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A contribution to the study of local river-bed scour around bridge piers Van der Gugten, Cornelis Adrianus 1972

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A CONTRIBUTION TO THE STUDY OF LOCAL RIVER-BED SCOUR AROUND BRIDGE PIERS by CORNELIS ADRIANUS van der GUGTEN B . A . S c , U n i v e r s i t y of B r i t i s h Columbia,  1967.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department • of CIVIL ENGINEERING  Vie accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY" OF BRITISH COLUMBIA September, 1972.  In  presenting this  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  in p a r t i a l  fulfilment of  the U n i v e r s i t y of B r i t i s h  make i t  freely available  that permission  for  the requirements f o r  Columbia,  I agree  r e f e r e n c e and  for extensive copying o f  study.  this  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department by h i s of  this  written  representatives. thesis  It  for financial  gain shall  not be allowed without my  C.A. van der Gugten  Civil  Engineering  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  Columbia  September 29, 1972.  or  i s understood that c o p y i n g o r p u b l i c a t i o n  permission.  Department o f  that  ABSTRACT  This Thesis presents a review of the reported research on l o c a l sandbed scour around bridge p i e r s , describes the mechanics of l o c a l scour with p a r t i c u l a r reference to the horseshoe vortex, describes experiments on the e f f e c t of the v e r t i c a l v e l o c i t y d i s t r i b u t i o n of the approach flow on l o c a l scour, and reports the r e s u l t s of these experiments. The review of the previous research on l o c a l scour shows that the p i e r s i z e i s the most important parameter a f f e c t i n g the e q u i l i b r i u m depth of l o c a l scour, while the p i e r shape i s of secondary importance. parameters are found to be important;  Two flow  the flow v e l o c i t y and the flow depth,  although t h e i r . r e l a t i v e importance depends on the regime of the flow being considered. The primary scouring agent;is seen to be the horseshoe yortex system, which i s a system of linlced v o r t i c e s that a r i s e s out of the v o r t i c i t y always  i n  present i n shear flows, due to the i n t e r a c t i o n of the p i e r and the flow. The experiments that were done consisted of observations  and  measurements o f - v e r t i c a l -velocity d i s t r i b u t i o n s , scour depths, and vortex patterns, f o r a c i r c u l a r c y l i n d e r i n a laboratory flume.  These experiments  showed that the v e r t i c a l : v e l o c i t y d i s t r i b u t i o n of the boundary layer flow approaching the p i e r a f f e c t s the structure of the horseshoe vortex system and the e q u i l i b r i u m depth of scour at the p i e r nose. The Thesis concludes with a Summary and Conclusions, recommendations f o r f u r t h e r research.  r  including  iv.  TABLE OF CONTENTS  Page LIST OF TABLES  viii.  LIST OF FIGURES  ix.  LIST OF SYMBOLS  xi i i .  ACKNOWLEDGEMENTS  XX.  INTRODUCTION  1. •'  LITERATURE REVIEW  5.  A.  EARLY INVESTIGATORS  5.  B.  TISON AND ISHIHARA  6.  1. 2.  6. 7.  C.  Tison Ishihara  INGLIS AND REGIME THEORY ADHERENTS  9.  1. 2. 3.  9. 9. 10.  Inglis Blench Varzeliotis  D.  LAURSEN  11.  E.  CHABERT AND ENGELDINGER  14.  V.  Chapter II  III.  Page F.  BATA AND KNEZEVIC  16.  1.  Bata  16.  2.  Knezevic  16.  G.  TARAPORE  18.  H.  MOORE AND MASCH  20.  I.  BREUSERS: DELFT HYDRAULICS LABORATORY 1. Oosterschelde Bridge Model Studies  22. 22.  2.  24.  Scour Around D r i l l i n g Platforms  J.  MAZA AND SANCHEZ  24.  K.  SHEN et a l : COLORADO STATE UNIVERSITY 1. Introduction 2. D e s c r i p t i o n of the Mechanics of Local Scour 3. Theoretical Analysis 4. Experimental Results 5. R e c o n c i l i a t i o n of Divergent Concepts  26., 26.. 26.. 28. 31.$, 33.  6.  34.^*  Methods of Reducing Scour  L.  CARSTENS  35.  M.  TANAKA AND YANO, AND THOMAS 1. Tanaka and Yano  37. 37.  2.  38.  Thomas  N.  SCHNEIDER  39.  0.  COLEMAN  43.  THE MECHANICS OF LOCAL SCOUR  45.  A.  INTRODUCTION  45.  B.  THE BASIC VORTEX MECHANISM  46.  C.  ORIGIN OF THE HORSESHOE VORTEX 1. V o r t i c i t y 2. V o r t i c i t y i n the Flow near a C y l i n d e r (a) The Approach Flow. (b) Flow near the C y l i n d e r .  47. 47. 48. 48. 49.  D.  E.  CHARACTERISTICS OF THE HORSESHOE VORTEX 1. 2.  Structure Formation  3.  Strength  SUGGESTED RESEARCH  EXPERIMENTAL WORK A.  PURPOSE AND SCOPE  B.  EQUIPMENT AND EXPERIMENTAL ARRANGEMENT 1. General Arrangement 2. Sand Bed 3. Test P i e r 4. V e l o c i t y - D i s t r i b u t i o n Control Gate 5. Current Flowmeter 6. Dye I n j e c t o r EXPERIMENTAL METHODS AND PROCEDURES  C.  1. S t a r t i n g - u p 2. Generation of V e r t i c a l V e l o c i t y P r o f i l e s 3. V e l o c i t y Measurements 4. Flow Patterns 5. Scour Hole Development and E q u i l i b r i u m Depth of Scour EXPERIMENTAL RESULTS A.  GENERAL  B.  APPROACH FLOW VELOCITY PROFILES  C.  EQUILIBRIUM SCOUR DEPTH 1. 2. 3.  Introduction Results Comparison w i t h Results of Others (a) Laursen (b) Tarapore (c) Breusers (d) Maza and Sanchez (e) Larras Cf ] Shen ert aj_. Cg) Coleman  vii. Chapter V.  Page D.  VORTEX PATTERNS AND SCOUR HOLE DEVELOPMENT  72.  1. 2.  72.  3. 4. VI.  The Horseshoe vortex System Scour Hole Development: Beginning of Scour Approach Flow V e l o c i t y P r o f i l e and Vortex Structure Transport out of the Scour Hole  73. 75. 76.  SUMMARY AND CONCLUSIONS  78.  A.  PREVIOUS INVESTIGATIONS  78.  B.  MECHANISM OF LOCAL SCOUR  80.  C.  EXPERIMENTAL RESULTS  81.  I-. 2. 3. D.  Introduction E q u i l i b r i u m Depth of Scour Horseshoe Vortex Flow Patterns  81. 81. 82.  RECOMMENDATIONS  83.  1. 2.  83. 84.  P r e d i c t i n g Scour Depths Further Research  BIBLIOGRAPHY  85.  TABLES  98.  FIGURES  101.  vii i .  LIST OF TABLES  Table I  II  Ill  Page The v a r i a t i o n of e q u i l i b r i u m scour depth with average v e l o c i t y , f o r d i f f e r e n t p i e r diameters ( c i r c u l a r p i e r ) and d i f f e r e n t bed sediments, as reported by Breusers. Values of the c o e f f i c i e n t 1C used i n the equation of J a r o s l a v t s i e v . Summary of Scour Experiments.  98 q g  100  ix.  LIST OF FIGURES  Figure  Page  1.  Representation of curvature of the flow near an o b s t r u c t i o n , as proposed by Tison.  1Q1  2.  Laursen's non-dimensional p l o t of e q u i l i b r i u m scour depth (d ) versus flow depth (H), f o r a rectangular p i e r of width b, at an angle of attack of 3 0 ° .  102  Schematic diagram showing v a r i a t i o n of the e q u i l i b r i u m scour depth ( d ) with average flow v e l o c i t y Cu), as found by Chabert and Engeldinger ( f o r any p i e r ) .  103  4.  V e l o c i t y d i f f u s i o n i n t o the scour h o l e , according to Tarapore.  104  5.  Schematic i l l u s t r a t i o n of scour hole at e q u i l i b r i u m c o n d i t i o n s , according to Tarapore.  105  6.  Approach flow v e l o c i t y d i s t r i b u t i o n and stagnation pressure on the plane of symmetry in f r o n t of a c i r c u l a r c y l i n d e r .  106  s e  3.  s e  7.  Values of the c o e f f i c i e n t IC^ to be used i n the Maza and Sanchez v e r s i o r r of J a r o s l a v t s i e v s equation. 1  107  Figure 8.  9. 10.  11.  Page Idealized representation of the flow on the plane of symmetry i n f r o n t of a c i r c u l a r c y l i n d e r ; Shen e_t al_.  108  Control volume on the stagnation plane i n f r o n t of a c i r c u l a r c y l i n d e r , Shen ejt al_.  109  Schematic representation of the v e l o c i t y d i s t r i b u t i o n and flow pattern i n the scour hole on the plane of symmetry i n f r o n t of a c i r c u l a r c y l i n d e r ; Shen et al_.  110  E q u i l i b r i u m scour depth versus p i e r Reynolds number CR, ) f o r c i r c u l a r c y l i n d r i c a l p i e r s ; Shen  eta].. 12.  *  HI  E q u i l i b r i u m scour depth versus p i e r Reynolds number CR,), f o r c i r c u l a r c y l i n d r i c a l piers of d i f f e r e n t s i z e s ; Shen ejt al_.  112  E q u i l i b r i u m scour depth versus p i e r Reynolds number CR,)* f o r c i r c u l a r c y l i n d r i c a l piers and d i f f e r e n t g r a i n s i z e s ; Shen e_t al_.  113  14.  V e l o c i t y d i s t r i b u t i o n along the v e r t i c a l , i n the experiments of Tanaka and Yano.  114  15.  Types of c i r c u l a r c y l i n d r i c a l p i e r s studied by Tanaka and Yano.  115  16.  E q u i l i b r i u m depth of scour f o r d i f f e r e n t c i r c u l a r c y l i n d r i c a l p i e r types; Tanaka and Yano.  116  17.  Scour depth versus d i s c p o s i t i o n , f o r a c i r c u l a r c y l i n d r i c a l p i e r ; Thomas.  117  18.  The scour regions of Schneider ( f o r any p i e r ) .  118  19.  Sketch of the vortex s t r u c t u r e on the plane of symmetry i n f r o n t of a c i r c u l a r c y l i n d e r i n a laminar boundary l a y e r , from a photograph of an experiment by Gregory and Walker, published i n thwaitesC125l.  119  Sketch of laboratory flume c r o s s - s e c t i o n general arrangement.  120  13.  20.  showing  xi. Figure  Page  21.  Flume sand g r a i n - s i z e d i s t r i b u t i o n .  121  22.  View of v e l o c i t y - c o n t r o l gate from a p o s i t i o n downstream of the p i e r (gate i s r e f l e c t e d i n glass w a l l s on e i t h e r s i d e ) .  122  23.  V e l o c i t y p r o f i l e s , s e r i e s 1.  123  24.  V e l o c i t y p r o f i l e s , s e r i e s 2-A.  124  25.  V e l o c i t y p r o f i l e s , s e r i e s 2.  125  26.  V e l o c i t y p r o f i l e s , s e r i e s 3 and 4.  126  27.  V e l o c i t y p r o f i l e , s e r i e s 5.  127  28.  S t a b i l i t y of v e l o c i t y p r o f i l e , s e r i e s 1-A.  128  29.  S t a b i l i t y of v e l o c i t y p r o f i l e , s e r i e s 2-A3.  129  30.  Scour hole development with time, s e r i e s 1.  130  31.  Scour hole development with time, s e r i e s 2-A.  131  32.  Scour hole development with time, s e r i e s 2.  132  33.  Scour hole development with time, s e r i e s 3.  133  34.  Scour hole development with time, s e r i e s 4.  134  35.  Scour hole development with time, s e r i e s 5.  135  36.  Vortex p a t t e r n , f l a t bed with low flow velocity.  136  37.  Vortex p a t t e r n , beginning of scour s e c t i o n a l view).  137  38.  Scour hole development with time, at 9 = 30° and e = 0 ° , s e r i e s 1-A.  138  39.  Vortex p a t t e r n , beginning of scour (plan view).  139  40.  Vortex p a t t e r n s ; p a r t l y developed scour h o l e ; showing dependence of the i n d i v i d u a l v o r t i c e s on the v e r t i c a l v e l o c i t y p r o f i l e of the approach f l o w .  140  (cross-  Figure 41.  Vortex p a t t e r n s ; p a r t l y developed scour hole showing e f f e c t of the v e r t i c a l v e l o c i t y p r o f i l e of the approach flow.  42.  Scour hole development, showing motion of sand g r a i n s .  43.  Views of f u l l y - d e v e l o p e d scour hole: Ca) looking downstream; (b) looking upstream  44.  View of f u l l y developed scour hoVj&i looking d i a g o n a l l y upstream.  xi i i .  LIST OF SYMBOLS  A corner of the control volume ABCD (see Figure 9 ) . v  Area of the horseshoe vortex core. Points on a v e r t i c a l i n the flow near the f r o n t of an o b s t r u c t i o n Csee Figure 1). Cylinder radius Parameter varying with type of flow: a = 0.6 f o r main flow channel; a^^ = 1.0 f o r f l o o d p l a i n (Maza and Sanchez). A corner of the control volume ABCD (see Figure 9 ) . Diameter of a c i r c u l a r d i s c f i t t e d around a c y l i n d r i c a l p i e r (see Figure 15). Points on a v e r t i c a l i n the flow i n the region which remains unaffected by the pressure of an o b s t r u c t i o n (see Figure 1 ) . P i e r width or diameter. P i e r width projected onto a plane perpendicular to the approach flow d i r e c t i o n . A constant c o e f f i c i e n t .  XIV.  Symbol C  A corner of the control volume ABCD (see  C  Experimental constant  C  A c o e f f i c i e n t proportional to channel roughness and  1  D  u'/H  Figure'9).  (Ishihara).  Clshihara).  A corner of the control volume ABCD (see Figure 9 ) .  D  c  C r i t i c a l depth of flow  D  L  Lacey regime depth  D 3  =  =  (q /g) 2  1 / 3  0.47 ( Q / f ) m  L  1 / 3  , in feet.  Total scoured depth, measured from the water s u r f a c e , in feet.  dA  Element of surface area.  ds  Element of l e n g t h .  d  Blench's zero f l o o d depth.  d  g  C h a r a c t e r i s t i c grain s i z e of bed sediment used by Carstens.  d  g  Depth of l o c a l scour below normal bed l e v e l .  d s e  E q u i l i b r i u m or l i m i t i n g value of d , f o r a given sediment, p i e r geometry, and flow c o n d i t i o n s 3  Maximum possible value of d  d d 0 0  , f o r a given p i e r geometry.  The grain s i z e such that 00% of the material i s f i n e r , by weight.  e  The base of natural logarithms = 2.718.  F  The flow Froude number  F, b  Blench bed f a c t o r ,  F, bo  Blench zero bed f a c t o r ,  F M  Total d i s t u r b i n g force on a bed sand-grain  F  Total r e t a r d i n g force on a bed sand-grain  R  =  U/j/gH*  (Carstens). (Carstens).  XV.  Function of Lacey s i l t f a c t o r .  A c c e l e r a t i o n of  gravity.  Approach flow depth. Height of p i e r m o d i f i c a t i o n above normal bed l e v e l Csee Figure 15). P i e r shape c o e f f i c i e n t ( L a r r a s ) . A constant c o e f f i c i e n t i n a uniform flow r e l a t i o n (Shen). P i e r shape c o e f f i c i e n t  (Jaroslavtsiev).  A c o e f f i c i e n t which i s a f u n c t i o n of H/b  (see Table  1  A c o e f f i c i e n t which i s a f u n c t i o n of U /gb Csee Figure 7 ) .  II).  and H/b  P i e r shape c o e f f i c i e n t CLaursen). A coefficient  =  0.53 ( U / g b ) " ° ' 2  1  3 2  CMaza and Sanchez).  C o e f f i c i e n t f o r the angle of attack of the approach flow to the p i e r alignment A c o e f f i c i e n t representing the rate of v e l o c i t y i n t o the scour hole CTarapore). C h a r a c t e r i s t i c length of flow  diffusion  obstruction.  Distance along a flow streamline, s t a r t i n g at edge.  scour-hole  A constant exponent i n a uniform flow r e l a t i o n (Shen). Carstens' sediment number The c r i t i c a l value of N  s  =  u"//ts-l)gd  g  .  at which l o c a l scour i s i n i t i a t e d .  xv i .  A point on the edge of the scour hole (see Figure 5). Hydrostatic Stagnation  pressure. pressure.  Flow discharge. C r i t i c a l yalue of Q at which l o c a l scour i s  initiated.  Maximum f l o o d discharge, i n cubic f e e t per second. Time rate of sediment transport i n t o scour hole (volume). Time rate of sediment transport out of scour hole (volume). Time rate at which sediment i s removed from the scour h o l e , i n being deepened (weight). Value of Q' at the beginning of the l o c a l scour i e . at t = 0.  process;  s  Time rate of sediment transport i n t o scour hole (weight). = q'  s  .  m  b  (Schneider).  Time rate of sediment transport out of scour hole (weight). Flow discharge per u n i t width. The value of q f o r the c e n t r a l approach flow j u s t upstream of the p i e r , i n cubic f e e t per second per f o o t . Rate of bed-load sediment transport per u n i t width (Straub). Time rate of sediment transport i n t o the scour h o l e , per u n i t width (weight). P i e r Reynolds number  =  Ub/u.  Radius of curvature of flow streamlines. Scour force at p i e r nose ( I s h i h a r a ) .  xvi i .  S p e c i f i c g r a v i t y of bed sediment. Time, from the beginning of the l o c a l scour Time constant  process.  (Schneider).  Time at which d  reaches d  s  . se  Average v e l o c i t y of approach flow. The value of U at which l o c a l scour i s i n i t i a t e d . The value of U at which general bed-load transport initiated. The l o c a l v e l o c i t y i n the x - d i r e c t i on. Local y e l o c i t y at the boundary  (Tarapore).  Local p o t e n t i a l v e l o c i t y (Tarapore). Total v e l o c i t y vector (Roper). Volume of scour hole. Local v e l o c i t y i n the y - d i r e c t i on. Local v e l o c i t y i n the x - d i r e c t i on. Local y e l o c i t y i n the y - d i r e c t i on. Local v e l o c i t y i n the z - d i r e c t i o n . Stream or channel w i d t h . Scour hole w i d t h . Local v e l o c i t y i n the z - d i r e c t i o n . Axes of orthogonal co-ordinate system. E l e y a t i o n head.  is  xviii.  Included angle of p i e r nose. Circulation  =  (jl V" • 3s~  S p e c i f i c weight of water. Boundary l a y e r t h i c k n e s s . Constant exponent i n a uniform flow equation (Shen). Void r a t i o of bed sediment. Kinematic v i s c o s i t y of water. Upward distance that the scour hole a f f e c t s the main flow (Jarapore). 3.1416 Density of water. Density of bed sediment. Standard d e y i a t i o n . Bed shear  stress.  C r i t i c a l yalue of T f o r the i n i t i a t i o n of bed sediment transport. Angle of repose of the bed sediment. A parameter  =  3  t  tan 0 / TT(1-A)P gb s  (Schneider).  Sediment t r a n s p o r t c h a r a c t e r i s t i c (Straub). Flow streamline on the plane of symmetry of the approach flow (see Figure 5). Flow streamline passing through point P on the edge of the scour hole (see Figure 5). Total y o r t i c i t y vector  =  V x V.  V o r t i c i t y with axis of r o t a t i o n i n the x - d i r e c t i on. V o r t i c i t y w i t h axis of r o t a t i o n i n the y - d i r e c t i o n .  xix.  V o r t i c i t y w i t h axis of r o t a t i o n i n the z - d i r e c t i on. Angular v e l o c i t y of the horseshoe vortex core.  Vector operator. A s u p e r s c r i p t i n d i c a t i n g that the parameter i s to be evaluated at the surface of the flow. A s u p e r s c r i p t i n d i c a t i n g that the parameter i s to be evaluated at the bed.  XX.  ACKNOWLEDGEMENTS  The author wishes to express h i s sincere thanks to his Professor E.S.  Supervisor,  P r e t i o u s , f o r h i s h e l p f u l advice, constant encouragement,  and enduring patience throughout t h i s study, e s p e c i a l l y during the l a s t months, when he was already r e t i r e d from the U n i v e r s i t y . Thanks are due to Professor H.W. Shen, of Colorado State U n i v e r s i t y , f o r his suggestions and encouragement given to the author during a v i s i t to the C.S.U. campus i n the summer of 1968.  Special thanks are due to  Dr. Verne Schneider, then a post-graduate student at Colorado S t a t e , f o r h i s w i l l i n g n e s s to spend considerable time with the author i n d i s c u s s i n g and demonstrating various aspects of the l o c a l scour phenomenon during this v i s i t . The f i n a n c i a l support of the National Research Council of Canada i s g r a t e f u l l y acknowledged.  The t r i p to Colorado was made possible through  the N.R.C. operating grant to Professor P r e t i o u s .  CHAPTER I  INTRODUCTION  The successful foundation design of a structure founded i n the sandbed of a flowing stream, i n order to safe-guard against excessive loss of bearing c a p a c i t y , must include reasonably accurate methods f o r p r e d i c t i n g the t o t a l amount of bed degradation at the foundations. Degradation of the bed of a stream at a p a r t i c u l a r c r o s s - s e c t i o n can occur due to changes i n the general sediment-transport capacity of the flow at that s e c t i o n .  For example, an increase i n the v e l o c i t y of  the f l o w , due to increased discharge or reduced channel w i d t h , increases the capacity of the flow to carry sediment, and degradation r e s u l t s . S i m i l a r l y , trapping or removing some of the sediment load of the flow (eg. by a dam) causes the stream to recover i t s f u l l sediment load by scouring of the bed.  2.  In a d d i t i o n to general degradation of the bed at a channel s e c t i o n , l o c a l bed-level changes can occur due t o : (1)  meandering of the main flow channel (thalweg) between the banks of the stream,  (2)  the downstream movement of bed dunes,  C3)  l o c a l scour.  Local scour i s the scour of the stream bed i n the immediate v i c i n i t y of a s t r u c t u r e Csuch as a groyne, t r a i n i n g w a l l , wharf, bridge p i e r , or abutment) founded i n that bed.  