph (Fig. 3.3.), measured data on Bear Creek at Morrison (Jarrett, 1984) has been plotted against four velocity profiles developed in the present work (Sections 3.1., 3.2., 3.3 and 3.4). In addition, Prandtl's logarithmic velocity profile has also been plotted, using the equation: 2.3«. y « = " « + — l ° g f (3-2 9-) The measurement was taken on June, 07, 1983 with the following characteristics: total depth of flow 7 was equal to 0.701 m, slope S was equal to 0.022, and mean grain size D50 was equal to 0.08 m, while D84=0.29 m. The maximum velocity for the measured velocity profile occurred at 0.8 of the total depth, measured from the bottom. The measured velocity profile has a backward looking curve in the vicinity of the water surface (Fig. 3.3.). None of the equations developed in this present work can follow such a trend. A two dimensional equation to describe this phenomena would require an increase of a negative shear stress near the surface and such equation does not exist, as far as the author is aware. However, such a velocity profile is a reality recognized by several authors in the past (e.g. Nezu and Nakagawa, 1993). Equation 3.12. was developed with the assumption that the mixing length decreases, while in equation 3.15. the mixing length increases. It is clearly shown in Figure 3.3. that equations 3.72. and 3.15. do not give the required shape for the velocity profile This clearly justifies the decision to abandon these two approaches. 56 Equation 3.9., where the mixing length was kept constant, gave a shape for the velocity profile, which was too straight. The parameter a in this case was equal to 0.75. However, the main reason for abandoning this approach was that a unique value for the parameter a for different measurements could not be found, as discussed in Section 3.1. Prandtl's equation also did not follow the measured profile. Another form of his equation that uses the mean instead of the maximum velocity would lead to a higher maximum velocity and bigger difference in the upper half of the measured and predicted velocity profile, but would give a better fit to the measured data in the lower half of the profile. The modified logarithmic low (equation 3.22.) described the measured profile the best. Nevertheless, it did not follow the measured profile in the surface zone, because neither this equation nor any of the other equations give a reversal of velocity gradient near the surface. Despite the difference in the upper limb of the velocity curve, equations 3.22. and 3.28. can provide a fairly good approximation of the mean velocity and the discharge. 57 58 Chapter 4 Experimental Investigation The experimental investigation was carried out in the Hydraulics Laboratory of the Department of Civil Engineering at the University of British Columbia. Tests were performed to determine the influence of slope and roughness elements on the velocity profile and Manning's roughness coefficient. Turbulence was measured, which provided data for Reynolds' stresses and shear velocity calculations. In the second series of tests, additional roughness elements were placed into the flume, and their impact on the cross-flows, velocity profile and roughness coefficient was investigated. 4.1. The First Experimental Set-up The big flume, shown on Figure 4.1., is 15.15 meters long, 0.51 meters wide and 1.0 meter deep. The system is a recirculating flume supplied by two pumps, which drain water from a reservoir in the basement of the Hydraulics Laboratory. The pumps have a capacity of 0.17 m3/s and 0.113 m3/s (6 cfs and 4 cfs), giving a maximum combined flow of 0.28 m3/s. On the end of the flume, there is an outlet tank, which discharges to the reservoir. At entry to the flume, there is a system of meshes to stabilize the flow, while at outlet, there is a gate to control the flow, but it has not been used in this investigation. The flume has an adjustable slope from 0 to 2.1%. 59 <L> ti-ro ro I CO 00 so I S S 3 1 60 For roughness elements, river gravel was used. The supplier classified it as "River gravel 2-6 inches". For this experimental set-up, the gravel was resorted into a more uniformly sized gravel, having diameter DSa=5.0 cm (size ranging from 3.5 to 7.0 centimeters) (Fig.4.2.). This was done visually. The rocks were rounded, oval shaped, having, in general, three unequal principal dimensions. Figure 4.2. The first experimental set-up with uniform gravel size, Di0=5 cm.. For velocity and turbulence measurements, the Acoustic Doppler Velocimeter (ADV) was used and will be discussed in more detail in the Section 4.3. The A D V was placed toward the end of the flume (2.75 meters from the outlet) and in the middle of the flume, allowing a complete establishment of the flow. A simple holder designed for this 61 experiment, allowed vertical and horizontal positioning for the probe head. Measurements were taken on one vertical for the following fractions of the total depth, measured from the surface: 0.05, 0.10, 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.85, 0.90, and 0.95. The experimental investigation was performed for two different slopes. One was the maximum achievable slope for this system, equal to 2.1%, the other was chosen to be equal to 1.0%. For the slope of 2.1%, the maximum depth of flow that could be achieved before the rocks started to move, was 22 cm. For higher discharge, and therefore bigger depth of flow, the gravel bed was no longer stable and the rocks started to move downstream. Consequently, the threshold of motion for selected gravel size and slope was the limiting factor for the depth of flow. For the slope of 1.0%, several tests were performed, with depths varying from 26 cm to a maximum of 35 cm. It has to be noted here that these depths were measured from the bottom of the flume, not the top of the roughness elements. 4.2. The Second Experimental Set-up In this experimental set-up, additional roughness elements were placed into the flume in a zigzag pattern, with a distance of 1.0 meter between them. The largest rocks from the supplied material (15 cm or 6 inches in diameter) were chosen, and placed in groups of three or four (Fig. 4.3.). With this change, the flow became tumbling flow. The test was run for the maximum slope of 2.1%, and the maximum depth of flow for this set-up was 26 cm, before the rocks started to move. 62 Figure 4.3. The second experimental set-up with additional roughness elements placed in zigzag pattern. The velocity and turbulence measurements were taken close to the outlet of the flume using A D V . Five cross-sectional areas were examined with fifteen velocity profiles in total, three for each cross-section. For each cross-section, the verticals were placed 15 cm from the side walls, and in the middle of the flume. The longitudinal distance between the cross-sections was 25 cm. The velocity measurements were taken at same depths for each vertical as described in Section 4.1. 63 4.3. Measuring Equipment Velocity measurements were taken using the 3D ADV instrument, manufactured by Sontek, Inc., C A USA. There are several reasons for this selection: Measurements of all three velocity components can be taken simultaneously. Velocity fluctuations in all three directions can be measured. The human error in readings is minimized by the computerization of the process. <$• The equipment set-up is simple. Results compare very well with LDA measurements (Kraus, Lohrmann and Cabrera, 1994). The stem of the probe is of small dimensions and does not disturb the flow. The equipment does not need frequent calibration. All these advantages made the decision in selecting the instrument easy. Pitot tubes and current meters cannot measure turbulence. Hot-film anemometers can measure turbulence, but have to be calibrated for each set of measurements, which requires additional equipment. LDA can measure turbulence with the highest accuracy, however, it needs a complicated set-up, and for simultaneous measurement of all three velocity components, the five-beam method has to be used, which has not yet been done for open channel flows (Nezu and Nakagawa, 1993). The ADV has three different probe heads, which allow taking measurements downward, upward, or sideways looking. In this research the downward and upward looking probe heads were used (Fig. 4.4.), allowing measurements very close to the bed or 64 to the free surface, thus covering the total depth of flow. Details of the ADV probe, its orientation and the principle of operation, are given in Section 2.4.5. The probe head was connected to the signal processing module installed in a 33MHz - 386 Personal Computer. The computer is used to operate the ADV, and to store data, using the ADV Software Version 2.3. This program has three modes of operation: the Set-up Mode, the Boundary Adjustment Mode, and the Data Acquisition Mode. In the Set-up Mode, a number of parameters have to be specified for the proper operation of the ADV. Some of the important parameters are: 65 •> Water temperature and salinity, which are required to calculate the speed of sound (temperature was measured before each experiment and salinity was equal to 0.0 parts per thousand); Sampling rate, which determines the rate at which data are recorded (can be set between 0.1 Hz and 25 Hz; 25 Hz was used, giving 25 velocity measurements per second for each of the three directions); Velocity range, which is necessary as the Doppler noise (statistical uncertainty of the velocity measurements) is proportional to the maximum velocity range that ADV can measure (can be set at 3, 10, 30, 100 or 250 cm/s; 250 cm/s was used as a minimum value that covers the range of velocity fluctuations expected in these experiments); External synchronization, as ADV can be synchronized with an external data acquisition system or remote trigger (can be set as: disabled, start on sync and sample on sync; disabled was used); In Boundary Adjustment Mode, the region in front of the probe head is continuously scanned. The presence and the distance from the tip of the probe to a solid or a surface boundary are detected. If the boundary is within 25 cm, the distance will be displayed, otherwise, the program would display a "Not detected" message. The distance from the sampling volume to the detected boundary and the effective velocity range are also displayed. From this mode, the Data Acquisition Mode can be entered. 66 In Data Acquisition Mode, the data collection will start when ENTER key is pressed, allowing a precise timing for the first data sample. Several groups of information are displayed on the screen (Fig.4.5.): -v* Current status of data acquisition; Filtered values of the last few data for: three velocity components, signal-to-noise ratio, signal correlation coefficient, and standard deviations for all three velocities; -$> Graphical representation of the real-time trace for three velocity components; and Menu keys and their functions. Figure 4.5. The experimental set-up, S=0.01, Y=26 cm, measurement taken on Aug. 17. 