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UBC Theses and Dissertations

Shear stresses under waves and currents Kingston, Kristopher William 1985

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SHEAR STRESSES UNDER WAVES AND CURRENTS by KRISTOPHER WILLIAM KINGSTON THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as co n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF A p r i l , © KRISTOPHER WILLI BRITISH COLUMBIA 1985 AM KINGSTON, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the THE UNIVERSITY OF BRITISH COLUMBIA, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d . t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: A p r i l , 1985 ABSTRACT T h i s s t u d y s e t out t o i n v e s t i g a t e the shear s t r e s s b e h a v i o u r a t the bed under combined wave and c u r r e n t a c t i o n . The i n t e n t i o n of the st u d y was t o make e x p e r i m e n t a l measurements t o dete r m i n e how wave and c u r r e n t shear s t r e s s e s combine, so t h a t t h e o r e t i c a l models d e s c r i b i n g the combined f l o w c o n d i t i o n c o u l d be proposed. Two t y p e s of experiment were c o n d u c t e d , and t h e o r e t i c a l models f o r the combined f l o w were a s s e s s e d . One s e t of e x p e r i m e n t s a t t e m p t e d t o use a shear p l a t e t o make d i r e c t measurements of the combined f l o w shear s t r e s s , and of the shear s t r e s s e s f o r the component waves and s t e a d y c u r r e n t s . T h i s approach f a i l e d because the l a r g e c o r r e c t i o n terms i n t r o d u c e d by the non-uniform wave p r e s s u r e f i e l d c o u l d not be a c c u r a t e l y e s t i m a t e d . The second s e t of e x p e r i m e n t s used a l a s e r d o p p l e r anemometer t o make d e t a i l e d v e l o c i t y p r o f i l e measurements over f l a t sediment beds. The onset of sediment motion was used as a c r i t e r i o n t o c a r e f u l l y c o n t r o l the e x p e r i m e n t s . I t i s assumed t h a t the t h r e s h o l d of sediment motion r e p r e s e n t s a s p e c i f i c shear s t r e s s i n t e n s i t y a t the bed f o r sediments of narrow s i z e r anges. As t h e shear s t r e s s e s can be d e t e r m i n e d from the v e l o c i t y f i e l d s under waves and c u r r e n t s , t h e i r a d d i t i v e n a t u r e under combined f l o w c o n d i t i o n s c o u l d be i n v e s t i g a t e d . For each sediment s i z e range, i t i s shown t h a t the same maximum v e l o c i t y v e r y near t h e bed can be used t o s p e c i f y i i the t h r e s h o l d of sediment motion c o n d i t i o n f o r a l l f l o w t y p e s , be they under waves, c u r r e n t s , or combined waves and c u r r e n t s . I t i s a l s o shown t h a t the near-bed v e l o c i t y under a l a b o r a t o r y wave can be p r e d i c t e d a c c u r a t e l y from second o r d e r wave t h e o r y and t h a t the v e l o c i t y under a c u r r e n t can be p r e d i c t e d from c o m b i n i n g Manning's r e l a t i o n w i t h the u n i v e r s a l l o g v e l o c i t y law. I t i s f u r t h e r shown t h a t the near-bed v e l o c i t y under a combined wave and c u r r e n t can be d e s c r i b e d by the v e c t o r i a l a d d i t i o n of the maximum component wave v e l o c i t y and the average component c u r r e n t v e l o c i t y . The shear s t r e s s f o r the onset of motion i s c a l c u l a t e d f o r the s t e a d y c u r r e n t u s i n g Manning's r e l a t i o n , f o r the wave by combining the o s c i l l a t o r y shear s t r e s s f o r m u l a w i t h Kamphuis's rough t u r b u l e n t f r i c t i o n f a c t o r r e l a t i o n , and f o r the combined wave and c u r r e n t by the s i m p l e v e c t o r i a l a d d i t i o n of the component shear s t r e s s e s , and i s shown t o be comparable w i t h S h i e l d s ' s t h r e s h o l d c r i t e r i o n f o r n e a r l y a l l c o n d i t i o n s t e s t e d . T a b l e of C o n t e n t s ABSTRACT i i LIST OF TABLES i x LIST OF FIGURES x SYMBOLS x i i i ACKNOWLEDGEMENTS x x i 1. INTRODUCTION 1 1.1 Overview of Research 1 1.2 O b j e c t i v e s of Research 2 1.3 Proposed E x p e r i m e n t s 3 1.3.1 D i r e c t Shear F o r c e Measurement 4 1.3.2 D i r e c t V e l o c i t y P r o f i l e Measurement 5 1.4 Review of S c a l e and L a b o r a t o r y E f f e c t s 6 1.4.1 I n t r o d u c t i o n 6 1 .4.2 S c a l e E f f e c t s 7 1.4.3 L a b o r a t o r y E f f e c t s 8 1.4.3.1 Wave G e n e r a t i o n 8 1.4.3.2 Wave A t t e n u a t i o n 9 1.4.3.3 Wave D i s t o r t i o n . 10 1.4.3.4 A d d i t i o n a l D i f f i c u l t i e s 12 1.4.3.5 C o n c l u s i o n s 12 1.5 D e s c r i p t i o n of Wave Flume 13 2. BOUNDARY LAYER THEORY 16 2.1 Concept of the Boundary La y e r 16 2.2 Laminar Boundary Layer 17 2.2.1 I n t r o d u c t i o n 17 2.2.2 N a v i e r - S t o k e s E q u a t i o n s f o r Laminar Flow...18 i v 2.2.3 S o l u t i o n s t o the N a v i e r - S t o k e s E q u a t i o n s f o r Laminar Flow 21 2.2.4 C o n c l u s i o n s ' 22 2.3 T u r b u l e n t Boundary L a y e r 23 2.3.1 I n t r o d u c t i o n 23 2.3.2 A P h y s i c a l Argument 24 2.3.3 N a v i e r - S t o k e s E q u a t i o n s f o r T u r b u l e n t Flow 25 2.3.4 T h e o r e t i c a l Assumptions f o r E s t a b l i s h i n g T u r b u l e n t Shear S t r e s s S o l u t i o n 28 2.3.4.1 V e l o c i t y P r o f i l e D e s c r i p t i o n 32 2.3.4.2 An A l t e r n a t e V e l o c i t y P r o f i l e D e s c r i p t i o n 33 2.3.5 E m p i r i c a l Method f o r D e t e r m i n i n g T u r b u l e n t Shear S t r e s s V a l u e s 35 2.3.6 C o n c l u s i o n s ....36 2.4 T r a n s i t i o n 37 2.4.1 P r e s s u r e G r a d i e n t E f f e c t s 38 2.4.2 T u r b u l e n t E f f e c t s 39 2.4.3 Roughness E f f e c t s 39 2.4.3.1 S i n g l e C y l i n d r i c a l Roughness Elements 40 2.4.3.2 D i s t r i b u t e d Roughness 41 2.4.4 C o n c l u s i o n s 42 2.5 C o n c l u s i o n s 43 3. GOVERNING EQUATIONS FOR OSCILLATORY FLOW 44 3.1 Laminar Flow V e l o c i t y P r o f i l e 44 3.2 T u r b u l e n t F l o w " V e l o c i t y P r o f i l e 48 3.3 T r a n s i t i o n 56 3.3.1 I n t r o d u c t i o n 56 v 3.3.2 S e l e c t e d T h e o r e t i c a l and E x p e r i m e n t a l A n a l y s e s 57 3.3.3 C o n c l u s i o n s 64 3.4 Shear S t r e s s E q u a t i o n s 65 3.4.1 Laminar Shear S t r e s s E q u a t i o n s 66 3.4.2 Rough T u r b u l e n t Shear S t r e s s E q u a t i o n s ....67 3.4.3 Smooth T u r b u l e n t Shear S t r e s s E q u a t i o n s ...70 3.4.4 C o n c l u s i o n s ..70 3.5 C o n c l u s i o n s 71 4. WAVE THEORIES 72 .4.1 G o v e r n i n g E q u a t i o n s 72 4.1.1 I n t r o d u c t i o n 72 4.1.2 S m a l l A m p l i t u d e Wave Theory ....74 4.1.3 S t o k e s F i n i t e A m p l i t u d e Wave Theory 75 4.1.4 N o n l i n e a r S h a l l o w Water T h e o r i e s 77 4.1.5 Stream F u n c t i o n Theory 79 4.1.6 C o n c l u s i o n s 80 4.2 Comparison of Wave T h e o r i e s 8 1 4.2.1 I n t r o d u c t i o n 8 1 4.2.2 Comparisons Based on Theory 82 4.2.3 Comparison Based on Experiment 83 4.2.4 C o n c l u s i o n s 86 4.3 Mass T r a n s p o r t Under Waves 86 4.3.1 I n t r o d u c t i o n 86 4.3.2 S t o k e s Theory ..86 4.3.3 L o n g u e t - H i g g i n s Theory 87 4.3.4 E x p e r i m e n t a l A n a l y s e s 88 v i 4.3.5 C o n c l u s i o n s 91 4.4 Combined Waves and C u r r e n t s 92 4.4.1 I n t r o d u c t i o n 92 4.4.2 A p p l i c a b l e E q u a t i o n s 92 4.4.3 Waves t o be Generated 94 4.5 C o e f f i c i e n t of R e f l e c t i o n 94 4.6 C o n c l u s i o n s 96 INITIATION OF SEDIMENT MOTION 98 5.1 I n t r o d u c t i o n 98 5.2 A n a l y s i s of the S h i e l d s C r i t e r i o n ...99 5.2.1 I n t r o d u c t i o n 99 5.2.2 E x e r t e d Shear S t r e s s D i s t r i b u t i o n 101 5.2.3 C r i t i c a l Shear S t r e s s D i s t r i b u t i o n 103 5.2.4 T h r e s h o l d of M o t i o n 105 5.2.5 C o n c l u s i o n s 107 5.3 Bagnold's O s c i l l a t i n g P l a t e Experiment 108 5.3.1 D e s c r i p t i o n of E x p e r i m e n t a l P r o c e d u r e ....108 5.3.2 D e s c r i p t i o n of E x p e r i m e n t a l R e s u l t s 110 5.4 Comparison of U n i d i r e c t i o n a l and O s c i l l a t o r y C r i t i c a l C o n d i t i o n s 112 5.4.1 I n t r o d u c t i o n 112 5.4.2 A n a l y s i s of F o r c e s .112 5.4.3 A n a l y s i s of T h r e s h o l d Measurements 118 5.4.4 Other T h r e s h o l d Measurements 123 5.5 C o n c l u s i o n s 126 SHEAR PLATE EXPERIMENT 128 6.1 I n t r o d u c t i o n 128 v i i 6.2 G e n e r a l D e s c r i p t i o n of Shear P l a t e 128 6.3 Assumptions For Use of Shear P l a t e 132 6.4 E x p e r i m e n t a l Setup and Proc e d u r e 133 6.5 E x p e r i m e n t a l R e s u l t s .....135 6.5.1 O b s e r v a t i o n 135 6.5.2 N u m e r i c a l R e s u l t s 135 6.6 C o n c l u s i o n s 137 7. LASER DOPPLER ANEMOMETRY EXPERIMENT 138 7.1 I n t r o d u c t i o n 138 7.2 E x p e r i m e n t a l Equipment and Proc e d u r e 140 7.2.1 F a c i l i t y 140 7.2.2 I n s t r u m e n t a t i o n 141 7.3 Measurements Taken 145 7.3.1 F i r s t Set of E x p e r i m e n t s 146 7.3.1.1 E x p e r i m e n t a l R e s u l t s 148 7.3.2 Second Set of E x p e r i m e n t s 150 7.3.2.1 E x p e r i m e n t a l R e s u l t s 152 7.3.3 Comparison of Steady C u r r e n t R e s u l t s w i t h Other V e l o c i t y P r o f i l e s 163 7.3.4 Comparison of the Measured and T h e o r e t i c a l O s c i l l a t o r y Flows 165 7.3.5 R e s u l t i n g C a l c u l a t e d Shear S t r e s s e s 166 7.4 C o n c l u s i o n s 172 8. CONCLUSIONS 174 BIBLIOGRAPHY 177 v i i i LIST OF TABLES T a b l e T i t l e Page 4.1 ( M o d i f i e d a f t e r Sarpkaya and I s a a c s o n , 1981). R e s u l t s of l i n e a r wave t h e o r y 76 4.2 ( M o d i f i e d a f t e r Sarpkaya and I s a a c s o n , 1981). R e s u l t s of S t o k e s second o r d e r t h e o r y 78 7.1 Measured and t h e o r e t i c a l f o r w a r d and r e v e r s e v e l o c i t i e s 149 7.2 Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 0.30 t o 0.85 mm 153 7.3 Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 0.85 t o 1.16 mm 154 7.4 Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 1.16 t o 1.70 mm 155 7.5 Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 1.70 t o 2.00 mm 156 7.6 Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 2.00 t o 2.30 mm 157 7.7 Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 1.16 t o 1.70 mm. Adverse c u r r e n t 158 k_U* 7.8 V a l u e s of — 5 — and p r e d i c t e d f l o w type f o r a l l sediment s i z e s 165 7.9 E x p e r i m e n t a l shear s t r e s s e s as c a l c u l a t e d from v a r i o u s f o r m u l a e , compared w i t h t h e S h i e l d s c r i t e r i o n 1 68 i x LIST OF FIGURES F i g u r e T i t l e Page 1.1 D e f i n i t i o n s k e t c h of wave flume 13 2.1 ( A f t e r S c h l i c h t i n g , 1979). S k e t c h of boundary l a y e r on a f l a t p l a t e i n p a r a l l e l f l o w a t z e r o i n c i d e n c e 16 2.2 ( A f t e r S c h l i c h t i n g , 1979). T r a n s p o r t of momentum due t o t u r b u l e n t v e l o c i t y f l u c t u a t i o n .24 2.3 ( A f t e r S c h l i c h t i n g , 1979). Measurement of f l u c t u a t i n g t u r b u l e n t components i n a wind t u n n e l , a t maximum v e l o c i t y U=l00cm/s 28 2.4 ( M o d i f i e d a f t e r S c h l i c h t i n g , 1979). Measurement of f l u c t u a t i n g t u r b u l e n t components i n a c h a n n e l 29 2.5 ( A f t e r S c h l i c h t i n g , 1979). Roughness f u n c t i o n B i n k„U* terms of s , f o r N i k u r a d s e ' s sand roughness 33 v 2.6 ( A f t e r S c h l i c h t i n g , 1979). Boundary l a y e r t h i c k n e s s p l o t t e d a g a i n s t the l e n g t h R e y n o l d s number f o r a f l a t p l a t e i n p a r a l l e l f l o w a t z e r o i n c i d e n c e , as measured by Hansen ( 1930) 37 3.1 ( M o d i f i e d a f t e r K n i g h t , 1978). I n s t a n t a n e o u s v e l o c i t y p r o f i l e s i n l a m i n a r o s c i l l a t o r y f l o w , 0d=8.O 46 3.2 ( M o d i f i e d a f t e r K n i g h t , 1978). Laminar o s c i l l a t o r y f l o w - e x p e r i m e n t a l d a t a f o r v e l o c i t y a m p l i t u d e v a r i a t i o n 46 x F i g u r e T i t l e Page 3.3 ( M o d i f i e d a f t e r K n i g h t , 1978). Laminar o s c i l l a t o r y f l o w - e x p e r i m e n t a l d a t a f o r v e l o c i t y phase v a r i a t i o n 47 4.1 D e f i n i t i o n s k e t c h f o r a p r o g r e s s i v e wave t r a i n 72 4.2 ( A f t e r Sarpkaya and I s a a c s o n , 1981). Ranges of wave t h e o r i e s g i v i n g the b e s t f i t t o the dynamic f r e e s u r f a c e boundary c o n d i t i o n , a f t e r Dean ( 1970) 83 4.3 ( A f t e r Sarpkaya and I s a a c s o n , 1981). Ranges of s u i t a b i l i t y f o r v a r i o u s wave t h e o r i e s as s u g g e s t e d by Le Mehaute ( 1976).. 