I t i s p r i m a r i l y caused by the flow  disturbance generated by such a s t r u c t u r e . I t was the purpose of t h i s t h e s i s to i n v e s t i g a t e some of the aspects of the l o c a l scour phenomenon, with the aim of improving our understanding of the mechanics of the scour process and thus provide a sounder basis f o r p r e d i c t i n g bed changes due to l o c a l  scour.  The l i t e r a t u r e on l o c a l scour was reviewed f a i r l y comprehensively, and i s summarized i n Chapter II.  This review o u t l i n e s the h i s t o r i c a l  development o f , and provides a basis f o r , the understanding of the l o c a l scour phenomenon.  Although a l l of the references i n the Bibliography  were examined, only the more important and representative studies and f i n d i n g s are reported i n the review. Chapter III  describes the mechanics of l o c a l scour of a sand-bed at  a v e r t i c a l c y l i n d e r , based on the f i n d i n g s summarized i n Chapter II observations reported i n the f i e l d of f l u i d dynamics.  and on  The o r i g i n and  c h a r a c t e r i s t i c s of the horseshoe vortex are discussed, p a r t i c u l a r l y the various flow parameters that influence the strength of the horseshoe  3.  vortex system. A f t e r considering a l l of the a v a i l a b l e information on l o c a l developed in Chapters II  and I I I ,  scour  i t was decided to i n v e s t i g a t e the  influence of the approach flow v e l o c i t y d i s t r i b u t i o n .  An o u t l i n e of the  experimental work done, and a d e s c r i p t i o n of the equipment and methods which were used, i s given i n Chapter IV.  The experiments consisted of  observations and measurements of v e l o c i t y d i s t r i b u t i o n s , scour depths, and flow patterns in a laboratory flume. Chapter V presents the r e s u l t s of the experimental work, and Chapter VI gives the summary and conclusions. Symbols are defined where they f i r s t appear i n the t e x t , and also i n the L i s t of Symbols, p . x i i i . B i b l i o g r a p h i c a l references are i n d i c a t e d by r a i s e d , bracketed numerals.  Footnotes are i n d i c a t e d by r a i s e d , unbracketed  numerals, and are located at the end of each chapter. The author was f i r s t encouraged to engage i n studies i n the f i e l d of l o c a l r i v e r - b e d scour by h i s t h e s i s s u p e r v i s o r , Professor E.S.  Pretious,  who had p r e v i o u s l y , i n a consulting c a p a c i t y , c a r r i e d out p r i v a t e i n v e s t i g a t i o n s of r i v e r - b e d scour at bridge p i e r s .  These i n v e s t i g a t i o n s  were mainly concerned with f i n d i n g the most adverse scour patterns that could occur at the p i e r s of actual proposed or e x i s t i n g bridges i n B r i t i s h Columbia.  These included bridges over the Columbia River at  T r a i l , the Columbia River at K i n n a i r d , the Kootenay River near Creston, the Fraser River at Agassiz, the Fraser River at Oak Street  (Vancouver),  and Morey Channel at Sea Island (Richmond). The procedure used i n these p r i v a t e i n v e s t i g a t i o n s was to carry out a  4.  h y d r a u l i c laboratory study using scale model piers and a moveable, sandbed, and to compare the r e s u l t s thus obtained with scour depths predicted by various formulas.  I t was found that often the predicted  scour and the laboratory t e s t r e s u l t s d i d not agree very w e l l .  I t was  t h i s lack of a quick, r a t i o n a l , and safe method f o r p r e d i c t i n g r i v e r - b e d scour around flow o b s t r u c t i o n s , t h a t motivated the present study.  5.  CHAPTER IT  LITERATURE REVIEW  A.  EARLY INVESTIGATORS The f i r s t r e p o r t e d model studies i n v o l v i n g bridge piers were 1  c a r r i e d out i n Germany i n the 1890's by H. Engels at the Technical (3l)  U n i v e r s i t y of Dresden^  .  These studies showed that the maximum scour  of the bed occurs at the upstream end or nose of the p i e r .  They also  showed that riprap should be placed around the p i e r f l u s h with the normal bed l e v e l , rather than on top of the bed.  No other model studies  were reported u n t i l the e a r l y 1920's, when T.H. Rehbock at the Technical U n i v e r s i t y of Karlsruhe did some t e s t s and found that the maximum depth of scour Cwhich occurred at the p i e r nose) v a r i e d with flow v e l o c i t y , bed m a t e r i a l s , p i e r shape, and duration of f l o w , but the nature of these v a r i a t i o n s was not reported i n d e t a i l ^ ^ . 9 8  The i n f l u e n c e of the depth of  flow was apparently not i n v e s t i g a t e d .  The scour at the p i e r nose was  a t t r i b u t e d to cross-currents s e t up there.  B.  TISON AND ISHIKARA 1.  Tison The f i r s t attempt at a t h e o r e t i c a l approach to l o c a l scour was o  made by L . J . Tison i n 1937 . i n yarious places  '  L  His work has subsequently been republished  *  .  He considered the flow near the f r o n t  of an o b s t r u c t i o n i n an open channel to be analogous to the flow of an i r r o t a t i o n a l vortex.  On t h i s basis he obtained an expression f o r a  d i f f e r e n c e i n piezometric head between the surface and the bottom of the flow close to the f r o n t of the o b s t r u c t i o n :  PA, Y *  "  °  C  B' .2 o u  A'  9 = ds'  °  B  a±  ds-]  (i)  r' o " o where z = the e l e v a t i o n head, p = the s t a t i c pressure, y = the s p e c i f i c A  r  weight of water, g = the a c c e l e r a t i o n of g r a v i t y , u = the l o c a l  approach  flow v e l o c i t y , r = the radius of curvature of the flow s t r e a m l i n e s , ds = an element of length taken along an orthogonal to the flow streamlines from A to B ,A represents points on a v e r t i c a l near the f r o n t of the o o' o o b s t r u c t i o n , B represents points on a v e r t i c a l i n the flow which remains Q  uninfluenced by the o b s t r u c t i o n , s.ingle-primed symbols are f o r the flow at the s u r f a c e , and double-primed symbols are f o r the flow at the bottom (see Figure 1 ) .  7.  For the case of an o b s t r u c t i o n of uniform c r o s s - s e c t i o n , r' and ds' = d s " .  = r"  I f , as commonly occurs i n streams and laboratory flumes,  the flow i s not uniform, but a v e l o c i t y gradient e x i s t s on the v e r t i c a l plane, the surface v e l o c i t y u' w i l l be greater than the bottom v e l o c i t y u " , and the right-hand side of equation Cl) w i l l be p o s i t i v e .  Thus there  w i l l be a pressure d i f f e r e n c e between surface and bottom, and as a r e s u l t the flow w i l l acquire a downward component i n f r o n t of the p i e r and attack the bed, causing scour.  Tison concluded that the greater the  d i f f e r e n c e between u' and u " , the greater would be the scour.  In  a d d i t i o n , smaller Values of r ' would also increase the scour.  Tison  v e r i f i e d each of these conclusions w i t h laboratory t e s t s .  He used p i e r  shapes ranging from square to l e n t i c u l a r , and obtained decreasing values of scour depth as the p i e r shape became more streamlined and permitted a more gentle curvature of the flow.  By using a very coarse material f o r  the upstream bed, he was able to change the v e l o c i t y d i s t r i b u t i o n so that u" became less and u  1  became g r e a t e r , thus increasing the d i f f e r e n c e  between u' and u " , and obtained a greater scour hole depth.  However, h i s  r e s u l t s are only q u a l i t a t i v e i n nature and he did not present any s p e c i f i c r e l a t i o n s h i p i n v o l v i n g the depth of scour. 2.  Ishihara S h o r t l y a f t e r Tison f i r s t published his work, Ishihara performed  an extensive s e r i e s of t e s t s of scour at bridge piers ^ ^ . 4  I t was found  that the scour depth at the nose was mainly governed by the shape of the nose - a sharper nose producing less scour - and not at a l l by the shape of the p i e r t a i l or the length of the p i e r , f o r piers aligned with the  8.  flow d i r e c t i o n .  The scour depth was found to increase w i t h increased  skewness of the p i e r s ; the more so f o r a sharper p i e r nose.  The e f f e c t  of c o n s t r i c t i o n of the f l o w , expressed i n terms of the r a t i o of stream width W to p i e r width b, was found to be s m a l l , f o r values of W/b from about 6 to 10, and n i l f o r values of W/b greater than about 15.  I t was  observed that scour decreased with decreasing flow depth, but that the rate of t h i s decrease depended on the c h a r a c t e r i s t i c s of the sand, the p i e r shape and the p i e r s i z e . In a d d i t i o n to his experimental work, Ishihara developed a theory f o r l o c a l scour at p i e r s by assuming the flow near a p i e r to be s i m i l a r to the flow i n the bend of a r i v e r .  He obtained an expression f o r the  secondary downward flow component i n a r i v e r bend by considering the main flow there to be analogous to the flow of an i r r o t a t i o n a l v o r t e x , much a f t e r the manner of Tison Csee above).  By assuming that the "scour f o r c e "  was proportional to the value of the downward flow component, and applying the r e s u l t s f o r the r i v e r bend d i r e c t l y to the case of flow around an o b s t r u c t i o n , he obtained the expression: A S  F  where S C  2  =  w  C  1 2 C  2  ^ B o  d  s  = the scour force at the p i e r nose, C  (  2  )  = an experimental constant,  = a parameter which increases with increasing channel roughness and  decreasing value of H/u', U = the average v e l o c i t y of the approach f l o w , and H = the flow depth. of Tison Cequation C l ) ) .  This expression i s s u b s t a n t i a l l y s i m i l a r to that  9.  C.  INGLIS AND REGIME THEORY ADHERENTS 1.  Inglis The f i r s t r e l a t i o n s h i p e x p l i c i t l y i n v o l v i n g the depth of scour  at a bridge p i e r was formulated by S i r Claude I n g l i s Cwith A.R. Thomas and D.V. Joglekar) i n 1939 .  On the basis of model s t u d i e s , using  round-nosed p i e r s of s i m i l a r geometry but d i f f e r i n g s i z e , he obtained:  ^  =  where D  $  1.70  I-|  J 0.78  (3)  = the t o t a l scoured depth measured from the water surface (in  f e e t ) , b = the p i e r width (in f e e t ) , and q  c  = the central approach-flow  discharge per u n i t width O n square f e e t per second). I n g l i s i n 1949 presented a second r e l a t i o n s h i p , t h i s time on the basis of a considerable number of f i e l d data as well as the experimental data mentioned above. D  = 2D  s  where D  g  He obtained: C4)  L  i s the t o t a l scoured depth as defined above, and  i s the Lacey  regime depth: D  =  L  0.47  \ * f  where Q  m  /  (5)  3  L  = the maximum f l o o d discharge i n cubic f e e t per secondhand f^ =  the Lacey s i l t f a c t o r . 2.  Blench L a t e r , B l e n c h ^ ^ , i n 1957, reported the r e s u l t s of a p l o t , by 11  Andru, using data of I n g l i s , Laursen, and others, of D ^  1/3  vs. q ,  10.  without any attempt to d i f f e r e n t i a t e between d i f f e r e n t types of obstructions.  This data included scour at bridge p i e r s , guide banks,  spur noses, downstream of bridges, e t c . , and produced a "best f i t " l i n e of: 1 / 3 D  s b  =  F  where D  $  q  0 74  (6)  = the t o t a l scour depth below the water surface ( i n f e e t ) , F 2  fa  =  the Blench bed-factor = U /H (U = average approach flow v e l o c i t y i n f e e t per second, H = approach flow depth i n f e e t ) , and q = the approach flow discharge per u n i t width ( i n square f e e t per second). In the same work, Blench proposes a r e l a t i o n s h i p f o r maximum scour at bridge p i e r s which i s the same as equation (4) of I n g l i s , except that instead of the Lacey depth,  , Blench proposes a "zero f l o o d depth,"  d f , which i s supposed to represent the required regime depth of a canal 0  having a bed f a c t o r corresponding to zero charge ( i e . bed-load charge i s 2 3 zero:  = F^  Q  = q / d^  ), with the discharge at the maximum, and  Q  using the reduced obstructed w i d t h . 4  3.  Varzeliotis In 1960, V a r z e l i o t i s did a laboratory study of l o c a l scour at  bridge p i e r s ^  1 3 4  \  Using arguments based on regime theory to arrange  his data, he obtained:  D  r  C  1  U  D F,  C DO  J  11.  O  where D  c  1  / o  = the c r i t i c a l depth of flow (= Iq /g] ' ).  Although a b e t t e r  " f i t " to the data i s obtained using an index of 0.28, V a r z e l i o t i s used h i n the b e l i e f that natural laws u s u a l l y f o l l o w simple i n d i c e s . D.  LAURSEN I t was not u n t i l the e a r l y 1950's that experimental work d i r e c t e d  towards the establishment of a design r e l a t i o n f o r l o c a l scour at bridge piers was s t a r t e d on a s i g n i f i c a n t s c a l e , by E.M. L a u r s e n ^  59,  6 0 ,  6 1  Laursen also formulated a number of concepts with respect to l o c a l  ^. scour  which are useful in e s t a b l i s h i n g a framework f o r the understanding of the l o c a l scour problem. (a)  These concepts can be enumerated as f o l l o w s :  The rate of scour equals the d i f f e r e n c e between the sediment  transport rate into the scour hole and the sediment transport rate out of the scour hole. 4£ dt where cW dt  =  Q s  x  Symbolically:  . - Q out s  . in  y  v  (8) '  _ the time rate of change of volume of the scour h o l e , Q "  .  the time rate at which sediment i s c a r r i e d out of the scour hole ( i n cubic f e e t per second), and Q  $  -  = the time rate at which sediment i s  n  transported i n t o the scour hole ( i n cubic f e e t per second) . 5  (b)  The rate at which the volume of the scour hole increases  decrease as the hole gets (c)  will  bigger.  There w i l l be some l i m i t i n g s i z e of scour hole ( f o r any given  geometry and flow c o n d i t i o n ) . to be at e q u i l i b r i u m , and Q  When t h i s occurs the scour depth i s said n n +  = Q  c  12.  (d)  This l i m i t i n g s i z e w i l l be approached a s y m p t o t i c a l l y .  On the basis of the f i r s t concept, Laursen d i s t i n g u i s h e d between three d i f f e r e n t scour cases: No scour.  Ca)  This c o n d i t i o n occurs when the v e l o c i t y of the flow  i s too low to cause any l o c a l scour, and i n i t i a l conditions are ^s i n = 9s out = ° ' "Clear-water" scour .  Cbl  6  The flow disturbance due to the  o b s t r u c t i o n i s strong enough to cause some scour, but sediment transport by the undisturbed approach flow does not occur.  The i n i t i a l  conditions  are: Q s  Q  .  s out  Q  .  ' ^s out  0.  >  Scour with general sediment motion.  Cc] x  = 0,  .  s in  >  x  Q  s  .  in  >  The i n i t i a l conditions are  0.  A l l of Laursen's e a r l i e r work was done with flows which were capable of general sediment transport Cease (c) above).  He i n v e s t i g a t e d the  e f f e c t s of bed sediment s i z e , average v e l o c i t y of the approach f l o w , and flow depth, on the e q u i l i b r i u m depth of scour at a p i e r .  Although his  data showed some s c a t t e r , he found no e f f e c t of bed sediment s i z e or average flow v e l o c i t y , and only the flow depth was found to have an e f f e c t on the e q u i l i b r i u m scour depth.  He was able to present the  influence of the flow depth i n terms of a graph, using co-ordinates non-dimensionalized on the basis of the p i e r width (see Figure 2 ) ^ ^ . 6 1  Laursen Cas had Posey^ ^ before him) observed that the basic 93  scouring agent was a r o l l e r or v o r t e x , with a h o r i z o n t a l a x i s , which formed i n f r o n t of the p i e r nose.  In order to e x p l a i n the observed  13.  e f f e c t s of the ayerage flow v e l o c i t y , flow depth, and sediment s i z e , Laursen reasoned along the f o l l o w i n g l i n e s ^ ' 5 9  6 1  ^.  At any given e q u i l i b r i u m scour c o n d i t i o n i n the general sediment transport region Cease Cc) above), the rate at which the r o l l e r moves sediment out of the hole i s j u s t balanced by the rate at which the approach flow moyes sediment i n t o the hole.  If the average flow  v e l o c i t y i s increased, the angular v e l o c i t y of the r o l l e r could be expected to increase i n a proportional way, and thus Q increase.  s  t  would  Howeyer, the v e l o c i t y of the approach flow near the bed  would also increase, and the net e f f e c t would be no change i n the difference Q ^ . - Q ^s out  . . in  S i m i l a r l y , i f the sediment s i z e would be "*'  increased, the transport rates i n t o and out of the scour hole would decrease, but i n the same p r o p o r t i o n , so that there would be no change i n the e q u i l i b r i u m scour depth. When the flow depth i s i n c r e a s e d , however, the angular v e l o c i t y of the r o l l e r i s presumed to remain the same as long as the average flow y e l o c i t y remains constant; thus Q  s  t  does not increase.  However, the  y e l o c i t y of the approach flow near the bed would be decreased somewhat, so that Q . would decrease, and the net r e s u l t would be an increase i n ^s m the e q u i l i b r i u m scour depth. Laursen concluded from h i s studies that the e q u i l i b r i u m depth of scour, f o r flows below the c r i t i c a l  CFroude number < 1) and capable of  general bed-load t r a n s p o r t , depends only on the flow depth, p i e r s i z e , p i e r shape, and the angle of attack of the approach flow.  This dependence  was presented i n the form of design curves and t a b l e s ^ ^ .  These design  6 1  14.  c r i t e r i a can be reduced to the expression:  IT where d  = s g  K  s  K a  • i-  5 0  ^°*  <>  3  9  = the e q u i l i b r i u m depth of scour below normal bed 1 eve!, K  a f a c t o r f o r p i e r shape, and K  a  i s a f a c t o r f o r angle of attack.  is  $  For a  c i r c u l a r c y l i n d r i c a l p i e r , equation (9) reduces t o : %  =  1.35  [g-J ' 0  (10)  3  Laursen l a t e r t r i e d to incorporate l o c a l bed scour at bridge p i e r s , f o r both the " c l e a r - w a t e r " case and the case of general sediment t r a n s p o r t , i n t o a general theory of scour at bridge c r o s s i n g s ^  6 2 ,  6 3 ,  6 5  '  6 6 ,  6 7  ^.  He d i d t h i s by f i r s t developing an equation f o r the depth of scour i n a long c o n t r a c t i o n , based on the Manning formula, his own sedimentconcentration formula, and considerations of c o n t i n u i t y of both the flow and the sediment discharges.  Laursen then r e l a t e d the depth of l o c a l  scour at a bridge p i e r to the general scour i n a long c o n t r a c t i o n by introducing a special c o e f f i c i e n t to account f o r the l o c a l nonuniformity of the flow i n the former, and by assuming that the contracted width could be represented by 2.75 d 2.75 d  g e  + 0.5b.  s g  and the uncontracted width by  Comparison of his theory to experimental data of I n g l i s  and Chabert and Engeldinger Crefs. 65 and 191, r e s p e c t i v e l y , i n Karaki and H a y n i e ^ h , showed q u a l i t a t i v e agreement only. 51  E.  CHABERT AND ENGELDINGER In 1956 Chabert and Engeldinger, i n France, reported a large s e r i e s  of t e s t s of l o c a l scour at bridge p i e r s . 7  Their study involved scour i n  15.  both the region of " c l e a r - w a t e r " scour and the region of general sediment transport.  Their main c o n t r i b u t i o n was that they found the e q u i l i b r i u m  scour depth to increase roughly l i n e a r l y with the bed shear  stress  throughout the region of " c l e a r - w a t e r " scour, and that the maximum e q u i l i b r i u m depth of scour occurred i n the t r a n s i t i o n region between " c l e a r - w a t e r " scour and scour w i t h general sediment transport - i e . when the average flow v e l o c i t y i s at about the c r i t i c a l f o r general bed-load transport Csee Figure 3 i . Chabert and Engeldinger also tested d i f f e r e n t s i z e s of bed m a t e r i a l . They obseryed that the maximum e q u i l i b r i u m scour depth f o r a given p i e r increased with increasing sediment s i z e , f o r median grain diameters 0.26 mm, 0.52 mm, and 1.50 mm.  of  However, f o r a sand of median diameter  = 3.00 mm, the maximum e q u i l i b r i u m scour depth was less than that f o r the 1.50 mm sand . 