1995. in the big flume of the Hydraulics Laboratory at UBC Pressing ALT F10 the data acquisition is stopped. All the collected data is recorded in compressed binary files. Several data conversion programs are given by the manufacturer, which extract data from binary files and generate them in ASCII format 67 suitable for further analysis with software packages such as Microsoft Excel ®, Lotus 1-2-3, Quatro, etc. These conversion programs are: GETCTL (for configuration and set-up information), GETVEL (for velocity data), GETAMP (for relative signal strength data), GETSNR (for signal-to-noise ratio data), and GETCOR (for signal correlation data). The time length for data acquisition at one point on the vertical was chosen to be 180 seconds. When the velocity was measured, two trends were observed: instantaneous velocity fluctuations, and a longer cycle wavelike velocity changes. All these changes were accounted for in the average values, if the timing was long enough. A progressive velocity averaging is a parameter that shows the required length of data collection, and it was checked for several cases. u [cm/s] 195 190 185 20 40 (y is measured from surface) y/Y=0.40 60 80 Time [s] -+-100 120 140 160 Umean=171.27 CITl /S Figure 4.6. Progressive velocity averaging for case of uniform roughness elements, when S=0.021, 7=78.5 cm, D=3.5 cm. 68 Figure 4.6. shows one of the extreme cases, when the time to achieve a steady average value was long (more than 120 s), and it was therefore decided that the time for taking a point measurements should not be less than 3 minutes. Even though that data was collected and averaged over a period of 180 s, only the first 160 s are represented on the Figure 4.6., because the plotting software, Microsoft Excel® can only present 4000 data points. Note that y is measured from the water surface. 69 Chapter 5 Experimental Results and Comparison With Theory First Experimental Set-up: Uniform Roughness Elements In this section the experimental results for the first experimental set-up described in Chapter 4, Section 4.1., will be discussed. The following measurements and calculations will be addressed: Flow rate per unit width (17), Manning's n, Froude number (Fr), and Reynolds number (Re) 4- Instantaneous velocities (u, v, and w); 4- Standard deviations for three velocity components (Su, Sv, and Sw). 4- Streamwise (u), vertical (v) and transverse (w) velocity measurements; and 4- Reynolds stresses (-u'v'u'w' and -v'w'). The experimental set-up has been discussed in the previous Chapter (Section 4.1.). The roughness elements were uniformly distributed, the mean diameter D50 was equal to 5.0 cm (ranging from 3.5 to 7.0 cm). However, for the following text and graphs, the symbol D will be used to represent the height of the rock that lay underneath the point of measurements. 70 5.1. Flow Rate, Manning's n, Froude Number and Reynolds Number Before discussing the turbulent nature of the flow, the following parameters which describe the overall nature of the flow will be presented and discussed: the flow rate, Manning's roughness coefficient, Froude number and Reynolds number. To calculate these parameters, the measured mean velocity in a streamwise direction will be used. This velocity was calculated by averaging over the depth where velocity measurements were taken, and this velocity is taken as a representative mean velocity for a vertical and is used in further calculations. The flow rate per unit width (q), Manning's roughness coefficient (n), Froude number (Fr) and Reynolds number (Re) were calculated using following formulae: q = uY, (5.1.) n = lY2nsV2> ( 5 2 ) u Fr = -¥=, and (5.3.) uY Re = —. (5.4.) v In the above equations the wide river approximation was used, as the large roughness elements on the bottom of the channel made much more resistance to the flow than the smooth glass side walls. Table 5.1. display the results of these calculations. 71 Conditions u [cm/s] q [cm2/s] n Fr Re* 10s S=0.021; Y=18.5 cm; D=3.5 cm 155.07 2868.79 0.0303 1.15 2.39 S=0.021; Y=20.5 cm; D=3.5 cm 143.98 2951.59 0.0350 1.02 2.46 S=0.01; Y=22.5 cm; D=3.5 cm 141.78 3190.05 0.0261 0.95 2.66 S=0.01; Y=32.0 cm; D=3.5 cm 176.34 5642.88 0.0265 0.99 4.70 Table 5.1. Calculated flow parameters for the first experimental set-up From Table 5.1., it can be seen that the mean velocity does not increase with slope, in fact, there is even a slight decrease. This near constant velocity observation indicates that energy dissipation is increasing considerably as the channel slope increases, probably due to shock losses, as the flow accelerates and decelerates over the rough elements. Manning's n increases if the slope increases. For the same slope Manning's n increases slightly with an increase in flow rate. Froude number indicates that the state of flow is very close to critical in all four cases, with values that are just a little bit lower or higher than unity. According to Jarrett (1989), supercritical flow in high gradient natural channels generally does not occur, except for short distances, because high energy losses are caused by the extreme roughness of the channel bed. The present measurements show that even doubling the channel slope from 0.01 to 0.021 only causes a slight increase in Froude number. As would be expected, Reynolds number indicates that the flow is in the zone of fully rough turbulent flow. 72 5.2. Instantaneous Velocities At each point on the vertical, for all four sets of measurements, twenty-five instantaneous velocities were recorded in a second, over a period of 180 seconds. Figures 5.1., 5.2. and 5.3. are representing instantaneous velocities (w, v, and w) for twelve different points on a vertical, measured for following experimental parameters: S=0.021, Y=20.5 cm, and D=3.5 cm. However, only the first 160 s of recording are represented on the graphs, because of the limitations of the used software package (Microsoft Excel ® can plot 4000 data points). Similar figures for other three experimental set-ups will not be presented here, as velocity fluctuations behave in similar manner for all of them. Thus, Figures 5.1., 5.2. and 5.3. can be taken as representative for all four sets of measurements. On each of the graphs, data about depth (y/Y) where the measurement was taken, and the mean point velocity (U, V, and W), which were calculated from instantaneous velocities at that depth, can be found. Velocity fluctuations (u\ v\ w') are deviations from the mean velocity at that point, and can be calculated using the following equations: u'=u-U, v'=v-V, w'=w-W, (5.11.) which means that they can have positive and negative values. The following inequalities can be written for the velocity fluctuations: u'>w'>v'. (5.12.) 73 This is true for most of the cases, but by parallel examination of Figures 5.1. and 5.3., it can be observed that velocity fluctuations in transverse direction, w', do not differ significantly from velocity fluctuations in streamwise direction, u', for the points close to the bottom, i.e. for y/Y=0.9-1.0. In other words, instantaneous velocities in transverse direction, w, are of the same order of magnitude as the instantaneous velocities in streamwise direction, u, in this region. This increase in instantaneous velocities in transverse direction can be explained by existence of secondary currents and flow separation around large roughness elements. When examining Figures 5.1. through 5.3., the following similarities are observed for all three velocity components: -v> Very close to the bed (y/Y=0.9-J.O), velocity fluctuations are suppressed by the solid boundary; The largest velocity fluctuations are experienced at the depth y/Y=0.9, y being measured from the surface. Near the bed, velocity fluctuations and turbulence are the strongest, due to the direct effect of large roughness elements; 0- As the measurements are taken farther from the bottom, the velocity fluctuations are decreasing. The free surface has effect on suppressing the velocity fluctuations; There is a slight increase in velocity fluctuations u' and w' near the free surface, as the vertical turbulence intensity is more affected and suppressed there, than the 74 other two. This means that there must be flow exchange between the three velocity components in order to satisfy continuity. Each of the Figures (5.1. through 5.3.) contains four pages, where on each page instantaneous velocities for three different points on the vertical are illustrated. The largest deviations from the mean point velocity in u direction occurred at y/Y=0.9 and the peak values are approximately equal to ±350 cm/s, however, the turbulence intensity (standard deviation) is equal to 102 cm/s (see Fig. 5.5.). For vertical velocities (v), the largest deviations from the mean occurred at the same depth and the peak values are approximately equal to ±75 cm/s, while the turbulence intensity is equal to 20 cm/s (Fig. 5.5.). In the transverse direction (w) the peak values for instantaneous velocities are approximately equal to ±300 cm/s, and the turbulence intensity is equal to 104 cm/s (Fig. 5.5.). At this depth the point mean velocities were: U=22. 7 cm/s, V=l.l cm/s and W=3.7 cm/s. 75 76 77 78 79 80 81 1 ? H—I—h H—I—h o o o o o o o o o o o o o o o o o t CO N CS O f [S/IU3] A 1 o II .1 I I I o o o o o o o o o o o o o — — — — *t m cs o o o o [S/UK>] A 1 f H—I—h H—I—h i o o o o o o o o o o o o o o o o o « n M - « M m f [g/UI3] A 83 84 85 86 87 5.3. Standard Deviations Standard deviation, (known in literature as turbulence intensity, or root mean square) is a statistical parameter, which illustrates how widely the examined values are dispersed from the mean value. Standard deviations were calculated for each point on the vertical, and their values represent deviations of measured instantaneous velocities from the mean point velocity. The "biased" or "n" method was applied when calculating standard deviations: Sa n 2 -1=1 ™ 2 n n=4500, (5.13.) where: Sa n a standard deviation for the examined velocity (Su, Sv, and Sw); the number of measurements in a point; the examined velocity (u, v, or w). Figures 5.4. through 5.7. represent root mean square values for four sets of measurements, where on Figures 5.6. and 5.7. two verticals were recorded. The experimental set-ups are explained in detail later in this Chapter (Sections 5.4.1.1. through 5.4.1.4.). Standard deviations for all three velocity components are presented on the same graph, so that the values can be compared. 88 89 Figure 5.6. Standard deviations for S=0.01, Y=22.5 cm, D=3.5.cm Figure 5.7. Standard deviations for S=0.01, Y=32 cm, D=3.5 cm 90 For all four graphs (Figures 5.4. through 5.7.), the following observations are made: •> Su>Sw>Sv throughout the whole region of the flow, which means that turbulent velocity fluctuations and deviations from the mean velocity are the highest in the streamwise direction, and the lowest in the vertical direction; > For all three velocity components, the standard deviations are the highest at about y/Y=0.9, y being measured from the free surface. This illustrates that instantaneous velocities deviate more from the mean, and that the velocity fluctuations are bigger here, than in other regions of the flow; -v- Standard deviations for velocities in streamwise and transverse directions (u and w) have the same pattern and are close in magnitude, while standard deviations for velocities in the vertical direction (v) are smaller and almost linear; Standard deviations decrease towards the surface, but a sudden increase occurs at y/Y=0.1, which is caused by increased turbulence induced by surface waves. Figure 5.6. illustrates standard deviations for two measured verticals, that are only 6 cm apart. The corresponding standard deviations are practically overlapping each other, which means that there is a little change in the flow between the two verticals, and this will be also confirmed with other examined flow parameters discussed later in this Chapter (Section 5.4.). On Figure 5.7., the two verticals are farther apart (60 cm), and the differences in corresponding standard deviations are somewhat more evident, which indicates certain changes in flow pattern itself (also discussed later in Section 5.4.). 91 5.4. Velocity Profiles For all measured velocity profiles, each point on the graph represents the average value of the velocity measurements recorded over a period of three minutes at that depth (4500 data points). 