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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.Sc, Univers ity 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 in the Department • of CIVIL ENGINEERING Vie accept th i s thesis as conforming to the required standard THE UNIVERSITY" OF BRITISH COLUMBIA September, 1972. In presenting th i s thes i s in p a r t i a l f u l f i lmen t o f the requirements f o r an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r e e l y ava i l ab l e for reference and study. I fu r ther agree that permission for extens ive copying of th i s thes i s for scho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i c a t i on of th i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. C.A. van der Gugten Department of C i v i l Engineering The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date September 29, 1972. ABSTRACT This Thesis presents a review of the reported research on loca l sand-bed scour around bridge p ie r s , describes the mechanics of loca l scour with pa r t i cu la r reference to the horseshoe vortex, describes experiments on the ef fect of the ve r t i ca l ve loc i ty d i s t r i bu t i on of the approach flow on local scour, and reports the results of these experiments. The review of the previous research on local scour shows that the pier s ize i s the most important parameter af fect ing the equi l ibr ium depth of local scour, while the p ier shape i s of secondary importance. Two flow parameters are found to be important; the flow ve loc i ty 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 vo r t i ce s that ar ises out of the v o r t i c i t y always i n present in shear f lows, due to the interact ion of the p ier and the flow. The experiments that were done consisted of observations and measurements o f - ve r t i c a l -velocity d i s t r i bu t i on s , scour depths, and vortex patterns, fo r a c i r c u l a r cy l inder in a laboratory flume. These experiments showed that the ve r t i ca l : v e l o c i t y d i s t r i bu t i on of the boundary layer flow approaching the p ier affects the structure of the horseshoe vortex system and the equi l ibr ium depth of scour at the p ier nose. The Thesis concludes with a Summary and Conclusions, including recommendations for further research. r i v . TABLE OF CONTENTS Page LIST OF TABLES v i i i . LIST OF FIGURES i x . LIST OF SYMBOLS x i i i . ACKNOWLEDGEMENTS X X . INTRODUCTION 1. •' LITERATURE REVIEW 5. A. EARLY INVESTIGATORS 5. B. TISON AND ISHIHARA 6. 1. Tison 6. 2. Ishihara 7. C. INGLIS AND REGIME THEORY ADHERENTS 9. 1. Ingl i s 9. 2. Blench 9. 3. Va r ze l i o t i s 10. D. LAURSEN 11. E. CHABERT AND ENGELDINGER 14. V . Chapter Page II F. BATA AND KNEZEVIC 16. 1. Bata 16. 2. Knezevic 16. G. TARAPORE 18. H. MOORE AND MASCH 20. I. BREUSERS: DELFT HYDRAULICS LABORATORY 22. 1. Oosterschelde Bridge Model Studies 22. 2. Scour Around D r i l l i n g Platforms 24. J . MAZA AND SANCHEZ 24. K. SHEN et a l : COLORADO STATE UNIVERSITY 26., 1. Introduction 26.. 2. Descr ipt ion of the Mechanics of Local Scour 26.. 3. Theoretical Analysis 28. 4. Experimental Results 31.$, 5. Reconc i l iat ion of Divergent Concepts 33. 6. Methods of Reducing Scour 34.^ * L. CARSTENS 35. M. TANAKA AND YANO, AND THOMAS 37. 1. Tanaka and Yano 37. 2. Thomas 38. N. SCHNEIDER 39. 0. COLEMAN 43. I I I . THE MECHANICS OF LOCAL SCOUR 45. A. INTRODUCTION 45. B. THE BASIC VORTEX MECHANISM 46. C. ORIGIN OF THE HORSESHOE VORTEX 47. 1. V o r t i c i t y 47. 2. V o r t i c i t y in the Flow near a Cyl inder 48. (a) The Approach Flow. 48. (b) Flow near the Cyl inder. 49. D. CHARACTERISTICS OF THE HORSESHOE VORTEX 1. Structure 2. Formation 3. Strength E. SUGGESTED RESEARCH EXPERIMENTAL WORK A. PURPOSE AND SCOPE B. EQUIPMENT AND EXPERIMENTAL ARRANGEMENT 1. General Arrangement 2. Sand Bed 3. Test P ier 4. Ve loc i t y -D i s t r i bu t i on Control Gate 5. Current Flowmeter 6. Dye Injector C. EXPERIMENTAL METHODS AND PROCEDURES 1. Starting-up 2. Generation of Ver t i ca l Ve loc i ty P ro f i l e s 3. Ve loc i ty Measurements 4. Flow Patterns 5. Scour Hole Development and Equi l ibr ium Depth of Scour EXPERIMENTAL RESULTS A. GENERAL B. APPROACH FLOW VELOCITY PROFILES C. EQUILIBRIUM SCOUR DEPTH 1. Introduction 2. Results 3. Comparison with Results of Others (a) Laursen (b) Tarapore (c) Breusers (d) Maza and Sanchez (e) Larras Cf ] Shen ert aj_. Cg) Coleman v i i . Chapter Page V. D. VORTEX PATTERNS AND SCOUR HOLE DEVELOPMENT 72. 1. The Horseshoe vortex System 72. 2. Scour Hole Development: Beginning of Scour 73. 3. Approach Flow Ve loc i ty P r o f i l e and Vortex Structure 75. 4. Transport out of the Scour Hole 76. VI. SUMMARY AND CONCLUSIONS 78. A. PREVIOUS INVESTIGATIONS 78. B. MECHANISM OF LOCAL SCOUR 80. C. EXPERIMENTAL RESULTS 81. I-. Introduction 81. 2. Equi l ibr ium Depth of Scour 81. 3. Horseshoe Vortex Flow Patterns 82. D. RECOMMENDATIONS 83. 1. Predict ing Scour Depths 83. 2. Further Research 84. BIBLIOGRAPHY 85. TABLES 98. FIGURES 101. v i i i . LIST OF TABLES Table Page I The va r i a t i on of equi l ibr ium scour depth with average v e l o c i t y , fo r d i f f e ren t p ier diameters ( c i r c u l a r p ier) and d i f f e ren t bed sediments, as reported by Breusers. 98 II Values of the coe f f i c i en t 1C used i n the q g equation of Jaros lavt s iev . Ill Summary of Scour Experiments. 100 i x . LIST OF FIGURES Figure Page 1. Representation of curvature of the flow near an obst ruct ion, as proposed by Tison. 1Q1 2. Laursen's non-dimensional p lot of equi l ibr ium scour depth (d ) versus flow depth (H), for a rectangular s e p ier of width b, at an angle of attack of 30°. 102 3. Schematic diagram showing va r i a t i on of the equi l ibr ium scour depth ( d s e ) with average flow ve loc i t y Cu), as found by Chabert and Engeldinger ( for any p i e r ) . 103 4. Ve loc i ty d i f f u s i on into the scour hole, according to Tarapore. 104 5. Schematic i l l u s t r a t i o n of scour hole at equi l ibr ium condit ions, according to Tarapore. 105 6. Approach flow ve loc i t y d i s t r i bu t i on and stagnation pressure on the plane of symmetry in f ront of a c i r c u l a r cy l inder . 106 7. Values of the coe f f i c i en t IC^ to be used in the Maza and Sanchez vers io r r of Ja ro s l av t s i ev 1 s equation. 107 Figure Page 8. Idealized representation of the flow on the plane of symmetry in f ront of a c i r c u l a r cy l inder ; Shen e_t al_. 108 9. Control volume on the stagnation plane in f ront of a c i r c u l a r cy l i nder , Shen ejt al_. 109 10. Schematic representation of the ve loc i t y d i s t r i bu t i on and flow pattern in the scour hole on the plane of symmetry in f ront of a c i r c u l a r cy l i nder ; Shen et al_. 110 11. Equi l ibr ium scour depth versus p ier Reynolds number CR, ) for 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].. * HI 12. Equi l ibr ium scour depth versus p ier 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 ren t s i z e s ; Shen ejt al_. 112 13. Equi l ibr ium scour depth versus p ier 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 ren t grain s i ze s ; Shen e_t al_. 113 14. Ve loc i ty d i s t r i bu t i on along the v e r t i c a l , in 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 piers studied by Tanaka and Yano. 115 16. Equi l ibr ium depth of scour fo r d i f f e ren t c i r c u l a r c y l i n d r i c a l p ier types; Tanaka and Yano. 116 17. Scour depth versus disc po s i t i on , fo 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 ( for any p i e r ) . 118 19. Sketch of the vortex structure on the plane of symmetry in f ront of a c i r c u l a r cy l inder in a laminar boundary l ayer , from a photograph of an experiment by Gregory and Walker, published in thwaitesC125l. 119 20. Sketch of laboratory flume cross-sect ion showing general arrangement. 120 x i . Figure Page 21. Flume sand gra in- s ize d i s t r i b u t i o n . 121 22. View of ve loc i t y -cont ro l gate from a pos i t ion downstream of the p ier (gate i s re f lected i n glass walls on e i ther s ide ) . 122 23. Ve loc i ty p r o f i l e s , series 1. 123 24. Ve loc i ty p r o f i l e s , ser ies 2-A. 124 25. Ve loc i ty p r o f i l e s , series 2. 125 26. Ve loc i ty p r o f i l e s , series 3 and 4. 126 27. Ve loc i ty p r o f i l e , ser ies 5. 127 28. S t a b i l i t y of ve l oc i t y p r o f i l e , series 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 , ser ies 2-A3. 129 30. Scour hole development with time, ser ies 1. 130 31. Scour hole development with time, ser ies 2-A. 131 32. Scour hole development with time, ser ies 2. 132 33. Scour hole development with time, series 3. 133 34. Scour hole development with time, ser ies 4. 134 35. Scour hole development with time, ser ies 5. 135 36. Vortex pattern, f l a t bed with low flow ve l oc i t y . 136 37. Vortex pattern, beginning of scour (cross-sect ional view). 137 38. Scour hole development with time, at 9 = 30° and e = 0 ° , ser ies 1-A. 138 39. Vortex pattern, beginning of scour (plan view). 139 40. Vortex patterns; par t ly developed scour hole; showing dependence of the ind iv idua l vort ices on the v e r t i c a l ve loc i t y p r o f i l e of the approach f low. 140 Figure 41. Vortex patterns; par t ly developed scour hole showing e f f ec t of the v e r t i c a l ve l oc i t y p r o f i l e of the approach flow. 42. Scour hole development, showing motion of sand grains. 43. Views of ful ly-developed scour hole: Ca) looking downstream; (b) looking upstream 44. View of f u l l y developed scour hoVj&i looking diagonally upstream. xi i i . LIST OF SYMBOLS A corner vof the control volume ABCD (see Figure 9). Area of the horseshoe vortex core. Points on a v e r t i c a l in the flow near the f ront of an obstruction Csee Figure 1). Cylinder radius Parameter varying with type of f low: a = 0.6 fo r main flow channel; a^ ^ = 1.0 for f lood p la in (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 sc f i t t e d around a c y l i n d r i c a l p ier (see Figure 15). Points on a v e r t i c a l in the flow in the region which remains unaffected by the pressure of an obstruction (see Figure 1 ) . P ier width or diameter. P ier width projected onto a plane perpendicular to the approach flow d i r e c t i o n . A constant c oe f f i c i e n t . X I V . Symbol C A corner of the control volume ABCD (see F igure ' 9 ) . C Experimental constant ( I shihara). C A coe f f i c i en t proportional to channel roughness and 1 u'/H Clshihara). D A corner of the control volume ABCD (see Figure 9). D c C r i t i c a l depth of flow = ( q 2 / g ) 1 / 3 D L Lacey regime depth = 0.47 ( Q m / f L ) 1 / 3 , in feet . D Total scoured depth, measured from the water surface, in 3 feet. dA Element of surface area. ds Element of length. d Blench's zero f lood depth. d g Character i s t ic grain s ize of bed sediment used by Carstens. d g Depth of loca l scour below normal bed l e v e l . d Equi l ibr ium or l i m i t i n g value of d , for a given sediment, s e p ier geometry, and flow cond i t ions 3 d Maximum possible value of d , fo r a given p ier geometry. d The grain s i ze such that 00% of the material is f i n e r , by 0 0 weight. e The base of natural logarithms = 2.718. F The flow Froude number = U/j/gH* F, Blench bed factor , b F, Blench zero bed factor , bo F Total d isturbing force on a bed sand-grain (Carstens). M F Total retarding force on a bed sand-grain (Carstens). R XV. Function of Lacey s i l t factor . Accelerat ion of grav i ty . Approach flow depth. Height of p ier modif icat ion above normal bed level Csee Figure 15). Pier shape coe f f i c i en t (Larras). A constant c oe f f i c i en t in a uniform flow re l a t i on (Shen). P ier shape coe f f i c i en t ( Ja ros lav t s iev ) . A coe f f i c i en t which i s a function of H/b1 (see Table II). A coe f f i c i en t which i s a function of U /gb and H/b Csee Figure 7). P ier shape coe f f i c i en t CLaursen). A coe f f i c i en t = 0.53 ( U 2 / g b 1 ) " ° ' 3 2 CMaza and Sanchez). Coef f i c ient fo r the angle of attack of the approach flow A coe f f i c i en t representing the rate of ve loc i t y d i f fu s i on into the scour hole CTarapore). Character i s t i c length of flow obstruct ion. Distance along a flow streamline, s ta r t ing at scour-hole A constant exponent in a uniform flow re l a t i on (Shen). Carstens' sediment number = u"//ts-l)gd . to the p ier alignment edge. g The c r i t i c a l value of N at which loca l scour i s i n i t i a t e d . s xv i . A point on the edge of the scour hole (see Figure 5). Hydrostatic pressure. Stagnation pressure. Flow discharge. C r i t i c a l yalue of Q at which loca l scour i s i n i t i a t e d . Maximum flood discharge, in cubic feet per second. Time rate of sediment transport into scour hole (volume). Time rate of sediment transport out of scour hole (volume). Time rate at which sediment i s removed from the scour hole, in being deepened (weight). Value of Q' at the beginning of the loca l scour process; i e . at s t = 0. Time rate of sediment transport into scour hole (weight). = q ' . b (Schneider). s m Time rate of sediment transport out of scour hole (weight). Flow discharge per un i t width. The value of q for the central approach flow ju s t upstream of the p i e r , in cubic feet per second per foot. Rate of bed-load sediment transport per un i t width (Straub). Time rate of sediment transport into the scour hole, per un i t width (weight). P ier Reynolds number = Ub/u. Radius of curvature of flow streamlines. Scour force at p ier nose ( I sh ihara). xvi i . Spec i f i c grav i ty of bed sediment. Time, from the beginning of the loca l scour process. Time constant (Schneider). Time at which d reaches d . s se Average ve l oc i t y of approach f low. The value of U at which loca l scour i s i n i t i a t e d . The value of U at which general bed-load transport i s i n i t i a t e d . The loca l ve loc i t y in the x - d i r e c t i on. Local ye l oc i t y at the boundary (Tarapore). Local potent ia l ve l oc i t y (Tarapore). Total ve l oc i t y vector (Roper). Volume of scour hole. Local ve l oc i t y in the y - d i r e c t i on. Local ve l oc i t y in the x -d i r ec t i on. Local ye loc i t y in the y - d i r e c t i on. Local ve l oc i t y in the z -d i rec t i on . Stream or channel width. Scour hole width. Local ve loc i t y i n the z - d i r e c t i on . Axes of orthogonal co-ordinate system. Eleyation head. x v i i i . Included angle of p ier nose. C i rcu la t ion = (jl V" • 3s~ Spec i f i c weight of water. Boundary layer thickness. Constant exponent in a uniform flow equation (Shen). Void r a t i o of bed sediment. Kinematic v i s co s i t y of water. Upward distance that the scour hole a f fects the main flow (Jarapore). 3.1416 Density of water. Density of bed sediment. Standard dey iat ion. Bed shear s t ress . 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 s gb (Schneider). Sediment transport cha rac te 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 rotat ion in the x - d i r e c t i on. V o r t i c i t y with axis of rotat ion in the y - d i r e c t i on . x i x . V o r t i c i t y with axis of rotat ion in the z - d i r e c t i on. Angular ve loc i t y of the horseshoe vortex core. Vector operator. A superscr ipt ind icat ing that the parameter i s to be evaluated at the surface of the f low. A superscr ipt ind icat ing that the parameter i s to be evaluated at the bed. XX. ACKNOWLEDGEMENTS The author wishes to express his s incere thanks to his Supervisor, Professor E.S. Pret ious, fo r his helpful advice, constant encouragement, and enduring patience throughout th i s study, e spec ia l l y during the l a s t months, when he was already r e t i r ed from the Univers i ty . Thanks are due to Professor H.W. Shen, of Colorado State Un iver s i ty , f o r his suggestions and encouragement given to the author during a v i s i t to the C.S.U. campus in the summer of 1968. Special thanks are due to Dr. Verne Schneider, then a post-graduate student at Colorado State, fo r his wi l l ingness to spend considerable time with the author in discussing and demonstrating various aspects of the loca l scour phenomenon during th i s v i s i t . The f i nanc ia l support of the National Research Council of Canada i s g ra te fu l l y acknowledged. The t r i p to Colorado was made possible through the N.R.C. operating grant to Professor Pret ious. CHAPTER I INTRODUCTION The successful foundation design of a structure founded in the sand-bed of a flowing stream, in order to safe-guard against excessive loss of bearing capacity, must include reasonably accurate methods for predict ing the tota l amount of bed degradation at the foundations. Degradation of the bed of a stream at a par t i cu la r cross-section can occur due to changes in the general sediment-transport capacity of the flow at that sect ion. For example, an increase in the ve loc i ty of the f low, due to increased discharge or reduced channel width, increases the capacity of the flow to carry sediment, and degradation resu l t s . S im 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 addit ion to general degradation of the bed at a channel sect ion, loca l bed-level changes can occur due to : (1 ) meandering of the main flow channel (thalweg) between the banks of the stream, (2) the downstream movement of bed dunes, C3) loca l scour. Local scour i s the scour of the stream bed in the immediate v i c i n i t y of a structure Csuch as a groyne, t ra in ing w a l l , wharf, bridge p i e r , or abutment) founded in that bed. I t i s pr imar i ly caused by the flow disturbance generated by such a s t ructure. I t was the purpose of th i s thesis to invest igate some of the aspects of the loca l scour phenomenon, with the aim of improving our understanding of the mechanics of the scour process and thus provide a sounder basis fo r predict ing bed changes due to loca l scour. The l i t e r a t u r e on loca l scour was reviewed f a i r l y comprehensively, and i s summarized in Chapter II. This review out l ines the h i s t o r i c a l development of , and provides a basis f o r , the understanding of the loca l scour phenomenon. Although a l l of the references in the Bibliography were examined, only the more important and representative studies and f indings are reported in the review. Chapter III describes the mechanics of loca l scour of a sand-bed at a v e r t i c a l cy l i nder , based on the f indings summarized in Chapter II and on observations reported in the f i e l d of f l u i d dynamics. The o r i g in and character i s t i c s of the horseshoe vortex are discussed, pa r t i c u l a r l y the various flow parameters that influence the strength of the horseshoe 3. vortex system. After considering a l l of the ava i lable information on loca l scour developed in Chapters II and I I I , i t was decided to invest igate the influence of the approach flow ve loc i t y d i s t r i b u t i o n . An out l ine of the experimental work done, and a descr ipt ion of the equipment and methods which were used, is given in Chapter IV. The experiments consisted of observations and measurements of ve l oc i t y d i s t r i b u t i on s , scour depths, and flow patterns in a laboratory flume. Chapter V presents the resu lts of the experimental work, and Chapter VI gives the summary and conclusions. Symbols are defined where they f i r s t appear in the tex t , and also in the L i s t of Symbols, p . x i i i . B ib l iograph ica l references are indicated by ra i sed, bracketed numerals. Footnotes are indicated by ra i sed , unbracketed numerals, and are located at the end of each chapter. The author was f i r s t encouraged to engage in studies in the f i e l d of loca l r iver-bed scour by his thesis supervisor, Professor E.S. Pret ious, who had previously, in a consulting capacity, carr ied out pr ivate invest igat ions of r iver-bed scour at bridge p ie r s . These invest igat ions were mainly concerned with f ind ing the most adverse scour patterns that could occur at the piers of actual proposed or ex i s t ing 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 innaird, 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 in these pr ivate invest igat ions was to carry out a 4. hydraul ic laboratory study using scale model piers and a moveable, sand-bed, and to compare the resu l t s thus obtained with scour depths predicted by various formulas. I t was found that often the predicted scour and the laboratory tes t resu l t s did not agree very w e l l . I t was th i s lack of a quick, r a t i o n a l , and safe method fo r predict ing r iver-bed scour around flow obstruct ions,that motivated the present study. 