9  L a t e r , Larras obtained a r e l a t i o n f o r l o c a l scour based on the data of Chabert and Engeldinger, as well as f i e l d d a t a : 1 0  d  sem  =  1  A  Z  K  b  ° *  7  5  C l l )  where d i s the maximum e q u i l i b r i u m depth of scour below normal bed sem n  r  l e v e l , i n f e e t , and K i s a f a c t o r to account f o r p i e r shape: c i r c u l a r p i e r s , K = 1.4 f o r rectangular p i e r s . measured i n f e e t .  i  K = 1.0 f o r  The p i e r width b i s  16.  F.  BATA AND KNEZEVIC 1.  Bata In 1960 Bata reported a study of the problem of l o c a l scour at  bridge p i e r s using f i e l d measurements, laboratory t e s t s , and t h e o r e t i c a l analysis^,  tie applied p o t e n t i a l flow analysis to an assumed  logarithmic v e l o c i t y d i s t r i b u t i o n of the flow approaching a c i r c u l a r pier.  This analysis showed that v e r t i c a l l y downward v e l o c i t y components  are present i n f r o n t of the p i e r , and have a magnitude approximately one-half of the magnitude of the average approach flow v e l o c i t y , f o r the region which extends upstream about three or four p i e r r a d i i in f r o n t of the p i e r .  These v e r t i c a l v e l o c i t y components were thought to be the  main cause of l o c a l  scour.  Bata p l o t t e d his laboratory and f i e l d data as d / H versus U /gH ge  Cthe flow Froude number), and obtained a l i n e a r r e l a t i o n s h i p .  This was  noted to be s i m i l a r to the formula of Jaroslavcev ( J a r o s l a v t s i e v ) , which has d  s e  all  ( t h i s formula i s discussed below i n connection w i t h the  work of Maza and Sanchez). 2.  Knezevic Knezevic i n 1960 also reported a study of the l o c a l  scour  ("53 \  problem  .  The relevant aspects of his study involved attempts to  determine the conditions required f o r the s t a r t of l o c a l scour, the e q u i l i b r i u m depth of scour, and the influence of several methods of reducing the maximum depth of scour. The r e s u l t s of the i n v e s t i g a t i o n i n t o the conditions required f o r l o c a l scour were presented i n terms of the c r i t i c a l discharge Q  just  17. large enough to s t a r t l o c a l scour. by e x t r a p o l a t i o n , to d various water depths.  $ e  I t appears that Q was determined c  = 0, of p l o t s of d  g e  versus discharge Q, f o r  Although the data are not s u f f i c i e n t to provide  accurate values for Q , they do i n d i c a t e that Q  i s s l i g h t l y less f o r a  p i e r w i t h a square nose than f o r a p i e r w i t h a round nose.  The data  also show a d e f i n i t e increase i n Q with i n c r e a s i n g sand-grain s i z e , f o r c  the sands used CdgQ = 0.285 mm., 2.4 mm., and 4.5 mm). Knezeyic arranged his data f o r e q u i l i b r i u m depth of scour to obtain a r e l a t i o n of the form;  d  ..--C'Cn;) ' 9  H ' 5  where C = a constant.  4  g  (12)  2  ^  The value of C i s supposed to vary only with the  p i e r shape, but a c t u a l l y there was s c a t t e r with respect to both sand-grain s i z e and flow depth.  Ayerage yalues were, f o r the c i r c u l a r nose, C =  8.7; f o r the rectangular nose, C = 9.8. Knezevic conceiyed the scour-causing  vortex at the base of the p i e r  to be due to the v e r t i c a l l y downward v e l o c i t y components i n f r o n t of the p i e r , as suggested by Tison (see above). reducing scour occurred to him.  On t h i s basis two methods of  One method consisted of a s p i r a t i n g the  downward flow i n f r o n t of the p i e r by means of a h o r i z o n t a l hole or s l o t cut through the pier i n the d i r e c t i o n of the main flow.  A definite  reduction i n the e q u i l i b r i u m depth of scour was observed, ranging from about 27% to 76%, depending on the shape of the p i e r and the s i z e of the bed sand-grains.  The s i z e , shape, and l o c a t i o n of the s l o t were not given.  A second method consisted of placing a s e r i e s of bands around the  18.  p i e r at various l e v e l s , i n order to d e f l e c t and r e t a r d the downward flow j u s t i n f r o n t of the p i e r .  Using three bands of s t e e l p l a t e 15 mm. wide  and 2 mm. t h i c k , placed around a p i e r 10 cm. wide, reductions of the order of 30% to 40% i n the e q u i l i b r i u m scour depth were obtained.  G.  TARAPORE Tarapore i n 1962 completed a Doctoral Thesis i n which he reported  laboratory measurements and presented a t h e o r e t i c a l method f o r determining the l o c a l scour at an o b s t r u c t i o n ^  1 1 9  '  1 2  °).  in his t h e o r e t i c a l  i n v e s t i g a t i o n he assumed that the i n i t i a l flow pattern was given by p o t e n t i a l flow theory.  As the scour hole developed, i t was assumed that  the free stream d i f f u s e d i n t o the scour hole i n a manner analogous to that of a mixing layer s i t u a t i o n , and the v e l o c i t y d i s t r i b u t i o n was given by: uCz)  =  uC-E} e  +E)/1  -k(z  (13)  where u(z) = the l o c a l v e l o c i t y at depth z , i n the x - d i r e c t i o n , u(-£) = the p o t e n t i a l v e l o c i t y , £ = the distance that the e f f e c t s of the scour hole have penetrated i n t o the main f l o w , 1 = distance along a streamline, s t a r t i n g at the scour hole edge, and k = a c o e f f i c i e n t representing  the  rate of v e l o c i t y d i f f u s i o n i n t o the scour hole (see Figure 4).  The rate  of bed load transport was assumed to f o l l o w S t r a u b ' s expression  (1935):  a  T(T  -  T  )  (14)  Y  which, f o r high rates of transport ( T » T ), can be s i m p l i f i e d t o : (15)  19.  where q  s  = the rate of transport of bed load ( i n cubic f e e t per hour per  foot w i d t h ) , $ = Straub's  t r a n s p o r t a t i o n c h a r a c t e r i s t i c , y = the s p e c i f i c  weight of water, T = the bed shear s t r e s s , x = the c r i t i c a l bed shear c s t r e s s . The assumption f o r the bed shear stress was: T a p JuCnlJ  (16)  2  where uCn) = the value of the ( d i f f u s e d ) v e l o c i t y a t the boundary, and p = the density of water.  When the scour hole i s a t the e q u i l i b r i u m  c o n d i t i o n , i t was assumed that the rate of transport i n t o the hole equaled the rate of transport out of the h o l e , thus the transport r a t e between the streamline on the plane of symmetry of the approach f l o w , ip , and Q  the streamline passing through the point P on the edge of the scour hoi e, \pp, was constant q dy  f  P  Csee  In terms of an equation:  q dy  V  =  s  Figure 5 ) .  (17)  s  Cx=o)  0  Gc—)  where the x and y coordinates are as shown i n Figure 5.  Substitution  of equations (15) and (16) y i e l d s :  K2  2 P  * \p 0 P  [u(n)J dy  Y  Cx=0l  4  = JL y '  p  P  2 Y  ,A U dy H  (18)  o  (x=-?)  Further s u b s t i t u t i o n o f equation C L 3 ) , with u C n ) = u ( z ) , and a d d i t i o n a l assumptions and manipulation, y i e l d s an i n t e g r a l equation which can be solved by numerical methods to y i e l d the p o s i t i o n of the point P Figure 5 ) .  Csee  The depth of scour can then be determined, i f the shape of  the scour hole i s known.  In a d d i t i o n , the values o f k and £/l have to be  20.  estimated i n advance from experimental data.  Since £/l represents the  rate at which the influence of the scour hole i s propagated towards the surface of the f l o w , t h i s term i s determined by p l o t t i n g the experimental data as d / b versus H/b, and noting where d / b i s no longer a f f e c t e d by sg  sg  the depth of f l o w .  The value of k i s obtained by s o l v i n g the equation  f o r a know experimental r e s u l t . Tarapore's p l o t s of d / b vs. H/b show considerable disagreement sg  between his own data and that of others which he included i n his p l o t , and his s e l e c t i o n of H/b = 1.15, as the value above which flow depth no longer influences scour depth, seems a r b i t r a r y .  Tarapore c a l c u l a t e d the  depth of scour at an e l l i p t i c a l c y l i n d e r , according to the method o u t l i n e d above, and obtained r e s u l t s which were about 10% to 15% less than the experimental r e s u l t s .  The d i f f e r e n c e i s explained as due to  improper estimation of the scour hole shape.  In a d d i t i o n , Tarapore  concluded that the maximum depth of scour (below normal bed l e v e l ) at a c i r c u l a r c y l i n d e r , f o r large depths of f l o w , i s equal to 1.35  H.  b.  MOORE AND MASCH In 1963 Moore and Masch published a paper i n which they attempted to  achieve an understanding of the secondary flow caused by the presence of a p i e r i n an otherwise undisturbed f l o w ^ ^ . 8 0  Their a n a l y s i s i s l a r g e l y  based on the f a c t that a pressure gradient e x i s t s along the stagnation l i n e formed by the i n t e r s e c t i o n of the plane of symmetry of the approach flow and the p i e r s u r f a c e , i f the approach flow v e l o c i t y i s not uniform (see Figure 6).  If the flow i s two-dimensional, then the stagnation  21.  pressure head, p/y, at any l o c a t i o n on the stagnation l i n e , i s j u s t equal to the v e l o c i t y head of the approach at that l e v e l , u /2g. i f u = u ( y ) , p = p ( y ) , where y i s the v e r t i c a l coordinate.  Thus,  Since the sum  of s t a t i c pressure head and e l e v a t i o n head remains a constant, a net pressure gradient e x i s t s along the stagnation l i n e .  This pressure  gradient induces an a c c e l e r a t i o n of the nearby f l u i d from the point of maximum pressure towards the regions of lower pressure.  If  viscous  e f f e c t s are neglected and one considers a streamline running downward along the stagnation l i n e , the conservation of energy p r i n c i p l e y i e l d s , f o r the secondary v e r t i c a l flow at any p o i n t , a v e l o c i t y head v /2g, which i s j u s t equal to the d i f f e r e n c e between the pressure head at that point and the maximum pressure head. vCy)  P* max.  2 =  2g  _  Thus:  p(y)  Y  Y  u max. max. 2g  2  u max.  2  "  iu(y) 2g  (19)  or: vCy)  2  =  -  u (y)  2  .....(20)  Moore and Masch contended t h a t the above concept provides the basis f o r the mechanism producing a strong v e r t i c a l l y downward flow j u s t i n f r o n t of the p i e r and thus accounts f o r the f a c t that the maximum depth of scour occurs at the nose of the p i e r rather than at the point of maximum breadth, as would be expected i f only the two-dimensional p o t e n t i a l flow pattern were considered.  This downward flow was thought  to s i g n i f i c a n t l y contribute to the formation and maintenance of the  22.  s p i r a l vortex i n the scour hole. On the basis of t h e i r understanding of the scour process, Moore and Masch b r i e f l y discussed four methods of reducing scour.  One method  c o n s i s t s of placing h o r i z o n t a l discs around the p i e r , below or above normal bed l e v e l , with p o s s i b l y a v e r t i c a l l i p around the outside edge to d e f l e c t the secondary flows upward away from the b e d . 1 1  Another  p r e v i o u s l y proposed method consists of p l a c i n g an a u x i l i a r y p i e r or p i l e 12  upstream of the p i e r to be protected  .  Masch and Moore thought t h a t  such a p i l e would "destroy" the v e l o c i t y gradient of the approach f l o w . The reduction i n scour depths achieved by sharpening the p i e r nose i s explained i n terms of the reduction i n the s i z e and strength of the downward flow component e f f e c t e d by such a nose.  A f i n a l method suggested  by Moore and Masch consists of a swept-back leading edge, p o s s i b l y required only near the bed, i n order f o r the d e f l e c t e d flow component which then deyelops to counteract the downward flow induced by the v e l o c i t y gradient of the approach f l o w . I.  BREUSERS; 1.  DELFT HYDRAULICS LABORATORY  Oosterschelde Bridge Model Studies In 1964, D e l f t Hydraulics Laboratory published a report of a  model study of scour around the p i e r s of the Oosterschelde Bridge, under the d i r e c t i o n of H.N.C. B r e u s e r s ^ ^ . 16  Experiments were c a r r i e d out using  c i r c u l a r c y l i n d r i c a l p i e r s i n order to i n v e s t i g a t e the i n f l u e n c e of the average approach flow y e l o c i t y , flow depth, p i e r diameter, and bed material.  I t was observed that with a bed sand of d g = 0.20 mm. 5  23.  (U v  .. =0.25 m./sec. = 0.81 f t . / s e c ) , maximum scour occurred when crit.  U/U  ..  reached a value of 1.4, and did not increase f o r greater  v e l o c i t i e s CU_„--  t  i s defined as the value of U at which general bed  load transport i s i n i t i a t e d ) .  I t was found that the r a t i o d  s e m  /b  decreased somewhat with increasing p i e r diameter, having a value of 1.5 f o r b = 11 cm. and 1.6  f o r b = 5 cm. Csee Table I ) .  This r e s u l t was  obtained f o r a flow depth H, f o r the 5 cm. p i e r , of 0.25 m. and a flow depth f o r the 11 cm. p i e r of 0.50 m., i n the understanding that s i m i l a r i t y would be b e t t e r approximated using s i m i l a r values of H/b rather than j u s t H alone.  The flume width was 0.95 m.  The influence of the flow depth on_the e q u i l i b r i u m depth of scour (below normal bed l e v e l ) was tested using the 11 cm. p i e r and flow depths of 0.15 m., 0.25 m., and 0.50 m., at the c o n d i t i o n U/U  = 1.4.  There  was no s i g n i f i c a n t d i f f e r e n c e i n the e q u i l i b r i u m scour depths f o r the two l a r g e r flow depths, while f o r the smallest flow depth, the e q u i l i b r i u m scour depth was somewhat l e s s .  I t i s i n t e r e s t i n g to note that at the  i n i t i a l stages of scour, at a time t = 15 min. from the s t a r t of scour, the smallest flow depth produced a l a r g e r scour depth than the two l a r g e r flow depths. Tests were also run w i t h a bed material of polystyrene spheres Cdensity = 1050 kg/m ) of diameter 1.5 mm. C U flume 3.5 m. wide. ing to d  c r i t  =0.09 m/sec.) i n a  For the 11 cm. p i e r , maximum scour depth correspond-  /b = 1.7 was reached, at U/U  J CHI  reached a steady value of 1.65 at U/U  ... = 1.2.  U l  .  I  The r a t i o d /b  w «  > 1.4 (see Table I ) .  JC  24. 2.  Scour around D r i l l i n g Platforms In 1965, Breusers reported the r e s u l t s of a p r i v a t e study of  scour around d r i l l i n g platforms^- ^. 17  d  c *  m  = ! -  4  He o b t a i n e d : ' C21)  D  sera J.  MAZA AND SANCHEZ Maza and Sanchez reported a study of l o c a l scour at bridge piers i n (79)  1964  '.  v  They reviewed the l i t e r a t u r e a v a i l a b l e to them and found that  there were b a s i c a l l y two d i f f e r e n t r e l a t i o n s . proposed by Laursen and i s given above. H/b, p i e r shape, angle of a t t a c k ) .  E s s e n t i a l l y i t was d  5 Q  s  g  = f(b,  The second r e l a t i o n s h i p i s due to  J a r o s l a v t s i e v , and can be formulated as d shape, d  The f i r s t of these was  g e  = f ( U , H/b , U /gb. , p i e r 1  , type of f l o w ) , where b represents the width of the p i e r 1  projected onto a plane perpendicular to the approach flow d i r e c t i o n . The r e s t r i c t i o n on Laursen's r e l a t i o n s h i p i s that i t i s v a l i d only f o r U > U .. , whereas J a r o s l a v t s i e v ' s r e l a t i o n s h i p i s v a l i d only f o r H/b > 1.5, and i n a d d i t i o n d^g i s included only i f i t exceeds 5 mm.  In terms  of a design equation, J a r o s l a v t s i e v ' s r e l a t i o n can be expressed as: d  Here  se  =  W  V V  f  i s a function of H/b  1  "  3  d  50  (  2  2  )  and decreases with increasing H/b^ up to a  value of H/b1 = 5 Csee Table I I ) .  In the i n t e r v a l 1.5 < H/b < 6, the  r e l a t i o n can be approximated by  = (ti/b^)' 7^.  x  The parameter  a1  varies from 0.6 f o r a p i e r i n a main flow channel to 1.0 f o r a p i e r on a 2  flood p l a i n .  The term Ky i s a function of the p i e r Froude number U / g b ^  25. O  A  O O  and can be expressed as  = 0.53(U / g b ) ~  c o e f f i c i e n t varies from  = 10 f o r c i r c u l a r p i e r s to K^. = 12.4 f o r  rectangular p i e r s .  1  .  The p i e r shape  A l l u n i t s are i n meters except f o r d^Q which i s i n  millimeters. Maza and Sanchez c a r r i e d out a number of experiments and compared t h e i r data to the two scour r e l a t i o n s .  They found that t h e i r data f i t t e d  the r e l a t i o n of J a r o s l a v t s i e y quite well f o r values of H/b > 1.5. percent of t h e i r data f e l l below Laursen's it.  Ninety  r e l a t i o n , and none exceeded  On the basis of t h e i r r e s u l t s , Maza and Sanchez proposed a  modified version of J a r o s l a v t s i e v ' s equation, i n the form: se BT" d  VttU  =  1  U g¥, 3  l  50 ~b~— 3  "  d  ( 2 3 )  l  with dj-g i n m i l l i m e t e r s and a l l other lengths i n meters. K  HU  The parameter  i s p r i m a r i l y a f u n c t i o n of U / g b , and secondarily of H/b 1  l a t t e r has l i t t l e i n f l u e n c e except f o r values of U /gb1 Figure 7).  <  1>  0.10  but the (see  For purposes of design, Maza and Sanchez recommend that the  smaller of the values given by equation (23) and Laursen's r e l a t i o n , equation ( 9 ) . be used, with the r e s t r i c t i o n that equation (23) i s v a l i d only f o r H/b1 d,-Q > 5 mm.  > 1.5, and the term with d  5 Q  be included only i f  26.  K.  SHEN et a l . : 1.  COLORADO STATE UNIVERSITY  Introduction Since 1962, workers at Colorado State U n i v e r s i t y , l a r g e l y  under the d i r e c t i o n of H.w*. Shen and S.S.  K a r a k i , under a contract with  the United States Bureau of P u b l i c Roads, have i n v e s t i g a t e d the problem of l o c a l bed scour at bridge piersCi09,no,ill,112,113}^  Experimental  work has involved the study of the v a r i a t i o n of the depth of scour with time, the dependence of the e q u i l i b r i u m depth of scour on the h y d r a u l i c parameters, the y e l o c i t y patterns of the flow i n and around the p i e r and scour h o l e , and the e f f e c t i v e n e s s of various methods of reducing scour. Theoretical work was aimed mainly at t r y i n g to understand the mechanics of the l o c a l scour phenomenon, e s p e c i a l l y with respect to the vortex at the p i e r base, and thus provide a conceptual framework f o r f u n c t i o n a l l y r e l a t i n g the e q u i l i b r i u m scour depth to the important h y d r a u l i c and other parameters of the problem. 2.  Description of the Mechanics of Local Scour Shen, e t . a_l_. gave a p r e l i m i n a r y d e s c r i p t i o n of the l o c a l  phenomenon, using a c i r c u l a r c y l i n d e r as a convenient e x a m p l e ^  110  scour  ^.  The  c y l i n d e r , by the pressure f i e l d i t induces, apprehends the v o r t i c i t y normally present i n the flow as i t i s swept towards the c y l i n d e r , and concentrates i t near i t s leading surface.  This process can also be  described as vortex tubes c o l l e c t i n g i n f r o n t of the c y l i n d e r , and being bent and stretched around the c y l i n d e r .  This process reaches a state of  approximate e q u i l i b r i u m when the rate of d i s s i p a t i o n of v o r t i c i t y at the boundaries (the c y l i n d e r surface and neighbouring bedj equals the rate at  27.  which v o r t i c i t y enters the region. The primary flow s t r u c t u r e associated with t h i s concentration of v o r t i c i t y i s the horseshoe vortex or h o r i z o n t a l r o l l e r which forms at the base of the p i e r , and i t i s the basic scouring agent.  If the  adverse pressure gradient induced by the p i e r i s s u f f i c i e n t l y s t r o n g , i t causes separation of the three-dimensional boundary l a y e r upstream of the p i e r .  This separated boundary layer r o l l s up to form the horseshoe  vortex system (see Figure 8 ) .  In a d d i t i o n , due to the mechanism  described by Moore and Masch (see above)>  a downward flow e x i s t s near  the c y l i n d e r leading surface. The horseshoe vortex system i s i n general grossly unsteady, and can include a secondary vortex i n a d d i t i o n to the primary vortex.  