5.4.1. Streamwise Velocity Measurements. - Streamwise velocity profiles for four different sets of measurements will be discussed. These are the u velocities, the velocities parallel to the main direction of the flow. The measured velocity profiles will be compared with two theoretical velocity profiles, calculated using the equation 3.28. of this thesis (Chapter 3.4.), and Prandtl's equation: For the theoretical velocity profiles, the mean velocity, u, is calculated from the measured velocities, and the shear velocity, u*, is calculated from the known values of Y and S for each set of measurements, will be used. There is a slight disagreement between the input mean velocity and the mean velocity calculated afterwards from the theoretical curves, because of the numerical methods used in these calculations. Also, in order to get smoother theoretical curves without visible "corners" at the points where the velocities were calculated, a finer step for y/Y value was used in calculations than in measurements, which also contributed to the discrepancy between values of the two mean velocities. However, these errors are of the order of 1% or less of the measured mean velocity. The (5.5.) 92 mean velocities for all profiles (measured and theoretical) were calculated using the method of weighted averaging. For one of the following sets of measurements, different values of the von Karman's constant were tested, but generally, a value of 0.41 was used for all theoretical curves. The measured velocity profiles will not be compared with other equations derived in this thesis (Chapter 3: Sections 3.1., 3.2. and 3.3.), because the velocity profiles calculated using those equations do not follow the measured velocity profile in the natural stream or in the artificial channel (discussed in Section 3.5. of this thesis). 5.4.1.1. Streamwise Velocity Measurements for S=0.021, Y=18.5 cm, D=3.5 cm. -This set of measurements was taken for the highest possible slope to which this flume can be adjusted. One pump was used, and the average depth of flow above the rocks was equal to 18.5 cm. The flow was kept below the threshold of motion. Measurements were taken with ADV and Ott-meter, so that the two devices could be compared. Figure 5.8. shows the measured and the theoretical velocity profiles. It can be seen on the graph that measurements with the Ott-meter could not be taken so close to the bottom or to the free surface as with ADV, because the Ott-meter propeller has a diameter of 3 cm. In general, the velocity profile measured with both devices follows the same pattern. However, it can be observed from the graph, that the Ott-meter measured slightly lower values in the lower half of the flow (i.e. close to the bottom) and slightly higher values in the upper half of the flow than the ADV. There are several possible explanations for such result. Human error in vertical positioning of the propeller 93 94 meter would contribute to the difference in readings for two devices. Also, positioning of the propeller meter in the streamwise direction is very important, as it is sensitive to the vertical and transverse velocity components, which are not detected separately from the velocities in the main flow direction, but rather partially superimposed with it. The Ott-meter responds to approximately cosine of cross-flows. The fine structure of turbulence cannot be measured with the Ott-meter. Therefore, the Ott-meter should be used with caution and the results should be taken with reserve. On the other hand, tests reported by Kraus, Lohrmann and Cabrera (1994.), which were carried out with both, LDA and ADV, showed that the results between the two are in close agreement (as discussed in Chapter 2, Section 2.4.5. of this thesis). Hence, it was assumed that the ADV gives reliable results. The mean velocities for the profile measured with ADV, and for the theoretical profiles applying equations 3.28. of this thesis and Prandtl's equation (5.5.) are 155.1 cm/s, 155.3 cm/s and 155.7 cm/s, respectively, while, the mean velocity for the profile measured with Ott-meter is 166.5 cm/s. The mean velocity occurred at y/Y equal to 0.63 for the profile measured with ADV, at 0.67 for the theoretical profile (equation 3.28.) and at 0.63 for the Prandtl's profile, y being measured from the surface. For the velocity profile measured with propeller meter, the mean velocity occurred closer to the surface, at y/Y being 0.48, which agrees with Jarrett's observations that the mean velocity occurs at 0.5 of the total depth (Jarrett, 1989.). This is of no surprise, if having in mind that Jarrett's measurements were carried out with cup type meters. 5.4.1.2. Streamwise Velocity Measurements for S=0.021, Y=20.5 cm, D=3.5 cm. -This set of measurements was made for the largest possible slope for this experimental 95 flume. Two pumps were used, that could provide 0.28 m3/s (10 cfs) of flow in total. However, the maximum depth of flow of 20.5 cm above the rocks was the highest that could be achieved before a considerable movement of rocks occurred. Some movement of rocks on the bottom of the flume was observed during the experiment, but the rock underneath the probe head and the rocks around it were stable. The surface was very wavy and reliable measurement could not be taken closer to the surface than 0.1 of the total depth. Figure 5.9. illustrates the profile measured with ADV vs. theoretical profiles. The measured and theoretical values show a reasonable agreement except in mid-depth region, where a local "S" shape was measured with ADV, which could not be followed with the theoretical curve. At depth where this "S' shape occurred, the downward looking probe head was changed to the upward looking probe head. Even though, a similar procedure was carried out for all other experiments, a similar problem was not experienced. This "S" shape in the measured profile consists of higher velocities at y/Y of 0.5 to 0.6 and slightly lower velocities at y/Y of 0.4 to 0.5. Two theoretical curves using equation 3.28. have been plotted, for one the Von Karman constant K was equal to 0.41 (same as in all other calculations), for the other K was equal to 0.26. Visually, the curve calculated with K=0.26, followed the measured profile much better. However, that curve would verge upon y/Y=l for high negative velocities. The theoretical profile using Prandtl's equation (5.5.) was also plotted on the graph and it has quite a good fit, even though the correct shear stress distribution was ignored when developing the equation. 96 97 The mean velocity for the profile measured with ADV was equal to 144.0 cm/s and occurred at 0.71 of the total depth measured from the surface. The mean velocity for the theoretical curve calculated with K=0.41 was equal to 145.2 cm/s and occurred at 0.64 of the total depth, while the curve calculated with K=0.26 had the mean velocity equal to 152.8 cm/s, which occurred at 0.64. The mean velocity calculated from Prandtl's profile was equal to 147.4 cm/s and it occurred at 0.61 of the total depth. When examining Figure 5.9., it can be seen that in the upper half of the flow (surface region), the velocity gradient du/dy has lower values for equation 3.28., than for Prandtl's equation. For this set of measurements, the velocity profile calculated using equation 3.28. differs from the measured profile in the upper region of the flow more than Prandtl's velocity profile. It can be said in this case, that Prandtl's profile has a better fit to the measured data, because it is less curved, indicating that u should increase more at the surface. 5.4.1.3. Streamwise Velocity Measurements for S=0.01, Y=22.5 cm, D=3.5 cm. - For this set of measurements, the slope was reduced to 0.01. One pump that can provide a flow of 0.17 m3/s (6 cfs) was used and the maximum depth that could be achieved with this set-up was 22.5 cm above the rocks. As a much higher rock (6 cm high) lay downstream of the point where the measurements were taken, another set of measurements was taken above that rock to evaluate its effects on the original set of measurements. The first profile was measured 2.75 m from the flume outlet and above a rock that was 3.5 cm high. The second profile was measured 6 cm downstream from this 98 99 point, or 2.69 m from the flume outlet and above the rock that was 6 cm high. During the experiment, movement of rocks was not observed. Figure 5.10. illustrates the two measured velocity profiles, and the two theoretical profiles (equation 3.28. and 3.29.), that were calculated using the mean velocity from the first measured profile. Both measured velocity profiles follow the same pattern, having a backward looking curve in the surface region (i.e. negative velocity gradient du/dy), and have very close agreement at the same depths. The mean velocity for the first measured profile is equal to 141.8 cm/s and it occurs at 0.66 of the total depth. The mean velocity for the second measured profile is somewhat higher and is equal to 150.4 cm/s, while occurs at 0.60 of the total depth. The explanation for this difference would be that lower velocities that were measured on the bottom of the first profile did not exist in the second profile, because of the presence of the large rock. Other than this observation, a significant influence of the bigger rock on the first profile cannot be reported. The two theoretical velocity profiles closely follow both measured profiles. Yet, comparing velocity profiles calculated using equation 3.28. and Prandtl's equation, it can be observed that the equation 3.28. gives lower values for the velocity gradient du/dy than Prandtl's equation for the upper limb of the curve (i.e. close to the surface), and therefore, agrees better with the measured velocity profile if a backward looking curve is experienced. The mean velocities calculated from the two theoretical curves are matched with the mean velocity calculated from the first measured profile, which was used as an input to the equations. The mean velocity for the profile calculated using equation 3.28. occurs at 0.66 of the total depth and is equal to 142.6 cm/s, while the mean velocity calculated using Prandtl's equation occurs at 0.62 of the total depth and is equal to 142.7 cm/s. 100 5.4.1.4. Streamwise Velocity Measurements for S=0.01, Y=32 cm, D=3.5 cm. - For this set of measurements the maximum flow that two pumps can provide, was used. The slope was set to be equal to 0.01, and the maximum depth achieved at the threshold of movement was equal to 32 cm. Two profiles were recorded with ADV and they are illustrated together with the two theoretical curves (equations 3.28. and 5.5.) on the Figure 5.11. The first profile was measured 2.75 m from the flume outlet and above the 3.5 cm high rock. The second profile was measured 60 cm downstream from this point, or 2.15 m from the flume outlet and above the 4.0 cm high rock. Both profiles follow the same pattern and have a backward looking curve in the upper limb of the profile, which could not be described with the present theoretical curves. The point velocity values of the second profile are approximately 10% lower than the values for the first profile. This means that, in order to satisfy the law of continuity, the streamwise velocities across the flume have to be distributed differently for the two cross-sections. This can be considered as an illustration of the three dimensional nature of the rough turbulent flow in which such deviations are to be expected. The mean velocity for the first profile occurred at 0.73 of the total depth measured from the surface, and is equal to 176.3 cm/s. The downstream profile has a mean velocity equal to 165.6 cm/s at 0.67 of the total depth. The theoretical curves were calculated using the mean velocity and the shear velocity of the first measured profile. Both theoretical curves lay in between the two measured profiles in the lower region of the flow, and differ from them in the upper region (from the surface to 0.4 of the total depth measured from the surface). In the surface region, the velocity profile plotted using equation 3.28. fits somewhat better to the measured profiles than the 101 velocity profile plotted using Prandtl's equation, as equation 3.28. gives lower values for the velocity gradient, du/dy. The mean velocity calculated from the profile that used equation 3.28. is equal tol77.0 cm/s and occurs at 0.66 of the total depth. The mean velocity from Prandtl's profile is equal to 177.7 cm/s and occurs at 0.62 of the total depth. The general conclusion for measurements described so far, would be that the mean velocity occurs between 0.60 and 0.73 of the total depth measured from the surface. The mean velocity for the theoretical curve (equation 3.28.) occurs approximately at 0.66, while for Prandtl's curve at approximately 0.62 of the total depth, measured form the surface. The theoretical velocity profile plotted using equation 3.28. gives better fit to the measured profile if it has negative du/dy in the surface region , than Prandtl's velocity profile. 