5. CHAPTER IT LITERATURE REVIEW A. EARLY INVESTIGATORS The f i r s t reported 1 model studies involv ing bridge piers were carr ied out in Germany in the 1890's by H. Engels at the Technical (3l) Univers i ty of Dresden^ . These studies showed that the maximum scour of the bed occurs at the upstream end or nose of the p ie r . They also showed that r iprap should be placed around the p ier f lush with the normal bed l e v e l , rather than on top of the bed. No other model studies were reported un t i l the ear ly 1920's, when T.H. Rehbock at the Technical Univers i ty of Karlsruhe did some tests and found that the maximum depth of scour Cwhich occurred at the p ier nose) varied with flow v e l o c i t y , bed mater ia l s , p ier shape, and duration of f low, but the nature of these var iat ions was not reported in d e t a i l ^ 9 8 ^ . The inf luence of the depth of flow was apparently not invest igated. The scour at the p ier nose was att r ibuted to cross-currents set up there. B. TISON AND ISHIKARA 1. Tison The f i r s t attempt at a theoret ica l approach to loca l scour was o made by L .J . Tison in 1937 . His work has subsequently been republished in yarious places L ' * . He considered the flow near the f ront of an obstruction in an open channel to be analogous to the flow of an i r r o t a t i ona l vortex. On th i s basis he obtained an expression fo r a dif ference in piezometric head between the surface and the bottom of the flow close to the front of the obstruct ion: Y * " C ° PA, 9 = B' .2 o u ds' A' r' o B° a± ds-] A o r " ( i ) where z = the elevat ion head, p = the s t a t i c pressure, y = the s p e c i f i c weight of water, g = the accelerat ion of g rav i ty , u = the loca l approach flow v e l o c i t y , r = the radius of curvature of the flow streamlines, ds = an element of length taken along an orthogonal to the flow streamlines from A to B ,A represents points on a ve r t i c a l near the front of the o o' o obstruct ion, B Qrepresents points on a v e r t i c a l in the flow which remains uninfluenced by the obst ruct ion, s.ingle-primed symbols are for the flow at the surface, and double-primed symbols are f o r the flow at the bottom (see Figure 1 ) . 7. For the case of an obstruction of uniform cross - sect ion, r ' = r" and ds' = ds". I f , as commonly occurs in streams and laboratory flumes, the flow i s not uniform, but a ve l oc i t y gradient ex i s t s on the v e r t i c a l plane, the surface ve loc i t y u' w i l l be greater than the bottom ve l o c i t y u " , and the right-hand side of equation Cl) w i l l be pos i t i ve . Thus there w i l l be a pressure difference between surface and bottom, and as a re su l t the flow w i l l acquire a downward component in front of the p ier and attack the bed, causing scour. Tison concluded that the greater the di f ference between u' and u", the greater would be the scour. In add i t ion, smaller Values of r ' would also increase the scour. Tison v e r i f i e d each of these conclusions with laboratory tes t s . He used p ier 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 ier shape became more streamlined and permitted a more gentle curvature of the flow. By using a very coarse material fo r the upstream bed, he was able to change the ve loc i t y d i s t r i b u t i on so that u" became less and u 1 became greater, thus increasing the dif ference between u' and u " , and obtained a greater scour hole depth. However, his resu l t s are only qua l i t a t i ve i n nature and he did not present any s p e c i f i c re lat ionsh ip involv ing the depth of scour. 2. Ishihara Shortly a f ter Tison f i r s t published his work, Ishihara performed an extensive series of tests 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 ier t a i l or the length of the p i e r , fo r piers aligned with the 8. flow d i r ec t i on . The scour depth was found to increase with increased skewness of the p ie r s ; the more so for a sharper p ier nose. The e f fec t of const r i c t ion of the f low, expressed in terms of the r a t i o of stream width W to p ier width b, was found to be smal l , f o r values of W/b from about 6 to 10, and n i l for values of W/b greater than about 15. I t was observed that scour decreased with decreasing flow depth, but that the rate of th i s decrease depended on the character i s t i c s of the sand, the p ier shape and the p ier s i z e . In addit ion to his experimental work, Ishihara developed a theory fo r loca l scour at p iers by assuming the flow near a p ier to be s im i l a r to the flow in the bend of a r i v e r . He obtained an expression f o r the secondary downward flow component in a r i ve 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 na l vortex, much a f te r the manner of Tison Csee above). By assuming that the "scour fo rce " was proportional to the value of the downward flow component, and applying the resu lts fo r the r i v e r bend d i r e c t l y to the case of flow around an obst ruct ion, he obtained the expression: SF = C 1 C 2 A 2 ^ d s ( 2 ) B o where S w = the scour force at the p ier nose, C = an experimental constant, C 2 = a parameter which increases with increasing channel roughness and decreasing value of H/u', U = the average ve loc i t y of the approach f low, and H = the flow depth. This expression i s substant ia l l y s im i l a r to that of Tison Cequation C l ) ) . 9. C. INGLIS AND REGIME THEORY ADHERENTS 1. Ing l i s The f i r s t re lat ionsh ip e x p l i c i t l y involv ing the depth of scour at a bridge p ier was formulated by S i r Claude Ingl i s Cwith A.R. Thomas and D.V. Joglekar) in 1939 . On the basis of model s tudies, using round-nosed piers of s im i l a r geometry but d i f f e r i n g s i z e , he obtained: ^ = 1.70 I-| J 0.78 (3) where D$ = the tota l scoured depth measured from the water surface (in f e e t ) , b = the p ier width (in f e e t ) , and q c = the central approach-flow discharge per un i t width O n square feet per second). Ing l i s in 1949 presented a second re l a t i on sh ip , th 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. He obtained: D s = 2D L C4) where Dg i s the tota l scoured depth as defined above, and i s the Lacey regime depth: D L = 0.47 \ * / 3 (5) f L where Qm = the maximum flood discharge i n cubic feet per secondhand f^ = the Lacey s i l t f ac tor . 2. Blench Later, B l ench^ 1 1 ^ , in 1957, reported the resu l t s of a p l o t , by 1/3 Andru, using data of I ng l i s , Laursen, and others, of D ^ vs. q, 10. without any attempt to d i f f e ren t i a te between d i f f e ren 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 ne of: 1 / 3 0 74 D s F b = q (6) where D$ = the tota l scour depth below the water surface ( in f e e t ) , Ffa = 2 the Blench bed-factor = U /H (U = average approach flow ve loc i t y in feet per second, H = approach flow depth in f e e t ) , and q = the approach flow discharge per un i t width ( in square feet per second). In the same work, Blench proposes a re lat ionsh ip for maximum scour at bridge piers which i s the same as equation (4) of I ng l i s , except that instead of the Lacey depth, , Blench proposes a "zero f lood depth," d f 0 , which i s supposed to represent the required regime depth of a canal having a bed factor corresponding to zero charge ( i e . bed-load charge i s 2 3 zero: = F^Q = q / d^ Q ), with the discharge at the maximum, and using the reduced obstructed w i d t h 4 . 3. Va r ze l i o t i s In 1960, Va r ze l i o t i s did a laboratory study of loca l scour at bridge p i e r s ^ 1 3 4 \ Using arguments based on regime theory to arrange his data, he obtained: Dr 1 U D F, J C C DO 1 1 . O 1 / o where D c = the c r i t i c a l depth of flow (= Iq /g] ' ). Although a better " f i t " to the data i s obtained using an index of 0.28, Va r ze l i o t i s used h in the be l i e f that natural laws usual ly fo l low simple ind ices . D. LAURSEN I t was not unt i l the ear ly 1950's that experimental work directed towards the establishment of a design re la t i on for loca l scour at bridge piers was started on a s i g n i f i c an t s ca le , by E.M. Lau r sen^ 5 9 , 6 0 , 6 1 ^ . Laursen also formulated a number of concepts with respect to loca l scour which are useful in estab l i sh ing a framework fo r the understanding of the loca l scour problem. These concepts can be enumerated as fo l lows: (a) The rate of scour equals the dif ference between the sediment transport rate into the scour hole and the sediment transport rate out of the scour hole. Symbol ical ly: 4 £ = Q . - Q . (8) dt x s out y s i n v ' where cW _ the time rate of change of volume of the scour hole, Q . dt " the time rate at which sediment i s carr ied out of the scour hole ( in cubic feet per second), and Q$ - n = the time rate at which sediment i s transported into the scour hole ( in cubic feet per second) 5. (b) The rate at which the volume of the scour hole increases w i l l decrease as the hole gets bigger. (c) There w i l l be some l i m i t i n g s ize of scour hole (for any given geometry and flow condit ion). When th i s occurs the scour depth i s said to be at equ i l ibr ium, and Q n n + = Qc 12. (d) This l im i t i n g s i ze w i l l be approached asymptotical ly. On the basis of the f i r s t concept, Laursen dist inguished between three d i f fe rent scour cases: Ca) No scour. This condit ion occurs when the ve loc i t y of the flow i s too low to cause any loca l scour, and i n i t i a l conditions are ^s in = 9s out = ° ' Cbl "Clear-water" scour 6 . The flow disturbance due to the obstruction 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 . = 0 , Q . > 0 . s s i n ' ^s out Cc] Scour with general sediment motion. The i n i t i a l conditions are Q . > Q . > 0 . x s out x s i n 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 invest igated the e f fect s of bed sediment s i z e , average ve l oc i t y of the approach f low, and flow depth, on the equi l ibr ium depth of scour at a p ie r . Although his data showed some sca t te r , he found no e f f ec t of bed sediment s i ze or average flow ve l o c i t y , and only the flow depth was found to have an e f f e c t on the equi l ibr ium scour depth. He was able to present the influence of the flow depth in terms of a graph, using co-ordinates non-dimensionalized on the basis of the p ier width (see Figure 2 ) ^ 6 1 ^ . Laursen Cas had Posey^ 9 3^ before him) observed that the basic scouring agent was a r o l l e r or vortex, with a horizontal a x i s , which formed i n f ront of the p ier nose. In order to explain the observed 13. ef fects of the ayerage flow v e l o c i t y , flow depth, and sediment s i z e , Laursen reasoned along the fol lowing l i n e s ^ 5 9 ' 6 1 ^ . At any given equi l ibr ium scour condit ion in 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 into the hole. If the average flow ve loc i t y i s increased, the angular ve loc i t y of the r o l l e r could be expected to increase in a proportional way, and thus Q s t would increase. Howeyer, the ve loc i t y of the approach flow near the bed would also increase, and the net e f f ec t would be no change in the di f ference Q ^ . - Q . . S i m i l a r l y , i f the sediment s ize would be ^s out i n "*' increased, the transport rates into and out of the scour hole would decrease, but in the same proport ion, so that there would be no change i n the equi l ibr ium scour depth. When the flow depth i s increased, however, the angular ve l oc i t y of the r o l l e r i s presumed to remain the same as long as the average flow ye loc i t y remains constant; thus Q s t does not increase. However, the ye loc i t y of the approach flow near the bed would be decreased somewhat, so that Q . would decrease, and the net re su l t would be an increase in ^s m the equi l ibr ium scour depth. Laursen concluded from his studies that the equi l ibr ium depth of scour, fo r flows below the c r i t i c a l CFroude number < 1) and capable of general bed-load transport, depends only on the flow depth, p ier s i z e , p ier shape, and the angle of attack of the approach flow. This dependence was presented in the form of design curves and t a b l e s ^ 6 1 ^ . These design 14. c r i t e r i a can be reduced to the expression: I T = K s K a • i - 5 0 ^ ° * 3 <9> where d s g = the equi l ibr ium depth of scour below normal bed 1 eve!, K$ i s a fac to r fo r p ier shape, and K a i s a fac to r for angle of attack. 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 to : % = 1.35 [g-J 0 ' 3 (10) Laursen l a te r t r i ed to incorporate loca l bed scour at bridge p i e r s , fo r both the "c lear-water " case and the case of general sediment transport, into 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 did th i s by f i r s t developing an equation for the depth of scour in a long contract ion, based on the Manning formula, his own sediment-concentration formula, and considerations of cont inu i ty of both the flow and the sediment discharges. Laursen then related the depth of loca l scour at a bridge p ier to the general scour in a long contraction by introducing a special c oe f f i c i en t to account for the loca l non-uniformity of the flow in the former, and by assuming that the contracted width could be represented by 2.75 d s g and the uncontracted width by 2.75 d g e + 0.5b. Comparison of his theory to experimental data of Ing l i s and Chabert and Engeldinger Crefs. 65 and 191, respect ive ly , in Karaki and Haynie^ 5 1h , showed qua l i t a t i ve agreement only. E. CHABERT AND ENGELDINGER In 1956 Chabert and Engeldinger, in France, reported a large ser ies of tests of loca l scour at bridge p i e r s 7 . Their study involved scour in 15. both the region of "c lear-water " scour and the region of general sediment transport. Their main contr ibut ion was that they found the equi l ibr ium scour depth to increase roughly l i n ea r l y with the bed shear stress throughout the region of "c lear-water " scour, and that the maximum equi l ibr ium depth of scour occurred in the t r an s i t i on region between "c lear-water " scour and scour with general sediment transport - i e . when the average flow ve loc i t y i s at about the c r i t i c a l for general bed-load transport Csee Figure 3 i . Chabert and Engeldinger also tested d i f f e ren t s izes of bed mater i a l . They obseryed that the maximum equi l ibr ium scour depth fo r a given p ier increased with increasing sediment s i z e , for median grain diameters of 0.26 mm, 0.52 mm, and 1.50 mm. However, fo r a sand of median diameter = 3.00 mm, the maximum equi l ibr ium scour depth was less than that f o r the 1.50 mm sand 9. Later, Larras obtained a re l a t i on for loca l scour based on the data of Chabert and Engeldinger, as well as f i e l d d a t a 1 0 : dsem = 1 A Z K b ° * 7 5 C l l ) where d i s the maximum equi l ibr ium depth of scour below normal bed sem n r  l e v e l , in f ee t , and K i s a factor to account for p ier shape: K = 1.0 for c i r c u l a r p i e r s , K = 1.4 fo r rectangular p ie r s . The pier width b i s measured in feet . i 16. F. BATA AND KNEZEVIC 1. Bata In 1960 Bata reported a study of the problem of local scour at bridge piers using f i e l d measurements, laboratory t e s t s , and theoret ica l a n a l y s i s ^ , tie applied potent ia l flow analysis to an assumed logarithmic ve loc i t y d i s t r i bu t i on of the flow approaching a c i r c u l a r p ie r . This analysis showed that v e r t i c a l l y downward ve loc i t y components are present in front of the p i e r , and have a magnitude approximately one-half of the magnitude of the average approach flow ve l o c i t y , fo r the region which extends upstream about three or four p ier r ad i i in front of the p ie r . These v e r t i c a l ve loc i t y components were thought to be the main cause of loca l scour. Bata p lotted his laboratory and f i e l d data as d g e /H versus U /gH Cthe flow Froude number), and obtained a l i nea r re la t ionsh ip . This was noted to be s im i l a r to the formula of Jaroslavcev (Jaroslavts iev), which has d s e a l l ( th is formula i s discussed below in connection with the work of Maza and Sanchez). 