I t i s the  l a t t e r however which i s mainly responsible f o r the scouring a c t i o n . Scouring begins when the shear stresses at the periphery of the primary vortex reach the c r i t i c a l f o r sediment t r a n s p o r t .  Scouring i s a c t u a l l y  i n i t i a t e d somewhat downstream of the leading edge of the c y l i n d e r , i n the region where the free stream v e l o c i t y imposed on the vortex a c t i o n i s high. Shen, e_t al_. proposed a d i v i s i o n of the l o c a l scour phenomenon i n t o two basic cases: piers.  scour at blunt-nosed piers and scour at sharp-nosed  A blunt-nosed p i e r i s defined as one which induces an adverse  pressure gradient strong enough to cause the upstream boundary l a y e r to separate and r o l l up to form the horseshoe vortex. defined as one which lacks t h i s property.  A sharp-nosed p i e r i s  28. 3.  Theoretical Analysis The t h e o r e t i c a l treatment of the horseshoe vortex was done  mainly by A.T. Roper, and i s o u t l i n e d b e l o w ^  1 1 0  '  1 1 3  ^.  Consider a control  volume (of small thickness) ABCD s i t u a t e d on the plane of symmetry [stagnation p l a n e ) i n f r o n t of a c i r c u l a r c y l i n d e r i n a uniform, steady flow i n an open channel, as shown i n Figure 9.  The flow i n t h i s control  yolume i s supposed to be two-dimensional. The c i r c u l a t i o n , r , about the control volume ABCD i s defined to be:  r  =  (24)  i V • 3s  where V = the t o t a l v e l o c i t y v e c t o r , and ds = an i n f i n i t e s i m a l element of length along the boundary of ABCD.  For the flow r i g h t at the s o l i d  boundaries AD and CD, the " n o - s l i p " c o n d i t i o n a p p l i e s , i e . the flow v e l o c i t y there i s zero. boundaries i s zero.  Therefore the product V • cfs~ along these  Further, Roper s p e c i f i e d that AB i s f a r upstream of  the p i e r i n the region where the flow i s not a f f e c t e d by the presence of the p i e r .  Consequently, the vectors V and ds" are at r i g h t angles to one  another, and t h e i r dot product i s zero.  This leaves only one term  remaining, and we get:  r  =  udx  (25)  B  This s i m p l i f i c a t i o n i s possible regardless of the shape of the bed, thus, expression C25) i s y a l i d eyen i f a large scour hole i s present. The right-hand-side of equation (25) BC.  can be evaluated i f u i s known along  The l a t t e r requirement can be s a t i s f i e d i f BC i s s p e c i f i e d to be  29.  s i t u a t e d above, the region where the v e l o c i t y v a r i e s with depth (the shear l a y e r ) , and also well below the free surface.  The value of u  along BC i s given by p o t e n t i a l flow theory f o r flow around a c i r c u l a r c y l i n d e r (with co-ordinates as shown i n Figure 9) as: u(x)  =  ,2  -u  [1 - ^  (a+x)  0  where -u  -]  (26)  2  = the free stream v e l o c i t y at point B, and a = the c y l i n d e r  radius.  S u b s t i t u t i n g t h i s expression f o r u i n t o the right-hand-side  of  equation (25), and evaluating the i n t e g r a l , we get: C  udx  t x=6  = J  x-x  = u x  2  -u  dx  °  O -U  o o  -J  [1  [a - - f — ] a+x  o  (27)  o  S u b s t i t u t i n g t h i s back i n t o equation (25), and r e - a r r a n g i n g : ax =  r  u x o o  "  u  a+x  o  o  (28)'  I f AB i s s p e c i f i e d to be f a r upstream of the p i e r , so that X  q  »  a,  then (a + x ) = x , and: o o ax o  so that equation (28) r  =  u x o o  -  becomes; au  o  (29)  For the undisturbed flow without a p i e r , the c i r c u l a t i o n r = u x . o o  30.  Therefore, the net e f f e c t of the p i e r i s to reduce the c i r c u l a t i o n by an amount: A r pier  =  -au  C30)  Q  Roper assumed that the reduction in c i r c u l a t i o n due to the p i e r i s proportional to the strength of the horseshoe vortex core.  Roper f u r t h e r  assumed that the horseshoe vortex core rotates as a r i g i d body, and that i t s strength can be represented by the term co A v e l o c i t y , and A the a r e a , of the yortex core. c assumptions i n the form of an equation: aU  o <*  CuA  cW  £  where co i s the angular  Expressing  these  C31)  Roper argued that since flow separation i s a viscous e f f e c t , the kinematic v i s c o s i t y of the f l u i d , v , was an important v a r i a b l e . Incorporating  t h i s i n t o equation C31), and r e p l a c i n g the p i e r r a d i u s , a ,  by the p i e r diameter, b, a non-dimensional . ^  bu =  v  r e l a t i o n i s obtained:  f  (32)  T _ O - |  v  The term on the right-hand-side  of the equation i s the Reynolds number  based on the p i e r w i d t h , R^, so t h a t : ^  =  f IR J 5  (33)  Now, since the horseshoe yortex i s supposed to be the primary agent causing scour, the depth of scour should be r e l a t e d to the strength of the vortex core.  Thus;  31. where R  fa  = the p i e r Reynolds number, and d  $ e  = the e q u i l i b r i u m depth of  scour below normal bed l e v e l . This analysis implies that the p i e r a c t u a l l y reduces the v o r t i c i t y i n the control yolume.  Further, the concentration of v o r t i c i t y in the  horseshoe vortex means that v o r t i c i t y i s reduced elsewhere.  Such, a  reduction i s i n f a c t observed i n the region between the separation l i n e and the horseshoe yortex i t s e l f , where the flow i s r e l a t i v e l y quiescent. 4.  Experimental  Results  The experimental work, c a r r i e d out at Colorado State U n i v e r s i t y confirmed the findings of previous studies i n that the shape of the scour hole around a blunt-nosed p i e r approximates the frustum of an inverted cone, with i t s sides at about the angle of repose, and that the maximum depth occurs at the upstream edge of the p i e r .  For sharp-nosed  piers  aligned w i t h the flow d i r e c t i o n , i t was found that the maximum depth of scour occurred at the downstream end of the p i e r . A q u a l i t a t i v e representation of the flow patterns observed i n the scour hole i s shown i n Figure J O ^  1 0 9  '  1 1 0  proportional to the measured values.  ^.  The v e l o c i t y p r o f i l e s are  The dashed l i n e s i n d i c a t e the  dominant flow f e a t u r e s , which are i n general not f i x e d but unsteady i n nature. Shen and his co-workers examined a l l the data a v a i l a b l e to them from the l i t e r a t u r e , but considered as useable only the data of Chabert and Engeldinger, besides t h e i r own, as a l l other data were e i t h e r incomplete or were obtained from experimental arrangements differed in their significant details.  and methods which  These other data were however  32.  used f o r purposes of comparison. Shen p l o t t e d the useable data, f o r sands with d^Q < 0.52  mm., i n  terms of the e q u i l i b r i u m scour depth [long-term average v a l u e ) , d versus the p i e r Reynolds number, R^, as shov/n i n Figure  ll^ ^. 113  s g  ,  This  p l o t produced the r e l a t i o n :  d  = 0.00073 R  $ e  C35)  0 , 6 1 9 b  which forms an approximate envelope f o r a l l the data.  This r e l a t i o n s h i p  i s supported by the other data mentioned above. The s c a t t e r of the data a r i s e s not only from the d i f f e r e n c e s between the various experimental arrangements and methods used, and.inherent e r r o r s t h e r i n , but i s due to the shortcomings of the analysis i t s e l f , as enumerated by Shen i n the f o l l o w i n g p o i n t s . For a given p i e r geometry and sand s i z e , the curve of d  Ca)  s g  versus  R^ r i s e s r a p i d l y to some maximum, beyond which i t f a l l s o f f somewhat, as shown i n Figure Cb) Figure  12.  The curve described i n Ca) d i f f e r s f o r each p i e r , as shown i n  12.  Cc)  A d i f f e r e n t sand s i z e also gives a d i f f e r e n t curve, even i f  the p i e r geometry i s constant, as shown i n Figure  13.  Shen et aj_. therefore recommended that equation (35)  be used to  estimate the e q u i l i b r i u m depth of scour f o r the " c l e a r - w a t e r " scour case only.  The r e s u l t w i l l be on the safe s i d e .  A cayeat was added, however,  i n that the laboratory r e s u l t s haye to be extrapolated several orders of magnitude i n order to apply to the much higher p i e r Reynolds numbers of prototype c o n d i t i o n s .  For the case of scour v/here there i s general  33.  bedload t r a n s p o r t , Shen e_t al_. recommended that e i t h e r the r e l a t i o n proposed by Breusers: d  sem  -  -  :  4  b  ( 2 1 !  or the one of Larras; d  =  1.42 K b  0  ,  7  (11)  5  sem  '  v  be used to determine the maximum e q u i l i b r i u m depth of scour below normal bed l e v e l . 5.  R e c o n c i l i a t i o n of Diyergent Concepts.The p i e r Reynolds number r e l a t i o n (equation 35) was f u r t h e r  analyzed, and i t was found that by assuming a uniform flow r e l a t i o n of the type: U  =  K.' H  C36)  e  where K' and e are constants, equation (35) could be transformed to an expression of a form s i m i l a r to Laursen's equation ( 9 ) , or an expression of the form: %  =  C[F (^) J 2  3  m  (37)  where C and m are constants, and F i s the flow Froude number, U//gTT. Shen therefore concluded that arguments about whether the depth of flow or the average approach v e l o c i t y of the flow i s more important are of no great consequence, since the two parameters are r e l a t e d according to equation C 3 6 ) ^  110  ^.  \  34. 6.  Methods of Reducing  Scour  Shen and his co-workers also c a r r i e d out a number of experiments [apparently i n the bed-load transport range only) to t e s t various methods of reducing the depth of l o c a l scour around bridge p i e r s , the r e s u l t s of which can be summarized as f o l l o w s : Ca)  Sharp-nosed piers were made by fastening noses, of included  angle 8 = 15° and 6 = 3 0 ° , onto a standard rectangular p i e r .  The  l o c a t i o n , shape, and s i z e of the scour hole were quite v a r i a b l e and no c o r r e l a t i o n w i t h any of the usual parameters was e v i d e n t , except that some of the v a r i a t i o n seemed to be due to s l i g h t angles of attack e f f e c t e d by bed forms near the nose of the p i e r .  For some runs, i n which  the upper regime plane-bed condition obtained, no scour at a l l occurred. Cb)  Rectangular p i e r s on a p r o t r u d i n g , f l a t , pile-supported f o o t i n g  s i t u a t e d below normal bed l e v e l afforded reductions of between 0% and 55%, the higher values being achieved with higher v e l o c i t i e s . Cc)  An arrangement s i m i l a r to Cb) but with a v e r t i c a l l i p around  the edge of the footing y i e l d e d reductions of the order of 40%-50%. the f o o t i n g and l i p are s i t u a t e d low enough, the arrangement  When  presumably  traps the horseshoe vortex and prevents i t from a f f e c t i n g the e r o d i b l e bed. Cd)  A rectangular p i e r with roughness elements attached to the  f r o n t did not give any s i g n i f i c a n t reductions i n the scour depth; the vortex system seems to be merely displaced s l i g h t l y upstream.  The  roughness elements may have been too close together to r e t a r d any downward flow i n f r o n t of the p i e r .  35. (e)  A c y l i n d e r of one-half the width of the rectangular p i e r was  placed in f r o n t of the l a t t e r , and a maximum reduction of about 60% was achieved when the c y l i n d e r was placed a distance of about two c y l i n d e r diameters upstream of the rectangular p i e r . Cf)  A c y l i n d e r s p l i t in the d i r e c t i o n of flow ( i e . h a l f c y l i n d e r s  separated by a distance of from 1/3 to 2/3 the c y l i n d e r radius) gave reductions of the order of 25% to 40%.  I t was observed that the horse-  shoe y o r t e x , although s t i l l present, was weaker than f o r the s o l i d cylinder.  L.  CARSTENS Carstens attacked the l o c a l scour problem by separating the f l u i d ,  sediment, and flow parameters from the geometric v a r i a b l e s ^ ^ . 2 0  A  t h e o r e t i c a l analysis of the forces a c t i n g on a t y p i c a l bed p a r t i c l e , assuming a n e g l i g i b l y t h i n boundary l a y e r , y i e l d e d a r a t i o of d i s t u r b i n g f o r c e , F^, to retarding f o r c e , F^, o f : F  M  p-i r  =  f Csediment p a r t i c l e geometry) N  R  2  (38)  s  U"  where N s  =  , the "sediment number." /Cs-l)g  d  g  sediment s p e c i f i c g r a v i t y , and d diameter.  In the l a t e r , s = the  g  = the c h a r a c t e r i s t i c sediment grain  For the case of c l e a r water scour CQ  S  sediment transport out of the scour a r e a , Q  $  ^  i n  t  = 0 ) , the rate of was assumed to be a  f u n c t i o n of the force r a t i o and the geometry of the s i t u a t i o n . non-dimensional  sediment transport f u n c t i o n was hypothesized:  Thus a  36.  s out u Wd s g  _  f[(fl  _ N ) , s , o b s t r u c t i o n geometry, sc , L  2  2  s  L V  d  sediment p a r t i c l e geometry]  (39)  where W = the scour hole w i d t h , L = a c h a r a c t e r i s t i c length of the g  obstruction, N = the c r i t i c a l sediment number at which l o c a l scour i s sc i n i t i a t e d , u = a reference v e l o c i t y , and d  = the scour depth below  g  normal bed l e v e l . Approximating the scour hole shape at a c i r c u l a r c y l i n d e r of diameter b by the frustum of an inverted cone of base diameter b and side slope equal to the angle of repose, <j>, Carstens rewrites equation (8)  as: =  n x  dV  s out  =  (S  *  dt  +  b  tan<j> tan<}>  )  d  d  ' s  v  dt  ( d  v  ( 4 Q )  }  s'  S e l e c t i n g the data of run 204 from Chabert and Engeldinger, and d e f i n i n g u = U, the average approach flow v e l o c i t y , L = b, the p i e r diameter, and W = b + 2(_d /tancj>), Carstens constructs a p l o t o f : s  s  d  2  s  Q ./U(b + —r) s our tanj> w  v  (N  2 s  -  T  N  2 s c  )  5  /  y  d g  d s  u c  2  the index 5/2 of the term ( N  G  - N  S C  )  having been obtained from  experiments of Le Feuvre, who studied sediment transport rates out of a rigid-boundary depression i n a closed conduit.  —  Q $ o u t  2d  = i . 3 io~ CN 5  2  s  - N p/ 2  sc  2  This p l o t gave:  c  d s  r  3  ^-i  C41)  37.  Equation (41) can be s u b s t i t u t e d i n t o C40), and the r e s u l t i n t e g r a t e d , to y i e l d an expression f o r scour depth with time.  This expression shows  that the scour depth increases c o n t i n u a l l y with time, and although the rate of increase decreases, an e q u i l i b r i u m scour depth i s never a t t a i n e d .  M.  TANAKA AND YANO, AND THOMAS 1.  Tanaka and Yano Tanaka and Yano report a laboratory study i n v o l v i n g the e f f e c t ,  on l o c a l scour, of various devices and m o d i f i c a t i o n s i n conjunction with a circular cylinder^  1 1 8  ^.  They considered that the horseshoe vortex i s  generated by a combination of main flow separation upstream of the c y l i n d e r and the secondary downward flow along the f r o n t of the c y l i n d e r , and that the magnitude of l o c a l scour depended on the strength and s i z e of the v o r t e x . The experimental arrangement consisted of a 30 cm. wide flume, with a bed sand of d^Q = 0.4 mm. and a flow depth of 10 cm.  Flow conditions  were arranged to j u s t obtain s l i g h t r i p p l e s on the bed, and were the same for a l l tests.  The v e l o c i t y d i s t r i b u t i o n was measured at the t e s t  l o c a t i o n and i s shown i n Figure 14. c y l i n d e r of 3 cm. diameter.  The basic p i e r used was a c i r c u l a r  The d i f f e r e n t types tested are l i s t e d below  and are also i l l u s t r a t e d shceraatically i n Figure 15. Ca)  Type I:  a normal c y l i n d e r extending to aboye the water surface.  (b)  Type II:  a c y l i n d e r with a hole 1 cm. square or 2 cm. square  cut i n the d i r e c t i o n of the main flow.  38.  Cc) =  Type III:  a c y l i n d e r f i t t e d with a c i r c u l a r d i s c of diameter  9 cm., 12 cm., or 18 cm.  d Cd)  Type IV<  a c y l i n d e r submerged below the water surface.  Ce)  Type V:  a c y l i n d e r f l o a t i n g above the normal bed surface.  The p o s i t i o n s of the m o d i f i c a t i o n s of the type II  and type III  piers,  and the top of the type IV, and bottom of the type V p i e r s , were v a r i e d . The experimental r e s u l t s are summarized i n Figure 16. Tanaka and Yano noted that the e f f e c t of the discs was  pronounced  only w i t h i n the boundary l a y e r C<S = 5 cm.), and that the s i z e of the d i s c Cwhen placed at bed l e v e l ) required to completely e l i m i n a t e scour was of the order of the boundary l a y e r thickness.  They therefore concluded  that the s i z e of the vortex i s r e l a t e d to the boundary l a y e r e s p e c i a l l y since the vortex i s a boundary l a y e r e f f e c t .  thickness,  Thus, they  argued that the e f f e c t of the p i e r modifications can be v a l i d l y represented i n terms of an expression of the form: scour hole s i z e = f Cji  C42)  where h = the height of the m o d i f i c a t i o n above normal bed l e v e l , and 6 = the boundary l a y e r  thickness.  Tanaka and Yano considered the v e r t i c a l downflow i n f r o n t of the p i e r to have l i t t l e e f f e c t on l o c a l scour, f o r t h e i r experimental conditions. 2.  Thomas Thomas reports a study of the e f f e c t of a d i s c f i t t e d around a  c i r c u l a r c y l i n d e r , s i m i l a r to Tanaka and Yano's type I I T ^ ^ . 1 2  He used  d i s c s of 10 cm. and 15 cm. diameters around a 5 cm. diameter p i e r , placed  39.  i n a flume 35 cm. wide.  His r e s u l t s , shown i n Figure 1 7 , are not  s t r i c t l y comparable because he used a scour depth equal to the a r i t h metical average of the scour at the c y l i n d e r quarter p o i n t s : back, and two s i d e s .  front,  Further, no information i s giyen with respect to  the flow c o n d i t i o n s , and the scour depth f o r the normal c y l i n d e r i s not stated.  N.  SCHNEIDER Schneider, one of Shen's co-workers at Colorado State U n i v e r s i t y ,  r e c e n t l y completed a doctoral d i s s e r t a t i o n on the mechanics of l o c a l scour^  1 0 6  ^.  In i t , he giyes a d e s c r i p t i o n of the mechanics of l o c a l  scour, based on the accumulated information of previous s t u d i e s , as well as an o u t l i n e of the more important concepts and r e l a t i o n s by previous i n v e s t i g a t o r s .  proposed  His own t h e o r e t i c a l work was mainly d i r e c t e d  towards formulating a r e l a t i o n f o r l o c a l scour based on the sediment c o n t i n u i t y equation [equation C8)), and his experimental work consisted l a r g e l y of determining sediment transport rates out of the scour hole. Other considerations included the e f f e c t of c o n s t r i c t i o n of the f l o w , unsteadiness  of the horseshoe v o r t e x , design c r i t e r i a , and s a f e t y  factors. Making the same assumption as Carstens with respect to the scour hole shape Csee above), but transforming the rate of change of scour hole yolume to the r a t e at which s o l i d s are removed, an expression s i m i l a r to equation C40) i s obtained; 2 ,  TTC1-X)P g  «.  • -tssr-  d_  'tk  j +  b d  s i at C s' d  C 4 3 )  40. where Q ' = the rate at which the scour hole i s deepened, i n terms of weight of s o l i d s remoyed per u n i t time, A = the void r a t i o of the sediment in the scour h o l e , and p  g  = the sediment d e n s i t y .  Equation  (8)  can then be expressed as:  Cl -  \  Q ' 4. Q ' • (44) s s s out s in = the time rate of sediment transport out of the scour hole  dt where Q s out v  ^3P 9 = Q  A i M  1  3  '  s  =  x  v  x  1  r  (by w e i g h t ) , and Q ' ^  n  = the time rate of sediment t r a n s p o r t i n t o the  scour hole (by weight). Schneider assumed that the value of  could be s u f f i c i e n t l y well  described by an exponential decay f u n c t i o n of the i n i t i a l v a l u e , Q  V where t  %±  =  c  e _ t / t c  1  :  C 4 5 )  = a time constant.  Schneider f u r t h e r thought that Q ' .  