5.4.2. Vertical Velocity Measurements. - These are the velocities and velocity fluctuations measured in the vertical direction, perpendicular to the flow, denoted as v velocities. The magnitude of these velocities is approximately equal to 1/10 of the magnitude of the streamwise velocities. They can be positive or negative. On all of the following four figures, which represent four sets of measurements (discussed in Sections 5.4.1.1. through 5.4.1.4.), the vertical velocities are negative in the lower half of the flow, close to the bottom, and positive in the upper half of the flow. It appears that the flow must be supplied from one of the two other components in the middle of the flume, and than taken away at the boundaries, indicating the presence of secondary flows. Vertical velocities are suppressed at the boundaries, which is indicated 103 v[cm/s] -15 -10 -5 0 5 10 15 1 1 O r © - 1 1 0.1 0 • 0.2 - y/Y 0 • 0.3 - B 0.4 0.5 0 Eg 0 6 E 0.7 • a 0.8 0 ffl.9 Q 1 : 0 A D V Figure 5.12. Vertical velocity measurements: S=0.021, Y=18.5 cm, D=3.5 cm v[cm/s] -15 -10 -5 0 5 10 15 ; 1 1 M- 1 1 0.1 -0.2 - . y / Y B 0 • H 0.3 -0.4 -0.5 ' • • 0.6 -m 0.7 E9 0.8 -i_n_ a A D V Figure 5.13. Vertical velocity measurements: S=0.021, Y=20.5 cm, D=3.5 cm 104 -15 -10 —\— v[cm/s] -5 0 H 0^ 0-10 —r— 15 y/Y A E3 A 0.1 0.2 0.3 -|A 0.4*-o . * E 0.6 0.7 0.8 4^0-A R A S ADV A ADV: 6 cm downstream Figure 5.14. Vertical velocity measurements: S=0.01, Y=22.5 cm, D=3.5 cm -15 -10 H — v[cm/s] -5 0 5 -I M-10 —I— 15 A A 0.1 0 tf.2 8 0.3 4 B 0.4 E 0.5 0.6 0.7 0.8 0.9, k0-y/Y E ADV A ADV: 60 cm downstream Figure 5.15. Vertical velocity measurements: S=0.01, Y=32 cm, D=3.5 cm 105 with lower values in the region of the solid boundary and surface. However, surface waves increase the intensity of velocity fluctuations near the free surface, but wave induced velocities are not necessarily turbulence velocities. This phenomenon is the most evident on Figure 5.13., in which case the surface was very wavy. Yet, if reliable measurement could have been carried out closer to the surface, than the vertical turbulent fluctuations should be decreasing. For the first two sets of measurements (Figures 5.12. and 5.13.), the slope was adjusted to 0.021 and the measurements were taken 2.75 m from the flume outlet. For each of the second two sets of measurements (Figures 5.14. and 5.15.), vertical velocities on two verticals were recorded. The two vertical velocity profiles follow the same pattern in both cases. In the first case graphed on Figure 5.14, the first profile was recorded 2.75 m from the flume outlet, and the second profile 6 cm downstream from that point. On Figure 5.15, the first profile was measured 2.75 m from the flume outlet, and the second was 60 cm downstream from that point. 5.4.3. Transverse Velocity Measurements. - The velocities measured in transverse direction (w velocities across the flume) have lower intensities than the velocities in the streamwise direction, being about 1/10 - 1/5 of the streamwise velocities, and as much as about twice the vertical velocities. Positive and negative transverse velocities can be observed on the graphical representation. On the following four graphs, four sets of measurements (discussed in Sections 5.4.1.1 through 5.4.1.4.) will be presented. Common for all of them is that transverse velocities are positive in the lower half of the flow, i.e. close to the bottom, while they are negative in the upper half of the flow, i.e. close to the surface, indicating a strong secondary spiral flow. For all four cases 106 examined, the channel geometry did not change, i.e. the channel width and the configuration of the rocks on the bottom of the channel were the same. Therefore, the secondary flow was also always in the same direction. Figures 5.16. and 5.17. illustrate data recorded 2.75 m upstream from the flume outlet, for a steeper bed slope of 0.021. For both cases, transverse velocities have higher values indicating stronger secondary currents, than for the next two cases when the slope was adjusted to 0.01 (Figures 5.18. and 5.19.). For each of the later two cases, two profiles were recorded for the same experimental set-up. On Figure 5.18., the first profile was measured 2.75 m from the flume outlet and above the rock that was 3.5 cm high, while the second profile was measured 6 cm downstream from that point and above the rock that was 6 cm high. On the Figure 5.19., the first profile was recorded 2.75 m from the flume outlet and above a 3.5 cm high rock, while the second was recorded 60 cm downstream from that point and above a 4 cm high rock. It can be seen that for both cases, the profiles follow the same pattern and have similar values for transverse velocities. 107 w [cm/s] -50 -40 -30 -20 -10 0 10 20 30 40 50 1 1 1 1 O T O H 1 1 1- 1 H "0.1 -B • 0.2 -y/Y a 0.3 -| 0.4 -0.5 -B 0.7 • a 0.8 0 0.9 B B H0- B 0 ADV Figure 5.16. Transverse velocity measurements: S=0.021, 7=78.5 cm, D=3.5 cm w [cm/s] -50 -40 -30 -20 -10 0 10 20 30 40 50 1 1 1 1 0^0- 1 1 1 1 0.1 -y/Y 0.3 E i 0.4 - B 0.5 - H B 0.6 - 0 0.7 - E3 0.8 - B 0.9 - B 0 i n _ 0 S ADV Figure 5.17. Transverse velocity measurements: S=0.021, Y=20.5 cm, £>=3.5 CTM 108 w [cm/s] 50 -40 -30 -20 -10 0 10 20 30 40 50 1 1 1 1 — A *1 -1 1 1 1 A 0 A « 2 -y/Y A 0.3 - -E3 44 - B A).5°-A A 0.6 • A 0.7 - AQ A 0.8 • A E A Q 0.9 -^6 Q ADV A ADV: 6 cm downstream Figure 5.18. Transverse velocity measurements: S=0.01, Y=22.5 cm, D=3.5 cm Figure 5.19. Transverse velocity measurements: S=0.01, Y=32 cm, D=3.5 cm 109 5.5. Reynolds Stresses Three different Reynolds stresses will be discussed in the following Sections: the primary Reynolds stress -pu'v', and the two transverse Reynolds stresses -pu'w' and -pv'w'. In further writing, Reynolds stresses will be addressed as -u'v', -u'w', and -v'w', while p will be omitted. Figure 5.20. illustrates the convention for Reynolds stresses, which act in pairs and for which the following equalities can be written: u'v' = v'u', u'w' = w'u', v'w' = w'v'. (5.6.) V w (a) (b) Figure 5.20. Reynolds stresses: (a) 2D flow; (b) 3D flow. For all measured Reynolds stress profiles, each point on the graph represents the average value of the measurements recorded over a period of three minutes at that depth (4500 data points). The following formulae were used to calculate these values: 110 u'v' = - tf)(v - V)], n=4500 (5.7.) 1 ^ -u'w' = — Y,[(u-U)(w-W)], n=4500 (5.8.) 1 " - v' w' = - - Y f (v - F)0 - W)}, n=4500 (5.9.) and: U = u±u', V = v±v', W = w±w', (5. JO.) where: n the number of measurements in the point; u, v, w instantaneous velocities in streamwise, vertical and transverse direction; U, V, W average values calculated from instantaneous velocities in the point; u', vw' velocity fluctuations in streamwise, vertical and transverse direction; and -u'v' ,-u'w' ,-v'w' average Reynolds stresses in the point. The total bed shear stress was calculated using the conventional formula (equation 3.8. of this thesis) for open channel flow, using the wide river approximation: i l l 5.5.1. The Primary Reynolds Stress -u'v'. - This is a shear stress that acts in the u-w plane, parallel to the u velocity and perpendicular to the v velocity. Theoretically, for uniform steady flow, Reynolds stress linear distribution normal to the bed, has a maximum value at the bottom of the channel (equal to the total bed shear stress, T(/p), and minimum value at the free surface (equal to zero). As Reynolds stress decreases towards the surface, the velocity gradient, du/dy, also decreases, but both have positive values. However, if du/dy becomes negative, as in a case when the streamwise velocity profile has a backward looking curve in the surface region (Figures 5.10. and 5.11.), than the Reynolds stress -u'v', will have negative values in the same region (Figures 5.23. and 5.24.). Secondary currents and transverse Reynolds stress, -u'w', contribute to the difference between the primary Reynolds stress, -u'v', and the total bed shear stress, To (Nezu and Nakagawa, 1993). Another possible reason that the theoretical shear stress, r, is higher than the measured Reynolds stress - u'v1 is that the flow is not uniform. If the flow is not uniform, than the convective acceleration will cause additional forces of dp/dx, which in turn means that the sediment can be exposed to more shear than can be measured in the turbulent structure of the flow (i.e. -u'v'). Figures 5.21. through 5.24. are graphical representations of Reynolds stresses-u'v', for four sets of measurements previously described. Reynolds stress deviates from a linear distribution, which is due to secondary currents. The values for -u'v' are lower than the values for the corresponding shear stress, r, which is caused by secondary currents, and because the flow is not a simple 2D uniform steady flow. 112 -uV [cm2/s2] -150 -100 -50 0 50 100 150 200 250 300 350 400 450 1 1 9 -0.1 s. 1 1 1 1 1 1 1 1 0.2 • 0.3 -y/Y J 0.4 -0.5 • 0.6 -0.7 -0.8 -E ^ N . -co/p=381.12 cm2Is2 0.9 - E3 1 i_l a ADV Theoretical Figure 5.21. Reynolds Stress -u'v' for S=0.021, Y=18.5 cm, D=3.5 cm -uV [cm2/s2] -150 -100 -50 0 50 100 150 200 250 300 350 400 450 1 1—OtO-0.1 -s — 1 1 1 1 1 1 1 1 0.2 - ^ S ^ B gg 0.3 0.4 • 0.5 • 0.6 0.7 - a \s^o/pF422.32cm2/s2 0.8 -0.9 - El B hfy— Q ADV Theoretical Figure 5.22. Reynolds stress - u' v' for S=0.021, Y=20.5 cm, D=3.5 cm 113 -uV [cm2/s2] -150 -100 -50 0 50 100 150 200 250 300 350 400 450 —1 1 0.0 i — 1 1 1 —1 1 1 1 1 0.2 : C 0.3 - XI \ y/Y * J \ 0.4 -0.5 x 13. \ X \ 0.6 - EL, \ X ^ 0.7 • ex X c 0.8 a . \ j . to/p=220.73cm/s2 H0.9 - B \ 1-£- tea \ 0 A D V X A D V : 6 cm downstream Theoretical Figure 5.23. Reynolds stress -u'v' forS=0.01, Y=22.5 cm, D=3.5 cm •uV [cm2/s2] -150 - 450 0 A D V x A D V : 60 cm downstream Theoretical Figure 5.24. Reynolds stress -u'v' for S=0.01, Y=32 cm, D=3.5 cm 114 5.5.2. The Transverse Reynolds Stress - u'w'. - This is the shear stress that acts in the u-v plane, parallel to the u velocity and perpendicular to the w velocity. The transverse gradient of the transverse Reynolds stress -u'w' depends on the direction of vertical velocities, v (Nezu and Nakagawa, 1993): duwl&<0 for upflow (v>0), and duw/ <2r > 0 for downflow (v<0). This feature can be verified by parallel examination of graphs that represent vertical velocities (Figures 5.12. through 5.15.) and graphs that represent the transverse Reynolds stress - u'w' (Figures 5.25. through 5.28.). In all four cases, there is a zone of downflow (v<0) in the lower half of the flow (i.e. close to the bottom of the channel), which results in increase of the transverse gradient of the transverse Reynolds stress -u'w'. On the other hand, there is a zone of upflow (v>0) in the upper half of the flow (i.e. close to the surface), which results in decrease of the transverse gradient of the transverse Reynolds stress - u'w'. However, in the last two cases (Figure 5.27. and 5.28.), where two verticals were examined for the same flow conditions, it is more difficult to observe this decrease of the transverse gradient of the transverse Reynolds stress -u'w', as - u'w' values are almost constant and close to zero in the upper half of the flow. On these two graphs, it can also be seen that there are some deviations in the Reynolds stress -u'w' between the two verticals examined, which indicates that the secondary currents are not the same in the two cross-sections. 