2. Knezevic Knezevic in 1960 also reported a study of the loca l scour ("53 \ problem . The relevant aspects of his study involved attempts to determine the conditions required for the s t a r t of loca l scour, the equi l ibr ium depth of scour, and the influence of several methods of reducing the maximum depth of scour. The results of the invest igat ion into the conditions required f o r loca l scour were presented in terms of the c r i t i c a l discharge Q ju s t 17. large enough to s ta r t loca l scour. I t appears that Qc was determined by extrapo lat ion, to d $ e = 0, of plots of d g e versus discharge Q, for various water depths. Although the data are not s u f f i c i e n t to provide accurate values for Q , they do ind icate that Q i s s l i g h t l y less fo r a p ier with a square nose than fo r a p ier with a round nose. The data also show a de f i n i te increase in Qc with increasing sand-grain s i z e , f o r the sands used CdgQ = 0.285 mm., 2.4 mm., and 4.5 mm). Knezeyic arranged his data for equi l ibr ium depth of scour to obtain a r e l a t i on of the form; d ..--C'Cn;)9'2- (12) H 5 ' 4 g ^ where C = a constant. The value of C i s supposed to vary only with the p ier shape, but actual ly there was scatter with respect to both sand-grain s ize and flow depth. Ayerage yalues were, fo r the c i r c u l a r nose, C = 8.7; for the rectangular nose, C = 9.8. Knezevic conceiyed the scour-causing vortex at the base of the p ier to be due to the v e r t i c a l l y downward ve loc i t y components in f ront of the p i e r , as suggested by Tison (see above). On th i s basis two methods of reducing scour occurred to him. One method consisted of asp i rat ing the downward flow in f ront of the p ier by means of a horizontal hole or s l o t cut through the pier in the d i rec t i on of the main f low. A de f i n i t e reduction in the equi l ibr ium depth of scour was observed, ranging from about 27% to 76%, depending on the shape of the p ier and the s ize of the bed sand-grains. The s i z e , shape, and locat ion of the s l o t were not given. A second method consisted of placing a ser ies of bands around the 18. p ier at various l e ve l s , in order to def lect and retard the downward flow ju s t in f ront of the p i e r . Using three bands of steel plate 15 mm. wide and 2 mm. th i c k , placed around a p ier 10 cm. wide, reductions of the order of 30% to 40% in the equi l ibr ium scour depth were obtained. G. TARAPORE Tarapore i n 1962 completed a Doctoral Thesis in which he reported laboratory measurements and presented a theoret ica l method f o r determin-ing the local scour at an o b s t r u c t i o n ^ 1 1 9 ' 1 2 ° ) . in his theoret ica l invest igat ion he assumed that the i n i t i a l flow pattern was given by potential flow theory. As the scour hole developed, i t was assumed that the free stream diffused into the scour hole in a manner analogous to that of a mixing layer s i t u a t i o n , and the ve loc i t y d i s t r i bu t i on was given by: where u(z) = the local ve loc i t y at depth z, in the x - d i r e c t i o n , u(-£) = the potent ia l v e l o c i t y , £ = the distance that the ef fects of the scour hole have penetrated into the main f low, 1 = distance along a streamline, s ta r t i ng at the scour hole edge, and k = a c oe f f i c i en t representing the rate of ve loc i t y d i f fu s i on into the scour hole (see Figure 4). The rate of bed load transport was assumed to fol low Straub's expression (1935): uCz) = uC-E} e -k(z +E)/1 ( 13 ) a T ( T - T ) Y (14) which, fo r high rates of transport ( T » T ), can be s imp l i f i ed to : (15) 19. where q s = the rate of transport of bed load ( in cubic feet per hour per foot width), $ = Straub's transportation cha r ac te 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 ress , x = the c r i t i c a l bed shear c s t ress . The assumption fo r the bed shear stress was: T a p J u C n l J 2 (16) where uCn) = the value of the (diffused) ve l oc i t y at the boundary, and p = the density of water. When the scour hole i s at the equi l ibr ium cond i t ion, i t was assumed that the rate of transport into the hole equaled the rate of transport out of the hole, thus the transport rate between the streamline on the plane of symmetry of the approach f low, ipQ, and the streamline passing through the point P on the edge of the scour hoi e, \pp, was constant C s e e Figure 5). In terms of an equation: fP q s dy = Cx=o) 0 G c — ) V q s dy (17) where the x and y coordinates are as shown in Figure 5. Subst i tut ion of equations (15) and (16) y i e l d s : K 2 2 P * P [u(n)J 4dy = JL p 2 \p y Y0 ' P ,A UHdy (18) Yo Cx=0l ( x = - ? ) Further subst i tut ion of equation C L 3 ) , with u C n ) = u (z ) , and addit ional assumptions and manipulation, y ie ld s an integral equation which can be solved by numerical methods to y i e l d the pos i t ion of the point P C s e e Figure 5). The depth of scour can then be determined, i f the shape of the scour hole i s known. In add i t ion , the values of k and £/l have to be 20. estimated in 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 low, th i s term i s determined by p lo t t i ng the experimental data as d s g /b versus H/b, and noting where d s g /b i s no longer affected by the depth of f low. The value of k i s obtained by solv ing the equation for a know experimental r e su l t . Tarapore's plots of d s g /b vs. H/b show considerable disagreement between his own data and that of others which he included in his p l o t , and his se lect ion of H/b = 1.15, as the value above which flow depth no longer influences scour depth, seems a rb i t r a r y . Tarapore calculated the depth of scour at an e l l i p t i c a l c y l i nder , according to the method outl ined above, and obtained results which were about 10% to 15% less than the experimental re su l t s . The dif ference i s explained as due to improper estimation of the scour hole shape. In addit ion, Tarapore concluded that the maximum depth of scour (below normal bed leve l ) at a c i r c u l a r cy l inder , fo r large depths of f low, i s equal to 1.35 b. H. MOORE AND MASCH In 1963 Moore and Masch published a paper in which they attempted to achieve an understanding of the secondary flow caused by the presence of a p ie r in an otherwise undisturbed f l o w ^ 8 0 ^ . Their analysis i s l a rge ly based on the fact that a pressure gradient ex i s t s along the stagnation l i ne formed by the intersect ion of the plane of symmetry of the approach flow and the p ier surface, i f the approach flow ve loc 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 locat ion on the stagnation l i n e , i s j u s t equal to the ve loc i t y head of the approach at that l e v e l , u /2g. Thus, i f u = u(y), p = p(y) , where y i s the ve r t i c a l coordinate. Since the sum of s t a t i c pressure head and elevat ion head remains a constant, a net pressure gradient ex ists along the stagnation l i n e . This pressure gradient induces an accelerat ion of the nearby f l u i d from the point of maximum pressure towards the regions of lower pressure. I f viscous ef fects are neglected and one considers a streamline running downward along the stagnation l i n e , the conservation of energy p r i nc i p l e y i e l d s , fo r the secondary v e r t i c a l flow at any po int, a ve l oc i t y head v /2g, which i s j u s t equal to the d i f ference between the pressure head at that point and the maximum pressure head. Thus: vCy) 2 = P* max. _ p(y) 2g Y Y max. 2 i 2g " 2g u . u(y) (19) or : vCy) 2 = u max. 2 - u ( y ) 2 .....(20) Moore and Masch contended that the above concept provides the basis for the mechanism producing a strong v e r t i c a l l y downward flow jus t i n f ront of the p ier and thus accounts for the fac t that the maximum depth of scour occurs at the nose of the p ier rather than at the point of maximum breadth, as would be expected i f only the two-dimensional potent ia 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. sp i ra l vortex in the scour hole. On the basis of the 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 consists of placing horizontal discs around the p i e r , below or above normal bed l e v e l , with possibly a v e r t i c a l l i p around the outside edge to def lect the secondary flows upward away from the b e d 1 1 . Another previously proposed method consists of placing an aux i l i a r y p ier or p i l e 12 upstream of the pier to be protected . Masch and Moore thought that such a p i l e would "destroy" the ve loc i t y gradient of the approach f low. The reduction in scour depths achieved by sharpening the p ier nose i s explained i n terms of the reduction in the s ize and strength of the downward flow component effected by such a nose. A f i n a l method suggested by Moore and Masch consists of a swept-back leading edge, possibly required only near the bed, in order fo r the deflected flow component which then deyelops to counteract the downward flow induced by the ve l oc i t y gradient of the approach f low. I. BREUSERS; DELFT HYDRAULICS LABORATORY 1. Oosterschelde Bridge Model Studies In 1964, De l f t Hydraulics Laboratory published a report of a model study of scour around the piers of the Oosterschelde Bridge, under the d i rec t i on of H.N.C. Breusers^ 1 6 ^. Experiments were carr ied out using c i r c u l a r c y l i n d r i c a l piers in order to invest igate the inf luence of the average approach flow y e l o c i t y , flow depth, p ier diameter, and bed mate r i a l . I t was observed that with a bed sand of d 5 g = 0.20 mm. 23. (U .. =0.25 m./sec. = 0.81 f t . / s e c ) , maximum scour occurred when v c r i t . U/U .. reached a value of 1.4, and did not increase fo r greater ve 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 ier diameter, having a value of 1.5 fo r b = 11 cm. and 1.6 for b = 5 cm. Csee Table I ) . This re su l t was obtained for a flow depth H, for the 5 cm. p i e r , of 0.25 m. and a flow depth f o r the 11 cm. p ier of 0.50 m., in the understanding that s i m i l a r i t y would be better approximated using s im i l a r values of H/b rather than ju s t H alone. The flume width was 0.95 m. The influence of the flow depth on_the equi l ibr ium depth of scour (below normal bed leve l ) was tested using the 11 cm. p ier and flow depths of 0.15 m., 0.25 m., and 0.50 m., at the condit ion U/U = 1.4. There was no s i gn i f i c an t d i f ference i n the equi l ibr ium scour depths fo r the two larger flow depths, while for the smallest flow depth, the equi l ibr ium scour depth was somewhat les s . I t i s in terest ing 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 larger scour depth than the two larger flow depths. Tests were also run with a bed material of polystyrene spheres Cdensity = 1050 kg/m ) of diameter 1.5 mm. C U c r i t =0.09 m/sec.) in a flume 3.5 m. wide. For the 11 cm. p i e r , maximum scour depth correspond-ing to d /b = 1.7 was reached, at U/U ... = 1.2. The r a t i o d /b J CHI U l I w « J C reached a steady value of 1.65 at U/U . > 1.4 (see Table I ) . 24. 2. Scour around D r i l l i n g Platforms In 1965, Breusers reported the resu l t s of a pr ivate study of scour around d r i l l i n g platforms^- 1 7^. He obtained: ' d c * m = ! - 4 D C21) sera J . MAZA AND SANCHEZ Maza and Sanchez reported a study of local scour at bridge piers in (79) 1964 v ' . They reviewed the l i t e r a t u r e ava i lab le to them and found that there were ba s i ca l l y two d i f fe rent re l a t i on s . The f i r s t of these was proposed by Laursen and i s given above. E s sent ia l l y i t was d s g = f ( b , H/b, p ie r shape, angle of attack) . The second re lat ionsh ip i s due to Jaros lavt s iev , and can be formulated as d g e = f(U , H/b 1, U /gb. , p ie r shape, d 5 Q , type of f l ow) , where b 1 represents the width of the p ier 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 re lat ionsh ip i s that i t i s v a l i d only fo r U > U .. , whereas Ja ro s l av t s i ev ' s re lat ionsh ip i s va l i d only fo r H/b > 1.5, and in addit ion d^g i s included only i f i t exceeds 5 mm. In terms of a design equation, Ja ro s l av t s i ev ' s re l a t i on can be expressed as: d se = W V V f " 3 d50 ( 2 2 ) Here i s a function of H/b1 and decreases with increasing H/b^ up to a value of H/b1 = 5 Csee Table I I ) . In the interva l 1.5 < H/bx < 6, the re la t i on can be approximated by = (ti/b^)'7^. The parameter a1 varies from 0.6 for a p ier in a main flow channel to 1.0 for a p ier on a 2 f lood p l a i n . The term Ky i s a function of the p ier Froude number U /gb^ 25. O A O O and can be expressed as = 0.53(U /gb 1 )~ . The p ier shape coe f f i c i en t varies from = 10 for c i r c u l a r p iers to K^ . = 12.4 f o r rectangular p ier s . A l l units are in meters except for d^Q which i s in mi l l imeter s . Maza and Sanchez carr ied out a number of experiments and compared the i r data to the two scour re l a t i on s . They found that t he i r data f i t t e d the re l a t i on of Jaros lavts iey quite well f o r values of H/b > 1.5. Ninety percent of the i r data f e l l below Laursen's r e l a t i o n , and none exceeded i t . On the basis of the i r r e su l t s , Maza and Sanchez proposed a modified version of Ja ro s l av t s i ev ' s equation, in the form: d se U 3 d 50 BT" = VttU g¥, " ~ b ~ — ( 2 3 ) 1 3 l l with dj-g in mi l l imeters and a l l other lengths in meters. The parameter K H U i s pr imar i ly a function of U /gb 1 , and secondari ly of H/b 1 > but the l a t t e r has l i t t l e influence except fo r values of U /gb1 < 0.10 (see Figure 7). 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 for H/b1 > 1.5, and the term with d 5 Q be included only i f d,-Q > 5 mm. 26. K. SHEN et a l . : COLORADO STATE UNIVERSITY 1. Introduction Since 1962, workers at Colorado State Un iver s i ty , l a rge ly under the d i rect ion of H.w*. Shen and S.S. Karak i , under a contract with the United States Bureau of Publ ic Roads, have invest igated the problem of loca l bed scour at bridge piersCi09,no,ill,112,113}^ Experimental work has involved the study of the va r i a t i on of the depth of scour with time, the dependence of the equi l ibr ium depth of scour on the hydraul ic parameters, the ye loc i t y patterns of the flow in and around the p ier and scour hole, and the effectiveness of various methods of reducing scour. Theoretical work was aimed mainly at t ry ing to understand the mechanics of the loca l scour phenomenon, espec ia l l y with respect to the vortex at the p ier base, and thus provide a conceptual framework for funct iona l l y r e l a t i ng the equi l ibr ium scour depth to the important hydraul ic and other parameters of the problem. 2. Description of the Mechanics of Local Scour Shen, et. a_l_. gave a prel iminary descr ipt ion of the loca l scour phenomenon, using a c i r c u l a r cy l inder as a convenient example^ 1 1 0 ^. The cy l i nder , by the pressure f i e l d i t induces, apprehends the v o r t i c i t y normally present in the flow as i t i s swept towards the cy l i nder , and concentrates i t near i t s leading surface. This process can also be described as vortex tubes co l l e c t i n g in f ront of the cy l i nder , and being bent and stretched around the cy l inder . This process reaches a state of approximate equi l ibr ium when the rate of d i s s ipat ion of v o r t i c i t y at the boundaries (the cy l inder 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 structure associated with th i s concentration of v o r t i c i t y i s the horseshoe vortex or horizontal 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 ier i s s u f f i c i e n t l y strong, i t causes separation of the three-dimensional boundary layer upstream of the p ie r . This separated boundary layer r o l l s up to form the horseshoe vortex system (see Figure 8 ) . In add i t ion, due to the mechanism described by Moore and Masch (see above)> a downward flow ex i s t s near the cy l inder leading surface. The horseshoe vortex system i s in general grossly unsteady, and can include a secondary vortex i n addit ion to the primary vortex. It i s the l a t t e r however which i s mainly responsible fo r the scouring act ion. 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 transport. Scouring i s ac tua l l y i n i t i a t e d somewhat downstream of the leading edge of the cy l i nder , i n the region where the free stream ve loc i t y imposed on the vortex act ion i s high. Shen, e_t al_. proposed a d i v i s i on of the loca l scour phenomenon into two basic cases: scour at blunt-nosed piers and scour at sharp-nosed p ie r s . A blunt-nosed p ier i s defined as one which induces an adverse pressure gradient strong enough to cause the upstream boundary layer to separate and r o l l up to form the horseshoe vortex. A sharp-nosed p ier i s defined as one which lacks th i s property. 28. 