84 4.4 ( M o d i f i e d a f t e r Sarpkaya and I s a a c s o n , 1981). Non-dimensional mass t r a n s p o r t v e l o c i t y p r o f i l e s w i t h depth f o r v a r i o u s v a l u e s of the d e p t h parameter kd .....89 4.5 ( A f t e r I s a a c s o n , 1978). N o n - d i m e n s i o n a l mass t r a n s p o r t v e l o c i t y o u t s i d e the seabed boundary l a y e r . d e p a r t u r e of the c n o i d a l and S t o k e s wave t h e o r i e s from proposed c u r v e s 90 4.6 D e f i n i t i o n s k e t c h f o r a p r o g r e s s i v e wave t r a i n on a s t e a d y c u r r e n t 93 5.1 The S h i e l d s e n t r a i n m e n t f u n c t i o n 99 5.2 E x e r t e d shear s t r e s s p r o b a b i l i t y d i s t r i b u t i o n 103 5.3 C r i t i c a l shear s t r e s s p r o b a b i l i t y d i s t r i b u t i o n 103 x i F i g u r e T i t l e Page 5.4 P r o b a b i l i t y d i s t r i b u t i o n f o r bed a r m o r i n g 105 5.5 ( A f t e r W i l l i a m s and Kemp, 1971). P r o b a b i l i t y d i s t r i b u t i o n f o r c r i t i c a l movement.. 106 5.6 D e f i n i t i o n s k e t c h of f o r c e s a c t i n g on the p a r t i c l e 113 5.7 ( M o d i f i e d a f t e r V i n c e n t , 1958). Near-bed c r i t i c a l v e l o c i t y f o r sands of d i f f e r e n t s i z e under waves of v a r y i n g w avelength 125 6.1 D e f i n i t i o n s k e t c h of shear p l a t e 129 6.2 D e f i n i t i o n s k e t c h of c a n t i l e v e r system 130 6.3 C a l i b r a t i o n c u r v e f o r shear p l a t e o u t p u t s i g n a l d e t e r m i n e d from experiment 131 6.4 T y p i c a l t r a c e of wave p r o f i l e and c o r r e s p o n d i n g shear p l a t e o u tput s i g n a l 134 7.1 B l o c k diagram of the LDA s e t u p 143 7.2 T y p i c a l t r a c e of the near-bed water p a r t i c l e v e l o c i t y 145 7.3 D e f i n i t i o n s k e t c h f o r wave c o n d i t i o n s 147 7.4 Measured t h r e s h o l d v e l o c i t y p r o f i l e s 159 7.5 Measured and t h e o r e t i c a l wave v e l o c i t y p r o f i l e s . . . . 1 6 0 7.6 Measured and t h e o r e t i c a l c u r r e n t v e l o c i t y p r o f i l e s 161 7.7 Measured wave and c u r r e n t v e l o c i t y p r o f i l e s 162 x i i SYMBOLS GENERAL SYMBOLS t o t a l o s c i l l a t o r y water p a r t i c l e e x c u r s i o n a t the bed = 2a0m (see p. 55) m mean p a r t i c l e s i z e of bed roughness elements f r i c t i o n f a c t o r f o r o s c i l l a t o r y f l o w Froude number = e n t r a i n m e n t f u n c t i o n g r a v i t a t i o n a l c o n s t a n t wave h e i g h t h e i g h t of the " e q u i v a l e n t " wave, where the " e q u i v a l e n t " wave has same h e i g h t as the wave on the c u r r e n t which produces c r i t i c a l m otion maximum and minimum wave h e i g h t s measured i n a flume h e i g h t of r e f l e c t e d wave h e i g h t of p a r t i c l e f o r e q u i v a l e n t sand roughness r e f l e c t i o n c o e f f i c i e n t f o r flume waves = pj— w a v e l e n g t h p r e s s u r e a t a p o i n t i n the f l o w R e y n o l d ' s number = ^ x i i i U*D p a r t i c l e R e y n o l d ' s number = paddle s t r o k e a m p l i t u d e l e n g t h R e y n o l d ' s number = — — o s c i l l a t o r y a m p l i t u d e Reynolds U ° m a ° m number = — v U 6 t h i c k n e s s R e y n o l d s number = —up-time i n the C a r t e s i a n c o o r d i n a t e system p e r i o d of o s c i l l a t i o n , wave p e r i o d v e l o c i t y components i n the x,y,z d i r e c t i o n s r e s p e c t i v e l y f r i c t i o n v e l o c i t y = (— P f r i c t i o n v e l o c i t y a t the w a l l s t e a d y c u r r e n t v e l o c i t y c a r t e s i a n c o o r d i n a t e s boundary l a y e r t h i c k n e s s von Karman's e m p i r i c a l c o n s t a n t = 0.40 stream f u n c t i o n shear s t r e s s shear s t r e s s a t t h e w a l l maximum shear s t r e s s a t the w a l l a b s o l u t e f l u i d v i s c o s i t y k i n e m a t i c f l u i d v i s c o s i t y = — P f l u i d d e n s i t y a n g u l a r f r e q u e n c y = x i v BOUNDARY LAYER SYMBOLS a m p l i t u d e of h o r i z o n t a l p a r t i c l e motion a t the bed i n o s c i l l a t o r y f l o w (see p. 55) eddy v i s c o s i t y upper l i m i t h e i g h t of o v e r l a p l a y e r above bottom l a m i n a r s u b l a y e r t h i c k n e s s over a smooth boundary i n n e r l a y e r t h i c k n e s s over a rough boundary d i m e n s i o n l e s s stream f u n c t i o n h e i g h t of bed roughness element, wave number = ^ parameter r e f l e c t i n g w a velength of bed roughness = c r i t i c a l h e i g h t of roughness element c o n s t a n t i n K a j i u r a ' s r e l a t i o n s = 0.02 t u r b u l e n t eddy v i s c o s i t y m i x i n g l e n g t h e m p i r i c a l f a c t o r i n f r i c t i o n f a c t o r r e l a t i o n s Manning's roughness c o e f f i c i e n t h y d r a u l i c r a d i u s , p i p e r a d i u s l o n g i t u d i n a l s l o p e of water s u r f a c e xv t e m p o r a l mean v e l o c i t i e s i n x and y d i r e c t i o n s r e s p e c t i v e l y components of t u r b u l e n t v e l o c i t i e s i n x and y d i r e c t i o n s r e s p e c t i v e l y t e m p o r a l means of t u r b u l e n t v e l o c i t i e s v e l o c i t y a t the o u t s i d e edge of the boundary l a y e r = v e l o c i t y i n p o t e n t i a l f l o w ( f r e e stream f l o w ) maximum l o n g i t u d i n a l o s c i l l a t o r y water p a r t i c a l v e l o c i t y a t the o u t s i d e edge of the boundary l a y e r ( f r e e stream r e g i o n ) (see p. 55) v e l o c i t y a t the w a l l (see p. 55) maximum o s c i l l a t o r y water p a r t i c l e v e l o c i t y a t the w a l l (see p. 55) mean f l u i d v e l o c i t y i n p i p e f r i c t i o n v e l o c i t y a t w a l l a t p o s i t i o n of roughness element a m p l i t u d e of f r i c t i o n shear v e l o c i t y a t the bottom i n o s c i l l a t o r y f l o w c o n s t a n t f r e e stream v e l o c i t y i n s t e a d y f l o w f o r c e s per u n i t volume i n the x,y,z d i r e c t i o n s r e s p e c t i v e l y d i s t a n c e from t h e w a l l roughness l e n g t h boundary l a y e r t h i c k n e s s , S t o k es t h i c k n e s s = i = ( — )"*" P D boundary l a y e r momentum t h i c k n e s s wave d i s p l a c e m e n t t h i c k n e s s k i n e m a t i c eddy v i s c o s i t y d i m e n s i o n l e s s d i s t a n c e from the w a l l = ^ wavelength of bed roughness normal s t r e s s on a p l a n e t u r b u l e n t shear s t r e s s WAVE THEORY SYMBOLS e m p i r i c a l c o e f f i c i e n t s f o r c r i t i c a l motion under o s c i l l a t o r y f l o w a m p l i t u d e of h o r i z o n t a l p a r t i c l e motion o u t s i d e the boundary l a y e r i n o s c i l l a t o r y f l o w (see p. 55) wave speed wave c e l e r i t y r e l a t i v e t o the f i x e d c o o r d i n a t e frame wave c e l e r i t y f o r t h e g e n e r a t e d wave t h a t moves onto a c u r r e n t J a c o b i a n e l l i p t i c f u n c t i o n wave c e l e r i t y r e l a t i v e t o the c o o r d i n a t e frame moving w i t h the c u r r e n t u n d i s t u r b e d depth of f l u i d wave number = — x v i i wavelength of t h e g e n e r a t e d wave t h a t moves onto a c u r r e n t B e r n o u l l i c o n s t a n t h e i g h t above t h e bed = y+d p e r i o d of wave on c u r r e n t as seen from moving c o o r d i n a t e frame p e r i o d of wave on c u r r e n t as seen from s t a t i o n a r y c o o r d i n a t e frame l i n e a r wave t h e o r y water p a r t i c l e v e l o c i t y a t bed (see p. 55) maximum o r b i t a l v e l o c i t y a t the bed as p r e d i c t e d by f i r s t o r d e r t h e o r y (see p. 55) maximum o r b i t a l water p a r t i c l e v e l o c i t y a t bed (see p. 55) mass t r a n s p o r t v e l o c i t y mass t r a n s p o r t v e l o c i t y components x c o o r d i n a t e f o r r e f e r e n c e frame moving w i t h c u r r e n t v e l o c i t y stream f u n c t i o n c o e f f i c i e n t h e i g h t above s t i l l water l e v e l p e r t u r b a t i o n parameter v e r t i c a l d i s t a n c e above s t i l l water l e v e l t o f l u i d s u r f a c e v e l o c i t y p o t e n t i a l , phase a n g l e x v i i i SEDIMENT FORCE SYMBOLS drag c o e f f i c i e n t form drag c o e f f i c i e n t l i f t c o e f f i c i e n t hydrodynamic mass c o e f f i c i e n t s u r f a c e d rag c o e f f i c i e n t f i c t i t i o u s a c c e l e r a t i o n f o r c e d r a g f o r c e form d r a g f o r c e s u r f a c e d rag f o r c e g r a v i t y f o r c e hydrodynamic f o r c e l i f t f o r c e p r e s s u r e f o r c e volume f o r c e m u l t i p l e of the sum of the s t a n d a r d d e v i a t i o n s s e p a r a t i n g two mean v a l u e s uneven p a r t i c l e p o s i t i o n i n g c o e f f i c i e n t s p e c i f i c g r a v i t y of p a r t i c l e s parameter r e p l a c i n g R* p a r t i c l e d e n s i t y x i x mean c r i t i c a l shear s t r e s s n e c e s s a r y a t the bed t o move p a r t i c l e s mean e x e r t e d shear s t r e s s a t the bed SHEAR PLATE SYMBOLS damping c o e f f i c i e n t r e s u l t i n g from water r e s i s t a n c e n a t u r a l f r e q u e n c y of shear p l a t e system i n a i r f o r c i n g f u n c t i o n e x e r t e d on the shear p l a t e system shear p l a t e s p r i n g c o n s t a n t shear p l a t e mass d i s p l a c e m e n t of shear p l a t e i n d i r e c t i o n of motion v e l o c i t y of shear p l a t e i n d i r e c t i o n of motion a c c e l e r a t i o n of shear p l a t e i n d i r e c t i o n of motion damping r a t i o of shear p l a t e system xx ACKNOWLEDGEMENTS The a u t h o r i s v e r y g r a t e f u l f o r the guidance and encouragement g i v e n by h i s s u p e r v i s o r , Dr. M. C. Qui c k . The a u t h o r i s a l s o g r a t e f u l f o r the f i n a n c i a l s u p p o r t p r o v i d e d t h r o u g h Dr. Quick by the B r i t i s h Columbia D i s a s t e r R e l i e f Fund and the N a t i o n a l S c i e n c e and E n g i n e e r i n g Research C o u n c i l . The a u t h o r i s a p p r e c i a t i v e of the time spent by Dr. Ron N i n n i s i n i n s t r u c t i n g us i n the se t u p and the use of the equipment f o r the L a s e r D o p p l e r Anemometry s e c t i o n of t h i s t h e s i s . The a u t h o r wishes t o thank Mr. K u r t N i e l s e n f o r h i s t e c h n i c a l e x p e r t i s e and i n v a l u a b l e s u p p o r t i n the l a b o r a t o r y . The a u t h o r a l s o wishes t o thank Dr. M. de S t . Q. I s a a c s o n and Dr. W. K. Oldham f o r t h e i r h e l p f u l a d v i c e . The a u t h o r wishes t o acknowledge the a d d i t i o n a l l a b o r a t o r y d a t a o b t a i n e d by Mr. Shenglong L e i , v i s i t i n g C h i n e s e s c h o l a r , and Mr. Herman Kwan, r e s e a r c h a s s i s t a n t . The a u t h o r i s t h a n k f u l f o r t h e encouragement and a d v i c e g i v e n by h i s p a r e n t s , and by h i s f e l l o w g r a d u a t e s t u d e n t s . x x i 1. INTRODUCTION I.I OVERVIEW OF RESEARCH D u r i n g the next few y e a r s i t i s p r o b a b l e t h a t many l a r g e undersea s t r u c t u r e s w i l l be b u i l t on the sand bank r e g i o n s o f f the P a c i f i c and A t l a n t i c c o a s t s of Canada. In t h e s e a r e a s , l a r g e waves combine w i t h s t r o n g t i d a l c u r r e n t s t o produce s i g n i f i c a n t shear s t r e s s e s and sediment motion on the s h a l l o w sea bed. The p r e s e n t work i s p a r t of a c o n t i n u i n g study which i s b e i n g undertaken t o attempt t o e s t i m a t e sediment movement under these