115 2 2 -u'w' [cm Is ] -400 -300 -200 -100 0 100 200 300 400 1 1 1 Or©— (ft -0.2 -1 1 1 . H 0 E9 0.3 y/Y J 0.4 B 0.5 -o.ia-B 0.8 -Q 1 0.9 -rn Hi 1 trO—1 B ADV Figure 5.25. Reynolds stress-u'w' for S=0.021, Y=18.5 cm, D=3.5 cm -u'w" [cm2/s2] -400 -300 -200 -100 0 100 200 300 400 1 1 1 0^0— 0.1 -1 1 1 pa 0.2 0.3 • B y,Y o.4 - B 0.5 • B a • 0.6 • 00.7 B O.g -E3 0.9 -m la 1 i_Q-J B ADV Figure5.26. Reynolds stress-u'w' for S=0.021, Y=20.5 cm, D=3.5 cm 116 -400 -300 — I — -200 — I — -u'w" [cm2/s2] -100 100 200 — I — 300 — I — 400 y/Y o . i 0.2 ip 0.3: 0.4 *3.7 + 0.8 0.9 ADV x ADV: 6 cm downstream FigureS.27. Reynolds stress-u'w' for S=0.01, Y=22.5 cm, D=3.5 cm -400 -300 -200 2 2 -u'w* [cm /s ] -100 100 200 -0^ 0-y/Y 300 —I— 400 0Atm' 0.2 0.3 ft 0.4* 0.5 2j-0.6 * 8 0.9 X -4^0-ADV x ADV: 60 cm downstream Figure 5.28. Reynolds stress-u'w' forS=0.0J, Y=32 cm, D=3.5 cm 117 5.5.3. The Transverse Reynolds Stress -v'w'. - This is a shear stress that acts in a u-w plane, parallel to the v velocity and perpendicular to the w velocity. Together with the transverse Reynolds stress - u'w', the Reynolds stress - v'w' will cause the secondary motion. Secondary currents are generated by anisotropy between vertical (v') and transverse (w') velocity fluctuations, for which the gradient is of same order as the Reynolds stress -v'w'. Reciprocally, the Reynolds stress -v'w' will be affected by secondary current gradient. Figures 5.29. through 5.32. represent the graphical interpretation of Reynolds stresses -v'w', measured in four experiments explained in detail in Sections 5.4.1.1. through 5.4.1.4. of this thesis. These Reynolds stresses have much lower values, that are close to zero almost in the whole region of the flow, than the other two Reynolds stresses (-u'v' and -u'w'), discussed in detail in Sections 5.5.1. and 5.5.2. Consequently, these Reynolds stresses have less effect on the turbulent flow structure and on the velocity distributions, than the other two. On Figures 5.31. and 5.32., Reynolds stresses -v'w1 for measurements on two verticals in the same flow conditions are presented. There are no significant deviations in the Reynolds stress distribution or in the corresponding Reynolds stress intensities between the measurements on the two verticals. 118 -Vw' [cm2/s2] -200 -150 -100 -50 0 50 100 150 200 1 1 1 0^0-0 1 1 1 | I fo2 -H0.3 -1 10.4 - y/Y 0.5 -B »•<•-0).7 -0.8 • B E 0.9 • E3 M - EI B ADV Figure 5.29. Reynolds stress - V w' for S=0.021, 7=78.5 cm, D=3.5 cm 2 2 -v'w' [cm /s ] -200 -150 -100 -50 0 50 100 150 200 1 1 1 0^ 0— 1 1 1 0.1 -H^0.2 • 0 0.3 H 0.4 -y/Y 0 0.5 0.6 0 0.7 - B 0.8 - E3 B 0.9 - B L f l _ E) B ADV Figure 5.30. Reynolds stress-v'w' for S=0.021, Y=20.5 cm, D=3.5 cm 119 -Vw* [cm /s ] -200 -150 -100 -50 0 50 100 150 200 1 1 1 OrO- 1 1 1 » 0.1* B 0.2B 0.8* 0.#- y/Y K LT7_> | V 0.6 • 0.7 X o* a a X H 0.9 -1^ 0-0 0 a A D V JK A D V : 6 cm downstream Figure 5.31. Reynolds stress - VV for S=0.01, Y=22.5 cm, D=3.5 cm -200 -150 -100 — I — -Vw* [cm2/s2] -50 50 100 — I — 150 — I — 200 0.36 0 0 0 0 0.^ 0*9 I X y/Y a A D V x A D V : 60 cm downstream Figure 5.32. Reynolds stress -v'V forS=0.01, Y=32 cm, D=3.5 cm 120 Chapter 6 Experimental Results for the Second Experimental Set-up and Comparison with the Results of the First Experimental Set-up In the second experimental set-up, additional roughness elements were added into the flume, as described in Chapter 4., Section 4.2. One experiment was carried out with the maximum slope equal to 2.1%, two pumps were used to provide the highest possible flow, and the maximum total depth observed (measured from the bottom of the channel to the surface) was equal to 26 cm. The surface was very wavy and the depth was changing in the streamwise and in the spanwise direction, which caused some difficulties when measuring the depth. It is estimated that the error in depth measurements is of order ±1.0 cm, or 5%. All three velocity components were recorded for five cross-sections on fifteen verticals (Figure 6.1.), and from these measurements, the Reynolds' stresses and the standard deviations were calculated. The flow rate, Manning's n, Froude and Reynolds numbers are calculated to give the overall character of the flow. Results and calculations obtained from these velocity measurements will be examined and compared with experimental results for the first experimental set-up. An attempt will be made to determine the effect of the additional roughness elements on the flow and on the resistance to the flow. 121 0.15 M -big rocks 1.75 0.25 0.25 0.25 0.25 — s * | < >l< H< > Figure 6.1. Positions for additional rocks and for fifteen verticals where velocities were measured. For each cross section (denoted 1-^ 5 on Fig. 6.1.), one depth was measured from the bottom of the channel to the surface. For each vertical (denoted A - H O on Fig.6.1.), the height of the rock (ranging from 2.5 cm to 6.5 cm) was than subtracted from those measured depths, to obtain the clear depth of flow. Therefore, fifteen different depths of flow were calculated (ranging from 17.5 cm to 22.0 cm), and those values were averaged out to get an average depth of flow for the whole examined section. The average depth of flow was equal to Y = 20.23 cm, was taken as a representative depth for all graphs presented in this Chapter. Hence, some measurements close to the bottom will appear in the range ofy/7 =1.0+1.1. This occurs when the rock underneath the point of measurement is small. Each graph contains measurements for all three examined verticals on one cross-section. Davg denotes the average rock height for each particular cross-section and ranges from 3.67 cm to 4.67 cm. 122 6.1. Flow Rate, Manning's n, Froude Number and Reynolds Number Flow rate per unit width (q), total flow rate (Q), Manning's roughness coefficient (n), Froude (Fr) and Reynolds (Re) numbers will be calculated for the second experimental set-up. The equations used for these calculations are the same as given in Section 5.1. of this thesis. First, the calculations will be done for individual readings on each vertical, summed and averaged out to get these parameters for the whole section. Than, the parameters will be calculated with average values for the whole section (Y =20.23 cm and =117.87 cm/s), and the two will be compared. The comparison with the first experimental set-up will also take place in this Section. Cross-section Vert. Y [cm] u [cm/s] q [cm2/s] Q [cm3/sL n Fr ReHO 5 A 20.5 131.53 2696.33 0.0383 0.93 2.25 1 B 22.0 118.45 2605.79 127530.08 0.0446 0.81 2.17 C 21.5 104.67 2250.30 0.0497 0.72 1.88 D 21.5 130.62 2808.26 0.0398 0.90 2.34 2 E 21.0 120.52 2531.01 124974.24 0.0425 0.84 2.11 F 19.0 107.94 2050.93 0.0444 0.79 1.71 G 19.0 121.76 2313.39 0.0393 0.89 1.93 3 H 19.0 120.82 2295.52 108238.78 0.0396 0.88 1.91 I 18.0 102.30 1841.46 0.0452 0.77 1.53 J 22.0 108.30 2382.68 0.0488 0.74 1.99 4 K 20.0 117.72 2354.31 112068.08 0.0421 0.84 1.96 L 22.0 87.76 1930.79 0.0602 0.60 1.61 M 19.0 137.47 2611.85 0.0348 1.01 2.18 5 N 17.5 133.91 2343.51 131583.97 0.0339 1.02 1.95 O 21.5 124.23 2670.97 0.0419 0.86 2.23 Avg. values: 20.23 117.87 2379.14 120879 0.0430 0.84 1.98 Table 6.1. Calculated flow parameters for the second experimental set-up 123 In the Table 6.1., calculations for the parameters described above are given for each vertical. Column titled 'Q' represents the total flow rate for the given cross-section and it changes from one cross-section to another. This would imply that the law of continuity is not satisfied, however, the differences between the average Q (equal to 120879 cm3/s) and the highest (cross-section 5) or the lowest (cross-section 3) 'local' Q's are within ±10%, which can be attributed to the fact that the velocity profiles across the flume were measured in only three points (verticals). In the row named 'Average values', the value in each column represents the average of the fifteen calculations given above. When examining Manning's n, it can be observed that in cross-sections 1 and 5, n increases for the verticals that are closest to the two piles of big rocks (verticals C and M), because the resistance to the flow is probably the highest there. Furthermore, verticals that are aligned behind the first pile of big rocks (F, I, L and O) are influenced by the wake zone behind those rocks, and large eddies caused an increase in Manning's n. Manning's n is a parameter which is used to describe the average resistance to the flow for the river, or for some section of it, and it should not be looked at locally. It does not depend only on roughness size, but on the overall effects that cause resistance to the flow. However, these 'local' n values, calculated for each vertical, describe the local changes in the resistance, whereas the average value equal to 0.043 can be taken as a representative value for the whole examined section. The average Manning's n, equal to 0.043, for the second experimental set-up is significantly higher than the same parameter for the first experimental set-up, where the highest experienced value was equal to 0.035 (Section 5.1., Table 5.1.). This indicates 124 that additional big rocks in the flume caused much more turbulence and stronger three-dimensional effects in the flow, which, in turn, resulted in higher resistance to the flow and in increased values for Manning's n. Manning's n was also calculated using Strickler's and Jarrett's equation (equations 2.36. and 2.23., respectively). In Strickler's equation a uniform grain size distribution is assumed, n depends on roughness size alone, and only one value is obtained for each experimental set-up. These values are equal to 0.025 (for D50=5.0 cm) and 0.030 (for D5o=15.0 cm) for the first and the second experimental set-up, respectively, which is approximately 30% lower than values calculated in Tables 5.1. and 6.1. In Jarrett's equation, n is proportional to the slope and inversely proportional to the hydraulic radius, while the roughness size does not influence its value. For each experiment one value for n is calculated, and for the first experimental set-up, these values were approximately equal to 0.096 for the slope of 0.021, or 0.069 for the slope of 0.01, which is almost three times as high as values calculated in Table 5.1. For the second experimental set-up, Jarrett's equation gives a value of 0.095 for n, which is two times the value calculated in Table 6.1. These values are surprisingly high, and there is no increase in n for the second experimental set-up, even though it is confirmed with measurements and other calculated parameters that the resistance increased. All this indicates that Jarrett's conditions on natural streams in Colorado, must have been very different from laboratory conditions where the experiments were carried out, and therefore, his equation has to be applied with caution. Froude number also changes from one vertical to another, but generally, the energy losses are high enough to keep the flow in subcritical region. There are local 125 increases in the Froude number (as recognized by Jarrett, 1989), indicating supercritical flow, but such conditions last only for short distances. The Froude number is lower for the second experimental set-up than for the first one, indicating higher energy losses, which are due to