3. Theoretical Analysis The theoret ica l treatment of the horseshoe vortex was done mainly by A.T. Roper, and i s out l ined b e l o w ^ 1 1 0 ' 1 1 3 ^ . Consider a control volume (of small thickness) ABCD s ituated on the plane of symmetry [stagnation p lane) in f ront of a c i r c u l a r cy l inder in a uniform, steady flow in an open channel, as shown in Figure 9. The flow in th 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 = i V • 3s (24) where V = the tota l ve l oc i t y vector, and ds = an i n f i n i t e s ima l element of length along the boundary of ABCD. For the flow r i gh t at the s o l i d boundaries AD and CD, the " no - s l i p " condit ion appl ies , i e . the flow ve loc i t y there i s zero. Therefore the product V • cfs~ along these boundaries i s zero. Further, Roper spec i f ied that AB i s f a r upstream of the p ier in the region where the flow i s not affected by the presence of the p ie r . Consequently, the vectors V and ds" are at r i ght angles to one another, and the i r dot product i s zero. This leaves only one term remaining, and we get: r = udx (25) B This s imp l i f i c a t i on 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) can be evaluated i f u is known along BC. The l a t t e r requirement can be s a t i s f i ed i f BC i s spec i f ied to be 29. s ituated above, the region where the ve loc i t y var ies with depth (the shear l a ye r ) , and also wel l below the free surface. The value of u along BC i s given by potent ia l flow theory fo r flow around a c i r c u l a r cy l inder (with co-ordinates as shown in Figure 9) as: , 2 u(x) = -u [1 - ^ - ] (26) 0 (a+x) 2 where -u = the free stream ve loc i t y at point B, and a = the cy l inder radius. Subst i tut ing th i s expression for u into the r ight-hand-side of equation (25), and evaluating the i n t e g r a l , we get: C t x = 6 2 udx = -u [1 -J dx J x - x ° O = u x - U [a - - f — ] (27) o o o a+x o Subst i tut ing th i s back into equation (25), and re-arranging: ax r = u x " u (28)' o o a+x o o I f AB i s spec i f ied to be far 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) becomes; r = u x - au (29) o o o 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 fec t of the p ier i s to reduce the c i r c u l a t i o n by an amount: A r p ie r = -au Q C30) Roper assumed that the reduction in c i r c u l a t i o n due to the p ier i s proportional to the strength of the horseshoe vortex core. Roper further 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 £ where co i s the angular v e l o c i t y , and A the area, of the yortex core. Expressing these c assumptions in the form of an equation: a U o <* CuAcW C31) Roper argued that since flow separation i s a viscous e f f e c t , the kinematic v i s co s i t y of the f l u i d , v, was an important var iab le . Incorporating th i s into equation C31), and replacing the p ier radius, a, by the p ier diameter, b, a non-dimensional r e l a t i on i s obtained: . bu ^ = f T _ O - | (32) v v The term on the right-hand-side of the equation i s the Reynolds number based on the p ier width, R ,^ so that: ^ = f IR 5 J (33) Now, since the horseshoe yortex i s supposed to be the primary agent causing scour, the depth of scour should be re lated to the strength of the vortex core. Thus; 31. where Rfa = the pier Reynolds number, and d $ e = the equi l ibr ium depth of scour below normal bed l e v e l . This analysis implies that the p ier actua l l y reduces the v o r t i c i t y in 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 in fact observed in the region between the separation l i ne 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, carr ied out at Colorado State Univers i ty confirmed the f indings of previous studies in that the shape of the scour hole around a blunt-nosed p ier 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 with the flow d i r e c t i on , i t was found that the maximum depth of scour occurred at the downstream end of the p ie r . A qua l i t a t i ve representation of the flow patterns observed in the scour hole i s shown in Figure J O ^ 1 0 9 ' 1 1 0 ^ . The ve loc i t y p ro f i l e s are proportional to the measured values. The dashed l ines ind icate the dominant flow features, which are in general not f i xed but unsteady in nature. Shen and his co-workers examined a l l the data ava i lab le 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 the i r own, as a l l other data were e i ther incomplete or were obtained from experimental arrangements and methods which d i f fe red in t he i r s i gn i f i c an t d e t a i l s . These other data were however 32. used for purposes of comparison. Shen plotted the useable data, for sands with d^Q < 0.52 mm., in terms of the equi l ibr ium scour depth [long-term average va lue) , d s g , versus the p ier Reynolds number, R ,^ as shov/n in Figure l l ^ 1 1 3 ^ . This p lot produced the r e l a t i o n : d $ e = 0.00073 Rb 0 , 6 1 9 C35) which forms an approximate envelope for a l l the data. This re lat ionsh ip i s supported by the other data mentioned above. The scatter of the data ar ises not only from the differences between the various experimental arrangements and methods used, and.inherent errors t he r i n , but i s due to the shortcomings of the analysis i t s e l f , as enumerated by Shen in the fol lowing points. Ca) For a given p ier geometry and sand s i z e , the curve of d s g versus R^  r i ses rap id ly to some maximum, beyond which i t f a l l s o f f somewhat, as shown in Figure 12. Cb) The curve described in Ca) d i f f e r s for each p i e r , as shown in Figure 12. Cc) A d i f f e ren t sand s ize also gives a d i f f e ren t curve, even i f the p ier geometry i s constant, as shown in Figure 13. Shen et aj_. therefore recommended that equation (35) be used to estimate the equi l ibr ium depth of scour for the "c lear-water " scour case only. The resu l t w i l l be on the safe s ide. A cayeat was added, however, in that the laboratory results haye to be extrapolated several orders of magnitude in order to apply to the much higher p ier Reynolds numbers of prototype condit ions. For the case of scour v/here there i s general 33. bedload transport, Shen e_t al_. recommended that e i ther the re l a t i on proposed by Breusers: dsem - : - 4 b ( 2 1 ! or the one of Larras; d = 1.42 K b 0 , 7 5 (11) sem v ' be used to determine the maximum equi l ibr ium depth of scour below normal bed l e v e l . 5. Reconc i l iat ion of Diyergent Concepts.-The pier Reynolds number r e l a t i on (equation 35) was further analyzed, and i t was found that by assuming a uniform flow re l a t i on of the type: U = K.' H e C36) where K' and e are constants, equation (35) could be transformed to an expression of a form s im i l a r to Laursen's equation (9) , or an expression of the form: % = C [ F 2 ( ^ ) 3 J 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 ve loc i t y of the flow i s more important are of no great consequence, since the two parameters are related according to equation C36)^ 1 1 0 ^. \ 34. 6. Methods of Reducing Scour Shen and his co-workers also carr ied out a number of experiments [apparently in the bed-load transport range only) to test various methods of reducing the depth of loca l scour around bridge p ie r s , the resu l t s of which can be summarized as fo l lows: Ca) Sharp-nosed piers were made by fastening noses, of included angle 8 = 15° and 6 = 30°, onto a standard rectangular p ie r . The l oca t i on , shape, and s ize of the scour hole were quite var iable and no cor re la t ion with any of the usual parameters was evident, except that some of the var ia t ion seemed to be due to s l i g h t angles of attack effected by bed forms near the nose of the p ie r . For some runs, in which the upper regime plane-bed condition obtained, no scour at a l l occurred. Cb) Rectangular piers on a protruding, f l a t , pi le-supported foot ing s i tuated below normal bed level 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 im 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 ie lded reductions of the order of 40%-50%. When the footing and l i p are s i tuated low enough, the arrangement presumably traps the horseshoe vortex and prevents i t from af fect ing the erodible bed. Cd) A rectangular p ier with roughness elements attached to the front did not give any s i g n i f i c an t reductions in 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 retard any down-ward flow in f ront of the p ie r . 35. (e) A cy l inder of one-half the width of the rectangular p ier was placed in f ront of the l a t t e r , and a maximum reduction of about 60% was achieved when the cy l inder was placed a distance of about two cy l inder diameters upstream of the rectangular p ie r . Cf) A cy l inder s p l i t in the d i rec t i on of flow ( i e . ha l f cy l inders separated by a distance of from 1/3 to 2/3 the cy l inder radius) gave reductions of the order of 25% to 40%. I t was observed that the horse-shoe yortex, although s t i l l present, was weaker than for the s o l i d cy l inder . L. CARSTENS Carstens attacked the loca 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 theoret ica l analysis of the forces acting on a typ ica l bed p a r t i c l e , assuming a neg l i g ib ly th in boundary l ayer , y ie lded a r a t i o of d isturbing fo rce, F^, to retarding fo rce, F^, of: F M 2 p-i = f Csediment p a r t i c l e geometry) N (38) r R s U" where N = , the "sediment number." In the l a t e r , s = the s /Cs- l )g d g sediment s pec i f i c g rav i ty , and d g = the cha rac te r i s t i c sediment grain diameter. For the case of c lear water scour CQS i n = 0 ) , the rate of sediment transport out of the scour area, Q$ ^ t was assumed to be a function of the force r a t i o and the geometry of the s i t ua t i on . Thus a non-dimensional sediment transport function was hypothesized: 36. s out _ f[(fl 2 _ N 2 ) , d s , obstruct ion geometry, u W d L V s sc , s g L sediment p a r t i c l e geometry] (39) where Wg = the scour hole width, L = a cha rac te r i s t i c length of the obstruct ion, N = the c r i t i c a l sediment number at which loca 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 g = the scour depth below normal bed l e v e l . Approximating the scour hole shape at a c i r c u l a r cy l inder 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 = dV = * ( S + b ) d d ( d } ( 4 Q ) x s out dt tan<j> vtan<}> ' s dt v s ' Select ing the data of run 204 from Chabert and Engeldinger, and def in ing u = U, the average approach flow v e l o c i t y , L = b, the p ier diameter, and Ws = b + 2(_ds/tancj>), Carstens constructs a p lot of: 2 d s Q ./U(b + T—r) d d ws our v tanj>y g u c s ( N s 2 - N s c 2 ) 5 / 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 in a closed conduit. This p lot gave: Q $ o u t — = i . 3 io~5 CN 2 - N 2 p / 2 c d s r 3 C41) 2 d s sc ^-i 37. Equation (41) can be substituted into C40), and the re su l t integrated, to y i e l d an expression fo r scour depth with time. This expression shows that the scour depth increases cont inual ly with time, and although the rate of increase decreases, an equi l ibr ium scour depth i s never atta ined. M. TANAKA AND YANO, AND THOMAS 1. Tanaka and Yano Tanaka and Yano report a laboratory study involv ing the e f f e c t , on loca l scour, of various devices and modif icat ions in conjunction with a c i r c u l a r c y l i n d e r ^ 1 1 8 ^ . They considered that the horseshoe vortex i s generated by a combination of main flow separation upstream of the cy l inder and the secondary downward flow along the front of the cy l i nder , and that the magnitude of loca l scour depended on the strength and s ize of the vortex. 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 ju s t obtain s l i g h t r ipples on the bed, and were the same fo r a l l te s t s . The ve loc i t y d i s t r i bu t i on was measured at the test locat ion and i s shown in Figure 14. The basic p ier used was a c i r c u l a r cy l inder of 3 cm. diameter. The d i f f e ren 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 in Figure 15. Ca) Type I: a normal cy l inder extending to aboye the water surface. (b) Type II: a cy l inder with a hole 1 cm. square or 2 cm. square cut in the d i rec t ion of the main f low. 38. Cc) Type I I I : a cy l inder f i t t e d with a c i r c u l a r d i sc of diameter = 9 cm., 12 cm., or 18 cm. d Cd) Type IV< a cy l inder submerged below the water surface. Ce) Type V: a cy l inder f l oa t i ng above the normal bed surface. The posit ions of the modif icat ions of the type II and type III p i e r s , and the top of the type IV, and bottom of the type V p i e r s , were var ied. The experimental resu l t s are summarized in Figure 16. Tanaka and Yano noted that the e f fec t of the discs was pronounced only with in the boundary layer C<S = 5 cm.), and that the s ize of the d i sc Cwhen placed at bed leve l ) required to completely el iminate scour was of the order of the boundary layer thickness. They therefore concluded that the s ize of the vortex i s re lated to the boundary layer thickness, e spec ia l l y since the vortex i s a boundary layer e f f e c t . Thus, they argued that the e f fec t of the p ier modif ications can be v a l i d l y represented in terms of an expression of the form: scour hole s i ze = f Cji C42) where h = the height of the modif icat ion above normal bed l e v e l , and 6 = the boundary layer thickness. Tanaka and Yano considered the v e r t i c a l downflow in f ront of the p ier to have l i t t l e e f fec t on loca l scour, fo r the i r experimental condit ions. 2. Thomas Thomas reports a study of the e f fec t of a disc f i t t e d around a c i r c u l a r cy l i nder , s im i l a r to Tanaka and Yano's type I I T ^ 1 2 ^ . He used discs of 10 cm. and 15 cm. diameters around a 5 cm. diameter p i e r , placed 39. in a flume 35 cm. wide. His r e su l t s , shown in Figure 17, 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 cy l inder quarter points: f r on t , back, and two s ides. Further, no information i s giyen with respect to the flow condit ions, and the scour depth fo r the normal cy l inder i s not stated. N. SCHNEIDER Schneider, one of Shen's co-workers at Colorado State Un iver s i ty , recently completed a doctoral d i s se r ta t ion on the mechanics of loca l s c ou r^ 1 0 6 ^ . In i t , he giyes a descr ipt ion of the mechanics of loca l scour, based on the accumulated information of previous s tudies, as well as an out l ine of the more important concepts and re lat ions proposed by previous invest igators . His own theoret ica l work was mainly directed towards formulating a re l a t i on for local scour based on the sediment cont inuity equation [equation C8)), and his experimental work consisted la rge ly of determining sediment transport rates out of the scour hole. Other considerations included the e f fec t of cons t r i c t i on of the f low, unsteadiness of the horseshoe vortex, design c r i t e r i a , and safety factor s . 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 rate at which so l ids are removed, an expression s im i l a r to equation C40) i s obtained; 2 , TTC1-X)P g d_ j «. • - t s s r - ' t k + b d s i at Cds' C 4 3 ) 40. where Q ' = the rate at which the scour hole i s deepened, in terms of weight of so l ids remoyed per un i t time, A = the void r a t i o of the sediment in the scour hole, and p g = the sediment density. Equation (8) can then be expressed as: \ Cl - ^3P 9 = Q ' = Q ' 4. - Q ' • (44) dt v A i M s 3 s s x s out x s in v 1 where Q 1 = the time rate of sediment transport out of the scour hole s out r (by weight), and Q ' ^ n = the time rate of sediment transport into 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 function of the i n i t i a l value, Q 1 : V = %± e _ t / t c C 4 5 ) where t = a time constant. Schneider further thought that Q ' . , c 3 i n 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 in across a width equal to the p ier width, i e . Q ' . = q ' . b. Equation (45) i s n s i n n s 1n thus divided by the p ier width and then subst ituted into equation (43), which, when integrated, y i e l d s : ( IT ) 3 + ! C i T ) 2 t a n * = M l - e _ t / t c ] (46) 3Q' t tan2<J> where X = — — - o — • As time becomes large ( i e . t » t ) equation *(1-A)p sg b J c (46) becomes: <i 3 , d 2 t-f-) + | (Hp). tan* - X (47) 41. This equation implies that fo r a given p ier s ize and given sediment cha rac te r i s t i c s , the equi l ibr ium 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) fo r t = t c , and was found to be the time required fo r the r a t i o d s / d $ e to reach a value from 0.796 to 0.858, depending on the r e l a t i v e importance of d /b and tan<j>. The value of t can thus be estimated from d s versus t curves. Schneider decided to determine Q s l- 1 by measuring Q s ' o u t in the laboratory and r e l a t i ng the former to the l a t t e r by means of equations (44) and (45). Since these measurements were made fo r the i n i t i a l f la t -bed condi t ion, the exponential time term reduces to un i ty , and the value of tQ i s not required i n order to obtain QS1-1-Values of Q s ' 1 - n were obtained from ex i s t ing data for sediment transport in f lumes. ' Schneider 's measurements of Q ' . y ie lded a r e l a t i o n : ^s out J Q ' sbo u t = 0.0754CU-U c) 3 (48) where U c = the average approach flow ve loc i t y at which loca l scour i s j u s t i n i t i a t e d . This ve l oc 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 isp lay any systematic scatter with respect to sediment s i z e , p ier width, or . pos i t ion 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 -bed condi t ion) , fo r most flow conditions tested 0"e. low ve loc i t ies ) . . 