combined wave and c u r r e n t c o n d i t i o n s . There have been e x t e n s i v e s t u d i e s of sediment movement and t h r e s h o l d of sediment motion under the a c t i o n of s t e a d y c u r r e n t s a l o n e and of waves a l o n e . For t h e s e s i t u a t i o n s t h e r e i s c o n s i d e r a b l e u n c e r t a i n t y i n p r e d i c t i n g e x a c t t h r e s h o l d c o n d i t i o n s , and even more u n c e r t a i n t y i n p r e d i c t i n g sediment t r a n s p o r t r a t e s . The combined wave and c u r r e n t s i t u a t i o n has r e c e i v e d much l e s s s t u d y and t h e r e i s v e r y l i t t l e e x p e r i m e n t a l d a t a a v a i l a b l e w i t h which t o check t h e o r e t i c a l a s sumptions c o n c e r n i n g the c o m b i n a t i o n of wave and c u r r e n t v e l o c i t y and s t r e s s f i e l d s . T h i s study w i l l i n v e s t i g a t e e x i s t i n g t h e o r i e s f o r s t e a d y f l o w and f o r waves, and w i l l then proceed t o examine the combined wave and c u r r e n t work. E x p e r i m e n t s w i l l be d e s c r i b e d which attempt t o measure the combined bed shear s t r e s s e s i n d uced by waves and c u r r e n t s . L a s e r d o p p l e r 1 2 measurements w i l l be d e s c r i b e d which are used t o i n v e s t i g a t e the combined v e l o c i t y f i e l d s and make comparison w i t h the component c u r r e n t and wave v e l o c i t y f i e l d s . The b a s i c e x p e r i m e n t a l d e s i g n has been attempted i n two d i f f e r e n t ways. In the f i r s t s e r i e s of t e s t s , a shear p l a t e was used t o i n v e s t i g a t e t h e wave-current shear s t r e s s e s , but i t w i l l be shown t h a t t h i s method i s s u b j e c t t o c o n s i d e r a b l e e r r o r s . C o n s e q u e n t l y a second s e r i e s of t e s t s was made which u t i l i z e s the onset of motion of sediment as an i n d i c a t o r of s t r e s s l e v e l . T h i s approach has proved t o be more s u c c e s s f u l and has l e d t o the development of a t h e o r e t i c a l model of combined wave and c u r r e n t b e h a v i o u r . The e x p e r i m e n t s t h a t have been conducted s h o u l d not be thought of as model s t u d i e s , but as p r o t o t y p e sediment i n v e s t i g a t i o n s . Some a t t e n t i o n w i l l be g i v e n t o the q u e s t i o n of m o d e l l i n g , because i t i s q u i t e common t o use m o d e l l i n g t e c h n i q u e s t o p r e d i c t p r o t o t y p e sediment b e h a v i o u r . The p r e s e n t work may w e l l have a p p l i c a t i o n t o m o d e l l i n g s t u d i e s , but t h i s a p p l i c a t i o n i s not t h e p r i m a r y c o n c e r n . /. 2 OBJECTIVES OF RESEARCH There i s u n c e r t a i n t y as t o what a r e the c o r r e c t f o r c e s t o be m o d e l l e d i n c o a s t a l movable-bed models. I t i s g e n e r a l l y a c c e p t e d t h a t t h e shear s t r e s s i s r e s p o n s i b l e f o r p u t t i n g sediment i n t o m o t i o n . R e c e n t l y , t h e r e has been much debate as t o whether t h e c r i t i c a l shear s t r e s s as used on 3 the S h i e l d s diagram can be used t o p r e d i c t c r i t i c a l sediment motion under waves. T h i s c r i t e r i o n i s t o h o l d f o r both p r o t o t y p e and model c o n d i t i o n s . I f the onset of motion i s t o be p r o p e r l y p r e d i c t e d , then a d e f i n i t i v e c r i t e r i o n f o r t h a t onset s h o u l d f i r s t be e s t a b l i s h e d . I t i s one o b j e c t i v e of t h i s r e s e a r c h t o d e f i n e the shear s t r e s s c r i t e r i o n and c o r r e s p o n d i n g v e l o c i t y f i e l d f o r the c r i t i c a l o nset of sediment motion f o r wave and f o r u n i d i r e c t i o n a l c u r r e n t c o n d i t i o n s , and f o r combined wave and c u r r e n t c o n d i t i o n s . The a d d i t i v e n a t u r e of waves and c u r r e n t s must a l s o be i n v e s t i g a t e d . I t has been w i d e l y assumed t h a t the v e l o c i t i e s add v e c t o r i a l l y , such t h a t the v e l o c i t y of the combined c o n d i t i o n c o u l d be p r e d i c t e d from knowing the v e l o c i t y f o r each component. Here, i t i s the second o b j e c t i v e t o d e t e r m i n e how combined wave and c u r r e n t s can be d e s c r i b e d by c o m b i n a t i o n s of wave and c u r r e n t components. T h i s o b j e c t i v e e n t a i l s d e t a i l e d i n v e s t i g a t i o n of the v e l o c i t y component a d d i t i o n , and a l s o i n v e s t i g a t i o n of any s p e c u l a t i v e shear s t r e s s component a d d i t i o n . 1.3 PROPOSED EXPERIMENTS There a r e t h r e e e x p e r i m e n t a l methods a v a i l a b l e f o r d e t e r m i n i n g shear s t r e s s e s under waves and c u r r e n t s . The f i r s t i n v o l v e s the measurement of wave h e i g h t v a r i a t i o n a l o n g the l e n g t h of a flume, where the v a r i o u s f l o w r e s i s t a n c e s a r e p r e d i c t e d from a number of t h e o r i e s 4 based on wave h e i g h t d e c l i n a t i o n . The shear s t r e s s i s j u s t one of the r e s i s t i n g s t r e s s e s , which i n c l u d e such s t r e s s e s as w a l l s t r e s s e s , i n t e r n a l wave s t r e s s e s , and wave-current i n t e r a c t i o n s t r e s s e s . T h i s method i s s u b j e c t t o tremendous e r r o r s because of the unknown v a l u e s of the parameters j u s t mentioned, and of o t h e r v a r i a b l e s such as the degree of wave r e f l e c t i o n and of wave s e t u p or setdown. The second method i n v o l v e s the d i r e c t measurement of the shear f o r c e s t h e m s e l v e s , u s u a l l y by means of a shear p l a t e s e t i n t o a r e c e s s e d p o r t i o n of a flume f l o o r . T h i s method i n v o l v e s the measurements of a t o t a l f o r c e and of a l a r g e p r e s s u r e f o r c e , where the d i f f e r e n c e between these f o r c e s g i v e s the shear f o r c e . The f i n a l , most p o p u l a r , method i n v o l v e s the measurement of the f l o w v e l o c i t y p r o f i l e near the bed, and the use of an a p p r o p r i a t e v e l o c i t y ^ p r o f i l e f o r m u l a t o c a l c u l a t e the shear s t r e s s . T h i s method d i r e c t l y a l l o w s i n v e s t i g a t i o n of v e l o c i t y component a d d i t i o n . The l a t t e r two methods have been a t t e m p t e d i n o r d e r t o a c h i e v e t h e o b j e c t i v e s d e f i n e d i n the p r e v i o u s s e c t i o n . 1.3.1 DIRECT SHEAR FORCE MEASUREMENT A shear p l a t e was used t o attempt t o measure the shear f o r c e s under waves, under c u r r e n t s , and under combined waves and c u r r e n t s . The wave p e r i o d s were m o d i f i e d such t h a t the p l a t e e x p e r i e n c e d the same wave l e n g t h and p e r i o d f o r a l l wave and combined wave and c u r r e n t c o n d i t i o n s . The wave 5 h e i g h t was g i v e n a range of v a l u e s and the shear p l a t e o u t p u t was measured f o r each h e i g h t . The p r e s s u r e f o r c e s under the waves were e s t i m a t e d , and the shear f o r c e was c a l c u l a t e d . S t r e s s e s were c a l c u l a t e d f o r combined waves and c u r r e n t s f o r p a r t i c u l a r wave h e i g h t s , and f o r component waves and c u r r e n t s h a v i n g the same wave h e i g h t s and c u r r e n t v e l o c i t i e s , r e s p e c t i v e l y . In t h i s manner, a t t e m p t s were made t o d etermine how the component shear f o r c e s combine, t o meet p a r t of the second o b j e c t i v e . T h i s method d i d not work i n t h i s i n s t a n c e because the p r e s s u r e f o r c e s were not measured, nor c o u l d they be measured f o r the equipment used. The s t r a i g h t - f o r w a r d e s t i m a t i o n of the p r e s s u r e f o r c e s cannot be done f o r a shear p l a t e , as w i l l be d i s c u s s e d i n a l a t e r c h a p t e r . 1.3.2 DIRECT VELOCITY PROFILE MEASUREMENT A La s e r D o p p l e r Anemometer was used t o measure near-bed water v e l o c i t y p r o f i l e s over a number of sediment beds. Shear s t r e s s e s c o u l d be c a l c u l a t e d from t h e s e p r o f i l e s , . and compared w i t h t h e o r e t i c a l and e m p i r i c a l r e s u l t s . The f l o w c o n d i t i o n s chosen were waves, c u r r e n t s , and combined waves and c u r r e n t s . The c r i t e r i o n used was the t h r e s h o l d of sediment m o t i o n , i n order t o meet the f i r s t o b j e c t i v e . A l s o , waves h a v i n g i d e n t i c a l h e i g h t , and c u r r e n t s h a v i n g i d e n t i c a l v e l o c i t y as the combined wave and c u r r e n t c o n d i t i o n s had t h e i r p r o f i l e s measured t o i n v e s t i g a t e t h e i r a d d i t i v e n a t u r e . In t h i s way, the second o b j e c t i v e c o u l d be met, 6 namely the v e l o c i t y component a d d i t i o n s and the shear s t r e s s component a d d i t i o n s c o u l d be i n v e s t i g a t e d . 7. 4 REVIEW OF SCALE AND LABORATORY EFFECTS 1.4.1 INTRODUCTION The f i e l d s of open c h a n n e l f l o w and c o a s t a l e n g i n e e r i n g o f t e n p r e s e n t problems which a r e i n s o l u b l e by t h e o r y or by r e f e r e n c e t o s t a n d a r d e m p i r i c a l d a t a , and so r e q u i r e h y d r a u l i c models. There are many d i f f i c u l t i e s e n c o u n t e r e d when p e r f o r m i n g m o d e l l i n g s t u d i e s . Examples of some of t h e s e d i f f i c u l t i e s a r e g i v e n by Henderson (1966), Iwagaki and Noda (1962), C o l l i n s and Chesnutt (1975), De V r i e s and Van der Zwaard (1975), F o s t e r (1975), Griine and F i i h r b o t e r (1975), Kamphuis (1975), Mahmood (1975), O n i s h i e t . a l . (1975), Y a l i n and P r i c e (1975), and Sharp (1981). Two reasons f o r thes e d i f f i c u l t i e s a r e t h a t models a r e s u b j e c t t o scale e f f e c t s and laboratory e f f e c t s . S c a l e e f f e c t s a r e the u n d e s i r e d d i f f e r e n c e s between model and p r o t o t y p e c o n d i t i o n s caused by the p r a c t i c a l i m p o s s i b i l i t y of s i m u l t a n e o u s l y m o d e l l i n g i n the l a b o r a t o r y a l l t h e f o r c e s a c t i n g on the p r o t o t y p e . L a b o r a t o r y e f f e c t s a r e the u n d e s i r e d d i f f e r e n c e s between model and p r o t o t y p e c o n d i t i o n s caused by the p h y s i c a l c o n s t r a i n t s which e x i s t i n the l a b o r a t o r y but not i n the p r o t o t y p e . 7 1.4.2 SCALE EFFECTS I t i s a p r a c t i c a l i m p o s s i b i l i t y t o keep the Froude number, F , and the R e y n o l d s number, R e, the same i n the model and p r o t o t y p e , so t h e r e a r e s c a l e e f f e c t s when both ar e r e q u i r e d t o be f u l f i l l e d . The Froude number may be made the same, but then t h e model Reynolds number w i l l be much s m a l l e r than i n the p r o t o t y p e . As a r e s u l t , the drag on the l a b o r a t o r y s t r u c t u r e s w i l l be r e l a t i v e l y l a r g e r than they ar e i n the p r o t o t y p e , u n l e s s the model roughness i s g r e a t l y reduced beyond g e o m e t r i c s c a l i n g . A f u r t h e r problem i s t h a t i n a model of v e r y s m a l l s i z e , s u r f a c e t e n s i o n e f f e c t s may overpower the v i s c o s i t y or g r a v i t i o n a l e f f e c t s so p r e v a l e n t i n the p r o t o t y p e . Chesnutt (1975) d e s c r i b e d t h a t s c a l e e f f e c t s make i t n e c e s s a r y t o e s t a b l i s h m o r p h o l o g i c a l time s c a l e s e m p i r i c a l l y . T h i s can o n l y be done i f the r e p e a t a b i l i t y of the e x p e r i m e n t s f o r the r a t e of bottom e v o l u t i o n i s good. In h i s e x p e r i m e n t s , C h e s n u t t d e t e r m i n e d t h a t l a b o r a t o r y e f f e c t s i n movable-bed models make r e p e a t a b i l i t y v e r y low. A l a b o r a t o r y s t u d y of the s c a l e e f f e c t s i n beach p r o c e s s e s was conducted by Iwagaki and Noda (1962). The r a t i o of the wave h e i g h t t o the sediment d i a m e t e r was found t o be a v e r y s i g n i f i c a n t f a c t o r i n the problem of s c a l i n g and m o d e l l i n g . T h i s r a t i o was found t o have a c r i t i c a l v a l u e which d i v i d e d the model sediment movement i n t o o f f s h o r e d i r e c t e d and onshore d i r e c t e d r e gimes. 