in ,  3  rather than have i t vary with scour hole s i z e , could best be represented by the constant value given by the rate of transport coming i n across a width equal to the p i e r w i d t h , i e . Q ' . = q ' . b. s i n s 1n n  n  Equation (45)  thus divided by the p i e r width and then s u b s t i t u t e d i n t o equation  is  (43),  which, when i n t e g r a t e d , y i e l d s : (  IT  )3  ! iT  +  C  ) 2  3Q'  where X  =  t a n  *  t tan<J> ——o — *(1-A)p g b  =Ml-e  _ t / t  c]  (46)  2  • As time becomes large ( i e . t »  J  t ) equation c  s  (46)  becomes: <i  3  t-f-)  , +  |  d (Hp).  2 tan*  -  X  (47)  41.  This equation implies that f o r a given p i e r s i z e and given sediment c h a r a c t e r i s t i c s , the e q u i l i b r i u m scour depth depends only on the time constant and the i n i t i a l rate of removal of sediment from the scour hole.  The time constant i s determined by combining equations (47) and  (46) f o r t = t , and was found to be the time required f o r the r a t i o c  d /d s  to reach a value from 0.796 to 0.858, depending on the r e l a t i v e  $ e  importance of d from d  /b and tan<j>.  versus t curves.  s  measuring Q ' s  o u t  The value of t  can thus be estimated  Schneider decided to determine Q -  1  sl  by  i n the laboratory and r e l a t i n g the former to the l a t t e r  by means of equations (44) and (45).  Since these measurements were made  f o r the i n i t i a l f l a t - b e d c o n d i t i o n , the exponential time term reduces to u n i t y , and the value of t  Q  Values of Q ' s  1  n  i s not required i n order to obtain Q - 1  S1  were obtained from e x i s t i n g data f o r sediment transport  in flumes.' Schneider's measurements of Q ' . yielded a r e l a t i o n : ^s out J  Q ' s  o  u  t  b  where U  c  =  0.0754CU-U )  (48)  3  c  = the average approach flow v e l o c i t y at which l o c a l scour i s  just initiated.  This v e l o c i t y was estimated to be equal to about 0.4  f t . / s e c , based on Knezevic's data (see above).  The data did not d i s p l a y  any systematic s c a t t e r with respect to sediment s i z e , p i e r w i d t h , or . p o s i t i o n of sediment feeding device.  Values of q '  s  i n  were obseryed to  be about one order of magnitude lower than yalues of Q ' ^ / s  o u  D  Cat the  i n i t i a l f l a t - b e d c o n d i t i o n ) , f o r most flow conditions tested 0"e. low velocities)..  42.  The time constant t  was determined from data of Chabert and  Engeldinger, and Shen, et. al_. , versus U.  L l 1 3  ^ and p l o t t e d as graphs of l/t  c  S i g n i f i c a n t c o r r e l a t i o n with p i e r diameter is also e v i d e n t ,  although the influence of sediment s i z e is not c l e a r . measured values of t  A comparison of  w i t h values c a l c u l a t e d from the known data, using  equations [48), [44), [45), and (47), shows f a i r agreement, although a d e f i n i t e s c a t t e r is evident f o r higher values of t , with the c a l c u l a t e d c  values being mostly lower than the actual measured values.  Predicted  scour depths based on the c a l c u l a t e d values would thus tend to be greater than those that would a c t u a l l y occur. One of the more i n t e r e s t i n g r e s u l t s of Schneider's work involves the well-known observation that in the region of scour with general bed-load t r a n s p o r t , the e q u i l i b r i u m scour depth is independent of the v e l o c i t y U (see Figure 3). product (Q -' S1  A constant scour depth implies a constant value of the  • t ) , by equation (46). c  However, i t has been noted that  at i n i t i a l c o n d i t i o n s , at l e a s t f o r v e l o c i t i e s of about two f e e t per second or l e s s , Q ' s i  =  Vout'  changes in i n i t i a l values of Q ' s  s  i  o u t  n  c  e  Vin  -  1 0 %  o f  Vouf  T h u s  (due to changes in v e l o c i t y ) are  almost completely balanced by changes in t , and the actual incoming sediment supply has only a small e f f e c t on the e q u i l i b r i u m scour depth. The development of general sediment transport i s thus not the true basis f o r the f l a t t e n i n g of the d  s e  yersus U curve; r a t h e r , i t seeros to be  based on the mechanics of the yortex i t s e l f . Schneider therefore proposed a separation of the l o c a l scour phenomenon i n t o two regions.  The f i r s t r e g i o n , where the e q u i l i b r i u m  43.  scour depth increases w i t h v e l o c i t y , he c a l l e d the " v o r t e x - b e d i n t e r a c t i o n - l i r a i t e d depth of scour r e g i o n " , since the e q u i l i b r i u m scour depth i s l i m i t e d by the shear stress that the vortex can exert on the bed.  The f l a t region of the curve he c a l l e d the "vortex-mechanics-  l i r o i t e d depth of scour r e g i o n " , since the e q u i l i b r i u m scour depth i s l i m i t e d by the inherent i n a b i l i t y of the vortex to penetrate below a c e r t a i n depth.  The e q u i l i b r i u m scour depth i n t h i s l a t t e r region can be  a f f e c t e d somewhat by the incoming sediment transport. regions are shown schematically i n Figure 18.  The two scour  This f i n d i n g modifies the  r e s u l t of Chabert and Engeldinger as given i n Figure 3.  0.  COLEMAN Coleman reports the r e s u l t s of analyzing laboratory data of H.W. ( 2&)  Shen and of h i m s e l f  u L  .  He obtained several groups of v a r i a b l e s by  dimensional a n a l y s i s , and then used the experimental data to obtain a r e l a t i o n s h i p between them.  He gotan expression f o r e q u i l i b r i u m depth  of scour as f o l l o w s : d  se  1.49  b  9/10  ,,2 (-)  2g  1/10  (49)  44. NOTES TO CHAPTER II  1.  Timonoff^- ^ r e f e r s to studies by Minard (1856) and Durand-Claye 126  C1873), but c i t e s no s p e c i f i c works f o r these. 2.  Reported i n Karaki and H a y n i e ^ ^ .  3.  Reported i n Karaki and H a y n i e ^ \  51  5 1  Some presentation of t h i s work  by I n g l i s can be found Q n t e r a l i a ) i n Arunachalam, I n g l i s ^ , Joglekar^ 4.  4 6 )  , and  ' Chi t a l e  ,  Thomas^ *. 121  This condition corresponds to the c o n d i t i o n of " c l e a r - w a t e r " s c o u r , j u s t below the threshold of general sediment t r a n s p o r t , as o u t l i n e d by Laursen.  5.  Sediment transport rates are' here defined i n terms of volume, i n order to make the equation dimensionally homogeneous.  Void r a t i o s  would have to be allowed f o r . 6.  The term " c l e a r - w a t e r " does not imply that v i s i b i l i t y i s good, but i n d i c a t e s only that there i s no transport of bed sediment.  7.  Reported i n Karaki and Haynie, r e f . 191.  8.  Median grain diameter = d ^ = the grain s i z e such that 50% of the material (by weight) i s f i n e r .  a]/  113  ^.  9.  Reported by Shen, et  10.  Reported by Shen, et a l _ . ^ -  11.  This idea had already been tested by Schneible, as reported i n  113  ^ , and N e i l l ^ ^ . 8 3  Laursen and Toch^- ^. 61  12.  Tison seems to have been the f i r s t to suggest and t e s t t h i s method.  He understood the a u x i l i a r y p i l e to reduce the curvature  of the flow and thus reduce scour.  45.  CHAPTER  III  THE MECHANICS OF LOCAL SCOUR  A.  INTRODUCTION The purpose of t h i s chapter i s to a r r i v e at an understanding of the  mechanics of l o c a l scour, based on observations and f i n d i n g s of previous i n v e s t i g a t i o n s and experiments as summarized i n Chapter I I ,  and on the  r e s u l t s of other s t u d i e s , notably those i n the f i e l d of f l u i d dynamics. Such an understanding should then proyide some basis f o r assessing the v a l i d i t y of the various hypotheses and methods of a n a l y s i s of the l o c a l scour phenomenon, and also suggest some guidelines f o r f u r t h e r study.  46.  B.  THE BASIC VORTEX MECHANISM There can be l i t t l e doubt that the basic -mechanism of l o c a l scour i s  the h o r i z o n t a l r o l l e r or v o r t e x , shaped l i k e a horseshoe i n p l a n , that forms on the bed d i r e c t l y i n f r o n t of a p i e r .  This v o r t e x , due to the  high v e l o c i t y at which i t r o t a t e s , exerts shear stresses on the bed that are greater than the c r i t i c a l shear stress f o r transport of the bed-sand. Consequently, bed sand i s moved away from the p i e r , and a scour hole develops. The horseshoe vortex has been observed and reported by many investigators, including Posey^ ' 93  C 8 0 )  , Maza and S a n c h e z  and S c h n e i d e r ^  106  ^.  C 7 9 )  9  5  \  Laursen^ '  , Shen e V a l /  59  1 0 9  '  1 1 0  '  6  1 1 3 )  1  \  Moore and Masch  , Tanaka and Y a n o  ( l l 8 )  ,  Shen, Schneider, and Tanaka and Yano i n p a r t i c u l a r ,  have recognized the importance of the v o r t e x , and i n t h e i r analyses have considered the vortex mechanism. Any attempt to understand or e x p l a i n l o c a l scour without taking i n t o account t h i s basic vortex mechanism i s doomed to f a i l from the very outset. Thus, an approach l i k e that of Tarapore (see s e c t i o n II.  G above), which  i s based on p o t e n t i a l flow theory and a concept of the free stream flow d i f f u s i n g i n t o the scour h o l e , can never lead to a proper understanding of the l o c a l scour phenomenon (although his experimental r e s u l t s remain v a l i d i n themselves). Ishihara { s e c t i o n II.  S i m i l a r l y , the t h e o r e t i c a l analyses of Tison and B) and Carstens (section II.  L) are misleading.  47. C.  ORIGIN OF THE HORSESHOE VORTEX 1.  Vorticity B a s i c a l l y , the horseshoe vortex o r i g i n a t e s out of the v o r t i c i t y  which i s always present i n shear f l o w s , that i s , flows i n which a v e l o c i t y gradient e x i s t s . A shear flow may be v i s u a l i z e d as c o n s i s t i n g of t h i n layers of f l u i d , with each layer moving at a s l i g h t l y d i f f e r e n t v e l o c i t y than the adjacent layers.  This d i f f e r e n t i a l movement r e s u l t s i n the s l i d i n g or shearing  of  adjacent layers oyer each other, and i n so doing causes a r o t a t i o n of the f l u i d p a r t i c l e s on the plane of s l i d i n g .  V o r t i c i t y represents the rate of  r o t a t i o n or the angular v e l o c i t y of these f l u i d p a r t i c l e s . A l i n e drawn through the f l u i d so that i t everywhere corresponds  to  the l o c a l axis of r o t a t i o n of the f l u i d p a r t i c l e s i s c a l l e d a vortex l i n e . If adgacent vortex l i n e s are j o i n e d together to form a s u r f a c e , the r e s u l t i s a vortex tube.  The f l u i d w i t h i n t h i s tube i s c a l l e d a vortex  tube or f i l a m e n t , or simply a vortex. Mathematically, v o r t i c i t y i s a v e c t o r , and i s expressed 0  =  as:  V x v  (50)  Expressed i n terms of orthogonal 3v  3v  components, t h i s  becomes:  48.  A theorem (due to Helmholtz and K e l v i n ) states that f o r any giyen v o r t e x , the product of the v o r t i c i t y and the c r o s s - s e c t i o n a l vortex must remain constant.  area of the  An important consequence of t h i s theorem i s  that vortex l i n e s cannot begin or end anywhere w i t h i n the f l u i d i t s e l f , but must e i t h e r form closed loops or terminate at the flow boundaries. Another theorem states that vortex l i n e s move with the f l u i d .  An  e x c e l l e n t d e s c r i p t i o n of the c h a r a c t e r i s t i c s of v o r t i c i t y has been presented by L i g h t h i l l ^ - ^ . 7 1  2.  V o r t i c i t y in the Flow near a Cylinder Consider the case of a uniform steady flow i n an open channel,  past a v e r t i c a l c y l i n d e r located i n the middle of the channel, with the x - d i r e c t i on i n the d i r e c t i o n of f l o w , the y - d i r e c t i on across the d i r e c t i o n of f l o w , and the z - d i r e c t i o n along the v e r t i c a l . (a)  The Approach Flow  Consider f i r s t the approach flow upstream of the c y l i n d e r i n the central part of the channel, where neither the c y l i n d e r i t s e l f nor the side w a l l s have any e f f e c t ( i e . a two-dimensional f l o w ) . t h i s flow has an x-component only (v ) . x  The v e l o c i t y of  A l s o , a v e l o c i t y gradient e x i s t s  only i n the v e r t i c a l or z - d i r e c t i o n , due to the f r i c t i o n a l drag of the bed of the channel.  The t o t a l v o r t i c i t y of the f l o w , as expressed by  equations 51 to 53, i s therefore reduced to a s i n g l e term: a y  =  fVx_  ....(54)  9z  The v o r t i c i t y appears i n the flow as yortex tubes or f i l a m e n t s , with axes of r o t a t i o n aligned i n the y - d i r e c t i o n , a l l p a r a l l e l to one another.  49. The vortex tubes are concentrated at the bottom of the f l o w , near the bed, i n the region where the v e l o c i t y gradient i s high.  Since the flow  being considered i s steady and uniform, the d i s t r i b u t i o n of the v o r t i c i t y w i t h i n the flow (the v o r t i c i t y f i e l d ] does not change i n the downstream d i r e c t i o n , and the rate at which v o r t i c i t y i s being produced i s j u s t equalled by the rate at which i t i s being d i s s i p a t e d . Cb]  Flow near the Cylinder As the flow from upstream approaches the c y l i n d e r i t d i v i d e s to  pass around i t , and each vortex tube, being c a r r i e d downstream by the f l o w , i s stretched as the streamlines diverge.  This s t r e t c h i n g begins  somewhat  upstream of the nose of the c y l i n d e r due to the c y l i n d e r - i n d u c e d pressure f i e l d (blunt-nosed c y l i n d e r s inducing a stronger pressure f i e l d and thus a f f e c t i n g the flow f a r t h e r upstream, than sharp-nosed ones).  This  s t r e t c h i n g reduces the c r o s s - s e c t i o n of each vortex tube so that the vorticity  increases.  As the flow passes around the c y l i n d e r , the vortex tubes are also bent around the c y l i n d e r , and a c t u a l l y become trapped or caught i n f r o n t of the c y l i n d e r , since vortex tubes cannot be cut even by a sharp f r o n t . Thus not only do the vortex tubes continue to s t r e t c h and increase i n v o r t i c i t y , but new ones are constantly being added from upstream.  This  process r e s u l t s i n intense v o r t i c i t y and vortex motion at the f r o n t and sides of the c y l i n d e r .  Howeyer, the v o r t i c i t y does not increase without  l i m i t , since viscous and turbulence e f f e c t s cause the d i f f u s i o n and convection of v o r t i c i t y to take p l a c e d " *  1 1 0  ).  The process of v o r t i c i t y i n t e n s i f i c a t i o n by s t r e t c h i n g and accumulat i o n , together with viscous and turbulence e f f e c t s , accounts f o r the o r i g i n  50.  and formation of the horseshoe vortex system, and, as such, accounts q u a l i t a t i v e l y f o r a l l scour at c y l i n d e r noses, since no other mechanism e x i s t s f o r the transport of sediment i n t h i s region.  D.  CHARACTERISTICS OF THE HORSESHOE VORTEX 1.  Structure The horseshoe vortex i s not j u s t a simple and s o l i t a r y h o r i z o n t a l  vortex that forms at the nose of a c y l i n d e r or p i e r in open-channel f l o w , but a c t u a l l y consists of a number of l i n k e d v o r t i c e s that operate together to scour the bed around the p i e r . Shen e_t a l _ . ^  1 1 0  Figures 8 and 10.  ^ obseryed a secondary "counter" vortex as shown i n  Thwaites^  125  ^ presents a photograph of an experiment by  Gregory and Walker, of a c y l i n d e r i n a laminar boundary l a y e r , which shows at l e a s t three v o r t i c e s .  This has been sketched i n Figure 19.  Rainbird et a l . report the same s t r u c t u r e as t h i s , but with s t i l l vortex upstream of the other t h r e e ^ ^ . 9 6  Roper^ ^ 101  another  reports three v o r t i c e s ,  as does Thwaites, above, although his proposed sequence of vortex formation and shedding i s in e r r o r . 2.  Formation The actual formation of the horseshoe vortex system occurs as  follows.  The adverse pressure gradient generated by the c y l i n d e r causes  the approach flow to slow down and, due to the y e l o c i t y gradient i n the v e r t i c a l d i r e c t i o n , to acquire a downward component i n f r o n t of the cylinder.  Q u a l i t a t i v e l y , t h i s i s the mechanism described by Bata  Csection IT. F] and Moore and Masch ( s e c t i o n n .  H), although t h e i r  51.  q u a n t i t a t i v e analyses, based on two-dimensional i n v i s c i d f l o w , are not valid. This downward flow i n turn induces a flow i n the upstream d i r e c t i o n along the bed i n f r o n t of the p i e r nose, with subsequent flow separation upstream of the p i e r .  A large h o r i z o n t a l vortex i s thus induced, which,  by viscous i n t e r a c t i o n w i t h the boundaries and the approach f l o w , generates y e t other v o r t i c e s . Another way of d e s c r i b i n g what happens i s to say that the adverse pressure gradient induced by the p i e r causes the separation and r o l l i n g up of the boundary l a y e r i n f r o n t of the p i e r to form the horseshoe system.  vortex  The horseshoe yortex system i s thus also seen to be the area of  concentration of the v o r t i c i t y present i n the boundary l a y e r of the approach flow.  This i s supported by T h w a i t e s ^  125  ^ and B o l c s ^ ^ , who 1 4  both s t a t e that the horseshoe vortex system i s constantly being r e p l e n i s h ed by f l u i d from upstream. 3.  Strength The a c t i o n of l o c a l scour at a bridge p i e r can be d i v i d e d i n t o  two p a r t s : (a) the resistance of the channel bed to scour. (b) the scouring strength or power of the vortex. For non-cohesive m a t e r i a l , the bed resistance depends only on the grain c h a r a c t e r i s t i c s of the bed sediment, and thus c o n s t i t u t e s a minor part of the a n a l y s i s .  The major part i n understanding l o c a l scour has to do with  discovering the f a c t o r s that i n f l u e n c e the vortex.  Any procedure that  attempts to r e l a t e flow parameters d i r e c t l y to scour depth, without  52.  considering the intermediary r o l e of the y o r t e x , can at best give only a p a r t i a l understanding of the scouring process, even though i t may y i e l d a r e l a t i o n s h i p v a l i d f o r the conditions f o r which i t was derived. The works of I n g l i s , Blench, V a r z e l i o t i s , Knezevic, Coleman (see Chapter II)  Breusers, and  a l l f a l l i n t o t h i s category.  The strength of the horseshoe vortex system w i l l be influenced by the parameters that govern the generation of v o r t i c i t y i n the approach f l o w , the concentration and d i s t r i b u t i o n of v o r t i c i t y around the p i e r , the d i s s i p a t i o n of v o r t i c i t y at the flow boundaries, and i t s convection downstream past the p i e r .  These parameters, f o r the case of a s i n g l e  p i e r i n a wide channel, can be l i s t e d as f o l l o w s : (a)  The average v e l o c i t y of the approach f l o w .  (b)  The v e l o c i t y gradient of the approach flow.  Cc)  The p i e r geometry Csize and shape).  Cd)  The scour hole geometry.  Ce)  The kinematic v i s c o s i t y of the f l u i d .  The influences of roost of these parameters on the maximum depth of scour have been studied by previous i n v e s t i g a t o r s . The i n f l u e n c e of the approach flow v e l o c i t y has been studied by various researchers, notably L a u r s e n ^ ' ^ , and Chabert and Engeldinger 6 0  (see s e c t i o n II.  6 1  E above).  The i n f l u e n c e of p i e r s i z e has been studied by most researchers i n the f i e l d .  Again, Laursen i s one of the c h i e f c o n t r i b u t o r s i n t h i s area, (39)  having also studied the e f f e c t of the shape of the p i e r .  Hawthorne  has shown a n a l y t i c a l l y that the shape of the p i e r i s s i g n i f i c a n t .  v  53.  The scour hole geometry a f f e c t s only the instantaneous s t r e n g t h , and not the f i n a l maximum scour depth.  As scour  vortex progresses,  the scour hole becomes l a r g e r and deeper, and the horseshoe vortex system, expanding with the scour h o l e , becomes weaker.  