42. The time constant t was determined from data of Chabert and Engeldinger, and Shen, et. al_. , L l 1 3 ^ and plotted as graphs of l/tc versus U. S i gn i f i cant cor re la t ion with p ier diameter is also evident, although the influence of sediment s i ze is not c lea r . A comparison of measured values of t with values calculated from the known data, using equations [48), [44), [45), and (47), shows f a i r agreement, although a de f i n i t e scatter is evident fo r higher values of t c , with the calculated values being mostly lower than the actual measured values. Predicted  scour depths based on the calculated values would thus tend to be greater  than those that would actua l l y occur. One of the more in teres t ing resu l t s of Schneider 's work involves the well-known observation that in the region of scour with general bed-load transport, the equi l ibr ium scour depth is independent of the ve loc i t y U (see Figure 3). A constant scour depth implies a constant value of the product (QS1-' • t c ) , by equation (46). However, i t has been noted that at i n i t i a l condit ions, at least fo r v e l o c i t i e s of about two feet per second or l e s s , Q s i ' = V o u t ' s i n c e V i n - 1 0 % o f V o u f T h u s changes in i n i t i a l values of Q s ' o u t (due to changes in ve loc i t y ) are almost completely balanced by changes in t , and the actual incoming sediment supply has only a small e f fec t on the equi l ibr ium scour depth. The development of general sediment transport i s thus not the true basis f o r the f l a t ten ing of the d s e yersus U curve; rather, 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 loca l scour phenomenon into two regions. The f i r s t region, where the equi l ibr ium 43. scour depth increases with v e l o c i t y , he ca l led the "vortex-bed-in te ract ion - l i ra i ted depth of scour reg ion " , since the equi l ibr ium scour depth i s l im i ted by the shear stress that the vortex can exert on the bed. The f l a t region of the curve he ca l led the "vortex-mechanics-l i ro ited depth of scour reg ion " , since the equi l ibr ium scour depth i s l im i ted by the inherent i n a b i l i t y of the vortex to penetrate below a certa in depth. The equi l ibr ium scour depth in th i s l a t t e r region can be affected somewhat by the incoming sediment transport. The two scour regions are shown schematically in Figure 18. This f ind ing modifies the re su l t of Chabert and Engeldinger as given in Figure 3. 0. COLEMAN Coleman reports the resu 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 var iab les by dimensional ana lys i s , and then used the experimental data to obtain a re lat ionsh ip between them. He gotan expression fo r equi l ibr ium depth of scour as fo l lows: d se 1.49 b 9 / 1 0 ,,2 1 / 1 0 ( - ) 2g (49) 44. NOTES TO CHAPTER II 1. Timonoff^- 1 2 6^ refers to studies by Minard (1856) and Durand-Claye C1873), but c i tes no s pec i f i c works for these. 2. Reported in Karaki and Haynie^ 5 1 ^. 3. Reported in Karaki and H a y n i e ^ 5 1 \ Some presentation of th i s work by Ing l i s can be found Qnte r a l i a ) i n Arunachalam, ' Chi ta le , I n g l i s ^ , J o g l e k a r ^ 4 6 ) , and Thomas^ 1 2 1 *. 4. This condition corresponds to the condit ion of "c lear-water " scour , j u s t below the threshold of general sediment transport, as out l ined by Laursen. 5. Sediment transport rates are' here defined in terms of volume, i n order to make the equation dimensionally homogeneous. Void rat ios would have to be allowed f o r . 6. The term "c lear-water " does not imply that v i s i b i l i t y i s good, but indicates only that there i s no transport of bed sediment. 7. Reported in Karaki and Haynie, re f . 191. 8. Median grain diameter = d ^ = the grain s ize such that 50% of the material (by weight) i s f i n e r . 9. Reported by Shen, et a ] / 1 1 3 ^ . 10. Reported by Shen, et a l _ . ^ - 1 1 3 ^ , and N e i l l ^ 8 3 ^ . 11. This idea had already been tested by Schneible, as reported in Laursen and Toch^- 6 1^. 12. Tison seems to have been the f i r s t to suggest and test th i s method. He understood the aux 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 th i s chapter i s to ar r ive at an understanding of the mechanics of loca l scour, based on observations and f indings of previous invest igat ions and experiments as summarized in Chapter I I , and on the resu l t s of other studies, notably those in the f i e l d of f l u i d dynamics. Such an understanding should then proyide some basis fo r assessing the v a l i d i t y of the various hypotheses and methods of analysis of the loca l scour phenomenon, and also suggest some guidelines fo r further study. 46. B. THE BASIC VORTEX MECHANISM There can be l i t t l e doubt that the basic -mechanism of loca l scour i s the horizontal r o l l e r or vortex, shaped l i k e a horseshoe i n p lan, that forms on the bed d i r e c t l y in f ront of a p ie r . This vortex, due to the high ve l o c i t y at which i t ro ta tes , exerts shear stresses on the bed that are greater than the c r i t i c a l shear stress for 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 inves t i gator s , including Posey^ 9 3 ' 9 5 \ Lau r sen^ 5 9 ' 6 1 \ Moore and Masch C 8 0 ) , Maza and S a n c h e z C 7 9 ) , Shen e V a l / 1 0 9 ' 1 1 0 ' 1 1 3 ) , Tanaka and Y a n o ( l l 8 ) , and Schne ider^ 1 0 6 ^. Shen, Schneider, and Tanaka and Yano in p a r t i c u l a r , have recognized the importance of the vortex, and i n t he i r analyses have considered the vortex mechanism. Any attempt to understand or explain loca l scour without taking into account th 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 sect ion I I. G above), which i s based on potent ia l flow theory and a concept of the free stream flow d i f fu s i ng into the scour hole, can never lead to a proper understanding of the local scour phenomenon (although his experimental resu lts remain v a l i d in themselves). S i m i l a r l y , the theoret ica l analyses of Tison and Ishihara {sect ion II. B) and Carstens (section II. L) are misleading. 47. C. ORIGIN OF THE HORSESHOE VORTEX 1. V o r t i c i t y Ba s i c a l l y , the horseshoe vortex or ig inates out of the v o r t i c i t y which i s always present in shear f lows, that i s , flows in which a ve l oc i t y gradient ex i s t s . A shear flow may be v i sua l i zed as cons ist ing of th in 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 ren t ve l oc i t y than the adjacent layers. This d i f f e r e n t i a l movement resu l t s in the s l i d i n g or shearing of adjacent layers oyer each other, and in so doing causes a rotat ion of the f l u i d pa 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 rotat ion or the angular ve l oc i t y of these f l u i d p a r t i c l e s . A l i ne drawn through the f l u i d so that i t everywhere corresponds to the loca l axis of rotat ion of the f l u i d pa r t i c l e s i s ca l l ed a vortex l i n e . If adgacent vortex l ines are joined together to form a surface, the re su l t i s a vortex tube. The f l u i d with in th i s tube i s ca l l ed a vortex tube or f i lament, or simply a vortex. Mathematically, v o r t i c i t y i s a vector, and i s expressed as: 0 = V x v (50) Expressed in terms of orthogonal components, th i s becomes: 3 v 3v 48. A theorem (due to Helmholtz and Kelv in) states that fo r any giyen vortex, the product of the v o r t i c i t y and the cross-sect ional area of the vortex must remain constant. An important consequence of th i s theorem i s that vortex l i nes cannot begin or end anywhere with in the f l u i d i t s e l f , but must e i ther form closed loops or terminate at the flow boundaries. Another theorem states that vortex l ines move with the f l u i d . An excel lent descr ipt ion of the character 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 Cyl inder Consider the case of a uniform steady flow in an open channel, past a v e r t i c a l cy l inder located in the middle of the channel, with the x - d i r e c t i on in the d i rec t ion of f low, the y - d i r e c t i on across the d i rec t i on of f low, and the z -d i rec t ion 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 cy l inder in the central part of the channel, where neither the cy l inder i t s e l f nor the side wal ls have any e f fec t ( i e . a two-dimensional f low). The ve loc i t y of th i s flow has an x-component only (v x ) . A l so, a ve l oc i t y gradient ex 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 to ta l v o r t i c i t y of the f low, as expressed by equations 51 to 53, i s therefore reduced to a s ing le term: a = f V x _ ....(54) y 9z The v o r t i c i t y appears in the flow as yortex tubes or f i laments, with axes of rotat ion aligned in 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 low, near the bed, in the region where the ve loc i t y gradient i s high. Since the flow being considered i s steady and uniform, the d i s t r i bu t i on of the v o r t i c i t y with in the flow (the v o r t i c i t y f i e l d ] does not change in 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 ipated. Cb] Flow near the Cyl inder As the flow from upstream approaches the cy l inder i t div ides to pass around i t , and each vortex tube, being carr ied downstream by the f low, i s stretched as the streamlines diverge. This s t retching begins somewhat upstream of the nose of the cy l inder due to the cyl inder- induced pressure f i e l d (blunt-nosed cyl inders inducing a stronger pressure f i e l d and thus a f fect ing the flow farther upstream, than sharp-nosed ones). This s t retching reduces the cross-sect ion of each vortex tube so that the v o r t i c i t y increases. As the flow passes around the cy l i nder , the vortex tubes are also bent around the cy l i nder , and actua l ly become trapped or caught in f ront of the cy l i nde r , since vortex tubes cannot be cut even by a sharp f ront . Thus not only do the vortex tubes continue to stretch and increase i n v o r t i c i t y , but new ones are constantly being added from upstream. This process resu l t s in intense v o r t i c i t y and vortex motion at the front and sides of the cy l inder. Howeyer, the v o r t i c i t y does not increase without l i m i t , since viscous and turbulence ef fects cause the d i f f u s i on 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 en s i f i c a t i o n by stretching and accumula-t i o n , together with viscous and turbulence e f f e c t s , accounts fo 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 for a l l scour at cy l inder noses, since no other mechanism exists fo r the transport of sediment in th 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 hor izontal vortex that forms at the nose of a cy l inder or p ier in open-channel f low, but actua l l y consists of a number of l inked vort ices that operate together to scour the bed around the p i e r . Shen e_t a l _ . ^ 1 1 0 ^ obseryed a secondary "counter" vortex as shown in Figures 8 and 10. Thwai tes^ 1 2 5 ^ presents a photograph of an experiment by Gregory and Walker, of a cy l inder in a laminar boundary l ayer , which shows at least three vo r t i ce s . This has been sketched i n Figure 19. Rainbird et a l . report the same structure as t h i s , but with s t i l l another vortex upstream of the other t h r e e ^ 9 6 ^ . Roper^ 1 0 1 ^ reports three vo r t i ce s , as does Thwaites, above, although his proposed sequence of vortex formation and shedding is in e r ro r . 2. Formation The actual formation of the horseshoe vortex system occurs as fo l lows. The adverse pressure gradient generated by the cy l inder causes the approach flow to slow down and, due to the ye loc i t y gradient in the ve r t i c a l d i r e c t i o n , to acquire a downward component in f ront of the cy l inder. Qua l i t a t i v e l y , th i s i s the mechanism described by Bata Csection IT. F] and Moore and Masch (section n . H), although t he i r 51. quant i tat ive analyses, based on two-dimensional i n v i s c i d f low, are not v a l i d . This downward flow in turn induces a flow in the upstream d i rec t ion along the bed in front of the p ier nose, with subsequent flow separation upstream of the p ie r . A large horizontal vortex i s thus induced, which, by viscous in teract ion with the boundaries and the approach f low, generates yet other vo r t i ce s . Another way of descr ibing what happens i s to say that the adverse pressure gradient induced by the p ier causes the separation and r o l l i n g -up of the boundary layer in f ront of the p ier to form the horseshoe vortex system. 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 in the boundary layer of the approach f low. This i s supported by Thwai tes^ 1 2 5 ^ and B o l c s ^ 1 4 ^ , who both state that the horseshoe vortex system i s constantly being rep len i sh -ed by f l u i d from upstream. 3. Strength The action of loca l scour at a bridge p ier can be divided into two parts: (a) the resistance of the channel bed to scour. (b) the scouring strength or power of the vortex. For non-cohesive mate r i a l , the bed resistance depends only on the grain character i s t i c s of the bed sediment, and thus const itutes a minor part of the analys i s . The major part in understanding loca l scour has to do with discovering the factors that inf luence the vortex. Any procedure that attempts to re la te flow parameters d i r e c t l y to scour depth, without 52. considering the intermediary ro le of the yortex, can at best give only a pa r t i a l understanding of the scouring process, even though i t may y i e l d a re lat ionsh ip v a l i d fo r the conditions fo 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, Breusers, and Coleman (see Chapter II) a l l f a l l into th 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 in the approach f low, the concentration and d i s t r i bu t i on of v o r t i c i t y around the p i e r , the d i s s ipat ion of v o r t i c i t y at the flow boundaries, and i t s convection downstream past the p ie r . These parameters, fo r the case of a s ingle p ier i n a wide channel, can be l i s t e d as fo l lows: (a) The average ve loc i t y of the approach f low. (b) The ve loc i t y gradient of the approach flow. Cc) The p ier geometry Csize and shape). Cd) The scour hole geometry. Ce) The kinematic v i s co 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 invest igators . The inf luence of the approach flow ve loc i t y has been studied by various researchers, notably L a u r s e n ^ 6 0 ' 6 1 ^ , and Chabert and Engeldinger (see sect ion II. E above). The influence of p ier s i ze has been studied by most researchers in the f i e l d . Again, Laursen i s one of the chief contr ibutors in th i s area, (39) having also studied the e f fec t of the shape of the p ie r . Hawthorne v has shown a n a l y t i c a l l y that the shape of the p ier i s s i g n i f i c a n t . 53 . The scour hole geometry af fects only the instantaneous vortex strength, and not the f i n a l maximum scour depth. As scour progresses, the scour hole becomes larger and deeper, and the horseshoe vortex system, expanding with the scour hole, becomes weaker. This process continues un t i l the yortex strength f a l l s to a level such that the rate of sediment transport out of the scour hole i s j u s t equal to the rate of sediment transport into the scour ho le, and a l i m i t i n g equi l ibr ium depth of scour is reached. The kinematic y i s co s i t y of the f l u i d i s of importance in the generation and d i s s ipat ion of y o r t i c i t y and in determining the magnitude of shear stresses in the f low. It i s governed by the temperature of the f l u i d . Shen e_t al_. recorded the temperature in the i r tes t r u n s ^ 1 1 1 ^ and included v i s co s i t y as an i m p l i c i t yar iable i n the i r a n a l y s i s ^ 1 1 0 ^ . With the exception of Tison (see fo l lowing paragraph), the ve r t i c a l ve l oc i t y gradient of the approach flow has, to the author 's knowledge never been e x p l i c i t l y studied in connection with loca l scour at bridge p ie r s . It must, however, have some inf luence, because i t i s important in the generation of v o r t i c i t y and the formation of the horseshoe vortex system. Vinje has suggested^ 1 3 6 ^ that the ve loc i t y d i s t r i b u t i on of the (2) approach flow i s important, as has Ahmadv . Tison did some experiments in which he varied the ve loc i t y d i s t r i -bution by changing the bed roughness (see sect ion II. B above). However he did not recognize the horseshoe vortex mechanism, and his resu lts have qua l i t a t i ve 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 en -cing the vortex strength, because th i s i s included i n the ve l oc i t y 54. d i s t r i bu t i on of the f low. Various researchers have studied the inf luence of flow depth on the maximum depth of scour, Laursen again being foremost among them. The overa l l r e su l t of these invest igat ions i s that the depth of flow i s not important except fo r small depths. This suggests that the flow depth i s not important except when i t a f fects the ye l oc i t y p r o f i l e in the boundary layer of the approach f low. Shen et al_. have attempted to derive an expression for the horseshoe vortex strength based on the concept of v o r t i c i t y Csee section II. K above). They postulated that two-dimensional flow ana ly s i s - i s v a l i d fo r the flow in the plane of symmetry in f ront of the cy l inder . However, th i s must be considered as questionable, s ince, as Johnson points o u t ^ 5 0 ^ , th i s plane of symmetry i s en t i r e l y immersed in a three-dimensional flow f i e l d . E. SUGGESTED RESEARCH From the foregoing d i scuss ion, i t i s evident that there has been l i t t l e invest igat ion of the inf luence of the v e r t i c a l ve l oc i t y d i s t r i b u -t ion of the approach flow on the mechanics of loca l scour, although a de f i n i t e connection can be shown to e x i s t , from basic p r inc ip le s of f l u i d dynamics. I t was therefore decided to invest igate th i s connection, in a prel iminary and qua l i t a t i ve way. 5 5 . CHAPTER IV EXPERIMENTAL WORK A. PURPOSE AND SCOPE The main purpose of the experimental work was to inves t i gate, in a prel iminary and qua l i t a t i ve way, the influence of the ve loc i t y d i s t r i b u t i on of the approach flow on the loca l scour around a cy l i nder , p a r t i c u l a r l y the horseshoe vortex system and the equi l ibr ium 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 de l iberate ly varied were the ve loc i t y p r o f i l e of the approach flow and the ayerage flow ve l o c i t y . The other parameters, such as sand s i z e , p ier s i z e , p ier shape and so on, were held constant, except the flow depth, which varied 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 var ied from 56°F to 68°F. B. EQUIPMENT AND EXPERIMENTAL ARRANGEMENT -1 • General Arrangement The experimental work was carr ied out in a fo r ty - foot - long steel and glass flume in the C i v i l Engineering Hydraulics Laboratory at the Univers i ty of B r i t i s h Columbia. The flume was three feet deep and two and one-half feet wide, and consisted of ba s i ca l l y three sect ions: a f i f t een - f oo t steel approach sect ion , a f i f t een - f oo t glass-walled tes t sect ion, and a ten-foot steel end section (see Figure 20). Inflows to the flume were provided by a r ec i r cu l a t i n g system in which water from a sump was pumped up to a constant-head supply tank. Discharges from the tank were contro l led 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 late located upstream of the valve. The discharge ca l i b r a t i on curve for the o r i f i c e plate was obtained and checked using two large weighing tanks (20,000 lbs . each) in which the flume discharge, over a given time i n t e r v a l , could be co l lected and weighed. The water depth in 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 section were placed two sets of double metal-gr id screens and a set of vanes to straighten the f low, as wel l as a platform f l o a t to suppress waye action and surface disturbances. 