8 Much d i f f i c u l t y i s e n c o u n t e r e d i n s e l e c t i n g s u i t a b l e s c a l e s f o r movable-bed models i n the c o a s t a l e nvironment. D i s t o r t i o n i s f r e q u e n t l y n e c e s s a r y , and t h e r e i s a tendency f o r some model m a t e r i a l s t o move i n o p p o s i t e d i r e c t i o n s t o thos e found i n the f i e l d . 1.4.3 LABORATORY EFFECTS In the l a b o r a t o r y , waves a r e g e n e r a t e d by use of a wave p a d d l e , and propagate a c e r t a i n d i s t a n c e a l o n g a w a l l e d flume of some depth and w i d t h . The waves u s u a l l y pass t h r o u g h some type of f i l t e r and a r e r e f l e c t e d from a s l o p i n g beach. C u r r e n t s a r e g e n e r a t e d by r e c i r c u l a t i n g water from one end of the flume and i n t r o d u c i n g i t a t some p o s i t i o n a t the o t h e r end. From flume t o flume the t u r b u l e n c e i s then n e c e s s a r i l y as d i f f e r e n t as the i n l e t boundary c o n d i t i o n s a r e d i f f e r e n t . 1.4.3.1 Wave G e n e r a t i o n Waves a r e g e n e r a t e d by a s i n u s o i d a l l y moving wave pad d l e a t the g e n e r a t i n g end of the flume. U s u a l l y , the pad d l e moves i n a p i s t o n type of motion f o r the c r e a t i o n of s h a l l o w water waves, and i n a f l a p type of motion f o r deeper water waves. The p e r i o d of the wave i s the same as the p e r i o d of the p a d d l e , and the wave h e i g h t depends on the pad d l e s t r o k e . There a r e t h e o r e t i c a l r e l a t i o n s g i v e n by B i e s e l (1951) which r e l a t e the wave h e i g h t , H, t o the pa d d l e s t r o k e a m p l i t u d e , R 0. Madsen (1970) has found t h a t the a c t u a l 9 measured wave h e i g h t i s s m a l l e r than t h a t p r e d i c t e d by the t h e o r y . T h i s he a t t r i b u t e d t o l e a k a g e around the p a d d l e , as the p a d d l e cannot be made t o f i t the e x a c t d i m e n s i o n s of the flume. Ippen e t . a l . (1964) found t h a t t r i a l a d j u s t m e n t s f o r pad d l e speed and s t r o k e were n e c e s s a r y t o s e t a s p e c i f i c wave i n the flume. Sarpkaya and I s a a c s o n (1981) noted t h a t the t h e o r e t i c a l r e l a t i o n s as g i v e n by B i e s e l a r e v a l i d f o r s m a l l m o t i o n s , but t h a t e x p e r i m e n t a l measurements are' g e n e r a l l y r e q u i r e d t o g i v e the wave h e i g h t . G i l b e r t e t . a l . (1971) have g i v e n d e t a i l s on wave g e n e r a t o r d e s i g n . P r e t i o u s (1967) has g i v e n the r e q u i r e m e n t s f o r a wave g e n e r a t o r which i s t o o p e r a t e i n a s h o r t flume. 1.4.3.2 Wave A t t e n u a t i o n High f r e q u e n c y waves do not c a r r y w e l l down the l e n g t h of the flume, a c c o r d i n g t o Webber and C h r i s t i a n (1974) who conducted an i n - d e p t h a n a l y s i s of d e s i g n c o n s i d e r a t i o n s f o r a programmable wave g e n e r a t o r . T h i s h i g h f r e q u e n c y a t t e n u a t i o n was presumed t o o c c u r because of t u r b u l e n t d i s s i p a t i o n . They a l s o found t h a t B i e s e l ' s r e l a t i o n s work w e l l f o r wave p e r i o d s of 1.0 t o 2.0 seconds. I w a s a k i and Sato (1972) put waves on an opposing c u r r e n t of v e l o c i t y , V. The o b s e r v e d r a t e of wave h e i g h t a t t e n u a t i o n was comparable w i t h t h e o r e t i c a l v a l u e s . Some wave energy was sup p o s e d l y t a k e n from the c u r r e n t by r a d i a t i o n s t r e s s , and some was d i s s i p a t e d by i n t e r n a l and boundary shear s t r e s s e s . The e v a l u a t i o n of the energy d i s s i p a t i o n was t o o low, and the a u t h o r s suggested t h a t 10 t u r b u l e n c e i n the wave motion must p l a y a r o l e . They found t h a t the i n t e r n a l f r i c t i o n shear s t r e s s becomes remarkable as the o p p o s i n g c u r r e n t v e l o c i t y i s i n c r e a s e d . I t i s n e c e s s a r y t o measure t h e wave h e i g h t a t the p o i n t of e x p e r i m e n t because of wave energy d i s s i p a t i o n by c u r r e n t , and by i n t e r n a l and boundary shear s t r e s s e s . T h i s i s p a r t i c u l a r l y i m p o r t a n t w i t h h i g h f r e q u e n c y waves, which a re e s p e c i a l l y v u l n e r a b l e . 1.4.3.3 Wave D i s t o r t i o n A f u r t h e r problem i s e n c o u n t e r e d i n l a b o r a t o r y s i t u a t i o n s . G a l v i n (1968) o b s e r v e d t h a t l o n g waves ge n e r a t e d at one end of a h o r i z o n t a l c h a n n e l by a p i s t o n type s i n u s o i d a l wave g e n e r a t o r d e v e l o p i n t o forms t h a t a r e not p e r i o d i c i n space. Madsen e t . a l . (1970) noted t h a t the l e a d i n g f a c e of the wave becomes s t e e p e r and the t r a i l i n g f a c e becomes f l a t t e r as the wave t r a v e l s down the flume. A l s o , i t i s seen t h a t secondary c r e s t s a r e a dominant f e a t u r e . C e r t a i n waves g e n e r a t e d by a s i n u s o i d a l p i s t o n wavemaker a r e be s t d e s c r i b e d by a second o r d e r S t o k e s wave superimposed by a f r e e second harmonic wave, a c c o r d i n g t o Hansen and Svendsen ( 1 9 7 4 ) . The h e i g h t and phase v e l o c i t y of t h i s f r e e second harmonic wave a r e p r e d i c t e d by Fon t a n e t (1961). The wave p a t t e r n i s o b s e r v e d t o be s t a t i o n a r y i n the fl u m e , and i s caused by the f a i l u r e of the r i g i d p a d d l e t o e x a c t l y reproduce the p a r t i c l e motion v a r i a t i o n seen i n r e a l p r o g r e s s i v e waves. 11 Hansen and Svendsen found t h a t a n o n s i n u s o i d a l time v a r i a t i o n of the p i s t o n s u c c e s s f u l l y r e d u c e d the h e i g h t of the second harmonic wave. H u l s b e r g e n (1974) found t h a t a bed s i l l may e f f e c t i v e l y s u p p r e s s the s e c o n d a r y wave i f i t i s p r o p e r l y p o s i t i o n e d . F l i c k and Guza (1980) a l s o reduced t h e secondary waves by u s i n g s p e c i a l wave g e n e r a t o r motions or c e r t a i n s h o a l i n g c o n d i t i o n s . Ippen e t . a l . (1964) p l a c e d a f i l t e r made of expanded aluminum s h e e t s hung from a wooden frame some 2.5 m from the p i s t o n i n an attempt t o reduce t h e harmonic waves. F u r t h e r , a beach of 1:16 s l o p e h a v i n g w i r e mesh b a s k e t s f i l l e d w i t h aluminum wool was p l a c e d a t t h e r e c e i v i n g end t o reduce r e f l e c t i o n . The secondary harmonic waves may be formed by the wavemaker m o t i o n , by s h o a l i n g c o n d i t i o n s , wave b r e a k i n g c o n d i t i o n s and by r e f l e c t i o n from the downstream end. They r e s u l t i n f i x e d p r o f i l e s i n the flume which v a r y from the s i n u s o i d a l and can c o n f u s e the f l u i d p a r t i c l e m o t i o n s. They can be removed by s p e c i a l n o n s i n u s o i d a l p a d d l e motions, or by c o r r e c t p o s i t i o n i n g of an underwater s t r u c t u r e . S u p p r e s s i o n of t h e s e waves i s more of an a r t than a s c i e n c e , as no complete t h e o r y e x i s t s which can e x p l a i n t h e i r o r i g i n and n a t u r e . The wave p r o f i l e i s r e p o r t e d l y b e s t d e s c r i b e d as a second o r d e r Stokes wave superimposed by a f r e e second harmonic wave, a t l e a s t f o r c e r t a i n d e p t h t o wavelength r a t i o s . P r a c t i c e u s u a l l y d i c t a t e s use of t h e second o r d e r 1 2 S t o k e s t h e o r y t o d e s c r i b e the wave p r o f i l e i n the flume. In o r d e r t o d e s c r i b e the wave p r o f i l e , i t i s a d v i s a b l e t o use a wave r e c o r d e r t o make time t r a c e s of the wave p r o f i l e a t the t e s t s e c t i o n . 1.4.3.4 A d d i t i o n a l D i f f i c u l t i e s Most flumes a r e v e r y i n a d e q u a t e when d e a l i n g w i t h s t e a d y c u r r e n t s t r a v e l l i n g i n e i t h e r d i r e c t i o n . U s u a l l y the c o - c u r r e n t has f l o w i n t r o d u c e d downstream of the wave paddle w h i l e the adver s e c u r r e n t i n v o l v e s f l o w i n t r o d u c e d behind the wave a b s o r b i n g beach. The f l o w p r o f i l e s a r e then n e c e s s a r i l y d i f f e r e n t a t the t e s t s e c t i o n , u n l e s s a v e r y l o n g c h a n n e l i s used, and t h i s i s r a r e l y the c a s e . Water v i s c o s i t y i s i m p o r t a n t t o c o n s i d e r when c o n d u c t i n g movable-bed model s t u d i e s . Even s l i g h t water temperature v a r i a t i o n has a c o n s i d e r a b l e e f f e c t on the v i s c o s i t y and so on the p a r t i c l e d r a g . 1.4.3.5 C o n c l u s i o n s I t i s i m p o r t a n t when comparing r e s e a r c h e r s ' r e s u l t s t o c a r e f u l l y c o n s i d e r the flume d i m e n s i o n s , the pa d d l e t y p e , the n a t u r e of the wave shape i n c l u d i n g secondary waves, the bed forms i n c l u d i n g the presence of b a r s , s i l l s , and s l o p e s , and the i n l e t boundary c o n d i t i o n s , and the water t e m p e r a t u r e . 13 7.5 DESCRIPTION OF WAVE FLUME The wave flume, shown i n F i g u r e 1.1, i s 28 m l o n g , 0.60 m wide, and 0.75 m deep. I t has a h o r i z o n t a l l a m i n a t e d hardwood t y p e bottom and g l a s s w a l l s . One end of the c h a n n e l c o n t a i n s a p l a s t i c h a i r permeable mat wave a b s o r b e r of 1:8.5 s l o p e t h a t s e r v e s t o h e l p d i s s i p a t e wave energy. Beh i n d t h i s wave a b s o r b e r i s a 3 m square b a s i n s e r v i n g as a sump f o r the r e t u r n f l o w pumps which a l s o a i d s i n wave energy d i s s i p a t i o n . Waves a r e c r e a t e d a t the o t h e r end by a g e n e r a l purpose wave g e n e r a t o r . As d e s c r i b e d by P r e t i o u s (1967), the pa d d l e can move h o r i z o n t a l l y f o r s h a l l o w water wave g e n e r a t i o n and can r o t a t e f o r g e n e r a t i o n of deep water waves. The prime mover i s a 5 HP, 115 v o l t , 6 0 . c y c l e , s i n g l e phase U.S. E l e c t r i c a l Motors e l e c t r i c motor w i t h a v a r i - d r i v e attachment. The wave g e n e r a t o r i s c a p a b l e of P L A A J V I E W 3.8 m 4.1 + 4.2 -v3.Sm g e n e r b f o r F i g u r e 1.1. D e f i n i t i o n s k e t c h of wave flume 1 4 making waves of up t o b r e a k i n g h e i g h t w i t h p e r i o d s of between 1.0 and 3.0 seconds. The waves are passed t h r o u g h an aluminum r a c k wave f i l t e r i n o r d e r t o damp the h i g h e r f r e q u e n c y components. However, t h e r e are s t a n d i n g waves i n the flume caused by r e f l e c t i o n from the wave a b s o r b e r and r e - r e f l e c t i o n from the wave p a d d l e . The r e f l e c t i o n c o e f f i c i e n t has been d e f i n e d by Sarpkaya and I s a a c s o n (1981) as H -H . K max min r"H +H . max mm where H , H . a r e the maximum and minimum wave h e i g h t s max min 3 measured th r o u g h o u t the l e n g t h of the flume under c o n t i n o u s p a d d l e m o t i o n . I t was d e t e r m i n e d t o be l e s s than 0.08 f o r the l a r g e s t waves. The s t a n d i n g waves were found t o s h i f t i n the flume f o r waves of d i f f e r e n t h e i g h t , because of the d i f f e r e n t e x c u r s i o n of the wave p a d d l e . For waves g e n e r a t e d on a c u r r e n t i t was q u a l i t a t i v e l y o b s e r v e d t h a t the s t a n d i n g waves were not e a s i l y r e c o g n i z e d . T h i s f i n d i n g agrees w i t h t h a t of Quick (1982) who used the same wave flume f o r h i s sand r i p p l e e x p e r i m e n t s . I t i s s u r m i s e d t h a t the h i g h e r components are c a r r i e d down the flume by the added c u r r e n t . There a r e two r e c i r c u l a t i n g pumps which s e r v e t o move water i n the flume from the g e n e r a t i n g end t o the a b s o r b i n g end. There i s a s l i g h t change i n water depth caused by the pump head and t h i s cannot be i g n o r e d when a t t e m p t i n g t o g e n e r a t e waves of a s p e c i f i c h e i g h t . The f i r s t pump was used 15 t o g e n e r a t e a c u r r e n t of 0.20 m/s and was used m a i n l y because i t c r e a t e d a f l o w t h a t had a n e a r l y r e g u l a r water s u r f a c e . The second pump was used s p a r i n g l y t o add a c u r r e n t of 0.30 m/s as i t had the drawback of c r e a t i n g an e r r a t i c water s u r f a c e . T h i s o c c u r s because the f l o w i s i n t r o d u c e d a t the aluminum wave damper th r o u g h f i n e t u b i n g and much of the f l o w i s r e t u r n e d above the s t i l l water l e v e l , r e s u l t i n g i n a cascade of water s t r i k i n g the water s u r f a c e which c r e a t e s many w a v e l e t s . . The t e s t s e c t i o n was 4.9 m from the c o n v e r g i n g s e c t i o n of the flume. 