This  process  continues u n t i l the yortex strength f a l l s to a l e v e l such that the r a t e of sediment transport out of the scour hole i s j u s t equal to the rate of sediment transport i n t o the scour h o l e , and a l i m i t i n g e q u i l i b r i u m depth of scour i s reached. The kinematic y i s c o s i t y of the f l u i d i s of importance i n the generation and d i s s i p a t i o n of y o r t i c i t y and i n determining the magnitude of shear stresses fluid.  i n the flow.  I t i s governed by the temperature of the  Shen e_t al_. recorded the temperature i n t h e i r t e s t r u n s ^  included v i s c o s i t y as an i m p l i c i t y a r i a b l e i n t h e i r  analysis^  1 1 0  1 1 1  ^ and  ^.  With the exception of Tison (see f o l l o w i n g paragraph), the v e r t i c a l v e l o c i t y gradient of the approach flow has, to the a u t h o r ' s  knowledge  never been e x p l i c i t l y studied i n connection w i t h l o c a l scour at bridge piers.  It must, however, have some i n f l u e n c e , because i t i s important i n  the generation of v o r t i c i t y and the formation of the horseshoe system.  V i n j e has s u g g e s t e d ^ ^ 136  vortex  that the v e l o c i t y d i s t r i b u t i o n of the (2)  approach flow i s important, as has Ahmad  v  .  Tison did some experiments i n which he v a r i e d the v e l o c i t y d i s t r i bution by changing the bed roughness (see s e c t i o n II.  B above).  However  he did not recognize the horseshoe vortex mechanism, and his r e s u l t s have q u a l i t a t i v e value only. Depth of flow has not been l i s t e d e x p l i c i t l y as a parameter i n f l u e n cing the vortex s t r e n g t h , because t h i s i s included i n the v e l o c i t y  54.  d i s t r i b u t i o n of the flow.  Various researchers have studied the  i n f l u e n c e of flow depth on the maximum depth of scour, Laursen being foremost among them.  again  The o v e r a l l r e s u l t of these i n v e s t i g a t i o n s  that the depth of flow i s not important except f o r small depths.  is  This  suggests that the flow depth i s not important except when i t a f f e c t s the y e l o c i t y p r o f i l e in the boundary l a y e r of the approach flow. Shen et al_. have attempted to derive an expression f o r the  horseshoe  vortex strength based on the concept of v o r t i c i t y Csee section II. above).  K  They postulated that two-dimensional flow a n a l y s i s - i s v a l i d f o r  the flow i n the plane of symmetry i n f r o n t of the c y l i n d e r .  However, t h i s  must be considered as questionable, s i n c e , as Johnson points o u t ^ ^ , t h i s 5 0  plane of symmetry i s e n t i r e l y immersed i n a three-dimensional flow f i e l d .  E.  SUGGESTED RESEARCH From the foregoing d i s c u s s i o n , i t i s evident that there has been  l i t t l e i n v e s t i g a t i o n of the i n f l u e n c e of the v e r t i c a l v e l o c i t y d i s t r i b u t i o n of the approach flow on the mechanics of l o c a l scour, although a d e f i n i t e connection can be shown to e x i s t , from basic p r i n c i p l e s of f l u i d dynamics.  I t was therefore decided to i n v e s t i g a t e t h i s connection, i n a  p r e l i m i n a r y and q u a l i t a t i v e way.  55.  CHAPTER IV  EXPERIMENTAL WORK  A.  PURPOSE AND SCOPE The main purpose of the experimental work was to i n v e s t i g a t e , i n a  preliminary and q u a l i t a t i v e way, the influence of the v e l o c i t y d i s t r i b u t i o n of the approach flow on the l o c a l scour around a c y l i n d e r , p a r t i c u l a r l y the horseshoe vortex system and the e q u i l i b r i u m scour depth. The experiments consisted of observations  and measurements of flow  v e l o c i t i e s , scour depths, and flow patterns. In these t e s t s , the only parameters which were d e l i b e r a t e l y varied were the v e l o c i t y p r o f i l e of the approach flow and the ayerage flow velocity.  The other parameters, such as sand s i z e , p i e r s i z e , p i e r shape  and so on, were held constant, except the flow depth, which v a r i e d  56.  s l i g h t l y with the average flow v e l o c i t y , and the water temperature which v a r i e d from 56°F to 68°F.  B.  EQUIPMENT AND EXPERIMENTAL ARRANGEMENT - • 1  General Arrangement The experimental work was c a r r i e d out i n a f o r t y - f o o t - l o n g s t e e l  and glass flume i n the C i v i l Engineering Hydraulics Laboratory at the U n i v e r s i t y of B r i t i s h Columbia.  The flume was three f e e t deep and two  and one-half f e e t wide, and consisted of b a s i c a l l y three s e c t i o n s :  a  f i f t e e n - f o o t s t e e l approach s e c t i o n , a f i f t e e n - f o o t g l a s s - w a l l e d t e s t s e c t i o n , and a ten-foot s t e e l end s e c t i o n (see Figure 20). Inflows to the flume were provided by a r e c i r c u l a t i n g system i n which water from a sump was pumped up to a constant-head supply tank.  Discharges  from the tank were c o n t r o l l e d by a valve i n the flume i n l e t l i n e , and were measured by a mercury manometer connected across an o r i f i c e p l a t e located upstream of the valve.  The discharge c a l i b r a t i o n curve f o r the o r i f i c e  p l a t e was obtained and checked using two large weighing tanks (20,000 l b s . each) i n which the flume discharge, over a given time i n t e r v a l , could be c o l l e c t e d and weighed. The water depth i n the flume was regulated by an adjustable v e r t i c a l overflow gate at the downstream end of the flume. In the approach s e c t i o n were placed two sets of double m e t a l - g r i d screens and a set of vanes to s t r a i g h t e n the f l o w , as well as a platform f l o a t to suppress waye a c t i o n and surface disturbances.  57.  2.  Sand Bed A sand bed twelve inches deep was placed over the e n t i r e f i f t e e n -  foot length of the glass-walled s e c t i o n of the flume.  Bed t r a n s i t i o n  sections were constructed out of plywood and sheet metal to provide smooth, gradual t r a n s i t i o n s between the flume bottom and the surface of the sand bed at each end Csee f i g u r e 20].  In t h i s way the flow was  guided p a r a l l e l to the sand bed without unnecessary and undesirable turbulence. The same sand was used f o r a l l the t e s t s , and had a grain s i z e d i s t r i bution as shown i n Figure 21.  The median grain s i z e was 0.215 mm.  The  standard d e v i a t i o n a, defined by: n  _  1  a  -  j  84  .  U—  +  d r  . 50 was equal to 1.33. a  d  50 > -J— ) 16  (55)  d  The sand was obtained from s t o c k p i l e s of material dredged from the Fraser R i v e r , B.C.  Occasional pebbles and fragments of wood had to be  screened out before placement i n the flume. 3.  Test P i e r The p i e r used i n the t e s t s was a hollow c i r c u l a r c y l i n d e r of c l e a r  p l e x i g l a s s , of four inches outside diameter, which extended from the bottom to the top of the flume.  The p i e r was positioned i n the center of the flume  as shown i n Figure 20, and midway between the two s i d e s .  Its bottom end  was prevented from s h i f t i n g by s l i d i n g i t onto a c i r c u l a r d i s c f i x e d to the flume bottom.  The top end f i t t e d i n t o a V-shaped notch i n a board which  58.  was secured across the top of the flume, preventing movement of the p i e r i n the downstream d i r e c t i o n .  This arrangement also permitted r o t a t i o n of  the p i e r . The scour depth below l o c a l bed l e v e l could be read at any time by a s c a l e , graduated i n hundredths of a f o o t , which was attached to the i n s i d e surface of the p i e r .  A 150-watt floodlamp shining through the  glass wall of the flume was used to i l l u m i n a t e the scour hole a r e a , and f a c i l i t a t e d reading of the s c a l e . 4.  V e l o c i t y - D i s t r i b u t i o n Control Gate A v e l o c i t y - d i s t r i b u t i o n control gate was placed 2h f e e t upstream  of the nose of the p i e r .  This gate consisted of spaced 3/8-inch-diameter  aluminium rods extending h o r i z o n t a l l y across the flume, and secured at each end between two narrow aluminum bars, one of which had a rubber s t r i p attached to i t s i n s i d e surface to prevent slippage.  These bars  could be e i t h e r tightened, to clamp the rods i n p l a c e , or loosened, to allow the spacing of the rods to be e a s i l y changed.  Figure 22  shows a  view of the control gate from a p o s i t i o n downstream of the p i e r (the image of the gate i s r e f l e c t e d by the glass w a l l s of the flume). 5.  Current Flowmeter V e l o c i t y measurements were made with an Armstrong-Whitworth  Miniature Current Flowmeter (Type 176/1).  The measuring head, located at  the end of an 1 8 - i n c h probe, had a p r o t e c t i v e cage 1.5 cm. i n diameter, and b a s i c a l l y consisted of a f i y e - b l a d e d p l a s t i c r o t o r mounted on a hard stainless steel spindle.  The spindle terminated i n f i n e burnished conical  p i v o t s which ran i n jewels mounted i n an open s t e e l frame.  The pivots and  59.  jewels were shrouded to reduce the p o s s i b i l i t y of f o u l i n g .  The  r e v o l u t i o n s of the r o t o r were counted on three Dekatron counters over a period of 8.33 seconds.  The count was displayed f o r 6.67 seconds, and  J.67 seconds were required a f t e r that f o r a new count to begin.  The  v e l o c i t y was read from a c a l i b r a t i o n curve which r e l a t e d the count to the l o c a l flow v e l o c i t y . A t r a v e r s i n g mechanism enabled the measuring head to be positioned anywhere i n the flow. Since the flowmeter required a conducting l i q u i d i n order to f u n c t i o n p r o p e r l y , a small amount of sodium s i l i c a t e s o l u t i o n was added to the laboratory water supply. 6.  Dye I n j e c t o r A simple dye i n j e c t o r was made by bending the lower four inches  of a s e c t i o n of t h i n glass tubing at r i g h t angles to the main stem to form a horizontal leg.  This leg was then stretched at the t i p , using a  Buntzen burner, to form a f i n e nozzle.  The nozzle could be positioned  anywhere i n the f l o w , and aligned i n any d i r e c t i o n , by a simple t r a v e r s i n g mechanism b u i l t out of wood.  C.  EXPERIMENTAL METHODS AND PROCEDURES 1.  Starting-up With the flume i n i t i a l l y dry, the sand bed was c a r e f u l l y  levelled.  Then, with the desired depth set by the v e r t i c a l overflow  gate at the end of the flume, the flume was slowly f i l l e d from both ends so that the sand bed would not be d i s t u r b e d .  The flume i n l e t valve was  60.  then opened r a p i d l y to the desired discharge, and the stop-watch s t a r t e d . This procedure r e s u l t e d i n a rapid but smooth r i s e to the desired discharge, and did not produce any i r r e g u l a r d i s t u r b i n g waves. 2.  Generation of V e r t i c a l V e l o c i t y P r o f i l e s D i f f e r e n t v e r t i c a l v e l o c i t y p r o f i l e s were generated by the  v e l o c i t y - d i s t r i b u t i o n control gate, by varying the number and spacing of the i n d i v i d u a l bars of the gate.  The gate s e t t i n g f o r each v e l o c i t y  p r o f i l e used i n the t e s t runs was determined i n preliminary t e s t s by t r i a l and e r r o r . The v e l o c i t y - d i s t r i b u t i o n control gate was located 2h f e e t upstream of the f r o n t face of the t e s t p i e r .  This distance was selected as a  compromise between two c o n f l i c t i n g considerations.  On the one hand, the  gate had to be placed f a r enough upstream so that the l o c a l  disturbance  i n the f l o w , caused by the i n d i v i d u a l bars of the gate, would be d i s s i p a t e d , and, on the other hand, the gate had to be placed near enough so that the v e l o c i t y p r o f i l e created by the gate would not decay appreciably before i t reached the p i e r . The head drop across the gate was quite s m a l l , being of the order of a hundredth of a f o o t or l e s s . 3.  V e l o c i t y Measurements V e l o c i t y measurements were made with the Armstrong-Whitworth  Miniature Current Flowmeter.  Its c a l i b r a t i o n was c a r e f u l l y checked with  a miniature Ott p r o p e l l e r meter which had j u s t been f a c t o r y - c a l i b r a t e d . A l l v e l o c i t y measurements were made on the c e n t e r l i n e of the flume.  In  general, measurements were made at 0.05-foot I n t e r v a l s v e r t i c a l l y oyer the e n t i r e depth, except that the measurement nearest the bed was made 0.02 f e e t above the bed.  Each measurement consisted of taking the  ayerage of f i v e consecutive readings of the flowmeter, each reading i t s e l f being integrated over 8.33 seconds, with 8.33 seconds elapsing between each reading, as described i n Section B.5 above. A h o r i z o n t a l traverse across the flume two f e e t upstream of the p i e r was made to determine the h o r i z o n t a l v e l o c i t y p r o f i l e . In g e n e r a l , f o r each t e s t , the v e r t i c a l v e l o c i t y p r o f i l e was measured i n the f i r s t hour or so of the t e s t run, at a distance of 1.5 f e e t upstream of the p i e r f a c e .  A f t e r a l l the t e s t runs had been completed and the p i e r  removed, the v e l o c i t y p r o f i l e f o r each t e s t s e r i e s was measured at the l o c a t i o n of the p i e r c e n t e r l i n e i n order to check the s t a b i l i t y of the profile. 4.  Flow Patterns Flow patterns i n the scour hole and around the p i e r were studied  with the aid of dye, which was introduced i n t o the flow at the desired l o c a t i o n with the dye i n j e c t o r .  The discharge of the dye was c o n t r o l l e d  so that i t did not d i s t u r b the flow. The flow patterns so observed were recorded by making on-the-spot freehand sketches,  photographs of the flow patterns were not made,  p a r t l y due to oyersight on the part of the author i n the e a r l y stages of the work, and p a r t l y due to t e c h n i c a l d i f f i c u l t i e s which arose subsequently.  62.  5.  Scour Kole Development and E q u i l i b r i u m Depth of Scour The depth of scour below the bed l e v e l could be measured at any  time by the graduated scale attached to the i n s i d e of the p i e r (which could be r o t a t e d in any d i r e c t i o n ] .  Scour depths were recorded f o r each  t e s t r u n , at regular i n t e r v a l s , so that the scour hole development with time, of d i f f e r e n t t e s t s , could be compared. Each reported t e s t run was continued u n t i l there was no longer any increase of scour depth w i t h time, and t h i s f i n a l scour depth was taken to be the e q u i l i b r i u m scour depth f o r the conditions of that t e s t run. As the scour hole deyeloped, i t was observed that occasional coarser p a r t i c l e s were l e f t behind i n the scour hole to produce an armouring e f f e c t .  When t h i s occurred, the flow was momentarily stopped  and the offending p a r t i c l e s c a r e f u l l y removed. The v i s i b i l i t y i n the t e s t arrangement permitted a completely unobstructed view of the scouring process.  The motion of the i n d i v i d u a l  sand grains out of the scour hole was e s p e c i a l l y noted, and sketches were made to record these observations.  63..  CHAPTER V  EXPERIMENTAL RESULTS  A.  GENERAL The experimental r e s u l t s c o n s i s t of the f o l l o w i n g data:  1.  Observations of scour depth at p e r i o d i c i n t e r v a l s during the development of the scour h o l e , and the f i n a l e q u i l i b r i u m scour depth, f o r each of various average v e l o c i t i e s and v e r t i c a l v e l o c i t y d i s t r i b u t i o n s of the approach f l o w .  2.  Observations of the development of the scour h o l e , the scouring process, and the motion of the i n d i v i d u a l sand grains i n and out of the scour hole.  3.  Observations of the vortex patterns i n the scour hole f o r yarious stages of scour hole development. These r e s u l t s are described and discussed below.  64.  B.  APPROACH FLOW VELOCITY PROFILES Eight d i f f e r e n t v e r t i c a l v e l o c i t y p r o f i l e s were t e s t e d :  2-A, 2-B, 2-C, 2-D, 3 , 4 ,  and 5.  series  1,  Series I was tested at three d i f f e r e n t  ayerage v e l o c i t i e s ; series 3 and 4 were each tested at two d i f f e r e n t average v e l o c i t i e s .  Series 2-A was tested three d i f f e r e n t times at the  same average v e l o c i t y .  The v e l o c i t y p r o f i l e s f o r a l l the t e s t s are shown  i n Figures 23 to 27, i n c l u s i y e . There was some unsteadiness inherent i n the approach flow. v e l o c i t y measurements with the Armstrong-Whitworth  Repeated  flow meter at any one  spot i n the flow u s u a l l y gave y e l o c i t y v a r i a t i o n s of two or three per cent, and sometimes as much as f i v e per cent, e s p e c i a l l y near the bed. The v e l o c i t y p r o f i l e s of series 1 were the ones which occurred n a t u r a l l y i n the flume without the v e l o c i t y - d i s t r i b u t i o n control  gate,  while the p r o f i l e s of the other seven t e s t series were a r t i f i c i a l l y generated by the gate.  A l l of the p r o f i l e s generated by the gate tended  to decay i n t o the natural p r o f i l e of series 1.  I t was not possible  to  t e s t very steep v e l o c i t y g r a d i e n t s , as these were d i f f i c u l t to generate, and moreover they decayed too r a p i d l y while approaching the p i e r . Figure 28 shows the s t a b i l i t y of the natural p r o f i l e of series  1-A.  The p r o f i l e was measured at a l o c a t i o n JL.5 f e e t upstream of the p i e r (with the p i e r i n p l a c e ) , and then at the p i e r c e n t e r - l i n e (with the p i e r removed).  The greatest d i f f e r e n c e between the two measurements i s  about  f i v e per cent, which i s w i t h i n the range of unsteadiness of the approach flow.  65.  Figure 29 shows the extent of the decay of the p r o f i l e of s e r i e s 2-A3, from 1.5 f e e t upstream of the p i e r , to the l o c a t i o n of the p i e r center-line.  The s h i f t towards the natural ( s e r i e s l ] v e l o c i t y p r o f i l e  i s quite e v i d e n t , and i s representative of the behaviour of the other velocity profiles tested.  This i n s t a b i l i t y prevented any q u a n t i t a t i v e  analysis of the influence of the approach flow v e l o c i t y gradient from being c a r r i e d out. The experiments were designed to run w i t h i n the " c l e a r - w a t e r " scour range - i e . scour w i t h no sediment transport of the sand bed i n general (see above, section II.  D).  This was done so that the development of  the scour hole and the f i n a l e q u i l i b r i u m value of the scour depth would not be obscured by general sediment transport with r i p p l e s or dunes moving through the scour hole. procedure.  I t also s i m p l i f i e d the experimental  Therefore, the average v e l o c i t i e s used f o r the various |  tests  were a l l less than the c r i t i c a l v e l o c i t y f o r general sediment transport of the sand bed (U  ...  ).  The c r i t i c a l v e l o c i t y f o r general sediment transport was determined to be about 0.90 f t / s e c .  At t h i s v e l o c i t y , small r i p p l e s s t a r t e d to form  on the sand bed of the flume.  The actual v e l o c i t i e s used i n the e x p e r i -  ments varied from 0.65 f t . / s e c . to 0.89 f t . / s e c .  C.  EQUILIBRIUM SCOUR DEPTH 1.  Introduction The e q u i l i b r i u m scour depth, d  at w h i c h , f o r given conditions  s g  , i s defined as the scour depth  of p i e r and channel geometry and flow  66.  c o n d i t i o n s , the rate of sediment transport out of the scour hole equals the rate of sediment transport i n t o the scour hole.  In the case of  " c l e a r - w a t e r " scour, both of these transport terms equal zero f o r the equilibrium condition. For each t e s t r u n , the depth of scour was measured at regular i n t e r v a l s , s t a r t i n g from the beginning of each t e s t at t = 0, u n t i l there was no longer any increase of scour depth with time. was taken to be the e q u i l i b r i u m scour depth, d depth was f i r s t reached i s c a l l e d t  .  s e >  This f i n a l value  The time at which t h i s  The scour depth measurements have  been p l o t t e d i n Figures 30 to 35, i n c l u s i v e . O c c a s i o n a l l y , l a r g e r p a r t i c l e s would become uncovered i n the scour hole as i t deepened.  