57. 2. Sand Bed A sand bed twelve inches deep was placed over the ent i re f i f t e e n -foot length of the glass-walled sect ion of the flume. Bed t r an s i t i on sections were constructed out of plywood and sheet metal to provide smooth, gradual t rans i t ions between the flume bottom and the surface of the sand bed at each end Csee f igure 20]. In th i s way the flow was guided pa ra l l e l to the sand bed without unnecessary and undesirable turbulence. The same sand was used fo r a l l the te s t s , and had a grain s i ze d i s t r i -bution as shown in Figure 21. The median grain s i ze was 0.215 mm. The standard deviat ion a, defined by: n _ 1 r d 8 4 . d 50 > a - j U — + - J — ) (55) . a50 d16 was equal to 1.33. The sand was obtained from stockpi les of material dredged from the Fraser R iver, B.C. Occasional pebbles and fragments of wood had to be screened out before placement in the flume. 3. Test P ie r The p ier used in the tests was a hollow c i r c u l a r cy l inder of c lear p lex i g la s s , of four inches outside diameter, which extended from the bottom to the top of the flume. The p ier was positioned in the center of the flume as shown in Figure 20, and midway between the two s ides. Its bottom end was prevented from sh 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 isc f i xed to the flume bottom. The top end f i t t e d into a V-shaped notch in a board which 58. was secured across the top of the flume, preventing movement of the p ie r in the downstream d i r e c t i on . This arrangement also permitted rotat ion of the p ie r . The scour depth below loca l bed level could be read at any time by a sca le , graduated in hundredths of a foot , which was attached to the ins ide surface of the p ie r . A 150-watt floodlamp shining through the glass wal l of the flume was used to i l luminate the scour hole area, and f a c i l i t a t e d reading of the sca le. 4. Ve loc i t y -D i s t r i bu t i on 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 feet upstream of the nose of the p ie r . This gate consisted of spaced 3/8-inch-diameter aluminium rods extending hor i zonta 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 ins ide surface to prevent sl ippage. These bars could be e i ther t ightened, to clamp the rods in place, or loosened, to allow the spacing of the rods to be eas i l y changed. Figure 22 shows a view of the control gate from a pos i t ion downstream of the p ier (the image of the gate i s re f lected by the glass wal ls of the flume). 5. Current Flowmeter Ve loc i ty measurements were made with an Armstrong-Whitworth Miniature Current Flowmeter (Type 176/1). The measuring head, located at the end of an 18-inch probe, had a protect ive cage 1.5 cm. in diameter, and ba s i ca l l y consisted of a f iye-bladed p l a s t i c rotor mounted on a hard sta in less steel spindle. The spindle terminated i n f ine burnished conical pivots which ran in jewels mounted in an open steel frame. The pivots and 59. jewels were shrouded to reduce the p o s s i b i l i t y of fou l i ng . The revolutions of the rotor were counted on three Dekatron counters over a period of 8.33 seconds. The count was displayed for 6.67 seconds, and J.67 seconds were required a f te r that for a new count to begin. The v e l o c i t y was read from a ca l i b r a t i on curve which re lated the count to the local flow ve l oc i t y . A travers ing mechanism enabled the measuring head to be posit ioned anywhere in the flow. Since the flowmeter required a conducting l i q u i d in order to function properly, a small amount of sodium s i l i c a t e so lut ion was added to the laboratory water supply. 6. Dye Injector A simple dye i n jec to r was made by bending the lower four inches of a section of thin glass tubing at r i ght 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 ine nozzle. The nozzle could be positioned anywhere i n the f low, and aligned in any d i r e c t i o n , by a simple travers ing 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 ca re fu l l y l e v e l l e d . 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 disturbed. The flume i n l e t valve was 60. then opened rap id ly to the desired discharge, and the stop-watch s tar ted. This procedure resulted in a rapid but smooth r i s e to the desired discharge, and did not produce any i r r egu la r d isturbing waves. 2. Generation of Ve r t i ca l Ve loc i ty P ro f i l e s D i f ferent v e r t i c a l ve l oc i t y p ro 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 ind iv idual bars of the gate. The gate set t ing for each ve loc i t y p r o f i l e used in the test runs was determined in prel iminary tests by t r i a l and er ror . 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 feet upstream of the f ront face of the tes 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 fa r enough upstream so that the loca l disturbance in the f low, caused by the ind iv idua l bars of the gate, would be d i s s ipated, and, on the other hand, the gate had to be placed near enough so that the ve l oc i t y p r o f i l e created by the gate would not decay appreciably before i t reached the p ie r . The head drop across the gate was quite smal l , being of the order of a hundredth of a foot or le s s . 3. Ve loc i ty Measurements Veloc i ty measurements were made with the Armstrong-Whitworth Miniature Current Flowmeter. Its c a l i b r a t i on was ca re fu l l y checked with a miniature Ott propel ler meter which had jus t been fac to ry - ca l i b ra ted . A l l v e l oc i t y measurements were made on the center l ine of the flume. In general, measurements were made at 0.05-foot Intervals v e r t i c a l l y oyer the ent i re depth, except that the measurement nearest the bed was made 0.02 feet above the bed. Each measurement consisted of taking the ayerage of f i ve 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 in Section B.5 above. A horizontal traverse across the flume two feet upstream of the p ier was made to determine the horizontal ve l oc i t y p r o f i l e . In general, for each t e s t , the v e r t i c a l ve loc i t y p r o f i l e was measured in the f i r s t hour or so of the test run, at a distance of 1.5 feet upstream of the p ier face. After a l l the test runs had been completed and the p ier removed, the ve loc i t y p r o f i l e fo r each test ser ies was measured at the locat ion of the p ier center l ine in order to check the s t a b i l i t y of the p r o f i l e . 4. Flow Patterns Flow patterns in the scour hole and around the p ier were studied with the aid of dye, which was introduced into the flow at the desired locat ion with the dye i n jec to r . The discharge of the dye was contro l led so that i t did not disturb the flow. The flow patterns so observed were recorded by making on-the-spot freehand sketches, photographs of the flow patterns were not made, par t l y due to oyersight on the part of the author in the ear ly stages of the work, and part ly due to technical d i f f i c u l t i e s which arose subsequently. 62. 5. Scour Kole Development and Equi l ibr ium Depth of Scour The depth of scour below the bed level could be measured at any time by the graduated scale attached to the ins ide of the p ier (which could be rotated in any d i r e c t i on ] . Scour depths were recorded for each test run, 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 ren t t e s t s , could be compared. Each reported test run was continued un t i l there was no longer any increase of scour depth with time, and th i s f i n a l scour depth was taken to be the equi l ibr ium scour depth for the conditions of that test run. As the scour hole deyeloped, i t was observed that occasional coarser pa r t i c l e s were l e f t behind in the scour hole to produce an armouring e f f e c t . When th i s occurred, the flow was momentarily stopped and the offending pa r t i c l e s ca re fu l l y removed. The v i s i b i l i t y in the test arrangement permitted a completely unobstructed view of the scouring process. The motion of the ind iv idua l sand grains out of the scour hole was espec ia l l y noted, and sketches were made to record these observations. 63.. CHAPTER V EXPERIMENTAL RESULTS A. GENERAL The experimental resu l t s cons ist of the fol lowing data: 1. Observations of scour depth at per iodic in terva l s during the development of the scour hole, and the f i n a l equi l ibr ium scour depth, for each of various average ve l o c i t i e s and ve r t i c a l ve l oc i t y d i s t r ibut ions of the approach f low. 2. Observations of the development of the scour hole, the scouring process, and the motion of the ind iv idual sand grains in and out of the scour hole. 3. Observations of the vortex patterns in the scour hole for yarious stages of scour hole development. These resu l t s are described and discussed below. 64. B. APPROACH FLOW VELOCITY PROFILES Eight d i f fe rent v e r t i c a l v e l o c i t y p ro f i l e s were tested: series 1, 2-A, 2-B, 2-C, 2-D, 3 , 4 , and 5. Series I was tested at three d i f f e ren 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 ren t average v e l o c i t i e s . Series 2-A was tested three d i f f e ren t times at the same average ve l oc i t y . The ve l oc i t y p ro f i l e s fo r a l l the tests are shown in Figures 23 to 27, i nc lu s i ye . There was some unsteadiness inherent in the approach f low. Repeated ve l o c i t y measurements with the Armstrong-Whitworth flow meter at any one spot in the flow usual ly gave ye l oc i t y var iat ions of two or three per cent, and sometimes as much as f i ve per cent, e spec ia l l y near the bed. The ve loc i t y p ro f i l e s of series 1 were the ones which occurred natura l ly in 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 test series were a r t i f i c i a l l y generated by the gate. A l l of the p ro f i l e s generated by the gate tended to decay into the natural p r o f i l e of series 1. I t was not possible to tes t very steep ve loc i t y gradients, as these were d i f f i c u l t to generate, and moreover they decayed too rapid ly while approaching the p ie 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 locat ion JL.5 feet upstream of the p ier (with the p ier in p lace) , and then at the p ier center - l ine (with the p ier removed). The greatest d i f ference between the two measurements i s about f i ve per cent, which i s with in the range of unsteadiness of the approach f low. 65. Figure 29 shows the extent of the decay of the p r o f i l e of series 2-A3, from 1.5 feet upstream of the p i e r , to the locat ion of the p ier cente r - l i ne . The s h i f t towards the natural (ser ies l ] v e l o c i t y p r o f i l e i s quite evident, and i s representative of the behaviour of the other ve l oc i t y p ro f i l e s tested. This i n s t a b i l i t y prevented any quant i tat ive analysis of the influence of the approach flow ve loc i t y gradient from being carr ied out. The experiments were designed to run within the "c lear-water " scour range - i e . scour with no sediment transport of the sand bed in general (see above, section II. D). This was done so that the development of the scour hole and the f i n a l equi l ibr ium value of the scour depth would not be obscured by general sediment transport with r ipp les or dunes moving through the scour hole. It also s imp l i f i ed the experimental procedure. Therefore, the average v e l o c i t i e s | used for the various tests were a l l less than the c r i t i c a l ve l oc i t y for general sediment transport of the sand bed (U ... ). The c r i t i c a l ve loc i t y for general sediment transport was determined to be about 0.90 f t/sec. At th i s v e l o c i t y , small r ipp les started to form on the sand bed of the flume. The actual ve l o c i t i e s used i n the exper i -ments varied from 0.65 f t . / sec . to 0.89 f t . / sec . C. EQUILIBRIUM SCOUR DEPTH 1. Introduction The equi l ibr ium scour depth, d s g , i s defined as the scour depth at which,for given conditions of p ier and channel geometry and flow 66. condit ions, the rate of sediment transport out of the scour hole equals the rate of sediment transport into the scour hole. In the case of "c lear-water " scour, both of these transport terms equal zero fo r the equi l ibr ium condit ion. For each test run, the depth of scour was measured at regular i n t e r va l s , s ta r t ing from the beginning of each test at t = 0, u n t i l there was no longer any increase of scour depth with time. This f i n a l value was taken to be the equi l ibr ium scour depth, d s e > The time at which th i s depth was f i r s t reached i s ca l l ed t . The scour depth measurements have been plotted in Figures 30 to 35, i nc lu s i ve . Occasional ly, larger pa r t i c l e s would become uncovered in the scour hole as i t deepened. When th i s happened the flow was momentarily stopped and the larger pa r t i c l e s were ca re fu l l y removed to prevent armouring. These occasions show up as uneyen segments in the plots of scour depth versus time. A comprehensive summary of the tes t data i s given in Table I I I . 2. Results The accuracy of the resu l t s i s indicated by the resu l t s f o r the three d i f fe rent runs of series 2-A Csee Figures 24 and 31). Within the l im i t s of accuracy of the measuring equipment used, the flow conditions for these runs were a l l the same. The resu l t ing equi l ibr ium scour depths, however;, were measured to be 0.37, 0.36, and 0.38 f ee t , a ya r i a t i on of about 5 per cent. The inherent experimental error i s therefore considered to be at least 5 per cent. 67. A comparison of the results fo r the various y e r t i c a l ye l oc i t y p ro 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 fol lowing observations; Ca) For any giyen ve l oc i t y p r o f i l e , d g e increases with increasing average flow ve l oc i t y . This i s c l ea r l y demonstrated by the resu l t s for series 1, 3, and 4. In each of these ser ie s , a higher average flow ve l o c i t y resulted in a larger equi l ibr ium scour depth. Cb) The ve l oc i t y of the lower part of the f low, near the bed, i s more important, in determining d $ e , then the ve l oc i t y of the upper part of the flow. Series 3-B and 4-A both had an average flow ve loc i t y of 0.77 f t . / s e c . , and series 3-A had an average flow ve loc i t y of 0.85 f t . / s e c . , yet the equi l ibr ium scour depth for ser ies 4-A was greater than that for e i ther ser ies 3-A or 3-B. This i s because the ve l o c i t y of the lower part of the flow was greater in series 4-A than in e i ther of series 3-A or 3-B, even though these l a s t two had higher flow ve l o c i t i e s in the upper part of the flow (Figure 26). Cc) For s im i l a r v e r t i c a l ve l oc i t y p r o f i l e s , the equi l ibr ium scour depth decreases with decreasing ve l o c i t i e s in the lower part of the f low. This i s evident from the resu lts of ser ies 2. There was a continuous reduction in d_ as the ve l o c i t i e s in the lower parts of the flow se r decreased, from series 2-A to 2-D. Cd) The equi l ibr ium scour depth tends to increase with an increase i n the v e r t i c a l ye l oc i t y gradient. This i s not a d i r ec t observation but i s implied by the tes t resu lts of series 5 and 1-C, and ser ies 1-A and 2-A. 68. Series 5 and ser ies 1-C had about the same value of d s g Cwithin the l im i t s of accuracy of the experiment). 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 resulted in a larger value of d s g . The addit ional factor which needs to be considered i s the ve loc i t y gradient Cthe rate of change of ve l oc i t y with depth) of the approach f low, e spec ia l l y i n the lower part of the flow. For series 5, th 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 gh t at the bed, whereas in ser ies 1-C, there was a de f i n i t e ve l oc i t y gradient up to 0.30 f t . aboye the bed. This lack of ve l oc i t y gradient in the lower part of the flow in series 5 counter-acted the increase i n v e l o c i t y , resu l t ing in no net change in the equi l ibr ium scour depth. A s im i l a r e f fec t was obseryed with ser ies 1-A and 2-A. Again, the equi l ibr ium depth of scour was about the same fo r both se r i e s , even though in series 2-A the ve l oc i t y of the lower part of the flow was lower. However, the e f fec t of th i s lower ve loc i t y was counter-acted by the greater ve l oc i t y gradient of series 2-A, resu l t ing in the same equi l ibr ium scour depth as for ser ies 1-A. The flow aboye a height above the equal to one p ier diameter (0.33 f t ) has some influence on the equi l ibr ium depth of scour. For series 3-A and 4-B, the flow ve 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 th i s height, however, the flow ve loc i t y for series 3-A was greater, and the ve loc i t y gradient was greater. This was re f lected in an equi l ibr ium scour depth which was 0.05 f t . larger (0.21 f t . ) than that of ser ies 4-B (0.16 f t . ) . 