2. BOUNDARY LAYER THEORY 2.I CONCEPT OF THE BOUNDARY LAYER I d e a l f l u i d t h e o r y uses a f r i c t i o n l e s s and i n c o m p r e s s i b l e f l u i d , such t h a t a f l o w i n g f l u i d i s a l l o w e d t o s l i p over a w a l l w i t h o u t e x e r t i n g any f o r c e on i t . A r e a l f l u i d a c t u a l l y adheres t o the w a l l , meaning t h a t t h e r e must be f r i c t i o n f o r c e s r e t a r d i n g the f l u i d f l o w i n a t h i n near w a l l l a y e r . T h i s t h i n l a y e r i s termed the boundary layer. In t he boundary l a y e r , the f l u i d v e l o c i t y i n c r e a s e s from z e r o a t the w a l l t o the f r e e stream v a l u e a t the o u t e r edge. The v e l o c i t y d i s t r i b u t i o n i n the boundary l a y e r above a p l a t e p l a c e d p a r a l l e l t o the f l o w i s r e p r e s e n t e d by F i g u r e 2.1. The v e l o c i t y i s u n i f o r m ahead of the p l a t e ' s l e a d i n g edge, but the v e l o c i t y p r o f i l e i s changed upon r e a c h i n g the p l a t e . The v e l o c i t y a t the w a l l i n s t a n t l y becomes z e r o , and the f l u i d near the w a l l i s i n c r e a s i n g l y r e t a r d e d i n the downstream d i r e c t i o n because of f r i c t i o n f o r c e s i n t h e f l u i d . As a r e s u l t , the t h i c k n e s s 6 of the F i g u r e 2.1. ( A f t e r S c h l i c h t i n g , 1979). S k e t c h of boundary l a y e r on a f l a t p l a t e i n p a r a l l e l f l o w a t z e r o i n c i d e n c e . 16 17 r e t a r d e d l a y e r c o n t i n u o u s l y i n c r e a s e s w i t h i n c r e a s i n g d i s t a n c e x i n the downstream d i r e c t i o n from t h e l e a d i n g edge of the p l a t e . There i s a l a r g e v e l o c i t y g r a d i e n t a c r o s s the boundary l a y e r and t h e r e a r e a c c o r d i n g l y l a r g e f r i c t i o n a l shear s t r e s s e s , g i v e n by Newton's law of f r i c t i o n ' - 4 ? < 2 - i ) where r i s the shear s t r e s s u i s the s t a t i c v i s c o s i t y i s the v e l o c i t y g r a d i e n t i n the boundary l a y e r . A l o n g the w a l l , the shear s t r e s s T 0 i s g i v e n as V 0 = iu(-|y-)o • (2.2) where the s u b s c r i p t " 0 " denotes the v a l u e a t the w a l l . O u t s i d e the boundary l a y e r , the g r a d i e n t i s v e r y s m a l l , and t h e f r i c t i o n a l s t r e s s e s are a l s o s m a l l . T h i s s u g g e s t s t h a t the f l o w f i e l d can be d i v i d e d i n t o two r e g i o n s f o r low v i s c o s i t y f l u i d s : a t h i n boundary l a y e r r e g i o n near the w a l l , and an o u t e r r e g i o n where i d e a l f l u i d t h e o r y g i v e s a good d e s c r i p t i o n of the f l o w . 2. 2 LAMINAR BOUNDARY LAYER 2.2.1 INTRODUCTION Few f l o w s over sediment beds under p r o t o t y p e c o n d i t i o n s a r e l a m i n a r , y e t t h e most w e l l known t h e o r i e s a r e f o r 18 l a m i n a r f l o w . T h i s type of f l u i d motion i s w e l l behaved, and the v e l o c i t i e s can be e a s i l y d e s c r i b e d throughout the f l o w . B e f o r e we can a p p r e c i a t e the d i f f i c u l t i e s of e s t a b l i s h i n g a t h e o r y f o r t u r b u l e n t f l o w , i t i s w o r t h w h i l e t o review the w e l l known e q u a t i o n s f o r f l o w t h a t i s l a m i n a r . 2.2.2 NAVIER-STOKES EQUATIONS FOR LAMINAR FLOW The N a v i e r - S t o k e s e q u a t i o n s f o r an i n c o m p r e s s i b l e , v i s c o u s f l u i d a r e w e l l known, and a r e g i v e n by / 9u. 9u, 3u. 9u\ v 9p. , 9 2u, 9 2u. 3 2u> (^ r, < \ p ( 9 t + U 9 7 + V 9 7 + W 9 1 = "9x ] " { 2 ' 3 - ] a ) /9u, 9u, 9u, 9u x „ 9p, , 9 2 v , 9 2 v , 9 2 v x \ P ( 9 t + U 9x"+ v~dy ~dz ~ 9y 9 x ^ + 9 y ^ + 9 i ^ > -.(2.3.1b) ,9w^ 9w^ 3w^ 9w\ „ 9p^ /3 2w.3 2w.3 2w x , _ , N  p ( 9 t + u 9 3 T + v3y"+w9¥> = Z " + " ( 9 x ^ + 9 F 7 9^> ' ' (2 ' 3 ' 1 c > 9 u + 9 u + 9 w = 0 (2 3 2) where p i s the f l u i d d e n s i t y /i i s the c o e f f i c i e n t of s t a t i c v i s c o s i t y u,v, and w a r e the components of v e l o c i t y i n the x, y, and z d i r e c t i o n s r e s p e c t i v e l y p i s the p r e s s u r e a t a p o i n t i n the f l o w X, Y, and Z, a r e the f o r c e s per u n i t volume i n the x, y, and z d i r e c t i o n s r e s p e c t i v e l y . S c h l i c h t i n g (1979) r e p o r t e d t h a t the v a l i d i t y of the N a v i e r - S t o k e s e q u a t i o n s can h a r d l y be doubted on the b a s i s of the good agreement between experiment and t h e known 19 s o l u t i o n s . I f t he body f o r c e s a r e due o n l y t o p r e s s u r e , v i s c o s i t y , and i n e r t i a , as i s the case f o r a homogeneous f l u i d of c o n s t a n t d e n s i t y , then the f o r c e s X, Y, and Z i n e q u a t i o n s (2.3.1a,b,c) a r e z e r o . F u r t h e r , i f the f l o w i s taken t o be t w o - d i m e n s i o n a l , then the components i n the z d i r e c t i o n a r e o m i t t e d from the e q u a t i o n s (2.3.1) and (2 . 3 . 2 ) , which become / 9u, 3'U, 9Ux 9p. /9 2u.9 2U\ / o o i ' \ p ( y t + U ^ u 9 7 ) = " 9 7 + M ( 9 x ^ + 9 y ^ ) (2.3.1a ) idv, dv, 9ux 9p, / 3 2 y , 9 2 y x / 0 ,,,\ p ( 9 t + u 9 7 + i ; 9 ? ) = " 9 f + M { 9 3 r r + 9 y T ) (2.3.1b ) — + — = 0 (2 3 2') These e q u a t i o n s can be f u r t h e r s i m p l i f i e d by d r o p p i n g terms t h a t a r e of s m a l l o r d e r s of magnitude. S c h l i c h t i n g (1979) showed how the o r d e r of magnitude of each term i s e s t i m a t e d , such t h a t the e q u a t i o n s (2.3.1') and (2.3.2') now become 9u, 9u. 9u 19p, 9 2u / ~ , ,''\ -ri-+u-r—+ u-r—=- ^ +1>-~—T (2.3.1 ) 9t 9x 9y p9x 9y z |^+|H = 0 ( 2 . 3 . 2 " ) 9x 9y where v=- i s the k i n e m a t i c v i s c o s i t y of the f l u i d . P These s i m p l i f i e d N a v i e r - S t o k e s e q u a t i o n s , a l s o known as the P r a n d t l boundary l a y e r e q u a t i o n s , have f u l l y d e t e r m i n e d p h y s i c a l s o l u t i o n s when the boundary c o n d i t i o n s and the 20 i n i t i a l c o n d i t i o n s a r e known. The no s l i p c o n d i t i o n a t the w a l l must be s a t i s f i e d i f the f l u i d i s taken t o be v i s c o u s , and the v e l o c i t y a t the o u t e r edge of the boundary l a y e r must be the same as the v e l o c i t y i n the f r e e stream r e g i o n of the f l o w . The boundary c o n d i t i o n s t o e q u a t i o n s (2.3.1'') and ( 2 . 3 . 2 " ) a r e y=0 u=0,v=0 y = oo u=U(x,t) where U ( x , t ) i s the v e l o c i t y of the f r e e stream. A f l o w must a l s o be d e s c i b e d over the e n t i r e boundary l a y e r r e g i o n f o r the i n i t i a l i n s t a n t of t=0. For steady f l o w , the e q u a t i o n s a r e f u r t h e r s i m p l i f i e d t o 9u, 9u 1dp. 9 2u , ,*>'\ U-r—+ u-r—=- — r (2.3.1 ) 9x 9y pdx 9y 2 lH+3u =o ' (2 3 2 " ' ) s u b j e c t t o the boundary c o n d i t i o n s y=0 u=0,u=0 y = o= U = U(x) and t o the i n i t i a l c o n d i t i o n t h a t a t an i n i t i a l s e c t i o n x=x 0, a g i v e n p r o f i l e U ( x 0 , y ) i s d e s c i b e d . I t i s noted t h a t the d e r i v a t i v e of the p r e s s u r e i s e x p r e s s e d as and t h i s i s the case as the p r e s s u r e depends o n l y on x i n the steady f l o w c o n d i t i o n . F u r t h e r , the p r e s s u r e d e r i v a t i v e term i s known a t the o u t e r edge of t h e boundary l a y e r f o r a l l f l o w s . From e q u a t i o n (2.3.1'') and by 21 r e a l i z i n g t h a t and a r e s m a l l at the o u t e r edge of the boundary l a y e r , we have 9U + U9U =-1?_E (2 3 3) and f o r s t e a d y f l o w and f o r st e a d y f l o w w i t h a c o n s t a n t f r e e stream v e l o c i t y - 1 dP=n (2 3 3'') For s t e a d y f l o w w i t h a c o n s t a n t f r e e stream v e l o c i t y U o, the f l o w i s p o t e n t i a l so g^ = n# and the e q u a t i o n s become 9u, 9u 9 2 u / 0 - , i v s ^"W'W* ,.(2.3.1 ) •^+-^=0 (2 3 2 i v ) s u b j e c t t o the boundary c o n d i t i o n s y=0 u=0,u=0 y = ex> u = U 2.2.3 SOLUTIONS TO THE NAVIER-STOKES EQUATIONS FOR  LAMINAR FLOW These e q u a t i o n s ( 2 . 3 . 1 1 V ) and ( 2 . 3 . 2 1 V ) can be c o n v e r t e d i n t o a n o n l i n e a r t h i r d o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n by i n t r o d u c i n g a"stream f u n c t i o n \p(x,y) such t h a t i/>=f (TJ) U x U j * 2 " 22 where f(7j) deno t e s the d i m e n s i o n l e s s stream f u n c t i o n , and TJ=^ i s the new d i m e n s i o n l e s s c o o r d i n a t e such t h a t The s o l u t i o n t o the r e s u l t i n g o r d i n a r y d i f f e r e n t i a l e q u a t i o n , known as B l a s i u s ' s e q u a t i o n f f " + 2f " ' = 0 s u b j e c t t o the boundary c o n d i t i o n s 7 7 = 0 f = 0, f ' = 0 77 = 00 f ' = 1 g i v e s i?=5.0 f o r u=.99U o o, such t h a t the l o c a l shear s t r e s s a t a p o s i t i o n a d i s t a n c e x a l o n g the w a l l i s g i v e n by T x = 0.332M U £ d( — P (2.4) The boundary l a y e r t h i c k n e s s at t h a t p o s i t i o n i s g i v e n by 6 - ( g M (2.5) oo i f we d e f i n e t h e boundary l a y e r t h i c k n e s s as the d i s t a n c e t o where u=.99U o o. The v e l o c i t y p r o f i l e a t t h a t p o s i t i o n i s g i v e n as u = f ' ( r j ) U o o , where f ' ( ) has been d e t e r m i n e d n u m e r i c a l l y by Howarth (1 9 3 8 ) . 2.2.4 CONCLUSIONS The v e l o c i t y p r o f i l e s and the s e c t i o n a l shear s t r e s s e s a r e w e l l known f o r l a m i n a r f l o w moving over a f l a t p l a t e . The boundary l a y e r t h i c k n e s s can be d e f i n e d i n many ways, 23 but perhaps the most u s e f u l d e f i n i t i o n i s as g i v e n above. In l a m i n a r f l o w , the t h i c k n e s s i s w e l l d e s c r i b e d . 2. 3 TURBULENT BOUNDARY LAYER 2.3.1 INTRODUCTION Most f l o w s o c c u r r i n g i n p r a c t i c a l a p p l i c a t i o n s are t u r b u l e n t . T u r b u l e n t f l o w i s d i v i d e d i n t o a mean motion and a f l u c t u a t i o n , such t h a t f o r i n c o m p r e s s i b l e , t w o - d i m e n s i o n a l f l o w u=u+u' (2.6) where the o v e r b a r denotes the time average of the measured q u a n t i t y and the dash denotes the f l u c t u a t i o n of the measured q u a n t i t y . The time averages a re det e r m i n e d a t a s i n g l e p o s i t i o n over a s u f f i c i e n t t i m e , and the time average of the dashed terms i s n e c e s s a r i l y z e r o . A time-averaged c o n s i d e r a t i o n i s n e c e s s a r y because the c o m p l e x i t y of the superimposed v e l o c i t y f l u c t u a t i o n s make a complete t h e o r e t i c a l f o r m u l a t i o n i m p o s s i b l e . The t u r b u l e n t m i x i n g i n the f l u i d i s of such magnitude t h a t t h e e f f e c t s on the f l o w a r e the same as they would be i f t he f l u i d v i s c o s i t y was v e r y l a r g e . The v i s c o s i t y i n t u r b u l e n t f l o w i s then a p p a r e n t l y l a r g e r than f o r l a m i n a r f l o w , and t h i s i n c r e a s e d apparent v i s c o s i t y forms the c e n t r a l concept of a l l t h e o r e t i c a l c o n s i d e r a t i o n s ( S c h l i c h t i n g , 1979). 