big rocks added to the flume. As would be expected, when the Froude number decreases, Manning's n increases, which was also described by Jarrett (1989). The calculated Reynolds numbers are indicating that the flow is in the zone of fully rough turbulent flow. The values are lower than for the first experimental set up, which can be explained by the lower velocities and depths of flow in the second experimental set-up because of the increased resistance to flow. 6.2. Instantaneous Velocities Instantaneous velocities are the velocities that were actually measured at each point on every vertical. The detailed explanation about these velocities is given in Section 5.2. of this thesis. Significant difference is not observed in the behaviour of instantaneous velocities between the two experimental set-ups. Therefore, all comments given in Section 5.2. are valid for second experimental set-up as well. Hence, instantaneous velocities will not be plotted here. 6.3. Standard Deviations Standard deviation in one point, or the root mean square, represents the deviation of measured instantaneous velocities from the mean point velocity. The intensity of the 126 standard deviation does not appear to change with the roughness and this agrees with the findings of Nezu and Nakagawa (1993). Therefore, apparently the velocity fluctuations are of the same order of magnitude for different sizes of roughness elements, which is surprising. The same method for calculations is used as described in Section 5.3. of this thesis. The following five graphs (Fig. 6.2. through 6.6.) represent the calculated rms values for five cross-sections. On each graph, standard deviations for three velocity components (u, v, and w) measured on three verticals of one cross-section are presented. Standard deviations for one velocity component differ from one vertical to other, however, Su>Sw>Sv for all cases and all verticals, which is similar to findings for the first experimental set-up. All other comments given in Section 5.3. are valid for the second experimental set-up, too. However, the values of the highest standard deviations that occur close to the bottom are lower than the highest values for the first experimental set-up. 127 Standard Deviation [cm/s] 0 20 40 60 80 100 0.0 H 1 1 1 1 1 1.1 • S u - A - • - S u - B a Su-C o S v - A -- • — Sv -B « Sv -C A Sw-A - A — S w - B A Sw-C Figure 6.2. Standard deviations for cross-section 1: S=0.021; Y=20.23 cm; Davg=3.67 cm Standard Deviation [cm/s] 0 20 40 60 80 100 0.0 i 1 1 1 1 1 1.1 • Su-D - • — S u - E a Su-F o S v - D - • — S v - E 0 Sv-F A Sw-D - ± - S w - E A Sw-F Figure 6.3. Standard deviations for cross-section 2: S=0.021; Y=20.23 cm; Davg=4.17 cm 128 Standard Deviation [cm/s] 0 20 40 60 80 100 0.0 H 1 1 1 1 1 • Su-G -•—Su-H B Su-I o Sv-G Sv-H * Sv-I A Sw-G - Sw-H A Sw-I Figure 6.4. Standard deviations for cross-section 3: S=0.021; Y=20.23 cm; Davg=4.33 cm Standard Deviation [cm/s] 0 20 40 60 80 100 0.0 i 1 1 1 1 1 • Su-J -•—Su-K EI Su-L o Sv-J - • — S v - K o Sv-L A Sw-J - f t—Sw-K A Sw-L Figure 6.5. Standard deviations for cross-section 4: S=0.021; Y=20.23 cm; Davg=4.67 cm 129 Standard Deviation [cm/s] 0 20 40 60 80 100 0.0 H 1 1 1 1 1 1.1 • S u - M - • - S u - N • S u - 0 © S v - M - S v - N o S v - 0 A S w - M — • * — S w - N A Sw-0 Figure 6.6. Standard deviations for cross-section 5: S=0.021; Y=20.23 cm; Davg=4.67 cm 6.4. Velocity Profiles Velocity measurements for three velocity components (u, v, and w,) will be presented on the following graphs, on which each point represents the average value of measurements recorded over a period of three minutes at that depth (4500 data points). 6.4.1. Streamwise Velocity Measurements. - Five graphs (Figures 6.7. through 6.11.) for u velocities measured on fifteen verticals for five different cross-sections will be presented and discussed. These measured profiles will be compared with two theoretical profiles, one being calculated from equation 3.28. (Chapter 3.4.) developed in this thesis, the other one being Prandtl's equation (5.5.). 130 Using averaged velocities and depths, theoretical equations give theoretical velocity profiles, which are the same for all five cross-sections. For each measured profile, the mean velocity is calculated and appears on the corresponding graph. In the further text, the "first" and the "second" groups of big rocks are taken as a reference points. These are not the first and the second groups of big rocks in the channel, as such groups were laid down in the flume in a "zig-zag" pattern from its beginning, but they are the first and the second with respect to the section of the flume where measurements were taken, and where the five examined cross-sections were positioned. 6.4.1.1. Streamwise Velocity Measurements for Cross-section 1. - This is the cross-section positioned at the first group of big rocks, height 15 cm, and which were laid at the right wall of the flume, when following the flow direction. It can be observed on Figure 6.7., that in the lower half of the flow (close to the bottom), the velocities are increased for the verticals positioned in the middle of the channel (vertical B - shaded squares) and on the opposite side of the flume (vertical A - diamonds). At the same time, for the vertical C (triangles), which was the closest to the first group of big rocks, the velocities are decreased for y/Y =0.85 to 1.0, but are increased in the zone that would correspond to the top of the big rocks. For the lower half of the flow, the highest velocities are experienced on the vertical A, which was the farthest from the first group of big rocks, while in the upper half of the flow, the highest velocities occur in the middle of the channel, on vertical B. The three measured velocity profiles have a backward looking curve in the surface region. The measured data have somewhat better fit with equation 3.28., than with Prandtl's equation. 131 6.4.1.2. Streamwise Velocity Measurements for Cross-section 2. - This cross-section is positioned 25 cm downstream from the cross-section 1, and from the first group of big rocks. There is some increase in velocities in the lower half of the flow (Fig. 6.8.) for verticals D (diamonds) and E (shaded squares), which are farther from the first pile of rocks (Fig. 6.1.). Vertical F (triangles on the graph) is the closest to big rocks, and it can be observed on the graph (Fig.6.8.), that the region of decreased velocities is moved somewhat higher, i.e. for y/Y =0.5 to 0.9. This means that the wake zone and eddies created by the pile of big rocks have spread over a larger depth than at cross-section 1. For the upper half of the flow, velocities on vertical F are still somewhat lower than on other two verticals at this cross-section, but the difference is not significant. The velocity profile measured in the middle of the channel (vertical E - shaded squares) follows the theoretical curve (equation 3.28.) the best. Even though there are some discrepancies in the lower half of the flow between the measured profiles and the theoretical ones, it still seems that equation 3.28. has a better fit than Prandtl's equation. 6.4.1.3. Streamwise Velocity Measurements for Cross-section 3. - This cross-section is positioned 25 cm downstream from the cross-section 2, and the velocity measurements are represented on the Figure 6.9. The velocities measured on the two verticals, which are farther away from the first pile of big rocks (namely: H - shaded squares, and G -diamonds), have a good fit with the theoretical profiles. However, there is an increase in velocities close to the surface, in the middle of the channel (vertical H), and therefore, Prandtl's equation gives a better fit to the measured data in this region and for this vertical. The vertical I (triangles on the graph) is the closest to the first pile of big rocks. 132 It can be seen, when examining the velocities on this vertical, that the wake zone has moved higher in the flow, so that the decreased velocities occur even farther from the bottom than at the previous cross-section, i.e. for y/Y =0.35 to 0.7. 6.4.1.4. Streamwise Velocity Measurements for Cross-section 4. - This cross-section is now farther away, approximately 75 cm from the first pile of big rocks, or 25 cm from the cross-section 3. A significant influence of these rocks on the two verticals (J and K on Figure 6.10.), is not observed, both velocity profiles have a close fit with the theoretical profiles in the lower half of the flow. Closer to the surface, there is some decrease in velocities for all three measured profiles at this cross-section. However, in surface region, all three measured velocity profiles are very close to each other. When examining vertical L, which is the closest to the first group of big rocks, it can be observed that the wake zone moved closer to the surface, and decreased velocities now occur for y/Y =0.25 to 0.6. The theoretical curves agree with the measured data in the lower half of the flow, but disagree in the surface zone, where lower velocities were measured. 6.4.1.5. Streamwise Velocity Measurements for Cross-section 5. - This was the last investigated cross-section, which was positioned right after the second pile of big rocks. This second pile was placed on the opposite side of the flume to the first pile. The height of the rocks in this second pile was 14 cm. An increase in velocities (Fig. 6.11.) is observed in the lower half of the flow for the vertical which is the farthest from the second pile of big rocks (vertical O - triangles). This is in agreement with observations for the vertical A at cross-section 1, which was placed in a corresponding position at the first pile 133 of big rocks. The farthest vertical from the big rocks has the highest increase in velocities in the lower half of the flow, as if the water is pushed around the rocks. Vertical M, the closest to the second pile of big rocks (diamonds on the graph), has somewhat increased velocities in the zone which corresponds to the top of the big rocks, which is also in agreement with the observations for the vertical C at cross-section 1. Vertical N (shaded squares), positioned in the middle of the flume, has somewhat increased velocities in the surface zone, and a better fit to Prandtl's equation. If the mean velocities are examined one cross-section after another, it can be argued that continuity is not satisfied. However, flow conditions were highly three-dimensional, and the entire flow field across the flume could not be fully described with just three verticals on each cross-section. Therefore, the profiles with highest velocities may have been measured for some cross-sections, and not for others. 134 135 136 137 138 6.4.2. Vertical Velocity Measurements. - In this section measured vertical velocities (v) will be presented on five graphs (Figures 6.12. through 6.16.) for five cross-sections. The positions for the cross-sections and fifteen verticals were described previously in this Chapter. Negative vertical velocities indicate the zone of downflow, while positive velocities indicate the zone of upflow. For cross-section 1 (Fig. 6.12.), vertical C (triangles) is the closest to the first pile of big rocks, vertical B (shaded squares) is in the middle of the flume, and vertical A (diamonds) is on the opposite side of the flume (Fig. 6.1.). For vertical A, there is a zone of downflow in the lower half of the flow, and a zone of upflow in the upper half of the flow. This is similar behaviour as experienced in all four series of measurements in the first experimental set-up, without the additional roughness rock groups. Vertical B is closer to the first pile of big rocks and there, some upflow is observed on the bottom of the channel. For this vertical, there is a zone of downflow in the middle of the flow, and again a zone of upflow in the upper half of the flow. For vertical C, there is a zone of high upflow very close to the bottom of the channel, followed by a zone of downflow for the rest of the flow. For cross-section 2 (Fig. 6.13.), downflow is observed on all three verticals for the entire depth of flow. The exception is vertical F (the closest to the first pile of big rocks), on which a slight upflow exists in the surface zone. For cross-section 3 (Fig. 6.14.), upflow is observed on all three verticals for the entire depth of flow, with a small deviation for vertical H, which has a little downflow very close to the bottom. 