69. 3. Comparison With Results of Others The flow conditions fo r series 1-A (undisturbed ve l o c i t y p r o f i l e , U - U } were probably c loser to the flow conditions of experiments done by previous invest igators , than those for any of the other tes t ser ies reported herein. The data for ser ies 1-A are therefore used in the fol lowing selected equations as proposed by d i f f e ren t researchers in the past. The equi l ibr ium scour depth fo r ser ies 1-A was 0.37 f t . below normal bed l e v e l . The symbols used in the fo l lowing equations are defined in Chapter II where they f i r s t appear and in the L i s t of Symbols. (a) Laursen (see section II. D) Laursen proposed a design curve fo r conditions of general sediment transport, which for a c i r c u l a r p ier i s given by: For the conditions of ser ies 1-A, th i s g ives: d = 2.01 b se 0.67 f t . This re su l t i s rather high. A check of Laursen's test data showed that he only used flow depths in the range 0.2 f t . to 0.9 f t . , with flow ve l o c i t i e s from 1.00 f t . / sec . to 2.50 f t . / sec . ^ 6 1 ^ .W i th these flow condit ions, the ve l oc i t y probably var ied throughout the ent i re depth of flow ( i e . boundary layer thickens - depth of f low}, and Laursen's resu lts cannot be expected to hold for flow conditions where the flow depth i s considerably larger than the boundary layer thickness. 70. Cb) Tarapore (see section II. G) Tarapore foand that the flow depth had no appreciable inf luence on d s e beyond a value of H = 1.15 b. He proposed, for the case of a c i r c u l a r p ier with a large flow depth, and conditions of general sediment transport: d M = 1.35 b sem For the conditions of ser ies _1-A, th i s giyes; d o a = 0.45 f t . sem (c) Breusers (see section I I. I) Breusers reported the resu l t s of two studies; ( i ) Model study of the piers of the bridge across the Oosterschelde: Table I shows, f o r u ~ u r ^ - t 2 5 ° - 8 1 f t ./sec. (b = 0.36 f t . , H = 1.64 f t ) : d = 1.25 b se For the conditions of ser ies 1-A, th i s g ives: d s e = 0.42 f t . ( i i ) Pr ivate study of scour around d r i l l i n g platforms: d c * m = 1.4 b sem For the conditions of series 1-A, th i s g ives: d c o m = 0.47 f t . sera (d) Maza and Sanchez (see section II. J) Maza and Sanchez proposed a modified form of an equation by Ja ros lav t s iev , fo r the region of "c lear-water " scour, subject to the condit ion that H > 1.5 b: 7 1 . dse••_ , \ ) \ 3 d 50 (u \ b f HU For the conditions of series 1-A, th i s g ives: d $ e = 0.85 b = 0.28 f t . Ce) Larras Csee section II. E) Based on f i e l d data and the data of Chabert and Engeldinger, Larras suggested a design equation; d C f l m = 1.42 b 3 / 4 sera For the conditions of series 1-A, th i s g ives: d = o.62 f t . seni This re su l t i s rather high. This i s because Larras 1 equation i s dimensionally non-homogeneous; for small p ie r diameters i t w i l l pred ict scour depths that are too la rge, and for large p ier diameters, i t w i l l predict scour depths that are too smal l , ( f ) Shen et a l . (see sect ion II. K) Shen and his co-workers at Colorado State Univers i ty obtained the r e l a t i o n : d s e = .00073 (R b) 0.619 For the conditions of series 1-A, R^  = 2.53 x 1 0 4 , which gives: d = 0.39 f t . se Cgl Coleman (.see section II. 0) Coleman proposed the fol lowing re l a t i on for the equi l ibr ium depth of scour: . . d s e = 1.49 b 9 / 1 ° C U 2 / 2 g ) 1 / 1 0 72. For the conditions of series 1-A, (U 2/2g) 1 / l 0 = C0.0122]1/10 = 0.644, which gives: d s e = 0 .35 f t . The equations o f l n g l i s , Blench, and Va r ze 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, in which flow depths were f i xed a r b i t r a r i l y . D. VORTEX PATTERNS AND SCOUR HOLE DEVELOPMENT 1. The Horseshoe Vortex System The structure of the horseshoe vortex system, which forms the basic mechanism of loca l scour around a cy l i nde r , i s shown in Figure 36. This pattern was observed for tes t series 2-C, before the s t a r t of scour, at an average approach-flow ve l oc i t y of 0.30 f t . / sec . Vortex 1 i s the main or primary vortex, and i t does most of the work involved in the scouring process. I t has been observed by invest igators in the f i e l d of f l u i d dynamics as wel l as invest igators who s p e c i f i c a l l y studied the local scour phenomenon (see Chapters II and I I I ) . Vortices 2(a) and 2(b) are secondary vort ices which are much weaker than the primary vortex. They are deriyed mostly from the in teract ion of the primary vortex with the rest of the flow and the flow boundaries. These secondary yort ices haye been observed by Shen et_ a l_.^ 1 0 8 ^(see also Figures 8 and 10), and by yarious invest igators in the f i e l d of f l u i d (94 99 123) dynamicsv- ' \ 73. Vortex 3, has, to the author 's knowledge, not been reported previously in the l i t e r a t u r e . I t was observed by the author on d i f f e ren t occasions for d i f f e ren t tes t se r i e s , and for d i f fe rent stages of scour hole development. This vortex, although undoubtedly influenced by the primary vortex, nevertheless does not derive i t s energy from i t , but i s maintained by the incoming flow from a level ju s t above that which feeds the primary vortex. The whole vortex system i s constantly being fed by the incoming flow from upstream. It i s quite unstable, and the ind iv idua l vort ices are repeatedly swept away and being reformed. The turbulence generally present in the approach flow i s probably the main factor in t h i s . 2. Scour Hole Development: Beginning of Scour The development of the scour hole with time i s indicated by the plots of Figures 30 to 35. In these f i gures , the plotted scour depth i s the depth (below normal bed leve l ) of the deepest part of the scour hole at the time of measurement. Scour of the bed around the p ier was observed to begin at symmetric-a l l y - l o ca ted points on both sides of the p ier about 30° from the f ront center of the p ier face. 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 act ion. These two scour spots were observed to increase in s i ze u n t i l they met at the front centre of the p ie r . This occurred with in the f i r s t minute of the test fo r almost a l l the tes t ser ies . However, the 30°- points s t i l l remained the points 74. of deepest scour for some time - un t i l the scour hole had developed to such a s i ze that the depth a l l around the front part of the p i e r , including the 30° - po ints , was the same. A sketch of the vortex pattern and scour hole at the beginning of i t s development, for tes t ser ies 1-A, i s shown in Figure 37. The stage of scour shown i s for time t = 12 minutes. The depth of scour at the front center of the p ier at th 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° - po ints , 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 hole, while vortex 3 scoured the bed area r ight at the p ier face. The length of time required for the scour hole around the f ront of the p ier to develop to a uniform depth must depend at least par t ly on the re l a t i ve strengths of vort ices 1 and 3, which in turn depend on the shape of the ve loc i t y p r o f i l e of the flow approaching the p ie r (see Section 3 below). This i s demonstrated by the tes t results for ser ies 1 and 2 (the only series for which scour depths at both 0° and 30° were measured). The time required for the scour hole to develop to a uniform depth fo r the series 1 tests was about two hours, while the time required fo r th i s in the ser ies 2 tests varied among the ind iv idual tests from 6 minutes to 30 minutes. The development of scour at the 30° - po ints , and at the f ront center of the p i e r , i s shown graphica l ly for series 1<-A, in Figure 38. 75. The plan-view vortex pattern i s shown in Figure 39, fo r tes t ser ies 2-C at time t = 20 minutes. The horseshoe vo r t i ce s , which haye hor izontal axes, are swept around the cy l inder by the main flow. The domain of the main yortex i s between the two dashed l ines on the f i gu re , while the domain of vortex 3 i s between the dashed l i ne nearest the p ier and the p ier i t s e l f . The two-dimensional vor t ices shed from the sides of the p ier due to the action 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 for a considerable distance downstream along the wake. 3. Approach Flow Veloc i ty P r o f i l e and Vortex Structure The connection between the approach flow ve r t i c a l ve l oc i t y p r o f i l e and the structure of the horseshoe vortex system i s i l l u s t r a t e d in 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 . / sec . to observe the vortex patterns. The approach flow ve loc i t y p r o f i l e i s p lotted on the r ight hand side of the f i gure. Dye was ca re fu l l y in jected into the flow ju s t upstream of the scour hole at various leyels above the bed, in the plane of symmetry of the experimental arrangement. It was observed that each ind iv idual vortex . with in the vortex system was maintained by or deriyed from a s pec i f i c flow leyel with in the boundary layer ( i e . with in the depth range in which the flow ye loc 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 in Figure 40. From the bed up to a height of about 0.04 f t . , the flow was very slow and went mostly into a very v/eak, type 2(b) vortex. From 0.04 f t . to 0.15 f t . , most of the flow went into the primary vortex, with occasional surges into type 2(a) and type 3 yo r t i ce s . From 0.15 f t . to 0.22 f t . , most of the flow went into vortex 3, with some going into yortex 2(a), and occasional ly into the main vortex. From 0.22 f t . to 0.29 f t . , the flow went into vortex 3. Above a level of 0.29 f t . no flow was observed moving down towards the vortex system. This leve l corresponds approximately with the level at which the ve loc i t y of the approach flow becomes constant. The above observations are combined in 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 fo r various test ser ies . The sand travels in ba s i ca l l y two steps. F i r s t , sand i s moved along the slope of the scour hole to points A and B as shown in Figure 42. Points A and B mark the locations of two " l i n e s " or "avenues" of sediment transport out of the hole and downstream, which l i e approximately at r i gh t angles to the slope. Three vort ices are act iye in moving the sand towards points A and B. Vortex 1 moves sand up the slope of the scour hole to point A. Vortex 2(a) moves sand down the s lope, also to point A. Vortex 3 moves sand r i ght into the p ier face to point B. Vortex 2(b) i s too weak to haye any e f f e c t , and the sand in th i s area simply s l ides down towards point A by the act ion of grav i ty . 77. Transport l i ne A i s located along the outside edge of vortex 1 Csee Figure 391, and l i ne B i s located r i gh t along the surface of the p ie r . These l ines are quite narrow at the front of the scour hole, but widen towards the rear. 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 gu res , the slope of the scour hole i s not constant. The steepest part of the slope i s r i gh t under the main vortex, and i s p a r t i a l l y maintained by the shear stresses exerted by the vortex. When the flow i s stopped, th i s part of the slope slumps down. Several photos of the ful ly-developed scour hole are shown in Figures 43 and 44. 78. CHAPTER VI SUMMARY AND CONCLUSIONS A. PREVIOUS INVESTIGATIONS . The reported study of local bed scour at bridge piers goes back over the l a s t eighty years or so. During th i s time, d i f fe rent investigators t r i ed to understand how scour of an erodible bed at a bridge p ier or other flow obstruction occurred, and, on the basis of the i r various ideas and concepts, t r i ed to establ i sh relat ionships between the depth of loca l II scour and the other parameters of the problem,.usually by conducting experiments on scale models in a hydraulic laboratory. The parameters that were shown by these experiments to be most important, can be divided into two groups - the parameters describing the 79. p ier geometry, and the parameters describing the flow condit ions. The p ier geometry i s adequately described by two parameters. The most important one i s the p ier s i z e , expressed as the p ier width, or diameter, b. The second parameter i s the p ier shape, and i s usual ly expressed as a constant c oe f f i c i en t and applied as a d i rec t factor in scour equations. There are two main flow parameters which the various invest igators have found to be important. These are the flow ve loc i t y and the flow depth. However, there i s not universal agreement as to the i r r e l a t i v e importance. Some invest igators , espec ia l l y Laursen, stressed the importance of the flow depth, while discounting the flow ve l o c i t y . Others, such as Shen e_t al_., stressed the flow ve loc i t y as being more important. The r e l a t i ve importance of these two flow parameters depends on the regime of flow which i s being considered. For a flow regime in which there i s no general transport of bed sediment, the flow ve l oc i t y has a de f i n i t e e f fec t on the equi l ibr ium depth of scour, d s g . However, th i s e f fec t decreases rap id ly once conditions of general sediment transport are establ ished Csee Figures 3 and 18). I f the flow regime i s such that the flow depth i s smal l , and the ve loc i t y p r o f i l e of the boundary layer i s affected by changes in flow depth, then the equi l ibr ium depth of scour w i l l be affected also. However, i f the depth of flow i s la rge, and the boundary layer ye loc i t y p r o f i l e i s f u l l y deyeloped, then var iat ions in flow depth w i l l not a f fec t the equi l ibr ium depth of scour. This conclusion i s implied by the f indings of Tarapore Cp.20, above), Breusers (p. 23, 80. above), and Maza and Sanchez (p. 25, above). For any given p ier in an erodible bed, the equi l ibr ium depth of scour (d s eI increases with flow depth and flow v e l o c i t y , only up to a certa in 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 th i s point w i l l no longer a f fec t the depth of scour. This l i m i t i n g yalue i s known as the maximum equi l ibr ium depth of scour (d ), and i s n r sem governed only by the p ier s i ze and shape. This has been i m p l i c i t l y recognized by a number of invest igators who have proposed re lat ionsh ips f o r maximum equi l ibr ium scour depth ( d $ e m ) based only on p ier s i ze and shape Ceg. Tarapore, Breusers). B. MECHANISM OF LOCAL SCOUR The basic mechanism of loca l scour i s the horseshoe vortex. The horseshoe vortex i s formed by the action of the p ier in apprehending the v o r t i c i t y normally present in the f low, and concentrating i t near the bed at the p ier nose. The p ier induces an adverse pressure gradient in the f low, which causes i t to acquire a downward component in f ront of the p ie r . This in turn causes separation of the boundary layer in f ront of the p i e r , which then r o l l s up to form a large vortex with a hor izontal axis and shaped l i k e a horseshoe in plan. Interaction of th i s vortex with the boundaries and the approach flow generates yet other vo r t i ce s , so that a system of l inked yort ices deyelops. This i s ca l l ed the horseshoe vortex system. Any attempt to understand or explain loca l scour without considering the basic vortex mechanism cannot succeed. Thus, the analyses of Tison, I shihara, Tarapore, and Carstens, do more to confuse the s i tua t ion than they do to explain i t , although the i r experimental resu l t s are v a l i d in themselves. 81. C. EXPERIMENTAL RESULTS 1. Introduction A consideration of the factors that inf luence the strength of the horseshoe vortex led to the decis ion to invest igate the e f fec t of the ve r t i c a l ve l oc i t y d i s t r i bu t i on of the approach flow on the equi l ibr ium depth of scour, and on the horseshoe vortex flow patterns. Experiments were therefore carr ied out to do t h i s . A flow regime with a iarge depth of f low, and flow ve 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 transport, were used. 2. Equi l ibr ium Depth of Scour The main resu lts of the experiments on the equi l ibr ium depth of scour can be summarized as fo l lows. (a) The equi l ibr ium depth of scour increases with increasing average approach flow v e l o c i t y , for the range of flow ve l o c i t i e s used. (b) The equi l ibr ium depth of scour depends more on the ve loc i t y of the lower part of the flow than on the ve loc i t y of the upper part of the flow. Cc) The equi l ibr ium depth of scour increases with an increase in the gradient of the v e r t i c a l ye loc i t y p r o f i l e of the approach flow. The equi l ibr ium depth of scour obtained for an undisturbed ye loc i t y p r o f i l e at an average ye loc 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 for 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 calculated from various equations proposed by preyious invest igators . The equations of Laursen (equation 10} and Larras (equation 11} gave very high values fo r the equi l ibr ium scour depth, and were considered to be inapp l i cab le , fo r the reasons stated in Chapter V. Equations of Tarapore and Breusers, for the maximum equi l ibr ium depth of scour, d s e m , are also inapp l i cab le , since these were derived fo r the most severe flow conditions poss ib le, with a state of general sediment transport, and would include the e f fec t of such factors as dune troughs passing through the scour hole. The remaining equations, of Breusers, Maza and Sanchez, Shen et a l . , and Coleman, gave values for d g e of 0.42 f t . , 0.28 f t . , 0.39 f t . , and 0.35 f t . , respect ive ly. These four equations give an average of 0.36 f t . , which compares well with the 0.37 f t . actua l l y obtained. 