24 C o n s i d e r i n g an ar e a dA p e r p e n d i c u l a r t o the x - a x i s i n a t w o - d i m e n s i o n a l t u r b u l e n t f l o w h a v i n g v e l o c i t y components u and v, the s t r e s s e s a c t i n g on the pl a n e a r e -p(u 2+u' 2) i n the x d i r e c t i o n - p ( U ' T J + U ' v') i n the y d i r e c t i o n where the f i r s t s t r e s s i n each e q u a t i o n i s the normal s t r e s s and the second i s the shear s t r e s s . I t i s seen t h a t the t u r b u l e n t f l u c t u a t i o n s have g i v e n two a d d i t i o n a l s t r e s s e s on t h i s p l a n e ox=-pu'2; T x y ' = _ p u ' v ' which a r e c a l l e d the apparent stresses, or Reynolds stresses. 2.3.2 A PHYSICAL ARGUMENT C o n s i d e r a t i o n of the s i m p l e f l o w diagram i n F i g u r e 2.2 y i e l d s an u n d e r s t a n d i n g of the t u r b u l e n t v e l o c i t y p r o f i l e . x F i g u r e 2.2. ( A f t e r S c h l i c h t i n g , 1979). T r a n s p o r t of momentum due t o t u r b u l e n t v e l o c i t y f l u c t u a t i o n . 25 Supposing t h a t s m a l l p a c k e t s of f l u i d a r e moving about i n the f l u i d and c a u s i n g the v e l o c i t y f l u c t u a t i o n s , i t i s seen t h a t — t h e +u' terms a r e a r r i v i n g a t a p o i n t from a l a y e r lower i n the f l o w where the u terms are s m a l l e r , such t h a t t h e i r u' c o n t r i b u t i o n s t o t h e i r new l e v e l a r e n e g a t i v e — t h e -v' terms are a r r i v i n g a t t h a t p o i n t from a l e v e l h i g h e r i n the f l o w , such t h a t t h e s e p a c k e t s a r e c o n t r i b u t i n g +u' terms. The f a i r l y u n i f o r m v e l o c i t y p r o f i l e a c r o s s the f r e e stream r e g i o n of the t u r b u l e n t f l o w i s e x p l a i n e d by the momentum exchange between a d j a c e n t l a y e r s , as a l a y e r w i t h h i g h e r v e l o c i t y p a c k e t s i n c r e a s e s the v e l o c i t y of an a d j a c e n t s l o w e r l a y e r , w h i l e the s l o w e r i a y e r ' s p a c k e t s slow the v e l o c i t y of the f a s t e r l a y e r . I t i s e x p e c t e d t h a t the u'v' terms a r e n e g a t i v e such t h a t the shear s t r e s s g i v e n by T , = - p u ' v ' i s p o s i t i v e and so of the same s i g n as the l a m i n a r shear s t r e s s . 2.3.3 NAVIER-STOKES EQUATIONS FOR TURBULENT FLOW The same r e s u l t f o r the shear s t r e s s can be o b t a i n e d from a d e r i v a t i o n u s i n g t h e N a v i e r - S t o k e s e q u a t i o n s f o r i n c o m p r e s s i b l e t w o - d i m e n s i o n a l f l o w (2.3.1') and ( 2 . 3 . 2 ' ) . U s i n g the t u r b u l e n t v e l o c i t y as i n t r o d u c e d i n e q u a t i o n (2.6) and assuming a g a i n t h a t the time average of each f l u c t u a t i n g term i s z e r o , then the N a v i e r - S t o k e s e q u a t i o n s , upon a v e r a g i n g , become 26 p{u^+v^ )=-^+v{w^+w r)~p{~^r'^y~ ) (2- 3- 4a) ,—3u — 3 I K 3p. /3 2u J_3 2Ux , 3u u ^3i>' 2 x , „ _ N I # ° « . 3 . 5 . ) W * l f = ° <2-3.5b> The l e f t hand s i d e s of the e q u a t i o n s of motion (2.3.4) a r e the s t e a d y s t a t e N a v i e r - S t o k e s e q u a t i o n s , w h i l e the extreme r i g h t hand s i d e s of t h e s e e q u a t i o n s c o n t a i n terms dependent on t h e t u r b u l e n t f l u c t u a t i o n s i n the f l o w . The t u r b u l e n t s t r e s s e s a r e then e q u a l t o the extreme r i g h t hand s i d e s of the e q u a t i o n s , and a r e the same as o b t a i n e d e a r l i e r a =-pu T ,=-pu v X X y ^ (2.7) a y'=-pTT^ r x y , = -pu v The components of the mean v e l o c i t y of the t u r b u l e n t f l o w s a t i s f y the same e q u a t i o n s as do those of l a m i n a r f l o w , e x c e p t t h a t the t u r b u l e n t s t r e s s e s must be i n c r e a s e d beyond the l a m i n a r s t r e s s e s by the a d d i t i o n a l a p p a r e n t s t r e s s e s of e q u a t i o n ( 2 . 7 ) . G e n e r a l l y , the apparent s t r e s s e s are so much l a r g e r than the v i s c o u s s t r e s s e s i n t u r b u l e n t f l o w t h a t the v i s c o u s s t r e s s e s may u s u a l l y be n e g l e c t e d . F u r t h e r s i m p l i f y i n g t h e s e e q u a t i o n s (2.3.4) and (2.3.5) by d r o p p i n g terms t h a t a r e of s m a l l o r d e r s of magnitude, S c h l i c h t i n g o b t a i n e d 27 (2.3.6) 0 (2.3.7) s u b j e c t t o the boundary c o n d i t i o n s y=0 u=0,u=0,u'=0,u'=0 u=UU,t) The boundary c o n d i t i o n s a r e the same as t h o s e i n o r d i n a r y l a m i n a r f l o w i n t h a t a l l v e l o c i t i e s v a n i s h a t the w a l l . A l l t u r b u l e n t components a l s o v a n i s h a t the w a l l , and near the w a l l are so s m a l l t h a t v i s c o u s s t r e s s e s a r e r e g a r d e d t o be the o n l y s t r e s s e s a c t i n g t h e r e . Even i n v e r y t u r b u l e n t f l o w t h e r e i s a t h i n near w a l l l a y e r which behaves l i k e a l a m i n a r l a y e r , and i s termed the laminar sublayer. T h i s s u b l a y e r i s so s m a l l t h a t i t i s v e r y d i f f i c u l t t o observe under e x p e r i m e n t a l c o n d i t i o n s . Above t h i s i s a t r a n s i t i o n l a y e r where the v i s c o u s and the t u r b u l e n t shear s t r e s s e s a r e of comparable magnitude. F u r t h e r from the w a l l i s the t u r b u l e n t boundary l a y e r where the v i s c o u s s t r e s s e s a r e i n s i g n i f i c a n t i n comparison w i t h the t u r b u l e n t s t r e s s e s . Measurements by R e i c h a r d t (193.9) i n a wind t u n n e l show the f l u c t u a t i o n s of the t u r b u l e n t components of v e l o c i t y e s p e c i a l l y how they d i e out s h a r p l y near the w a l l . I t i s of note how u' i s much l a r g e r than v' near the w a l l as seen i n F i g u r e 2.3. The p r o d u c t u'v' was c a l c u l a t e d and compared w i t h the v a r i a t i o n of t h e square of the shear v e l o c i t y as d e t e r m i n e d from the measured p r e s s u r e d i s t r i b u t i o n i n the 2 8 F i g u r e 2 . 3 . ( A f t e r S c h l i c h t i n g , 1 9 7 9 ) . Measurement of f l u c t u a t i n g t u r b u l e n t components i n a wind t u n n e l , a t maximum v e l o c i t y U=l00cm/s. t u n n e l . The e f f e c t of l a m i n a r f r i c t i o n i s shown as the d i f f e r e n c e between the — and the -u v c u r v e s i n F i g u r e 2 . 4 2 . 3 . 4 THEORETICAL ASSUMPTIONS FOR ESTABLISHING TURBULENT SHEAR STRESS SOLUTION F i r s t i n t r o d u c e d by B o u s s i n e s q ( 1 8 7 7 , 1 8 9 6 ) , the m i x i n g c o e f f i c i e n t , A T , i s used f o r the Reynolds s t r e s s i n the r e l a t i o n — 7 — r , 9u 9u r = - p u „ = A 7 . _ = p e T _ ( 2 . 8 ) 9u where 4^ i s the mean v e l o c i t y p r o f i l e s l o p e . I t i s r e c o g n i z e d t h a t A T i s analag o u s t o the u term f o r l a m i n a r f l o w , so t h i s c o e f f i c i e n t i s c a l l e d the apparent v i s c o s i t y or eddy v i s c o s i t y and i s c a l l e d the kinematic eddy v i s c o s i t y . With t h i s a s s u m p t i o n , the N a v i e r - S t o k e s e q u a t i o n s become 29 -9u -9u 13p,9 // , i3uv (2.3.6') 9x 9y u (2.3.7' ) However, the eddy v i s c o s i t y depends not o n l y on the f l u i d , but a l s o on the mean v e l o c i t y , u. I f the dependence of A T on v e l o c i t y i s unknown, then t h e r e can be no c a l c u l a t i o n s p o s s i b l e , so r e l a t i o n s a r e n e c e s s a r y . P r a n d t l ' s m i x i n g l e n g t h concept (1926) makes a g r e a t s t e p i n f i n d i n g t h e s e r e l a t i o n s . C o n s i d e r i n g the case of p a r a l l e l f l o w i n the d i r e c t i o n of the x - a x i s w i t h u=u(y) ,"u=0 o n l y the shear s t r e s s t , i s nonzero. P r a n d t l v i s u a l i z e d xy the'movement of p a c k e t s of p a r t i c l e s from one l a y e r t o F i g u r e 2.4. ( M o d i f i e d a f t e r S c h l i c h t i n g , 1979). Measurement of f l u c t u a t i n g t u r b u l e n t components i n a c h a n n e l . 30 a n o t h e r over a d i s t a n c e 1. The mean v e l o c i t y i n the x - d i r e c t i o n d i f f e r s from l a y e r t o l a y e r , such t h a t a packet of p a r t i c l e s moving from one l a y e r t o an o t h e r has a d i f f e r e n t mean v e l o c i t y . Then the t u r b u l e n t v e l o c i t y components a r e assumed t o be caused by the movement of t h e s e p a c k e t s of p a r t i c l e s of d i f f e r e n t mean v e l o c i t y . C o n s i d e r i n g t h a t the f l u i d packet a t l a y e r ( y ^ l ) has a v e l o c i t y u ( y , - l ) t h a t can be expanded by T a y l o r s e r i e s t o U ( y ) - 1 ( | H ) + . . . and s i m i l a r l y t h a t the f l u i d p a c k e t a t l a y e r (yi+1) has v e l o c i t y u ( y j + l ) which i s expanded t o U ( y ) + 1 ( | H ) + . . . then the d i f f e r e n c e s i n v e l o c i t i e s between t h e s e p a c k e t s moving i n t o l a y e r y and the mean v e l o c i t y t h e r e can be e x p r e s s e d as f o l l o w s A u 1 = U ( y ) - U ( y - l ) - l ( d ^ ) A u = U ( y 2 + l ) - u ( y ) = - l ( d ^ ) The time average of the a b s o l u t e v e l o c i t y of the f l u c t u a t i o n u' i s then TTTT4<|Au1| + | A u 2 | ) = l | g | 31 The t r a n s v e r s e v e l o c i t i e s , v', may be s i m p l y r e g a r d e d as h a v i n g the same o r d e r of magnitude as the u' terms, such t h a t • 7iT^7=const -TuTJ where the p o s i t i v e u' terms c o r r e s p o n d w i t h the n e g a t i v e v' terms. I n c o r p o r a t i n g t h i s unknown c o n s t a n t i n t o the s t i l l u n v a l u e d m i x i n g l e n g t h term, we can o b t a i n the shear s t r e s s e q u a t i o n r=- Pinr=pi 2(|H)2 As t h e r e i s l i t t l e m i x i n g near the boundary and no m i x i n g a t the boundary, a l i n e a r r e l a t i o n i n the neighbourhood of the w a l l may be assumed as l = /cy where y i s the d i s t a n c e from t h e w a l l , 1 i s the P r a n d t l m i x i n g l e n g t h , and K i s von Karman's d i m e n s i o n l e s s c o n s t a n t which has been e x p e r i m e n t a l l y d e t e r m i n e d as /c=0.40. Then the shear s t r e s s becomes r = p K 2 y 2 ( d ^ ) 2 (2.9) P r a n d t l f u r t h e r assumed t h a t t h e shear s t r e s s remains c o n s t a n t t h r o u g h the f l o w , such t h a t the shear s t r e s s a t the w a l l i s the same as the s t r e s s i n t h e f l u i d above the w a l l . 32 2.3.4.1 V e l o c i t y P r o f i l e D e s c r i p t i o n Upon i n t e g r a t i n g t h i s e q u a t i o n ( 2 . 9 ) , we o b t a i n P r a n d t l ' s u n i v e r s a l v e l o c i t y d i s t r i b u t i o n law above the w a l l f o r t u r b u l e n t c h a n n e l f l o w u = — l n ( y ) + C (2.10) tc T 4-where U* i s the f r i c t i o n v e l o c i t y = (-) C depends on the roughness of the w a l l and r e f l e c t s the f i t t i n g of t h i s t u r b u l e n t v e l o c i t y d i s t r i b u t i o n t o t h a t i n the l a m i n a r s u b l a y e r . I t must be s t r e s s e d t h a t the u n i v e r s a l v e l o c i t y d i s t r i b u t i o n law i s f o r l a r g e Reynolds numbers, where-v i s c o u s e f f e c t s a re i m p o r t a n t o n l y i n the l a m i n a r s u b l a y e r . To s o l v e f o r C, we assume t h a t the v e l o c i t y u i s z e r o a t a h e i g h t y 0 above the w a l l , such t h a t u = U i l n ( J - ) (2.11) K y 0 T h i s h e i g h t y 0 i s of the same or d e r of magnitude as the t h i c k n e s s of the l a m i n a r s u b l a y e r , and i s assumed t o be k y 0=j^- i n many c o n s i d e r a t i o n s . For rough p i p e s , N i k u r a d s e (1933) has made t h i s datum h e i g h t y 0 p r o p o r t i o n a l t o the d i s t r i b u t e d roughness s i z e , k g. The u n i v e r s a l l o g law f o r a f l o w then becomes g 7=5.751og|-+B (2.12) u s i n g von Karman's c o n s t a n t K=0.40, and c h a n g i n g the n a t u r a l l o g a r i t h m t o a base 10 l o g a r i t h m . Here, 33 U*k B=5.5+5.751og(—-—) f o r h y d r a u l i c a l l y smooth f l o w B=8.5 f o r h y d r a u l i c a l l y rough f l o w . The v a l u e s of B i n the t r a n s i t i o n regime v a r y and the s e v a l u e s as c a l c u l a t e d by N i k u r a d s e a r e p l o t t e d i n F i g u r e 2.5. For a f l a t p l a t e , Schultz-Grunow (194T) o b t a i n e d measurements t h a t i n d i c a t e d i f f e r e n t c o e f f i c i e n t s f o r the l o g law e q u a t i o n than t h o s e used i n p i p e f l o w . Here we have £ _ = 5 . 