When t h i s happened the flow was momentarily stopped  and the l a r g e r p a r t i c l e s were c a r e f u l l y removed to prevent armouring. These occasions show up as uneyen segments i n the p l o t s of scour depth versus time. A comprehensive summary of the t e s t data i s given i n Table 2.  III.  Results The accuracy of the r e s u l t s i s i n d i c a t e d by the r e s u l t s f o r the  three d i f f e r e n t runs of s e r i e s 2-A Csee Figures 24 and 31).  Within the  l i m i t s of accuracy of the measuring equipment used, the flow conditions f o r these runs were a l l the same.  The r e s u l t i n g e q u i l i b r i u m scour depths,  however;, were measured to be 0.37, 0.36, and 0.38 f e e t , a y a r i a t i o n of about 5 per cent.  The inherent experimental e r r o r i s therefore considered  to be at l e a s t 5 per cent.  67.  A comparison of the r e s u l t s f o r the various y e r t i c a l y e l o c i t y p r o f i l e s and range of ayerage flow y e l o c i t i e s , t e s t e d , leads to the following Ca)  observations;  For any giyen v e l o c i t y p r o f i l e , d  g e  increases with  increasing  average flow v e l o c i t y .  This i s c l e a r l y demonstrated by the r e s u l t s  f o r series 1, 3, and 4.  In each of these s e r i e s , a higher average  flow v e l o c i t y r e s u l t e d i n a l a r g e r e q u i l i b r i u m scour depth. Cb)  The v e l o c i t y of the lower part of the f l o w , near the bed, i s more important, in determining d of the flow.  $ e  , then the v e l o c i t y of the upper part  Series 3-B and 4-A both had an average flow v e l o c i t y  of 0.77 f t . / s e c . , and series 3-A had an average flow v e l o c i t y of 0.85 f t . / s e c . , yet the e q u i l i b r i u m scour depth f o r s e r i e s 4-A was greater than that f o r e i t h e r s e r i e s 3-A or 3-B.  This i s  because  the v e l o c i t y of the lower part of the flow was greater i n series  4-A  than i n e i t h e r of series 3-A or 3-B, even though these l a s t two had higher flow v e l o c i t i e s i n the upper part of the flow (Figure Cc)  26).  For s i m i l a r v e r t i c a l v e l o c i t y p r o f i l e s , the e q u i l i b r i u m scour depth decreases with decreasing v e l o c i t i e s i n the lower part of the flow. This i s evident from the r e s u l t s of s e r i e s 2.  There was a continuous  reduction in d_ as the v e l o c i t i e s i n the lower parts of the flow se r  decreased, from series 2-A to 2-D. Cd)  The e q u i l i b r i u m scour depth tends to increase w i t h an increase the v e r t i c a l y e l o c i t y gradient.  This i s not a d i r e c t observation  i s implied by the t e s t r e s u l t s of series 5 and 1-C, and 2-A.  in  and s e r i e s  1-A  but  68.  Series 5 and s e r i e s 1-C had about the same value of d l i m i t s of accuracy of the experiment).  s g  Cwithin the  However, series 5 had a  higher flow v e l o c i t y near the bed, which, based on the previous observations, should have r e s u l t e d i n a l a r g e r value of d  s g  .  The  a d d i t i o n a l f a c t o r which needs to be considered i s the v e l o c i t y gradient Cthe r a t e of change of v e l o c i t y with depth) of the approach f l o w , e s p e c i a l l y i n the lower part of the flow.  For s e r i e s 5, t h i s  y e l o c i t y gradient was p r a c t i c a l l y zero except r i g h t at the bed, whereas i n s e r i e s 1-C, there was a d e f i n i t e v e l o c i t y gradient up to 0.30 f t . aboye the bed.  This lack of v e l o c i t y gradient i n the lower  part of the flow i n s e r i e s 5 counter-acted the increase i n v e l o c i t y , r e s u l t i n g i n no net change i n the e q u i l i b r i u m scour depth. A s i m i l a r e f f e c t was obseryed with s e r i e s 1-A and 2-A.  Again, the  e q u i l i b r i u m depth of scour was about the same f o r both s e r i e s , even though i n series 2-A the v e l o c i t y of the lower part of the flow was lower.  However, the e f f e c t of t h i s lower v e l o c i t y was counter-acted  by the greater v e l o c i t y gradient of s e r i e s 2-A, r e s u l t i n g i n the same e q u i l i b r i u m scour depth as f o r s e r i e s  1-A.  The flow aboye a height above the equal to one p i e r diameter (0.33 f t ) has some influence on the e q u i l i b r i u m depth of scour.  For s e r i e s  3-A  and 4-B, the flow v e l o c i t i e s up to a height of approximately 0.33 f t . above the bed were about the same.  Above t h i s height, however, the  flow v e l o c i t y f o r s e r i e s 3-A was g r e a t e r , and the v e l o c i t y gradient was greater.  This was r e f l e c t e d i n an e q u i l i b r i u m scour depth which  was 0.05 f t . l a r g e r (0.21 f t . ) than that of s e r i e s 4-B (0.16 f t . ) .  69.  3.  Comparison With Results of Others The flow conditions f o r series 1-A (undisturbed v e l o c i t y p r o f i l e ,  U - U  } were probably c l o s e r to the flow conditions of experiments  done by previous i n v e s t i g a t o r s , than those f o r any of the other t e s t s e r i e s reported herein.  The data f o r s e r i e s 1-A are therefore used i n  the f o l l o w i n g selected equations as proposed by d i f f e r e n t researchers the past.  in  The e q u i l i b r i u m scour depth f o r s e r i e s 1-A was 0.37 f t . below  normal bed l e v e l . The symbols used i n the f o l l o w i n g equations are defined i n Chapter II where they f i r s t appear and i n the L i s t of Symbols. (a)  Laursen (see s e c t i o n II.  D)  Laursen proposed a design curve f o r conditions of general  sediment  t r a n s p o r t , which f o r a c i r c u l a r p i e r i s given by:  For the conditions of s e r i e s 1-A, d  se  =  this  gives:  2.01 b 0.67 f t .  This r e s u l t i s rather high.  A check of Laursen's t e s t data showed that  he only used flow depths i n the range 0.2 f t . to 0.9 f t . , with flow v e l o c i t i e s from 1.00 f t . / s e c . to 2.50 f t . / s e c . ^ ^ . W i t h these flow 61  c o n d i t i o n s , the v e l o c i t y probably v a r i e d throughout the e n t i r e depth of flow ( i e . boundary layer thickens - depth of flow}, and Laursen's r e s u l t s cannot be expected to hold f o r flow conditions where the flow depth i s considerably l a r g e r than the boundary l a y e r thickness.  70.  Cb)  Tarapore (see s e c t i o n II.  G)  Tarapore foand that the flow depth had no appreciable i n f l u e n c e on d  s e  beyond a value of H = 1.15 b.  He proposed, f o r the case of a  c i r c u l a r p i e r with a large flow depth, and conditions of general sediment transport: d  M  =  1.35 b  sem For the conditions of s e r i e s _1-A, t h i s g i y e s ; d  o a  =  0.45 f t .  sem (c)  Breusers (see s e c t i o n II.  I)  Breusers reported the r e s u l t s of two s t u d i e s ; (i)  Model study of the piers of the bridge across the Oosterschelde: u  ~ r^-t u  d  se  25  °-  =  Table I shows, f o r 8 1  f t . / s e c . ( b = 0.36 f t . , H = 1.64 f t ) :  1.25 b  For the conditions of s e r i e s 1-A, t h i s g i v e s : d (ii)  s e  =  0.42 f t .  P r i v a t e study of scour around d r i l l i n g platforms: d  c*m  sem  =  1.4  b  For the conditions of s e r i e s 1-A, t h i s g i v e s : d (d)  sera c o m  =  0.47 f t .  Maza and Sanchez (see s e c t i o n II.  J)  Maza and Sanchez proposed a modified form of an equation by J a r o s l a v t s i e v , f o r the region of " c l e a r - w a t e r " scour, subject to the c o n d i t i o n that H > 1.5 b:  71.  d  se••_ b  (, \ ) \\  3  d  u  f HU  For the conditions of s e r i e s 1-A, d  Ce)  $ e  =  0.85 b  =  0.28 f t .  Larras Csee s e c t i o n II.  50  this  gives:  E)  Based on f i e l d data and the data of Chabert and Engeldinger, Larras suggested a design equation; d  =  sera C f l m  1.42  b  3 / 4  For the conditions of s e r i e s 1-A, t h i s  =  d  seni  gives:  o.62 f t .  This r e s u l t i s rather high.  This i s because L a r r a s  1  equation  i s dimensionally non-homogeneous; f o r small p i e r diameters i t w i l l p r e d i c t scour depths that are too l a r g e , and f o r large p i e r diameters, i t w i l l p r e d i c t scour depths that are too s m a l l , (f)  Shen et a l . (see s e c t i o n II.  K)  Shen and his co-workers at Colorado State U n i v e r s i t y obtained the r e l a t i o n : d  s e  =  .00073 (R )  0.619  b  For the conditions of s e r i e s 1-A, R^ = 2.53 x 1 0 , which gives: 4  d Cgl  se  =  0.39 f t .  Coleman (.see s e c t i o n II.  0)  Coleman proposed the f o l l o w i n g r e l a t i o n f o r the e q u i l i b r i u m depth of scour:  . d  s e  =  1.49 b  9 / 1  . ° CU /2g) 2  1 / 1 0  72. For the conditions of s e r i e s  (U /2g) 2  =  1/l0  d  s e  1-A,  C0.0122]  1/10  =  =  0.644, which g i v e s :  0.35ft.  The equations o f l n g l i s , Blench, and V a r z e l i o t i s were not used. These equations apply to natural channels flowing at regime depths, and thus could not be applied to the flume experiments reported here, i n which flow depths were f i x e d a r b i t r a r i l y .  D.  VORTEX PATTERNS AND SCOUR HOLE DEVELOPMENT 1.  The Horseshoe Vortex System The s t r u c t u r e of the horseshoe vortex system, which forms the  basic mechanism of l o c a l scour around a c y l i n d e r , i s shown i n Figure 36. This pattern was observed f o r t e s t series 2-C, before the s t a r t of scour, at an average approach-flow v e l o c i t y of 0.30 f t . / s e c . Vortex 1 i s the main or primary v o r t e x , and i t does most of the work involved i n the scouring process.  I t has been observed by i n v e s t i g a t o r s  in the f i e l d of f l u i d dynamics as well as i n v e s t i g a t o r s who s p e c i f i c a l l y studied  the l o c a l scour phenomenon (see Chapters II and  III).  Vortices 2(a) and 2(b) are secondary v o r t i c e s which are much weaker than the primary vortex.  They are deriyed mostly from the i n t e r a c t i o n of  the primary vortex with the r e s t of the flow and the flow boundaries. These secondary y o r t i c e s haye been observed by Shen et_ a l _ . ^ ^ ( s e e also 108  Figures 8 and 1 0 ) , and by yarious i n v e s t i g a t o r s i n the f i e l d of f l u i d (94 99 123) dynamics - ' \ v  73.  Vortex 3, has, to the author's knowledge, not been reported previously i n the l i t e r a t u r e .  I t was observed by the author on d i f f e r e n t  occasions f o r d i f f e r e n t t e s t s e r i e s , and f o r d i f f e r e n t stages of scour hole development.  This v o r t e x , although undoubtedly influenced by the  primary v o r t e x , nevertheless does not derive i t s energy from i t , but i s maintained by the incoming flow from a l e v e l j u s t above that which feeds the primary vortex. The whole vortex system i s constantly being fed by the incoming flow from upstream.  I t i s q u i t e unstable, and the i n d i v i d u a l v o r t i c e s are  repeatedly swept away and being reformed.  The turbulence generally present  i n the approach flow i s probably the main f a c t o r i n t h i s . 2.  Scour Hole Development:  Beginning of Scour  The development of the scour hole with time i s i n d i c a t e d by the p l o t s of Figures 30 to 35.  In these f i g u r e s , the p l o t t e d scour depth i s  the depth (below normal bed l e v e l ) of the deepest part of the scour hole at the time of measurement. Scour of the bed around the p i e r was observed to begin at symmetrica l l y - l o c a t e d points on both sides of the p i e r about 30° from the f r o n t center of the p i e r f a c e .  The shear stresses on the bed are apparently  greatest at these spots, due to the superposition of the two-dimensional free-stream flow on the horseshoe vortex a c t i o n .  These two scour spots  were observed to increase i n s i z e u n t i l they met at the f r o n t centre of the p i e r .  This occurred w i t h i n the f i r s t minute of the t e s t f o r almost  a l l the t e s t s e r i e s .  However, the 3 0 ° - points s t i l l remained the points  74.  of deepest scour f o r some time - u n t i l the scour hole had developed to such a s i z e that the depth a l l around the f r o n t part of the p i e r , i n c l u d i n g the 30° - p o i n t s , was the same. A sketch of the vortex pattern and scour hole at the beginning of i t s development, f o r t e s t s e r i e s 1-A, i s shown i n Figure 37. of scour shown i s f o r time t = 12 minutes.  The stage  The depth of scour at the  f r o n t center of the p i e r at t h i s stage was 0.12 f t . below bed l e v e l , as shown, whereas the maximum scour depth, at the 30° - p o i n t s , was 0.15 f t . The action of each vortex was observed to be quite d i s t i n c t :  vortex  1 scoured out the main part of the scour h o l e , while vortex 3 scoured the bed area r i g h t at the p i e r f a c e . The length of time required f o r the scour hole around the f r o n t of the p i e r to develop to a uniform depth must depend at l e a s t p a r t l y on the r e l a t i v e strengths of v o r t i c e s 1 and 3, which i n turn depend on the shape of the v e l o c i t y p r o f i l e of the flow approaching the p i e r (see Section 3 below).  This i s demonstrated by the t e s t r e s u l t s f o r s e r i e s 1 and 2  (the only s e r i e s f o r which scour depths at both 0° and 30° were measured). The time required f o r the scour hole to develop to a uniform depth f o r the series 1 tests was about two hours, while the time required f o r t h i s i n the s e r i e s 2 t e s t s varied among the i n d i v i d u a l t e s t s from 6 minutes to 30 minutes. The development of scour at the 30° - p o i n t s , and at the f r o n t center of the p i e r , i s shown g r a p h i c a l l y f o r s e r i e s 1<-A, i n Figure 38.  75.  The plan-view vortex pattern i s shown i n Figure 39, f o r t e s t s e r i e s 2-C at time t = 20 minutes.  The horseshoe v o r t i c e s , which haye h o r i z o n t a l  axes, are swept around the c y l i n d e r by the main flow.  The domain of the  main yortex i s between the two dashed l i n e s on the f i g u r e , while the domain of vortex 3 i s between the dashed l i n e nearest the p i e r and the pier i t s e l f . The two-dimensional v o r t i c e s shed from the sides of the p i e r due to the a c t i o n of the free stream flow around the p i e r , superimpose  themselves  on vortex 3, and cause strong bursts of turbulent eddying which carry sediment f o r a considerable distance downstream along the wake. 3.  Approach Flow V e l o c i t y P r o f i l e and Vortex Structure The connection between the approach flow v e r t i c a l v e l o c i t y p r o f i l e  and the s t r u c t u r e of the horseshoe vortex system i s i l l u s t r a t e d i n Figures 40 and 41.  A scour hole was developed by a flow of U = 0.85 f t . / s e c , to  a depth of d  g  = 0.20 f t .  The flow was then slowed to U = 0.45 f t . / s e c . to  observe the vortex patterns.  The approach flow v e l o c i t y p r o f i l e i s  p l o t t e d on the r i g h t hand side of the f i g u r e . Dye was c a r e f u l l y i n j e c t e d i n t o the flow j u s t upstream of the scour hole at various l e y e l s above the bed, i n the plane of symmetry of the experimental arrangement.  I t was observed that each i n d i v i d u a l vortex .  w i t h i n the vortex system was maintained by or deriyed from a s p e c i f i c flow l e y e l w i t h i n the boundary l a y e r ( i e . w i t h i n the depth range i n which the flow y e l o c i t y y a r i e d j .  76.  The boundary layer thickness of the obseryed flow was about 0.30 f t . , as shown i n Figure 40.  From the bed up to a height of about 0.04 f t . ,  the flow was very slow and went mostly i n t o a very v/eak, type 2(b) v o r t e x . From 0.04 f t . to 0.15 f t . , most of the flow went i n t o the primary v o r t e x , with occasional surges i n t o type 2(a) and type 3 y o r t i c e s .  From 0.15 f t .  to 0.22 f t . , most of the flow went i n t o vortex 3, with some going i n t o yortex 2 ( a ) , and o c c a s i o n a l l y i n t o the main vortex. 0.29 f t . , the flow went i n t o vortex 3.  From 0.22 f t . to  Above a l e v e l of 0.29 f t . no flow  was observed moving down towards the vortex system.  This l e v e l  corresponds  approximately with the l e v e l at which the v e l o c i t y of the approach flow becomes constant.  The above observations are combined i n a sketch i n  Figure 41. 4.  Transport out of the Scour Hole The transport of sand out of the scour hole v/as observed f o r  various t e s t s e r i e s .  The sand t r a v e l s i n b a s i c a l l y two steps.  First,  sand i s moved along the slope of the scour hole to points A and B as shown i n Figure 42.  Points A and B mark the l o c a t i o n s of two " l i n e s " o r  "avenues" of sediment transport out of the hole and downstream, which l i e approximately at r i g h t angles to the slope. Three v o r t i c e s are a c t i y e i n moving the sand towards points A and B. Vortex 1 moves sand up the slope of the scour hole to point A. 2(a) moves sand down the s l o p e , also to point A. i n t o the p i e r face to point B.  Vortex  Vortex 3 moves sand r i g h t  Vortex 2(b) i s too weak to haye any e f f e c t ,  and the sand i n t h i s area simply s l i d e s down towards point A by the a c t i o n of g r a v i t y .  77.  Transport l i n e A i s located along the outside edge of vortex 1 Csee Figure 391, and l i n e B i s located r i g h t along the surface of the p i e r . These l i n e s are quite narrow at the f r o n t of the scour h o l e , but widen towards the r e a r . The average slope of the scour hole was measured several times Csee f o r example Figure 41} and was found to be 30°.  However, as can be seen  from the f i g u r e s , the slope of the scour hole i s not constant.  The  steepest part of the slope i s r i g h t under the main v o r t e x , and i s p a r t i a l l y maintained by the shear stresses exerted by the vortex. the flow i s stopped, t h i s part of the slope slumps down. Several photos of the f u l l y - d e v e l o p e d scour hole are shown i n Figures 43 and 44.  When  78.  CHAPTER VI  SUMMARY AND CONCLUSIONS  A.  PREVIOUS INVESTIGATIONS  .  The reported study of l o c a l bed scour at bridge piers goes back over the l a s t eighty years or so.  During t h i s time, d i f f e r e n t i n v e s t i g a t o r s  t r i e d to understand how scour of an erodible bed at a bridge p i e r or other flow obstruction occurred, and, on the basis of t h e i r various ideas and concepts, t r i e d to e s t a b l i s h r e l a t i o n s h i p s between the depth of l o c a l II  scour and the other parameters of the problem,.usually by conducting experiments on scale models i n a hydraulic laboratory. The parameters that were shown by these experiments to be most important, can be divided i n t o two groups - the parameters describing the  79.  p i e r geometry, and the parameters d e s c r i b i n g the flow c o n d i t i o n s . The p i e r geometry i s adequately described by two parameters.  The  most important one i s the p i e r s i z e , expressed as the p i e r w i d t h , or diameter, b.  The second parameter i s the p i e r shape, and i s u s u a l l y  expressed as a constant c o e f f i c i e n t and applied as a d i r e c t f a c t o r i n scour equations. There are two main flow parameters which the various have found to be important. depth.  investigators  These are the flow v e l o c i t y and the flow  However, there i s not universal agreement as to t h e i r r e l a t i v e  importance.  Some i n v e s t i g a t o r s , e s p e c i a l l y Laursen, stressed the  importance of the flow depth, w h i l e discounting the flow v e l o c i t y .  Others,  such as Shen e_t al_., stressed the flow v e l o c i t y as being more important. The r e l a t i v e importance of these two flow parameters depends on the regime of flow which i s being considered.  For a flow regime i n which  there i s no general transport of bed sediment, the flow v e l o c i t y has a d e f i n i t e e f f e c t on the e q u i l i b r i u m depth of scour, d  s g  .  However, t h i s  e f f e c t decreases r a p i d l y once conditions of general sediment transport are e s t a b l i s h e d Csee Figures 3 and 18).  