3. Horseshoe Vortex Flow Patterns The main results of the observations of flow patterns in the horseshoe vortex system are as fo l lows. a) The horseshoe vortex system i s a system of l inked vo r t i ce s , and i s made up of a primary vortex (vortex 1 ) , several secondary vort ices (vort ices 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 vortex; the secondary vo r t i ce s do very 83. l i t t l e work. Vortex 3 scours the region of the bed r ight next to the p ier face. c) The whole yortex system i s cont inual ly being fed by the incoming flow from upstream. Approach flow turbulence i s therefore re f lec ted in the alternate collapse and reformation of the ind iv idua l vo r t i ce s . d) Each ind iv idual vortex with in the horseshoe vortex system i s derived from and maintained by a s pec i f i c flow level w i th in the boundary l ayer , and the flow over the ent i re thickness of the boundary layer i s used to supply the horseshoe vortex system. D. RECOMMENDATIONS 1 . Predict ing Scour Depths The present state of knowledge and understanding of loca l scour at bridge piers i s such that , in general, the scour depth fo r a given p ier 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 fo r these, the only re lat ionsh ip that has been establ ished with any confidence i s the re lat ionsh ip fo r the maximum depth of scour, d s e m - This seems to be adequately defined by Breusers* equation, d s g m =1.4 b Cequation 211. For pa r t i cu l a r flow condit ions, p i e r geometries, and bed sediment char-a c t e r i s t i c s be safely predicted only on the basis of hydraul ic laboratory 84. studies in which actual prototype conditions are accurately reproduced . on a small sca le. Further, models of several d i f f e ren t scales should be tested to check fo r scale e f fec t s . The experimental work reported here indicates that the v e r t i c a l ve l oc i t y d i s t r i bu t i on should also be considered as one of the flow character i s t i c s that need to be properly scaled. 2. Further Research The fol lowing areas are suggested as p a r t i c u l a r l y in need of further research and invest igat ion. Ca) Experimental data covering a broad range of v e r t i c a l ve l oc i t y d i s t r i bu t ions needs to be obtained, so that the inf luence of th i s flow parameter can be quant i ta t i ve l y determined. CbI The e f fec t of the depth of flow on the v e r t i c a l ve l oc i t y gradient, and thus on the equi l ibr ium scour depth, should be investigated further. Cc) Preyious experimental data should be reviewed in the l i g h t of the observed e f fect s of the v e r t i c a l ve loc i t y gradient of the approach f low. BIBLIOGRAPHY 85. BIBLIOGRAPHY Ahmad, M. "Experiments on Design and Behaviour of Spur Dykes," Proceedings, Minnesota International Hydraulics Convention, Minneapolis, Minnesota, August, 1953, pp. 145-159. Ahmad, M. [.Discussion of E.M. Laursen's "Scour at Bridge Cross ings " ] , Journal of the Hydraulics D i v i s i on , Am.Soc. of C i v i l Engrs., Vo l . 85, No. HY9 (.November 1960), pp. 144-151. A l l e n , J . "The Ef fect of Engineering Structures on the Regime of Waterways, with Special Reference to Bridge P i e r s , Sluice-Dams, and Sp i l lways, " Scale Models in Hydraulic Engineering, Chapter I I I. London; Longmans, Green and Co., 1947. Arunachalara, K. "Scour Around Bridge P i e r s , " Journal of Indian Roads  Congress, Vo l . 29, No. 2 (August 1965),pp. 189-210. Arunachalara, K. IDiscussion of C.R. Net 11 1s "Measurements of Bridge Scour and Bed Changes in a Flooding Sand-Bed R i y e r " J , Proceedings, I n s t i tu t i on of C i y i l Engineers, Vo l . 36, Feb., 1967, pp. 402-404. 86. 6. Baines, W.D. [Discussion of F.D. Masch and W.L. Moore "Drag Forces in Veloc i ty Gradient F low"] , Journal of the Hydraulics D i v i s i on , Am. Soc. of C i v i l Engrs., Vo l . 87, No. HY1 (January 1961), pp. 271-273. 7. Bata, G. Scour Around Bridge P ie r s . Unpublished t r an s l a t i on , Colorado State Un ivers i ty , Fort C o l l i n s , Colorado. Translated from the Serbian by Markovic. Orig inal en t i t l ed E roz i ja oko Novosadskog  Mostovskog Stuba, published by I n s t i t u t za vodoprivredu, Jaroslav Cerai Beograd, Yugoslavia, 1960. 8. Barr, D.I.H. [Discussion of CR . N e i l l ' s "Measurements of Bridge Scour and Bed Changes in a Flooding Sand-Bed R i v e r " ] , Proceedings, I n s t i tu t i on of C i v i l Engineers, Vo l . 36, Feb., 1967, pp. 404-406. 9. Barr, D.I.H., and Herbertson, J.G. [Discussion of S. Komura's "Equi l ibr ium Depth of Scour in Long Contract ions " ] , Journal of the  Hydraulics D i v i s i on , Am.Soc. of C i v i l Engrs., Vo l . 93, No. HY3, (May 1967), pp. 220-221. 10. Bauer, W.J. {Discussion of E.M. Laursen's "Scour at Bridge Cross ings" ] , Journal of the Hydraulics D i v i s i on , Am. Soc. of C i v i l Engrs., Vo l . 86, No. HY9 (November 1960), pp. 132-133. 11. Blench, T. "Problems Related to Breadth, Depth, and Slope," Regime Behaviour of Canals and Rivers, Chapter 7. 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"Scour Around D r i l l i n g Platforms," B u l l e t i n Hydraulic Research, International Assoc. fo r Hydraulic Research, Vol. 19, 1964 and 1965, p. 276. 87. 18'. Breusers, H.N.C. "Time Scale of Two-Dimensional Local Scour," Proceedings, Twelfth Congress of the International Assoc. f o r Hydraulic Research, Fort Co l l i ng s , Colorado, September, 1967, Vo l . 3, pp. 275-282. 19. Breusers, H.N.C. [Discussion of H.W. Shen ejt al_. "Local Scour Around Bridge P ier s " ] , Journal of the Hydraulics D i v i s i on , Am.Soc. of C i v i l Engrs., Vol . 96, No. HY7 ( Ju ly , 1970), pp. 1638-1639. 20. Carstens, M.R. " S i m i l a r i t y Laws for Local ized Scour," Journal of the Hydraulics D i v i s i on , Am. Soc. of C i v i l Engrs., Vo l . 92, No. HY3 (May 1966), pp. 13-36. 21. Carstens, M.R. .[Closure to Discussion of M.R. Carstens' " S i m i l a r i t y Laws fo r Local ized Scour"J , Journal of the Hydraulics D i v i s i on , Am. 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"Prevention of Scour at Bridge Abutments," Proceedings, Twelfth Congress of the International Assoc. fo r Hydraulic Research, Fort C o l l i n s , Colorado, September, 1967, Vol . 2, pp. 74-87. 42. I ng l i s , S i r Claude C. "Maximum depth of scour at heads of guide banks and groynes, p ier noses, and downstream of br idges," The  Behaviour and Control of Rivers and Canals, Part I I , Chapter 8. Poona: Government of India, Central Waterpower I r r i ga t i on and Navigation Research S ta t ion , 1949. 43. I ng l i s , S i r Claude C. {Discussion of CR . N e i l l ' s "Measurements of Bridge Scour and Bed Changes in a Flooding Sand-Bed R i v e r " ] , Proceedings, I n s t i tu t i on of C i v i l Engineers, Vol . 36, Feb., 1967 pp. 406-407. 44. I shihara, T. An Experimental Study of Scour Around Bridge P ier s . Unpublished t r an s l a t i on , Colorado State Un ivers i ty ; Fort C o l l i n s , Colorado. 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Engrs., Vol . 82, Series D (Journal of Basic Engineering, March, 1960), pp. 233-246. 49. Johnston, J.P. [Closure to Discussion of J.P. Johnston's "On the Three-Dimensional Turbulent Boundary Layer Generated by Secondary F low"] , Transactions, Am. Soc. of Mech. Engrs., Vo l . 82, Series D (Journal of Basic Engineering, March 1960), p. 248. 90. 50. Johnston, J.P. "The Turbulent Boundary Layer at a Plane of Symmetry in a Three-Dimensional Flow," Transactions, Am. Soc. of Mech. Engrs., Vo l . 82, Series D (Journal of Basic Engineering, September, 1960), pp. 622-628. 51. Karaki , S.S., and Haynie, R.M. Mechanics of Local Scour, Part I I , Bib!iography. Report No. CER 63SSK46, C i v i l Engineering Sect ion, Colorado State Un ivers i ty , Fort C o l l i n s , Colorado, November, 1963. 52. Kemp. P.H., and Grass, A . J . 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P l a te , E . J . , and Goodwin, CR . "The Influence of Wind on Open Channel Flow," Coastal Engineering Specialty Conference. Santa Barbara: Am. Soc. of C i v i l Engrs., October, 1965, pp. 391-423. 94. 93. Posey, C.J. "Why Bridges Fa i l in Floods," C i v i l Engineering (New York), Vol. 19, February, 1949, p. 42 and 90. 94. Posey, C.J. "Some Basic Requirements f o r Protect ion Against Erosion," Proceedings, Minnesota International Hydraulics Convention, Minneapolis, Minnesota, August 1953, pp. 85-88. 95. Posey, C.J. "Scour at Bridge P ie r s : 2. Protection of Threatened P ie r s , " C i v i l Engineering (New York), Vo l . 33, No. 5 (May 1963), pp. 48-49. 96. Rainbird, W.J., ejt aj_. "Some Examples of Separation in Three-Dimensional Flows," Canadian Aeronautics and Space Journa l , Vol . 12, 1966, pp. 409-423. 97. Rajaratnam, N. {Discussion of Z. Thomas' "An Interest ing Hydraulic Ef fect Occurring at Local Scour " ] , Proceedings, Twelfth Congress of the International Assoc. fo r Hydraulic Research, Fort C o l l i n s , Colorado, September 1967, Vo l . 5, p. 449. 98. Rehbock, T. "The River-Hydraul ic Laboratory of the Technical Univers i ty of Karlsruhe," Hydraulic Laboratory Pract ice. J.R. Freeman, ed., New York: Am. Soc. of Mech. Engrs., 1929, pp. 135, 137, 161, 201. . 99. Richardson, P.D. "The Generation of Scour Marks Near Obstacles," Journal of Sedimentary Petrology, Vo l . 38, No. 4, December, 1968, pp. 965-970. 100. Romita, P.L. {Discussion of E.M. Laursen's "Scour at Bridge Cross ings" ] , Journal of the Hydraulics D i v i s i on , Am. Soc. of C i v i l Engrs., Vol . 86, No. HY9 (November 1960), pp. 151-152. 101. Roper, A.T. A Cylinder in a Turbulent Shear Layer. Ph.D. D isserat ion, Colorado State Un ivers i ty , Fort C o l l i n s , Colorado, August, 1967. 102. Roper, A.T. {Discussion of H.W. Shen e_t al_. 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Report No. CER66HWS22, C i v i l Engineering Department, Engineering Research Center, Colorado State Un ivers i ty , Fort C o l l i n s , Colorado, June, 1966. 111. Shen, H.W., Schneider, V.R., and Karaki , S. Mechanics of Local Scour, Data Supplement. Report No. CER66-67 HWS27, C i v i l Engineering Department, Engineering Research Center, Colorado • State Un ivers i ty , Fort C o l l i n s , Colorado, June, 1966. 112. Shen, H.W., Schneider, V.R., and Karaki, S. Mechanics of Local Scour, Supplement, Methods of Reducing Scour. Report No. CER66HWS36, C i v i l Engineering Department, Engineering Research Center, Colorado State Un ivers i ty , Fort C o l l i n s , Colorado, June, 1966. 113. Shen, H.W., Schneider, V.R., and Karaki , S. "Local Scour Around Bridge P i e r s , " Journal of the Hydraulics D i v i s i on , Am. Soc. of C i v i l Engrs., Vol . 95, No. HY6 (November 1969), pp. 1919-1940. 114. Shen, H.W., Schneider, V.R., and Karaki , S. [Closure to Discussion of H.W. 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"On the Flow Character i s t ics of Vortices in Three-Dimensional Local Scour," Proceedings, Twelfth Congress of the International Assoc. fo r Hydraulic Research, Fort C o l l i n s , Colorado, September 1967, Vo 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 Type ediment (mm.) C r i t i c a l Velocity for Sediment transport U c r l t . (cm/sec.) Velocity Ratio U U c r i t . Scour Depth Ratio d se b Max. Scour Depth Ratio d sem b 11 50 sand 0.2 25 0.8 1.0 1.2 1.4 1.6 1.1 1.25 1.4 1.5 1.5 1.5 5 25 sand 0.2 25 0.8 1.0 1.2 1.4 1.6 1.3 1.4 1.5 1.6 1.6 1.6 11 50 poly-sty ren e 1.5 9 1.0 1.2 1.4 1.6 1.5 1.7 1.65 1.65 1.7 21 50 poly-sty re n I e 1.5 9 1.0 1.5 TABLE II VALUES OF THE COEFFICIENT KR USED IN THE EQUATION OF JAROSLAVTSIEV Flow Depth Ratio H/bx .6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 5.0 6.0 8.0 Coeff ic ient KH .92 .67 .46 .31 .22 .15 .10 .08 .05 .05 .05 TABLE III SUMMARY OF SCOUR EXPERIMENTS Test P ie r : c i r c u l a r cy l i nder , d i a . b = 0.33 f t . Bed Sand: d = 0.215 mm., U = 0.90 f t ./ sec . Test Series Average Veloc i ty U f t . / sec . Equi l ibr ium Scour Depth d se f t . Time to reach d se t se min. Depth of flow H f t . H b d se U Water Temp. b U c r i t . 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 63 2-A1 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 obstruction, as proposed by Tison. 3 2 6 o | | j ! i .' ! T i i i • c Mean )iameter Velocity (mm) (fps) — < 0.44 o 1.00 i 0.58 o 1.25 0.97 o 1.50 --|- 1.30 » 1.75 2.25 o 2.00 e 2 2 5 i e ! *i ! 4 r i « 2.50 ! | • • ; i ! : 0 1 2 3 4 5 Figure 2. Laursen's non-dimensional p lot of equi l ibr ium scour depth (d ) versus flow depth (H), fo r a rectangular s e p ier of width b, at an angle of attack of 30°. c/eah - u/a-feh SCOUh T scout- ujUh jenehz/ AvehagG appi-oach ve/oc/fy (J Figure 3. Schematic diagram showing var iat ion of the equil ibrium scour depth ( d g e ) with average flow ve loc i ty (U), as found by Chabert and Engeldinger ( for any p ie r ) . a Figure 4. Velocity d i f f u s i o n into 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 equilibrium conditions, 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 cylinder. 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 version of Jaroslavtsiev's equation. Figure 8. Idealized representation of. the flow on the plane of symmetry i n front of a c i r c u l a r cylinder; Shen e_t al.. b<Lcl cu//nJet-Z> Figure 9. Control volume on the stagnation plane i n front of a circu l a r cylinder Shen et a l . Essentially " Dead Water" region vortex is very weak The velocity profiles are in proportion The doshed lines only show approximate location. Small counter vortex Strong primary vortex The location of the primary vortex is not steady but tends, in general, to move up the slope. 14 12 1.0 0.8 0.6 Distance Upstream From t Of Pier In Feet 0.4 0.25 0.1 a. o x> c O Figure 10. Schematic representation of the velocity distribution and flow pattern in the scour hole on the plane of symmetry in front of a circular cylinder Shen et a l . o <1> o I) , n 0.5 Q . Q \_ z> o o 0.2 cr 1x1 o. 0.0 5 j i i o° 0 £ c - °a 0 c 0 0 i > 2, n o 1 0 D <"> O o 1 j i A A D AA A • d se = 0. 00073 R0/ 6 ' 9 Source d 5 o , mm o Shen et gl. 0.24(VA) A | Chabert and 0 , 2 6 • •> Engeldingsr Q 5 2 • ! j 1 0 i i I 10' I 0 : Figure 11. Equilibrium scour depth versus pier Reynolds number (R ) for c i r c u l a r c y l i n d r i c a l b piers; Shen et a l . F i g u r e 12. E q u i l i b r i u m s c o u r d e p t h v e r s u s p i e r Reyno ld s 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 o f d i f f e r e n t s i z e s ; Shen e t a l . Figure 13. Equilibrium scour depth versus for c i r c u l a r c y l i n d r i c a l piers Shen et a l . pie r Reynolds and different number (R^), grain s i z e s ; 114. /.o .1 .3 • 7 .6 .5 X •i .J .t ./ o T 1 m m\ 1 * * / t r 5 / W ,Z .3 .4 .5" .6 .7 .8 .f AO H =10 cm. • run 1 u = 27.7 cm./sec. x run 2 max Figure 14. Velocity 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 l l ' l \ I t i l I t i l l I I I I I I I I / / / ' / / / / ' I / t f i t t I I' I I / I 7 I I I 1 I i I I I I i I 11 / > / i n m / I 7 I 111 f I I TTTTTTTTTTTTTTT7 Figure 15. Types of c i r c u l a r c y l i n d r i c a l piers studied by Tanaka and Yano. 116. I ? . 8 ^ .6 M .2 - 0 o n © \ \ n _ Q —£J-= o \ V \ > JLY Ci" ** —.—. —\-\ \ A y f ^ \ i €T i > s > ^ / / / / / / / ' i ~f : ) o -.2 4 d = 5 cm. se(I) 3 cm. I = 5 cm. <J> I I , 1 cm. sq. hole O I I , 2 cm. sq. hole O I I I , B. /b = 3 a 0 I I I , B./b = 4 a O I I I , Bd/b = 5 O I I I , Bj/b = 6 a « IV « V Figure 16. Equilibrium 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. 117. Figure 17. Scour depth versus disc position, 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 ; Thomas. sem C L o Q =3 O O CO e CT For c Constant Pier and Sediment Size Position of curve ^ depeWs o« value of in Approach Velocity , U Figure 18. The scour regions of Schneider (for any p i e r ) . oo 119. Figure 19. Sketch of the vortex structure on the plane of symmetry i n front of a c i r c u l a r cylinder in a laminar boundary layer, 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 glass walls on either s i d e ) . - 1 -127. 7 . 2 -'3t -0 . 2 -measure J /.s 'feet...'." VfS+he.iXHi of f>ich face. 0.2. a. 4 o.6 o.S Vefdcf^y in -Zee/ pch Secoyi<J Figure. .27,. Velocity p r o f i l e , series 5. \ • 128. / 2 -0.8 . - • --<x4: az — \\-• -1 - -.. ._ i f of phi- g , cstHt 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 profile," series-2-A3 .-7-.... 1. "S-evie/i E - C , H = 0 . ^ 5 • , U 3 O . 3 o ft./***.. Vo^Kce-s 2 a . <wj 2. W 6 U - * . -^ov-wtej Figure 36. Vortex pattern, f l a t bed with low flow velocity. CO cls= o./2 fr-//oS -ro Sca./e. cU« 0.12 ^ Vortex pattern, beginning of scour (cross-sectional view). 5 e K , ^ i 2 - C ; //=/.?<///. 0 " 0.8S //./cet 6 * 2e> nun cc/cS e^2 e °-f scout- kole ou,| side ed^e MAO**. uovWy, V^JWJUV oid side edL ,^ VrtaAe*; 3 Figure 39. Vortex pattern, 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 ft./sec. and H = 1.24 f t . to a /)f>f>k>acJ, -/loop depth d = 0.20 f t . The flow was then slowed to // r, s H- /.oo //. U = 0.45 ft./sec. to observe vortex patterns. Q. Q <fs ft. J sec Figure 40. Vortex patterns; partly developed scour hole; showing dependence of the i n d i v i d u a l vortices on the v e r t i c a l velocity p r o f i l e of the approach flow. Figure 42. Scour hole development, showing motion of sand grains. t—* -p. ro (a) looking downstream Figure 43. Views of fully-developed scour hole: Ca) looking downstream; (b) looking upstream. -1^  Figure 44. View of fully-developed scour hole, looking diagonally upstream. 

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