8 5 1 o g ^ + 5 . 5 6 These c o e f f i c i e n t s a r e not much d i f f e r e n t from those s e t f o r t h by N i k u r a d s e f o r h y d r a u l i c a l l y smooth f l o w . 2.3.4.2 An A l t e r n a t e V e l o c i t y P r o f i l e D e s c r i p t i o n A f i n d i n g i n p i p e e x p e r i m e n t s i s t h a t o f t e n the v e l o c i t y d i s t r i b u t i o n i s r e p r e s e n t a b l e by the l / 7 t h power ©- / i / • *•• • • • • 9 , ,. i •'*•-• -rfr If > • <»• 1 ^  (h :. t ! sm j sition 1 00th " ff3fl con tuvyii i 02 OA 06 OS 10 1.2 1A 7.6 IS 2.0 23 2A 2£ 2£ 10 12 k U* l o g (-§—) F i g u r e 2.5. ( A f t e r S c h l i c h t i n g , 1979). Roughness f u n c t i o n B i n terms of k s u* , f o r N i k u r a d s e 1 s sand roughness. 34 law ^ = 8 . 7 5 ( ^ ) T (2.13) where R i s the p i p e r a d i u s . In a p i p e , the boundary t h i c k n e s s 8 i s the same as R, and the shear s t r e s s a t the w a l l i s g i v e n by TO=.0225pUj(j)^ (2.14) Assuming the v e l o c i t y d i s t r i b u t i o n i n the boundary l a y e r over a f l a t p l a t e t o be s i m i l a r t o t h a t i n a p i p e , we o b t a i n £-=(£F ..(2.15) 00 such t h a t the v e l o c i t y p r o f i l e i s g _ = 8 . 7 5 ( ^ ) ^ (2.13') and the shear s t r e s s i s T 0 = .0225pu'J"(^)'i" (2.14') T h i s i s d e v e l o p e d on f u l l a nalogy t o p i p e f l o w , but t h i s cannot be e n t i r e l y a c c u r a t e , because of the e x i s t e n c e of a p r e s s u r e g r a d i e n t i n p i p e f l o w , where t h e r e i s not one i n f r e e stream f l o w over a f l a t boundary. However, d i f f e r e n c e s i n the v e l o c i t y d i s t r i b u t i o n a r e not impo r t a n t f o r R e ynolds numbers l e s s than 1 0 6 , as the v e l o c i t y p r o f i l e measured over the p l a t e can be d e s c r i b e d f a i r l y w e l l by the l / 7 t h power law g i v e n by e q u a t i o n ( 2 . 1 3 ' ) , as shown by the e x p e r i m e n t a l r e s u l t s o b t a i n e d by Hansen (1928) and 35 Bu r g e r s (1924). As Reynolds numbers grow l a r g e r than 10 6 as would be the case over a n a t u r a l sand r i v e r bed, the u n i v e r s a l l o g law v e l o c i t y p r o f i l e e q u a t i o n s h o u l d be used i n s t e a d of the l / 7 t h power law f o r m u l a . 2.3.5 EMPIRICAL METHOD FOR DETERMINING TURBULENT SHEAR  STRESS VALUES In d e s c r i b i n g the v e l o c i t y p r o f i l e , we n o r m a l l y use the e q u a t i o n g*«5.751og^ +B T h i s i s seen t o be i d e n t i c a l t o e q u a t i o n (2.12) as s e t f o r t h k by N i k u r a d s e , where we assume t h a t Yo=-jQm However, i t i s i m p o r t a n t t o note t h a t t h e r e i s a p i t f a l l a s s o c i a t e d w i t h our manner of c h o o s i n g a v a l u e f o r a datum h e i g h t y 0 . The v a l u e of U* i s h i g h l y dependent on the assumed v a l u e of y 0 . For example, a 100% e r r o r i n y 0 i s easy t o make i n e x p e r i m e n t , and t h i s g i v e s an e r r o r i n U* of over 70%. The e r r o r i n U* r e s u l t s i n an e r r o r i n the p r e d i c t e d v e l o c i t y p r o f i l e f o r a l l o t h e r h e i g h t s of y above y 0 . •An a l t e r n a t i v e method f o r the e s t i m a t i o n of U* uses the Manning e q u a t i o n : uJ-R^s"2" (2.16) n where R i s the h y d r a u l i c r a d i u s S i s the l o n g i t u d i n a l s l o p e 36 where U*=(gRS) T (2.16') and n=.038DT" ( 2 . 1 6 " ) g i v i n g (2.17) where ( 2 . 1 6 " ) and (2.17) are f o r m e t r i c u n i t s . A l a r g e e r r o r i n the e s t i m a t e of the h y d r a u l i c r a d i u s R or of the sediment roughness D w i l l r e s u l t i n o n l y a v e r y s m a l l e r r o r i n the v a l u e o f . U * . For example, an e r r o r of R 100% i n the r a t i o ^ r e s u l t s i n o n l y a 12% e r r o r i n the v a l u e of U*. Hence, the Manning e q u a t i o n s h o u l d be used t o c a l c u l a t e the v a l u e of U*, and t h i s v a l u e used t o c a l c u l a t e the f l o w v e l o c i t y p r o f i l e u s i n g the u n i v e r s a l l o g law. 2.3.6 CONCLUSIONS The a c c u r a t e t u r b u l e n t f l o w p a t t e r n appears i m p o s s i b l e t o d e s c r i b e , but a time averaged c o n s i d e r a t i o n g i v e s s o l u t i o n s t h a t agree w e l l w i t h o b s e r v e d r e s u l t s . The shear s t r e s s and the v e l o c i t y p r o f i l e a r e g i v e n by the u n i v e r s a l l o g law, and the 1/7th power law. I t i s a d v i s e d t h a t U* be c a l c u l a t e d u s i n g the e m p i r i c a l Manning e q u a t i o n r a t h e r than be c a l c u l a t e d u s i n g the u n i v e r s a l l o g law or the l / 7 t h power law, and the e x p e r i m e n t a l v e l o c i t y p r o f i l e . The v e l o c i t y 37 p r o f i l e can then be c o n s t r u c t e d u s i n g t h i s v a l u e of U*, and the u n i v e r s a l low law, and can be compared w i t h the e x p e r i m e n t a l v e l o c i t y p r o f i l e . 2. 4 TRANSITION Flow i n a boundary l a y e r a l o n g a w a l l becomes t u r b u l e n t f o r l a r g e f l u i d v e l o c i t i e s , and the t r a n s i t i o n i s e a s i l y i d e n t i f i e d by sudden l a r g e i n c r e a s e s i n the boundary l a y e r t h i c k n e s s and i n the shear s t r e s s near the w a l l . Measurements by Hansen (1930) r e l a t i n g the l e n g t h Reynolds number t o the d i m e n s i o n l e s s boundary l a y e r t h i c k n e s s r e v e a l c l e a r l y the s h a r p i n c r e a s e i n boundary l a y e r t h i c k n e s s as F i g u r e 2.6. ( A f t e r S c h l i c h t i n g , 1979). Boundary l a y e r t h i c k n e s s p l o t t e d a g a i n s t the l e n g t h Reynolds number f o r a f l a t p l a t e i n p a r a l l e l f l o w a t z e r o i n c i d e n c e , as measured by Hansen (1930). 38 shown i n F i g u r e ( 2 . 6 ) . The l e n g t h Reynolds number i s R = =3.2-10 5 a t t r a n s i t i o n f o r a f l a t p l a t e , and S c h l i c h t i n g r e g a r d e d t h i s v a l u e as the lower l i m i t . L a r g e r v a l u e s have been reached u s i n g f l o w s t h a t a r e v e r y d i s t u r b a n c e f r e e . In the boundary l a y e r , the t r a n s i t i o n i s a f f e c t e d by many i m p o r t a n t p a r a m e t e r s , i n c l u d i n g the p r e s s u r e d i s t r i b u t i o n i n the e x t e r n a l f l o w , the i n t e n s i t y of t u r b u l e n c e i n the e x t e r n a l f l o w , and the w a l l roughness. Flows h a v i n g a f a v o u r a b l e p r e s s u r e g r a d i e n t a re more s t a b l e than t h o s e h a v i n g an a d v e r s e p r e s s u r e g r a d i e n t . Both h i g h l y t u r b u l e n t e x t e r n a l f l o w s and rough w a l l s more e a s i l y c r e a t e t u r b u l e n t boundary l a y e r s . 2.4.1 PRESSURE GRADIENT EFFECTS The p r e s s u r e g r a d i e n t a l o n g a w a l l has a g r e a t i n f l u e n c e on the t r a n s i t i o n i n the boundary l a y e r . In a r e g i o n of. d e c r e a s i n g p r e s s u r e , t h e boundary l a y e r remains l a m i n a r , whereas even a s l i g h t p r e s s u r e i n c r e a s e almost always c r e a t e s t r a n s i t i o n t o t u r b u l e n t f l o w . From the N a v i e r - S t o k e s e q u a t i o n s , we have !§-<"$!>. and i t i s seen t h a t the p r e s s u r e g r a d i e n t d e t e r m i n e s the shape of the v e l o c i t y p r o f i l e . The s t a b i l i t y of the f l o w i s dependent on the shape of the p r o f i l e , such t h a t a p r o f i l e w i t h o u t i n f l e c t i o n s i s s t a b l e w h i l e one w i t h i n f l e c t i o n s i s 39 n o t . A p o s i t i v e ( a d v e r s e ) p r e s s u r e g r a d i e n t then i s u n s t a b l e as i t n e c e s s a r i l y has a p o i n t of i n f l e c t i o n because the v e l o c i t y p r o f i l e i s a s y m p t o t i c t o the e x t e r n a l f l o w . An e x t e r n a l stream has a v e l o c i t y as a f u n c t i o n of the l e n g t h c o o r d i n a t e , as i t depends on the p r e s s u r e a c c o r d i n g t o f £ ~ < Combining e q u a t i o n s (2.18) and (2.19) shows t h a t an a c c e l e r a t e d f l o w i s more s t a b l e than one t h a t i s d e c e l e r a t e d . 2.4.2 TURBULENT EFFECTS S c h l i c h t i n g r e p o r t e d on many e x p e r i m e n t s i n v o l v i n g t r a n s i t i o n depending on the degree of f r e e stream t u r b u l e n c e . R e f e r r i n g m a i n l y t o the works of Schubauer and Skramstad (1947) and G r a n v i l l e (1953), he a m p l i f i e d how an i n c r e a s e d t u r b u l e n c e i n the e x t e r n a l f l o w d e c r e a s e s the c r i t i c a l l e n g t h Reynolds number f o r t r a n s i t i o n . 2.4.3 ROUGHNESS EFFECTS I t has not been p o s s i b l e t o t h e o r e t i c a l l y a n a l y z e the problem of the i n f l u e n c e of roughness on t r a n s i t i o n . I t has been d e t e r m i n e d e x p e r i m e n t a l l y t h a t an i n c r e a s e d boundary roughness f a v o u r s t r a n s i t i o n from l a m i n a r f l o w a t a lower Reynolds number than f o r o t h e r w i s e i d e n t i c a l c o n d i t i o n s over a smooth boundary. Added roughness elements a r e s u s p e c t e d of 40 s i m p l y c r e a t i n g a d d i t i o n a l f l o w d i s t u r b a n c e s i n the l a m i n a r l a y e r such t h a t the f l o w i s as i f i t were of a more d i s t u r b e d n a t u r e . S m a l l roughness i s ex p e c t e d t o c r e a t e such s m a l l t u r b u l e n c e t h a t i t s added e f f e c t i s n e g l i g i b l e 3 , whereas v e r y l a r g e roughness may be e x p e c t e d t o t r i g g e r t r a n s i t i o n a t the v e r y p o i n t of roughness. An e x t e n s i v e r e v i e w of many e x p e r i m e n t s has been made by S c h l i c h t i n g (1979), and a s h o r t summary i s p r e s e n t e d h e r e . 2.4.3.1 S i n g l e C y l i n d r i c a l Roughness Elements From o l d e r measurements, G o l d s t e i n (1936) d e t e r m i n e d t h a t the c r i t i c a l h e i g h t below which a s i n g l e , c y l i n d r i c a l roughness element has no e f f e c t on t r a n s i t i o n i s g i v e n by k = 7 — c r i t 'u* where ^ c r i t * s t* i e c r i t i c a l h e i g h t of the roughness element and U£ i s the f r i c t i o n v e l o c i t y a t the w a l l at the p o s i t i o n of the roughness element. Fage and P r e s t o n (1941) used k c r i t = 2 0 U * as the minimum h e i g h t above which t r a n s i t i o n o c c u r s a t the roughness element i t s e l f . For v a l u e s of k c r i t between the g i v e n c r i t i c a l h e i g h t s , the t r a n s i t i o n p o i n t moves downstream f o r d e c r e a s i n g roughness element h e i g h t . Kraemer (1961) p r e s e n t e d a more u s e f u l e q u a t i o n independent of shear s t r e s s w i t h 41 k c r i t ^ 0 0 U -00 where U o i s the f r e e stream v e l o c i t y . For any roughness element h e i g h t l a r g e r than k . , the t r a n s i t i o n o c c u r s somewhere downstream f o r f l o w of d i f f e r e n t p r e s s u r e g r a d i e n t s and d i f f e r e n t t u r b u l e n t i n t e n s i t i e s , such t h a t the e q u a t i o n can be c o n s i d e r e d t o be u n i v e r s a l . T r a n s i t i o n may o c c u r f o r s m a l l e r v a l u e s of k c r i t r but always o c c u r s f o r v a l u e s above the l i m i t g i v e n . 