140 The vertical velocities for cross-section 4 (Fig.6.15.) are again positive on all three verticals for the entire depth of flow. There is a slight downflow, or negative velocities, on the bottom of the channel for vertical J. Cross-section 5 (Figure 6.16.) is positioned after the second pile of big rocks, which are laid on the left hand side of the flume. Vertical M (diamonds) is the closest to this group of rocks, and vertical N (shaded squares) is in the middle of the flume, while vertical O (triangles) is on the opposite side of the channel. A large increase in negative velocities is observed on vertical M at a depth that corresponds to the top of the large rocks. For the entire flow, a zone of downflow is experienced on all three verticals, with a minimum exception of vertical O which has a small zone of upflow close to the bottom. For four cases examined for the first experimental set-up with uniform roughness elements, there was a zone of downflow in the lower half of the flow, and a zone of upflow in the upper half of the flow (Fig. 5.12. through 5.15.). From the experiment carried out for the second experimental set-up with additional roughness elements, it can be observed that such a behaviour is completely changed. For all five examined cross-sections, there is either an upflow or a downflow for the entire depth of flow. Such a behaviour is not fully understood at this stage, but it can be said that the additional roughness elements have introduced more three-dimensionality and the flow pattern is consistent with strong secondary spiral flows. 141 Figure 6.12. Vertical velocity measurements for cross-section 1: S=0.021; Y=20.23 cm; Davg=3.67 cm Figure 6.13. Vertical velocity measurements for cross-section 2: S=0.021; Y=20.23 cm; Davg=4.17 cm 142 Figure 6.14. Vertical velocity measurements for cross-section 3: S=0.021; Y=20.23 cm; Davg=4.33 cm Figure 6.15. Vertical velocity measurements for cross-section 4: S=0.021; Y=20.23 cm; Davg=4.67 cm 143 v [cm/s] -50 -40 -30 -20 -10 0 10 20 30 40 50 1 —1 1 1 O T O - 1 1 1 1 o a A ©OA 0.1 O OA © a 0.2 • © B 0.3 • © 1 3 V 4 -© 0 O . 5 © P.6A y/Y © PI o . l -© m l © • A © © 0.9 • A A 1.0... A A hi-© M E3 N A O Figure 6.16. Vertical velocity measurements for cross-section 5: 5=0.027; Y=20.23 cm; Davg=4.67 cm 6.4.3. Transverse Velocity Measurements. - The following five graphs (Fig. 6.17. through 6.21.) represent the measured transverse velocities (w) for five cross-sections. Transverse velocities indicate the existence of secondary currents in the flow. While for uniform roughness elements (the first experimental set-up), secondary currents had the same direction for all four experiments, it will be shown here that the additional roughness elements introduced considerable change in behaviour. For transverse velocities, a plus sign indicates that the velocities are pointed toward the right hand side of the flume, when looking in the direction of the main flow. Figure 6.17. illustrates the w velocities for cross-section 1. For verticals B and C, the direction of the secondary currents is to the left near the bed and at the surface, away from the first group of big rocks, but to the right in the mid-depth, above the top of the 144 big rocks. For the vertical A, the flow is to the right and is stronger at mid-depth. Transverse velocities on all three verticals at this cross-section, follow a somewhat similar pattern. Figure 6.18. represents the transverse velocities for cross-section 2, and it can be observed that they are all negative in the lower half of the flow (or to the left), and positive in the upper half of the flow. This is exactly the opposite from the observations for the first experimental set-up and indicates a strong secondary current. For the cross-section 3 (Fig. 6.19.), the transverse velocities mostly follow the same pattern as described for cross-section 2, again indicating a strong secondary current. Cross section 4 is positioned just upstream the second pile of big rocks, and the vertical J (diamonds) is closest to it. It can be observed on Figure 6.20., that the transverse velocities on vertical J are to the right, away from the big rocks that are laid just ahead, for the entire depth of flow. At the same time, the transverse velocities on the other two verticals are negative for the entire depth of flow, as if they are pushed towards the next pile of big rocks. The positive transverse velocities are even higher on the vertical M (diamonds), which is the closest to the second group of big rocks, but just downstream of it (Fig. 6.21: cross-section 5). For the vertical N (shaded squares), the transverse velocities are positive in the lower half of the flow and negative in the upper half of the flow, while for the vertical O (triangles), which is the farthest from the second pile of big rocks, the transverse velocities are negative for the entire depth of flow. Strong secondary currents existed at cross-sections 2 and 3, but transverse flows at 1 and 4 were more generally either to the left or to the right. 145 -60 -40 -20 w [cm/s] 0 20 o A B A C 40 60 1 H 0^0- i 1 A H V I -A 0 * 0.23 £ A 0 3 © g0.4 " A 0 5 0.6-0.7 i 0.8 • ta i A © 0 © . © o A i i Figure 6.17. Transverse velocity measurements for cross-section 1: S=0.021; Y=20.23 cm Davg=3.67 cm -60 -40 -20 w [cm/s] 0 20 a A~ o o . i 0.2 0.3 0.4 — 0.5 > y/Y J 0.6 0.7 -[ ft.8 0 ° 1.0 hA~ B» A E> A O A 0 o D A F 40 —h-60 Figure 6.18. Transverse velocity measurements for cross-section 2: S=0.021; Y=20.23 cm Davg=4.17 cm 146 w [cm/s] -60 -40 -20 0 20 40 60 1 1 1 0^0-j 1 1 1 1 t © 0.1 • & © A © 0.2 -.0 A © 0.3 - E3 A © Q A © 0.4 Q A — 0.5 - E3 . y/Y ra A 0 ° V -© A BO 0.&-a © »9 • 0 1-0— i-ri— © G 0 H A I Figure 6.19. Transverse velocity measurements for cross-section 3: S=0.021; Y=20.23 cm Davg=4.33 cm Figure 6.20. Transverse velocity measurements for cross-section 4: S=0.021; Y=20.23 cm Davg=4.67 cm 147 Figure 6.21. Transverse velocity measurements for cross-section 5: S=0.021; Y=20.23 cm; Davg=4.67 cm 6.4.4. Flow Patterns. - After examining the velocity profiles in all three directions and for all five cross-sections, an attempt is made to plot the flow patterns (Fig. 6.22.). The wake zone and larger scale eddies grow after the flow passes the first pile of rocks. This growth is recognized in all directions, i.e. in streamwise, spanwise, and vertical directions. Beside and above the large rocks, the streamlines get closer to each other, indicating higher velocities. There is also a slight dip in the surface level above the pile of rocks. When going farther away from those rocks, the streamlines are going apart, and the wake zone is getting smaller. At cross-section 4, the highest depth of flow is experienced, and the lowest velocities were measured on the verticals at this cross-section. It is expected that for the second pile of big rocks, the streamlines in vertical direction (side view), would 148 look similar to those plotted on Figure 6.22.(b), but they could not be shown on the same graph. 6.5. Reynolds Stresses The primary Reynolds stress, -u'v', and the two transverse Reynolds stresses, -u'w' and -v'w', calculated from measurements for the second experimental set-up, will be reviewed and compared with the results for the first experimental set-up. The equations for Reynolds stresses calculations are given in Chapter 5, Section 5.5. of this 149 thesis. The convection for directions in which Reynolds stresses are acting is given on Figure 5.20. At any depth, measurements were recorded over a period of three minutes (4500 data points), and each point on the following graphs represents the average value of those measurements. Each graph will represent measurements for one cross-section. 6.5.1. The Primary Reynolds Stress - u'v'. - These Reynolds stresses are acting in a plane parallel to the bed. For a uniform, steady flow, their distribution would be linear and similar to the theoretical shear stress distribution (r). On the following five graphs (Fig. 6.23. through 6.27.), the distribution of -u'v' will be compared with the distribution of T. The total bed shear stress, To, was calculated with the average depth of flow for the whole section (7 = 20.23 cm), using equation 3.8. of this thesis. Therefore, the value of r0 will be the same for all five cross-sections and equal to To/p=416.8 cm2/^. Similar to the results for the first experimental set-up, negative primary Reynolds stresses can be observed in the surface zone for the cases in which a negative velocity gradient, du/dy, was measured. Secondary currents are stronger than in the first experimental set-up, which will also be seen in increased values for transverse Reynolds stresses -u'w' (Section 6.5.2.). This will contribute to larger differences between the primary Reynolds stress -u'v' and shear stress r, than was experienced for the first experimental set-up. For the verticals where the secondary currents were stronger, deviations from the linear Reynolds stress distribution will be more significant. 150 -uV [cm2/s2] -600 -500 -4-rlr © A a B A C Theoretical Figure 6.23. Reynolds stress - u' v' for cross-section 1: S-0.021; Y=20.23 cm; Davg=3.67 cm -uV [cm2/s2] -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 -93429 D 2 2 To/pF=416.76 cm /s Theoretical Figure 6.24. Reynolds stress - u' v' for cross-section 2: S-0.021; Y=20.23 cm; Davg=4.17 cm 151 2 2 -u'v' [cm /s ] -600 -500 -400 -300 -200 -100 100 200 300 400 500 600 I 0.0 To/p=416.76 cm Is H Theoretical Figure 6.25. Reynolds stress - «' v' for cross-section 3: S-0.021; Y=20.23 cm; Davg=4.33 cm 2 2 -u'v' [cm/s ] -600 -500 -400 -300 -200 -100 100 200 300 400 500 600 K Theoretical Figure 6.26. Reynolds stress - u' v' for cross-section 4: S-0.021; Y=20.23 cm; Davg=4.67 cm 152 2 2 -u'v' [cm /s ] © M a N A O Theoretical Figure 6.27. Reynolds stress - u'v' for cross-section 5: S-0.021; Y=20.23 cm; Davg=4.67 cm 6.5.2. The Transverse Reynolds Stress -u'w'. - The most obvious differences between the results of the first and the second experimental set-up, are observable in the behaviour of transverse Reynolds stress -u'w'. Extreme values (either positive or negative) for - u'w' occur close to the bottom of the channel for all five cross-sections. There is no reasonable explanation for such a behaviour at this point, but further investigation and more experiments would lead to a better understanding of this phenomenon. In order to show more accurately the trend of -u'w', the most extreme values are represented out of scale, and because their values are written beside them. 153 According to theory (Nezu and Nakagawa, 1993), the transverse gradient of the transverse Reynolds stress, - u'w', should depend on the direction of vertical velocities, v, in a following manner: duw / dz < 0 for upflow (v>0), and duw / dz > 0 for downflow (v<0). This is not the case for all measurements and for all verticals. In a very turbulent, three dimensional flow, the direction of vertical velocities is probably not the only factor which would determine the transverse gradient of the transverse Reynolds stresses. The distribution of the transverse Reynolds stress -u'w', is changing from one vertical to another (Fig. 6.28. through 6.32.), which indicates that the flow field is very complex, and that the direction of secondary currents is changing rapidly. Figure 6.28. Reynolds stress - u'w' for cross-section 1: S-0.021; Y=20.23 cm; Davg=3.67 cm 154 -u'w* [cm2/s2] -500 -400 -300 -200 -100 0 100 200 300 400 500 157B.48 1 y/Y *© o o . i 0 0.9 AO 0.4 0.5 0.6 0°7 0.8 9 1.0 -4-4--+-• A © D 0 E A F Figure 6.29. Reynolds stress - u' w' for cross-section 2: S-0.021; Y=20.23 cm; Davg=4.17 cm -u'w* [cm2/s2] -500 -400 -300 -200 -100 -1936 -1760.98 i.98 -0^ 0-A A 0 1 0® 0% 0.9% •1-.0-100 200 300 400 500 •+-© © A© A © G B H A I 13.03 Figure 6.30. Reynolds stress -u'w' for cross-section 3: S-0.021; Y=20.23 cm; Davg=4.33 cm 155 -u'w' [cm2/s2] -500 -400 -300 -200 -100 100 200 300 400 500 -+--4965.52 B A 0.1 0.2E m — « 5 y/Y J 0.6 o om O 0.8 0.9 t 1.0 H — -0 A a A AO A> -f- -+-© © © J H K A L 18)60.6 Figure 6.31. Reynolds stress -u'w' for cross-section 4: S-0.021; Y=20.23 cm; Davg=4.67 cm -u'w' [cm2/s2] -500 -400 -300 -200 -100 100 200 300 400 500 -1001.95 -+- -0^ 0-A H(©1 A ®6 (ECA y/Y 0.5 © 0.7 0.8 « . 