I f the flow regime i s such that  the flow depth i s s m a l l , and the v e l o c i t y p r o f i l e of the boundary l a y e r i s a f f e c t e d by changes i n flow depth, then the e q u i l i b r i u m depth of scour w i l l be a f f e c t e d a l s o .  However, i f the depth of flow i s l a r g e , and the  boundary l a y e r y e l o c i t y p r o f i l e i s f u l l y deyeloped, then v a r i a t i o n s i n flow depth w i l l not a f f e c t the e q u i l i b r i u m depth of scour.  This conclusion  i s implied by the f i n d i n g s of Tarapore Cp.20, above), Breusers (p. 23,  80. above), and Maza and Sanchez (p. 25, above). For any given p i e r i n an e r o d i b l e bed, the e q u i l i b r i u m depth of scour ( d I increases with flow depth and flow v e l o c i t y , only up to a se  c e r t a i n l i m i t i n g value.  Increasing the flow depths and v e l o c i t i e s beyond  t h i s point w i l l no longer a f f e c t the depth of scour.  This l i m i t i n g  yalue i s known as the maximum e q u i l i b r i u m depth of scour (d ), and i s sem n  r  governed only by the p i e r s i z e and shape.  This has been i m p l i c i t l y  recognized by a number of i n v e s t i g a t o r s who have proposed r e l a t i o n s h i p s f o r maximum e q u i l i b r i u m scour depth ( d  $ e m  ) based only on p i e r s i z e and  shape Ceg. Tarapore, Breusers). B.  MECHANISM OF LOCAL SCOUR The basic mechanism of l o c a l scour i s the horseshoe vortex.  The  horseshoe vortex i s formed by the a c t i o n of the p i e r i n apprehending the v o r t i c i t y normally present i n the f l o w , and concentrating i t near the bed at the p i e r nose.  The p i e r induces an adverse pressure gradient i n  the f l o w , which causes i t to acquire a downward component i n f r o n t of the pier.  This i n turn causes separation of the boundary l a y e r i n f r o n t of  the p i e r , which then r o l l s up to form a large vortex w i t h a h o r i z o n t a l axis and shaped l i k e a horseshoe i n plan.  I n t e r a c t i o n of t h i s vortex with  the boundaries and the approach flow generates yet other v o r t i c e s , so that a system of l i n k e d y o r t i c e s deyelops.  This i s c a l l e d the horseshoe vortex  system. Any attempt to understand or e x p l a i n l o c a l scour without considering the basic vortex mechanism cannot succeed.  Thus, the analyses of T i s o n ,  I s h i h a r a , Tarapore, and Carstens, do more to confuse the s i t u a t i o n than they do to e x p l a i n i t , although t h e i r experimental r e s u l t s are v a l i d i n themselves.  81.  C.  EXPERIMENTAL RESULTS 1.  Introduction A consideration of the f a c t o r s that i n f l u e n c e the strength of  the horseshoe vortex led to the d e c i s i o n to i n v e s t i g a t e the e f f e c t of the v e r t i c a l v e l o c i t y d i s t r i b u t i o n of the approach flow on the e q u i l i b r i u m depth of scour, and on the horseshoe vortex flow p a t t e r n s . were therefore c a r r i e d out to do t h i s .  Experiments  A flow regime with a iarge depth  of f l o w , and flow v e l o c i t i e s below the c r i t i c a l f o r the beginning of general sediment t r a n s p o r t , were used. 2.  E q u i l i b r i u m Depth of Scour The main r e s u l t s of the experiments on the e q u i l i b r i u m depth of  scour can be summarized as f o l l o w s . (a)  The e q u i l i b r i u m depth of scour increases with increasing  average approach flow v e l o c i t y , f o r the range of flow v e l o c i t i e s used. (b)  The e q u i l i b r i u m depth of scour depends more on the v e l o c i t y  of the lower part of the flow than on the v e l o c i t y of the upper part of the flow. Cc)  The e q u i l i b r i u m depth of scour increases with an increase in  the gradient of the v e r t i c a l y e l o c i t y p r o f i l e of the approach flow. The e q u i l i b r i u m depth of scour obtained f o r an undisturbed y e l o c i t y p r o f i l e at an average y e l o c i t y s l i g h t l y less than the c r i t i c a l v e l o c i t y required f o r the beginning of general sediment transport Cseries 1-A], was  82.  equal to 0.37 f t . (1.12  b).  This yalue was compared to scour depths  c a l c u l a t e d from various equations proposed by preyious  investigators.  The equations of Laursen (equation 10} and Larras (equation 11} gave very high values f o r the e q u i l i b r i u m scour depth, and were considered to be i n a p p l i c a b l e , f o r the reasons stated i n Chapter V.  Equations of  Tarapore and Breusers, f o r the maximum e q u i l i b r i u m depth of scour, d  s e m  ,  are also i n a p p l i c a b l e , since these were derived f o r the most severe flow conditions p o s s i b l e ,  with a  state  of general sediment t r a n s p o r t , and  would include the e f f e c t of such f a c t o r s as dune troughs passing  through  the scour hole. The remaining equations, of Breusers, Maza and Sanchez, Shen et a l . , and Coleman, gave values f o r d 0.35 f t . , r e s p e c t i v e l y .  g e  of 0.42 f t . , 0.28 f t . , 0.39 f t . , and  These four equations give an average of 0.36 f t . ,  which compares well with the 0.37 f t . a c t u a l l y obtained.  3.  Horseshoe Vortex Flow Patterns The main r e s u l t s of the observations of flow patterns i n the  horseshoe vortex system are as f o l l o w s . a)  The horseshoe vortex system i s a system of l i n k e d v o r t i c e s ,  and i s made up of a primary vortex (vortex 1 ) , several  secondary  v o r t i c e s ( v o r t i c e s 2(a) and 2 ( b ) ) , and a t e r t i a r y yortex (vortex 3).  This t e r t i a r y yortex has not been reported before.  b)  The work, of moving the sand grains out of the scour hole i s  done mainly by the primary v o r t e x ; the secondary v o r t i c e s do very  83.  l i t t l e work.  Vortex 3 scours the region of the bed r i g h t next  to the p i e r f a c e . c)  The whole yortex system i s c o n t i n u a l l y being fed by the  incoming flow from upstream.  Approach flow turbulence i s  therefore r e f l e c t e d i n the a l t e r n a t e collapse and reformation of the i n d i v i d u a l v o r t i c e s . d)  Each i n d i v i d u a l vortex w i t h i n the horseshoe vortex system  i s derived from and maintained by a s p e c i f i c flow l e v e l w i t h i n the boundary l a y e r , and the flow over the e n t i r e thickness of the boundary l a y e r i s used to supply the horseshoe vortex system.  D.  RECOMMENDATIONS 1.  P r e d i c t i n g Scour Depths The present state of knowledge and understanding of l o c a l  scour  at bridge p i e r s i s such t h a t , i n general, the scour depth f o r a given p i e r geometry and flow regime cannot be predicted with confidence. More experimental work has been done with c i r c u l a r c y l i n d r i c a l piers than with any other type.  However, even f o r these, the only r e l a t i o n s h i p  that has been e s t a b l i s h e d w i t h any confidence i s the r e l a t i o n s h i p f o r the maximum depth of scour, d Breusers*  equation, d  s g m  s e m  -  =1.4  This seems to be adequately defined by b Cequation 211.  For p a r t i c u l a r flow c o n d i t i o n s , p i e r geometries, and bed sediment chara c t e r i s t i c s be s a f e l y predicted only on the basis of h y d r a u l i c laboratory  84.  studies i n which actual prototype conditions are accurately reproduced . on a small s c a l e .  Further, models of several d i f f e r e n t scales should be  tested to check f o r scale e f f e c t s .  The experimental work reported  here i n d i c a t e s that the v e r t i c a l v e l o c i t y d i s t r i b u t i o n should also be considered as one of the flow c h a r a c t e r i s t i c s that need to be properly scaled. 2.  Further  Research  The f o l l o w i n g areas are suggested as p a r t i c u l a r l y i n need of f u r t h e r research and i n v e s t i g a t i o n . Ca)  Experimental data covering a broad range of v e r t i c a l v e l o c i t y  d i s t r i b u t i o n s needs to be obtained, so that the i n f l u e n c e of t h i s flow parameter can be q u a n t i t a t i v e l y determined. CbI  The e f f e c t of the depth of flow on the v e r t i c a l v e l o c i t y  gradient, and thus on the e q u i l i b r i u m scour depth, should be investigated further. Cc)  Preyious experimental data should be reviewed i n the l i g h t  of the observed e f f e c t s of the v e r t i c a l v e l o c i t y gradient of the approach f l o w .  BIBLIOGRAPHY  85.  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" A n a l y t i c a l Approach to Local Scour," Proceedings, Twelfth Congress of the International Assoc. f o r Hydraulics Research, Fort C o l l i n s , Colorado, September, 1967, V o l . 3, pp. 151-161.  104.  Sanden, E.J. Scour at Bridge P i e r s and Erosion of River Banks. Department of Highways, A l b e r t a . Presented at Thirteenth Annual Conference of the Western Assoc. of Canadian Highway O f f i c i a l s , October 3, 1960.  95. 105.  Sarma, K.V.N. [Discussion of M.R. Carstens' " S i m i l a r i t y Laws f o r Localized S c o u r " ] , Journal of the Hydraulics D i v i s i o n , Am. Soc. of C i v i l Engrs., V o l . 93, No. HY2 (March 1967), pp. 67-71.  106.  Schneider, V.R. Mechanics of Local Scour. Ph.D. D i s s e r t a t i o n , Colorado State U n i v e r s i t y , Fort C o l l i n s , Colorado, December, 1968.  107.  Schraub, F.A., e_t al_. "Use of Hydrogen Bubbles f o r Q u a n t i t a t i v e Determination of Time-Dependent V e l o c i t y F i e l d s i n Low Speed Water Flows," Transactions, Am. Soc. of Mech. Engrs., V o l . 87, Series D [Journal of Basic Engineering, June, 1965), pp. 429-444.  108.  Shankarachar, D., and Chandrasekhara, T.R. .[Discussion of H.W. Shen e_t al_. "Local Scour Around Bridge P i e r s " ] , Journal of the Hydraulics D i v i s i o n , Am. Soc. of C i v i l Engrs., V o l . 96, No. HY8, (.August 1970), p. 1747.  109.  Shen, H.W., Ogawa, Y., and K a r a k i , S.S. "Time V a r i a t i o n of Bed Deformation near Bridge P i e r s , " International A s s o c i a t i o n f o r Hydraulic Research, Eleventh Congress, Leningrad, 1965, Volume I I I , paper no. 3.14.  110.  Shen, H.W., Schneider, V.R., and K a r a k i , S. Mechanics of Local Scour. 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J . {Discussion of E.M. Laursen's "Scour at Bridge C r o s s i n g s " ] , Journal of the Hydraulics D i v i s i o n , Am. Soc. of C i v i l Engrs., V o l . 86, No. HY9 (November 1960), pp. 134-137.  131.  T i s o n , L . J . "Local Scour i n R i v e r s , " Journal of Geophysical Research, V o l . 66, No. 12 (December 1961), pp. 4227-4232.  132.  Toomre, A. "The Viscous Secondary Flow Ahead of an I n f i n i t e Cylinder i n a Uniform P a r a l l e l Shear Flow," Journal of F l u i d Mechanics, V o l . 7, No. 1 (1960), pp. 145-155.  133.  Van Beesten, C. {Discussion of C R . N e i l l ' s "Measurements of Bridge Scour and Bed Changes i n a Flooding Sand-Bed R i v e r " ] , Proceedings, I n s t i t u t i o n of C i v i l Engineers, V o l . 36, February, 1967, pp. 400-401.  134.  V a r z e l i o t i s , A.N. Model Studies of Scour Around Bridge Piers and Stone Aprons. M.Sc. Thesis, U n i v e r s i t y of A l b e r t a , Edmonton, A l b e r t a , September, 1960.  135.  Veiga da Cunha, L. [Discussion of H.W. Shen et al_. "Local Scour Around Bridge P i e r s " ] , Journal of the HycFaulics D i v i s i o n , Am. Soc. of C i v i l Engrs., V o l . 96, No. HY8 (August 1970), pp. 17421747.  136.  V i n j e , J . J . "On the Flow C h a r a c t e r i s t i c s of V o r t i c e s i n ThreeDimensional Local Scour," Proceedings, Twelfth Congress of the International Assoc. f o r Hydraulic Research, Fort C o l l i n s , Colorado, September 1967, V o l . 3, pp. 207-217.  TABLES  TABLE I THE VARIATION OF EQUILIBRIUM SCOUR DEPTH WITH AVERAGE VELOCITY, FOR DIFFERENT PIER DIAMETERS (CIRCULAR PIER) AND DIFFERENT BED SEDIMENTS, AS REPORTED BY BREUSERS Pier Diameter b (cm.)  Flow Depth H (cm.)  Bed S ediment Type  (mm.)  C r i t i c a l Velocity f o r Sediment transport U  11  50  sand  0.2  0.2  sand  crit.  25  0.8 1.0 1.2 1.4 1.6  1.1 1.25 1.4 1.5 1.5  1.5  25  0.8 1.0 1.2 1.4 1.6  1.3 1.4 1.5 1.6 1.6  1.6  1.0 1.2 1.4 1.6  1.5 1.7 1.65 1.65  1.0  1.5  c r l t .  5  25  11  50  polysty ren e  1.5  9  21  50  polysty re n e  1.5  9  I  Max. Scour Depth Ratio  Scour Depth Ratio d se  (cm/sec.)  Velocity Ratio U U  b  d sem  b  1.7  TABLE II VALUES OF THE COEFFICIENT K USED IN R  THE EQUATION OF JAROSLAVTSIEV  Flow Depth Ratio  H/b  Coefficient  K  H  x  .6  1.0  1.5  2.0  2.5  3.0  3.5  4.0  5.0  6.0  8.0  .92  .67  .46  .31  .22  .15  .10  .08  .05  .05  .05  TABLE  III  SUMMARY OF SCOUR EXPERIMENTS Test P i e r :  Test Series  c i r c u l a r c y l i n d e r , d i a . b = 0.33 f t . Average Velocity U ft./sec.  Equilibrium Scour Depth d se ft.  Bed Sand:  d  = 0.215 mm., U  Time to reach d se t se min.  Depth of flow H ft.  H b  = 0.90  d  se b  U U  crit.  ft./sec. Water Temp.  1-A  0.89  0.37  720  1.26  3.8  1.12  0.99  63  1-B  0.85  0.33  690  1.24  3.7  1.00  0.95  58  1-C  0.77  0.31  1260  1.20  3.6  0.94  0.86  2-A1  63  0.85  0.37  1170  1.24  3.7  1.12  0.95  68  2-A2  0.85  0.36  1440  1.24  3.7  1.09  0.95  62  2-A3  0.85  0.38  1470  1.24  3.7  1.15  0.95  61  2-B  0.85  0.35  1320  1.24  3.7  1.06  0.95  65  2-C  0.85  0.31  1200  1.24  3.7  0.94  0.95  63  2-D  0.85  0.28  960  1.24  3.7  0.85  0.95  60  3-A  0.85  0.21  240  1.24  3.7  0.64  0.95  56  3-B  0.77  0.18  300  1.20  3.6  0.55  0.86  58  4-A  0.77  0.30  810  1.20  3.6  0.91  0.86  61  4-B  0.65  0.16  180  1.12  3.4  0.48  0.72  62  5  0.79  0.30  720  1.20  3.6  0.91  0.88  62  FIGURES  Ft on/  c^>c c //'o  n  Representation of curvature of the flow near an o b s t r u c t i o n , as proposed by Tison.  3  |  |  j  !  i  .'  ! 2  i T  6  i  Mean  i•  c)iameter  i  e<  ! i !  (mm)  0.44 0.58 0.97 |- 1.30 2.25  *i  4  i  o  0  •  Figure 2.  •  1  ;  i  2  Velocity (fps) — o o  1.00 1.25 o 1.50 -- r » 1.75 o 2.00 e 225 « 2.50 !  | 3  !  4  :  5  Laursen's non-dimensional p l o t of e q u i l i b r i u m scour depth (d ) versus flow depth (H), f o r a rectangular p i e r of width b, at an angle of attack of 3 0 ° . s  e  c/eah - u/a-feh  scout- ujUh  jenehz/  SCOUh  T  AvehagG  appi-oach  ve/oc/fy  (J  Figure 3 . Schematic diagram showing v a r i a t i o n of the e q u i l i b r i u m scour depth ( d ) with average flow v e l o c i t y (U), as found by Chabert g e  and Engeldinger ( f o r any p i e r ) .  a  Figure  4.  V e l o c i t y d i f f u s i o n i n t o the scour hole, according to Tarapo  Figure  5.  Schematic i l l u s t r a t i o n of scour hole at e q u i l i b r i u m c o n d i t i o n s , according to Tarapore.  106.  6.  Approach flow v e l o c i t y d i s t r i b u t i o n and stagnation pressure on the plane of symmetry i n front of a c i r c u l a r c y l i n d e r .  Figure  7.  Values of the c o e f f i c i e n t  to be used i n  the Maza and Sanchez v e r s i o n of J a r o s l a v t s i e v ' s equation.  Figure  8.  I d e a l i z e d representation of. the flow on the plane of symmetry i n f r o n t of a c i r c u l a r c y l i n d e r ; Shen e_t al..  cu//nJet-  b<Lcl  Figure  9.  Z>  Control volume on the stagnation plane i n front of a c i r c u l a r c y l i n d e r Shen et a l .  Essentially " Dead  Water"  vortex  is very  region weak  The  velocity  profiles are in  The  doshed  lines  approximate  only  proportion  show  location. Small  The the  10.  vortex  Strong primary  vortex  location of  steady  Figure  counter  but  the primary  a. o  xc> is  not  tends, in general, to move  vortex  up O  slope.  14  12  Distance  Upstream  1.0 From  0.8 t  Of  0.6 Pier  In  0.4  0.25  0.1  Feet  Schematic representation of the v e l o c i t y d i s t r i b u t i o n and flow pattern in the scour hole on the plane of symmetry i n front of a circular cylinder Shen et a l .  j i  i o  c  <1>  o , n  I)  0.5  o°  >  i  0  2,  Q.  Q \_  no  £  0  °a c  0  0  0  1  O o D  z> o o  <">  1  A  j 0.2  A  D  •  AA A  i  cr  1x1  Source dse 0. 0 0 0 7 3 R / ' 0  =  6  9  o  o. A  !  0  j 1  d  Shen et gl.  , mm  5 o  0.24(VA)  | Chabert and  0  • •> Engeldingsr i i  ,  Q  2  6  5  2  •  I  0.0 5 10'  Figure  -  I0  :  11. E q u i l i b r i u m scour depth versus pier Reynolds number (R ) for c i r c u l a r b p i e r s ; Shen et a l .  cylindrical  Figure  12.  Equilibrium scour circular  depth versus  cylindrical piers  of  pier  Reynolds  different  number  sizes;  Shen  (R^) et  for al.  Figure  13.  E q u i l i b r i u m scour depth versus p i e r Reynolds number (R^), f o r c i r c u l a r c y l i n d r i c a l p i e r s and d i f f e r e n t grain s i z e s ; Shen et a l .  114.  /.o .1  T  .3  1  •7  m  .6  X  m\  .5 •i  *  .J  ./  5  o ,Z  H u  14.  /  t  .t  Figure  1  *  .3  =10 max  .4  .5" .6  .7  cm.  = 27.7 cm./sec.  .8  r /W  .f  AO  •  run 1  x  run 2  V e l o c i t y d i s t r i b u t i o n along the v e r t i c a l , i n the experiments of Tanaka and Yano.  H A t  ll'l\ It i l I t i l l  I I I  i  Figure  15.  II II I / / / ' / / / / '  I/ t f i t  n  t I I' I I / I 7 I I I  1 I i I I I I i I 11  / > /  / I 7 I 111  m  Types of c i r c u l a r c y l i n d r i c a l p i e r s studied by Tanaka and Yano.  f I ITTTTTTTTTTTTTTT7  116.  I  -0  ? ©  .8  \\ \  ^  M  .6  n V  —.—. —\\\ i >  o **  JLY  \  €T i  Ci"  A >  y  f  ^  s  ^  /  /  /  /  /  .2  /  /  ~f  i)  -.2  se(I)  —£J-=  >  \  d  o  n  _ Q  '  :  o  4  I  3 cm.  = 5 cm.  = 5 cm.  <J>  I I , 1 cm. s q . h o l e  O  I I I , B /b  O  I I , 2 cm. s q . h o l e  O  I I I , Bj/b = 6 a  O  0  I I I , B. /b = 3 a I I I , B./b = 4 a  Figure  16.  d  «  IV  «  V  = 5  E q u i l i b r i u m depth of scour f o r d i f f e r e n t c i r c u l a r c y l i n d r i c a l p i e r types; and Yano.  Tanaka  117.  Figure  17.  Scour depth versus d i s c p o s i t i o n , c i r c u l a r c y l i n d r i c a l p i e r ; Thomas.  for a  For c Constant Pier and Sediment Size  Position  sem  of  curve  ^ depeWs o « value of  in  CL  o  Q  =3 O O CO  e  CT  Approach Velocity , U oo Figure  18. The scour regions of Schneider ( f o r any p i e r ) .  119.  Figure 19.  Sketch of the vortex s t r u c t u r e on the plane of symmetry i n front of a c i r c u l a r c y l i n d e r i n a laminar boundary l a y e r , from a photograph of an experiment of Gregory and Walker, published i n Thwaites  .  Figure 20.  Sketch of laboratory flume cross-section showing general arrangement.  View of v e l o c i t y - c o n t r o l gate from a p o s i t i o n downstream of the p i e r (gate i s r e f l e c t e d i n g l a s s w a l l s on e i t h e r side).  -1  -  127.  7.2  -  measure J VfS+he.iXHi  /.s 'feet...'." of f>ich face.  '3t  -  0.2-  0.2.  a. 4  o.6  o.S  Vefdcf^y in -Zee/ pch Figure. .27,.  Secoyi<J  V e l o c i t y p r o f i l e , s e r i e s 5. \•  128.  /2  \\-  -  • -1-  -  ... ._ i  0.8  f  .  -  of phi- g , cstHt  • --  <x4  :  az —  0.2  Figure .28,  S t a b i l i t y "of v e l o c i t y profiles;'. seri~e's"i-A'."  I  I.  i  Figure 297  S t a b i l i t y "of ."velocity p r o f i l e , " series-2-A3 .-7-  .... 1.  "S-evie/i  E-C  Vo^Kce-s  Figure  36.  ,  H  =  0.^5  2a.  <wj  • ,  U  3  O.3o  ft./***..  2. W 6 U - * . -^ov-wtej  Vortex pattern, f l a t bed with low flow v e l o c i t y . CO  cl = s  //oS  cU«  0.12  -ro  o./2  fr-  Sca./e.  ^  Vortex pattern, beginning of scour (cross-sectional view).  5eK,^i  2-  C  ;  //=/.?<///.  0 " 0.8S //./ce t  6  e^2 ou,| side oid side  Figure  39.  edL^,  e  *  2e> nun cc/cS  °-f scout- kole  ed^e  MAO**. uovWy, ^VJWJUV  VrtaAe*; 3  Vortex p a t t e r n , beginning of scour (plan view),  The p a r t i a l l y developed scour hole was formed by a flow of U =• 0.85 f t . / s e c . and H = 1.24 f t . to a depth d  = 0.20 f t . The flow was then slowed to s U = 0.45 f t . / s e c . to observe vortex patterns.  Figure  40.  /)f>f>k>acJ, -/loop //  HQ.  /.oo  approach flow.  //.  Q <fs ft. J sec  Vortex patterns; p a r t l y developed scour hole; showing dependence of the i n d i v i d u a l v o r t i c e s on the v e r t i c a l v e l o c i t y p r o f i l e of the  r,  Figure  42.  Scour hole development, showing motion of sand grains.  t—* -p. ro  (a)  Figure 43.  looking downstream  Views of fully-developed scour hole: Ca)  looking downstream; (b) looking  upstream.  -1^  Figure 44.  View of fully-developed scour hole, looking d i a g o n a l l y upstream.  

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