9 •l-.O-- © © A A © M N A O Figure 6.32. Reynolds stress -u'w' for cross-section 5: S-0.021; Y=20.23 cm; Davg=4.67 cm 156 6.5.3. The Transverse Reynolds Stress -v'w'. - The values of these Reynolds stresses are somewhat higher than in the case of uniform roughness elements in the first experimental set up. Some extreme values are measured close to the bottom of the channel (Fig. 6.33. through 6.37.), which was also experienced for the other transverse Reynolds stresses, -u'w'. The distribution of transverse Reynolds stresses -v'w', is changing from vertical to vertical, a general trend is not observed. Reynolds stresses -u'w' and -v'w' are rarely measured and described in literature, and it is difficult to examine the accuracy of these measurements. Figure 6.33. Reynolds stress-v'w' for cross-section 1: S-0.021; Y=20.23 cm; Davg=3.67 cm 157 Figure 6.34. Reynolds stress - v V for cross-section 2: S-0.021; Y-20.23 cm; Davg=4.17 cm Figure 6.35. Reynolds stress -v'w' for cross-section 3: S-0.021; Y=20.23 cm; Davg=4.33 cm 158 Figure 6.36. Reynolds stress -v'w' for cross-section 4: S-0.021; Y=20.23 cm; Davg=4.67 Figure 6.37. Reynolds stress - v' w' for cross-section 5: S-0.021; Y=20.23 cm; Davg=4.67 cm 159 Chapter 7 Discussion and Conclusions Better understanding of hydraulic processes in steeper mountain rivers has become important, as these areas are more in use for diversity of purposes (discussed in detail in Introduction). Chezy developed the first theory for open channel flow in 1768. Later, theories were developed for pipe flow, and then with some adjustments applied to an open channel flow. Many of these theories are still in use. Prandtl's boundary layer theory for the velocity profile prediction is the most frequently used theory. Many studies showed that some assumptions on which this theory is based, are not valid in open channel flow. For example the flow is taken to be two dimensional, when in reality, the flow is three dimensional; or, the shear stress is given a constant value throughout the depth and across the bed; or, the velocity fluctuations in streamwise (w') and vertical (v') directions are considered equal. These experiments and other previous studies, clearly demonstrate that these assumptions are not valid, however, in spite of these invalid assumptions, Prandtl's equation still fits the measured data fairly well. A modified logarithmic law based on Prandtl's work, but with somewhat different set of assumptions, was developed and evaluated in this work. The shear stress was assumed to have a linear distribution as predicted by standard open channel flow theory, and Prandtl's definition of the shear stress in turbulent flow (in which u '=v') was taken as valid. This two dimensional model was considered because of a three dimensional model 160 is far too mathematically complex and, without simplifications, cannot yield analytical results. However, the ADV measurements clearly show that the flow is highly three dimensional. Consequently, the usual two dimensional boundary layer theories can only be considered to be approximate. Further work should examine whether three dimensional analysis can be developed, which would link the distribution of bed roughness to the flow field and to the resulting channel resistance and velocity. Hence, this theoretical work is just a step towards a better equation. The value of the present work is that it gives a better understanding of the role of the three dimensional flow in the total behaviour of the channel resistance. A final modified logarithmic equation (3.28.) needs the following parameters as an input: depth of flow, slope, and mean velocity. These experiments show that a common assumption that a good estimate of the mean channel velocity can be measured in the middle of the channel at about 0.6 of the total depth measured from the surface is reasonable for the conditions of the present task. Velocity profile calculated using this equation does not differ by much from the Prandtl's profile in the lower half of the flow (close to the bed), but in the upper half of the flow, this equation gives a lower velocity gradient and a steeper profile, which gives a better fit to the data. However, near the surface, some of the measured profiles have a negative velocity gradient du/dy, i.e. a backward looking curve in the surface zone, and the two dimensional theory cannot account for this behaviour. Measurements done with 3D ADV indicate a very complex three dimensional turbulent flow structure for both experimental set-ups, with increased three dimensionality for the second experimental set-up. The transverse velocity fluctuations (w') are almost as 161 high as the streamwise velocity fluctuations («'), which is illustrated on graphs that represent standard deviations and instantaneous velocities. This behaviour is caused by cross flows and strong secondary currents, which in turn considerably affect the primary Reynolds stresses -u'v'. Therefore, the Reynolds stress distribution deviates from the shear stress distribution, r, especially in the zone close to the rough boundary at the bottom of the channel. Flow patterns plotted for the second experimental set-up (Section 6.1.4.), indicate zones of high and low velocities. In the zone of high velocities around the big rocks, the streamlines get close to each other, acceleration is experienced and energy is probably conserved. The term Bernoulli zone has been used to describe this region. In the zone of lower velocities downstream of the big rocks, streamlines move apart because of deceleration and loss of energy (shock losses). The mean velocity does not increase much with increase in slope, as would be expected, which indicates that the energy dissipation also increases, due to shock losses. Even though the slope is twice as steep in the second experimental set-up, the Froude number is much lower (indicating subcritical flow) than in the first experimental set-up, indicating much higher energy losses. There is a local increase in Froude number, indicating supercritical flow, which quickly turns back to subcritical regime, showing the presence of shock loss in this region. Manning's roughness coefficient increases in the second experimental set-up by more than 20%, clearly indicating increase in the resistance to the flow. This increase in resistance is due to big rocks placed into the flume, which introduced more three dimensionality to the turbulent flow structure. If the entire bottom of the channel would 162 be covered with those same big rocks, the average roughness size would be higher than in the first experimental set-up, but the increase in Manning's n would not be as high. This confirms that Manning's n does not depend only on the roughness size, but also on the uniformity or non-uniformity of size distribution across and along the channel, and on the other factors that affect the resistance to the flow. 163 Bibliography Barnes, H.H., Jr. 1967. Roughness characteristics of natural channels. U.S. Geological Survey Water-Supply Paper 1849, 213 p. Bathurst, J.C, 1985. Flow Resistance Estimation in Mountain Rivers. Journal of Hydrologic Engineering, III (4), Paper 19661. Bergeron, N.E., 1994. An Analysis of Flow Velocity Profiles, Stream Bed Roughness, and Resistance to Flow in Natural Gravel Bed Streams. Ph.D. Dissertation, Department of Geography, State University of New York at Buffalo, Buffalo, New York, 163 p. Daugherty, R.L., Franzini, J.B., Finnemore, E.J., 1985. Fluid Mechanics With Engineering Applications. McGraw-Hill, Inc., eight edition. Henderson, F.M., 1966. Open Channel Flow. Macmillan Publishing Co., Inc., New York. Hicks, D.M. and Mason, P.D., 1991. Roughness Characteristics of New Zealand Rivers. Water Resources Survey, DSIR Marine and Freshwater. Higginson, N.N.J., and Johnston, H.T., 1989. Estimation of Manning's Roughness Coefficient in Alluvial Streams. Proceedings of the International Conference on Channel Flow and Catchment Runoff: Centennial of Manning's Formula and Kuichling's Rational Formula. ASCE, University of Virginia, pp. 383-391. Von Karman T., 1930. Mechaniche Aehnlichkeitund Turbulenz (Mechanical Similarity and Turbulence). Proceedings of the 3rd International Congress of Applied Mechanics, Stockholm, vol.1, pp.85-92. 164 Kraus, N.C., Lohrmann, A., and Cabrera, R., 1994. New Acoustic Meter for Measuring 3D Laboratory Flows. Journal of Hydraulic Engineering, vol. 120, No. 3, pp. 406-413. Kumar S., 1989. An Analytical Method for Computation of Rough Boundary Resistance. Proceedings of the International Conference on Channel Flow and Catchment Runoff: Centennial of Manning's Formula and Kuichling's Rational Formula. ASCE, University of Virginia, pp. 410-427. Jarrett R.D., 1989. Hydraulics Research in Mountain Rivers, Proceedings of the International Conference on Channel Flow and Catchment Runoff: Centennial of Manning's Formula and Kuichling's Rational Formula. ASCE, University of Virginia, pp. 599-608. Manning R., 1891. On the Flow of Water in Open Channels and Pipes. Transactions, Institution of Civil Engineers of Ireland, Dublin vol. 20, pp. 161-207; supplement, 1895, vol. 24, pp. 179-207. Marchand J.P., Jarrett R.D., and Jones L.L., 1984. Velocity Profile, Water-Surface Slope, and Bed-Material Size for Selected Streams in Colorado. U.S. Geological Survey, Open-File Report 84-733. Nezu I., Nakagawa H., 1993. Turbulence in Open-Channel Flows. A.A. Balkema, Rotterdam, Netherlands, IAHR, pp. 15-17, 25-27, 31-47, 85-91. Prandtl L., 1926. Uber die ausgebildeteTurbulenz (On fully Developed Turbulence). Proceedings of the 2nd International Congress of Applied Mechanics, Zurich, pp. 62-74. 165 Quick M . C , 1991. Reliability of Flood Discharge Estimates. Canadian Jurnal of Civil Engineering, vol. 18, pp.624-630. Roberson, J.A., and Crowe, C.T., 1985. Engineering Fluid Mechanics. Houghton Mifflin Company, third edition, pp.512-519. Rouse H., 1938. Fluid Mechanics for Hydraulic Engineers. McGraw-Hill Book Company, Inc., New York. Schlichting H., 1968. Grenzschicht - Theorie (Boundary Layer Theory). McGraw-Hill, Inc., sixth English edition, pp. 544-557. Smith P.L., 1979. Splines as a Useful and Convenient Statistical Tool. The American Statistician, vol. 33, pp. 57-62. Son Tek, Inc., 1994. Reference Manual for ADV Software Version. 2.3. Ugarte, A., and Madrid, M., 1994. Roughness Coefficient in Mountain Rivers. Proceedings to the 1994 National Conference on Hydraulic Engineering. ASCE, Buffalo, New York, pp.652-656. Ven Te Chow, 1981. Open-Channel Hydraulics. McGraw-Hill, Inc., International student edition, pp. 192-194. Vennard, J.K., and Street, R.L., 1982. Elementary Fluid Mechanics. John Wiley & Sons, Inc., sixth edition, pp. 511-520. 166 

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