2.4.3.2 D i s t r i b u t e d Roughness For d i s t r i b u t e d roughness, F e i n d t (1957) de t e r m i n e d t h a t a u n i f o r m sand roughness has an i n f l u e n c e on the t r a n s i t i o n p o i n t f o r k s * 1 7 0 u T where k g i s the u n i f o r m sand p a r t i c l e s i z e U, i s the f l u i d v e l o c i t y i n the p i p e where the t e s t s were p e r f o r m e d . Above t h i s v a l u e of sand p a r t i c l e d i a m e t e r , t h e d i s t a n c e from the b e g i n n i n g of the roughness t o the p o i n t of t r a n s i t i o n d e c r e a s e s markedly w i t h an i n c r e a s e i n roughness s i z e . Below t h i s v a l u e , t h e r e i s no i n f l u e n c e on the d i s t a n c e t o the t r a n s i t i o n p o i n t . N i k u r a d s e (1953) c a r r i e d out measurements i n rough p i p e s c o v e r e d on the i n s i d e as t i g h t l y as p o s s i b l e w i t h sand of d e f i n i t e g r a i n s i z e , h a v i n g a roughness k . The roughness was found t o s p l i t the f l o w i n t o t h r e e regimes: the 42 h y d r a u l i c a l l y smooth regime, the t r a n s i t i o n regime, and the h y d r a u l i c a l l y rough regime. These t h r e e regimes a re k U* d e l i n e a t e d by c e r t a i n v a l u e s of — - — , as f o l l o w s : k QU* 0 <——< 5 k U* 5 <——<70 70< k U* s > (2.20) The smooth regime has roughness so s m a l l t h a t i t i s c o n t a i n e d w i t h i n the l a m i n a r s u b l a y e r . In the t r a n s i t i o n regime, some roughness e x t e n d s o u t s i d e t h e s u b l a y e r , and some form d r a g p r o v i d e s f o r a d d i t i o n a l r e s i s t a n c e . For the c o m p l e t e l y rough regime, the form drag dominates as the roughness e x t e n d s o u t s i d e the s u b l a y e r , and the r e s i s t a n c e law becomes q u a d r a t i c . 2.4.4 CONCLUSIONS The phenomenon of t r a n s i t i o n i s e a s i l y o b s e r v e d , but i t s p r e d i c t e d p o i n t of o c c u r r e n c e i s not s i m p l y d e s c r i b e d , because of i t s dependence on the p r e s s u r e g r a d i e n t , the f l o w t u r b u l e n c e , and the bedform roughness. There a r e some r e l a t i o n s p r e s e n t e d which g i v e some i d e a as t o when and where t r a n s i t i o n does o c c u r . 43 2. J CONCLUSIONS The v e l o c i t y p r o f i l e s and the s e c t i o n a l shear s t r e s s e s a r e w e l l known f o r l a m i n a r f l o w moving over a f l a t p l a t e . The boundary l a y e r t h i c k n e s s i s d e f i n e d i n many d i f f e r e n t ways, but the most e a s i l y c o n c e p t u a l i z e d i s t h a t as d e f i n e d here - the t h i c k n e s s i s t h a t d i s t a n c e from the w a l l t o the h e i g h t where u=.99U o o. For t u r b u l e n t f l o w , a complete u n d e r s t a n d i n g of the f l o w i s not p o s s i b l e because of i t s c o m p l i c a t e d n a t u r e . The t h e o r e t i c a l a s sumptions t h a t have been made d e s c r i b e the a c t u a l p r o f i l e s and f l o w s q u i t e w e l l , but the mechanics l e a d i n g t o t h e s e c o n d i t i o n s a r e not f u l l y u n d e r s t o o d . The t h e o r e t i c a l boundary l a y e r t h i c k n e s s i s unknown, but we assume a v a l u e f o r the l a m i n a r s u b l a y e r t h i c k n e s s . T h i s v a l u e may be q u e s t i o n a b l e because of the e x p e r i m e n t a l problems i n d e t e r m i n i n g the s i z e of the s u b l a y e r . The v e l o c i t y p r o f i l e p r e d i c t i o n f o r the shear v e l o c i t y i s h i g h l y dependent on y 0 i n e q u a t i o n ( 2 . 1 1 ) , and i t i s recommended t h a t e q u a t i o n (2.17) be used t o determine the shear v e l o c i t y i n s t e a d . S e v e r a l c o n d i t i o n s f o r t r a n s i t i o n have been d e s c r i b e d t o a i d i n the u n d e r s t a n d i n g of t h i s phenomenon. 3. GOVERNING EQUATIONS FOR OSCILLATORY FLOW 3. I LAMINAR FLOW VELOCITY PROFILE As f o r u n i d i r e c t i o n a l f l o w , the b a s i c d i f f e r e n t i a l e q u a t i o n used t o d e s c r i b e l a m i n a r o s c i l l a t o r y f l o w i s the g e n e r a l N a v i e r - S t o k e s e q u a t i o n ( S c h l i c h t i n g , 1979) 9u. 9u. 9u 1 9p, 9 2u , -> 1 X -5-r+u-^ — +v-~—=-——r (3.1) 9t 9x 9y p9x 9y z The s o l u t i o n t o t h i s e q u a t i o n g i v e s the v e l o c i t y a m p l i t u d e v a r i a t i o n and phase v a r i a t i o n w i t h i n the l a m i n a r o s c i l l a t o r y boundary l a y e r over a smooth f l a t s u r f a c e t o a good degree of a p p r o x i m a t i o n . The v e l o c i t y a t the o u t s i d e edge of the boundary l a y e r , U, i s g i v e n by the E u l e r e q u a t i o n _19p_9U p9x 9t where U=U c o s ( k x - o j t ) , i n which U i s the maximum m m l o n g i t u d i n a l v e l o c i t y i n the f r e e stream r e g i o n , k = ^ K L where L i s the wavelength i f the o s c i l l a t i o n i s d r i v e n by a wave, and where T i~s the p e r i o d of t h e o s c i l l a t i o n . At a g i v e n l o c a t i o n , x, the f l o w dynamics a r e s i m i l a r t o thos e of an i n f i n i t e f l u i d o s c i l l a t i n g over a s t a t i o n a r y s u r f a c e . 44 45 The v e l o c i t y a t the w a l l , U 0, i s s i m p l y z e r o because of the r e q u i r e m e n t f o r f l u i d adherence t o the w a l l . The boundary c o n d i t i o n s t o e q u a t i o n (3.1) a r e then y=0 u=0 y= 6 u=U c o s ( k x - c j t ) m I f the wavelength i s l a r g e i n comparison w i t h the boundary l a y e r t h i c k n e s s , then the c o n v e c t i v e terms i n e q u a t i o n (3.1) can be n e g l e c t e d , and the N a v i e r - S t o k e s e q u a t i o n becomes 9u,9U. 9 2u n - g t + g t ^ a y ^ 0 ( 3 - 2 ) T h i s e q u a t i o n w i t h the g i v e n boundary c o n d i t i o n s has an |>5 and. £ e x a c t l a m i n a r s o l u t i o n f o r ^  %<5 as g i v e n by Lamb (1932) u = U m ( c o s ( k x - c j t ) - e x p ( - T j ) c o s ( k x - w t - 7 ? ) ) (3.3) w i t h rj=Jr i n which 6 i s the Stokes thickness g i v e n by The o s c i l l a t o r y f l o w here i s a m o d i f i c a t i o n of the c l a s s i c a l S t o k e s problem of the o s c i l l a t i n g f l a t p l a t e i n a f l u i d a t r e s t . C e r t a i n f e a t u r e s of the l a m i n a r o s c i l l a t o r y f l o w a re shown i n F i g u r e s 3.1 t o 3.3. F i g u r e 3.1 shows the k i n d of v e l o c i t y p r o f i l e s t h a t a r e t y p i c a l of l a m i n a r o s c i l l a t o r y f l o w and a r e ge n e r a t e d by e q u a t i o n ( 3 . 3 ) . F i g u r e 3.2 shows how the v e l o c i t y p r o f i l e near t h e bed i s i n advance of the 46 • 05 - 03 - 02 - i — I — r 01 02 U II IS - i i -to -at - o i - 0 - 1 - 0 3 o 02 oi as a t i « 12 pcou (dp/dx) F i g u r e 3.1. ( M o d i f i e d a f t e r K n i g h t , 1978). I n s t a n t a n e o u s v e l o c i t y p r o f i l e s i n l a m i n a r o s c i l l a t o r y f l o w , 0d=8.O v e l o c i t y o u t s i d e the boundary l a y e r and i s of l a r g e r magnitude. We wi s h t o s u b s t a n t i a t e the g e n e r a l form of Lamb's s o l u t i o n f o r l a m i n a r o s c i l l a t o r y f l o w over a smooth f l a t bed. F i g u r e s 3.2 and 3.3 compare some r e s u l t s of S l e a t h (1970) and Horikawa and Watanabe (1968) w i t h the t h e o r e t i c a l v a r i a t i o n i n v e l o c i t y a m p l i t u d e and phase f o r v a l u e s of 0y down t o 0.5. The f i g u r e s i n d i c a t e t h a t when the boundary i s p l a n e and smooth, the b a s i c t h e o r y a c c o u n t s f o r the v a r i a t i o n i n v e l o c i t y and phase q u i t e w e l l . The da t a 02 04 0 6 0 8 10 12 DCJU "(dp/dx) F i g u r e 3.2. ( M o d i f i e d a f t e r K n i g h t , 1978). Laminar o s c i l l a t o r y f l o w — e x p e r i m e n t a l d a t a f o r v e l o c i t y a m p l i t u d e v a r i a t i o n . 4 7 0y 2 0 10 T i l l o SItath * • Horihawa ft Wat a nob* -1 - Theory ff* ^ ' 1 1 -40* SO' « ' 70* 10* 90* 100* —»- f I 0 < « r m I F i g u r e 3.3. ( M o d i f i e d a f t e r K n i g h t , 1978). Laminar o s c i l l a t o r y f l o w — e x p e r i m e n t a l d a t a f o r v e l o c i t y phase v a r i a t i o n . i n v o l v e s o b s e r v a t i o n s by Horikawa and Watanabe t o w i t h i n 0.2 mm of the boundary l a y e r . Horikawa and Watanabe u t i l i z e d t he hydrogen bubble t e c h n i q u e t o measure the f l u i d v e l o c i t y i n the v i c i n i t y of the boundary s u r f a c e . Measurements made over a smooth bottom under p r o g r e s s i v e waves g i v e d a t a which p l o t s i n c o n s i s t e n t agreement w i t h K a j i u r a ' s t h e o r e t i c a l c u r v e s f o r v e l o c i t y a m p l i t u d e and phase. The t h i c k n e s s Reynolds number f o r these o b s e r v a t i o n s i s e q u a l t o R § = — — =35, such t h a t the fl o w i s i n t r a n s i t i o n between l a m i n a r and t u r b u l e n t . In t h i s t r a n s i t i o n regime, t h e d i s c r e p a n c y between t h e l a m i n a r t h e o r y as g i v e n by Lamb and the smooth t u r b u l e n t t h e o r y as g i v e n by K a j i u r a i s v e r y s m a l l , a c c o r d i n g t o Horikawa and Watanabe. I t i s n o t i c e d t h a t the phase near the boundary s u r f a c e i s d i v e r g e n t from the t h e o r y , and i s c l o s e t o 30 degree s . S l e a t h (1970) measured the t e n s i o n i n a t h i n f i b r e w i r e produced by a f l u i d f l o w i n g normal t o the l e n g t h of the w i r e . The o s c i l l a t o r y f l u i d f l o w was produced by waves of 48 s m a l l h e i g h t and the bed was a smooth f l a t p l a t e . The agreement between the experiment and t h e o r y was v e r y good. Ramaprian and M u e l l e r (1980) measured the v e l o c i t y p r o f i l e i n the o s c i l l a t o r y boundary l a y e r i n a water t u n n e l by use of a L a s e r Doppler Anemometer. The f l o w type was t r a n s i t i o n a l , and the v e l o c i t y d i s t r i b u t i o n s were o n l y m i l d l y d i f f e r e n t from those p r e d i c t e d by the l a m i n a r t h e o r y . S l e a t h (1974b) examined the v e l o c i t y d i s t r i b u t i o n s c l o s e t o rough beds i n l a m i n a r o s c i l l a t o r y f l o w . The f l o w remained l a m i n a r , but v o r t e x f o r m a t i o n and sudden j e t s of f l u i d m o d i f i e d the v e l o c i t y c l o s e t o the boundary s u r f a c e . The l a m i n a r o s c i l l a t o r y f l a t bed s o l u t i o n does not work w e l l over the rough bed, though S l e a t h has p r e s e n t e d a p e r t u r b a t i o n s o l u t i o n which works w e l l f o r some i n s t a n c e s . For l a m i n a r f l o w over a f l a t smooth bed, e x p e r i m e n t a l e v i d e n c e s u p p o r t s the t h e o r y q u i t e w e l l e x c e p t f o r the phase v e r y near t o the boundary. 3. 2 TURBULENT FLOW VELOCITY PROFILE The o s c i l l a t i n g t u r b u l e n t f l o w may be c o n s i d e r e d t o be governed by 9u, 9u, 9u 19p. 9 2u /, .\ 9t 9x 9y p9x 9y z i n which e i s the k i n e m a t i c eddy v i s c o s i t y ( K n i g h t , 1978). The s o l u t i o n t o e q u a t i o n (3.4) i s c o m p l i c a t e d by the v a r i a t i o n of e a c r o s s the boundary l a y e r . T h i s v a r i a t i o n has l e d t o many assumptions i n v o l v i n g d i f f e r e n t models i n which 49 c i s r e g a r d e d as c o n s t a n t over c e r t a i n l a y e r s of the f l u i d . The t a k i n g of tempor a l and s p a t i a l averages of the eddy v i s c o s i t y i s d e b a t a b l e , but s i m p l i c i t y j u s t i f i e s i t s use i n t h i s manner. S o l u t i o n s t o e q u a t i o n (3.4) a r e o b t a i n e d i n an analogou s manner t o l a m i n a r o s c i l l a t o r y f l o w . E q u a t i o n (3.4) i s reduced t o 9u.9U, 9 2u n /, c x - 9 t + 9t + e 9 y ^ = 0 .....(3.5) f o r a t h i n boundary l a y e r . The s i m p l e s t t h e o r i e s a r e those whose s o l u t i o n s a re d i r e c t l y analogous w i t h the s i n g l e l a y e r s o l u t i o n f o r l a m i n a r o s c i l l a t o r y f l o w . The s o l u t i o n of K a l k a n i s (1964) f o r the v e l o c i t y i s ^ — = c o s ( k x - w t ) - f , ( y ) c o s ( k x - w t - f 2 ( y ) ) (3.6) m where f i ( y ) and f a ( y ) are de t e r m i n e d e x p e r i m e n t a l l y f o r d i f f e r e n t o s c i l l a t o r y f l o w s . E q u a t i o n s f o r f ^ y ) and f 2 ( y ) a r e d e t e r m i n e d f o r d i f f e r e n t boundary roughnesses, but a r e v a l i d o n l y over s p e c i f i c r anges, and they a l s o v i o l a t e the boundary c o n d i t i o n s a t y=0. The s o l u t i o n proposed by S l e a t h (1970) a l s o i s analogou s w i t h the s i n g l e l a y e r s o l u t i o n f o r l a m i n a r o s c i l l a t o r y f l o w . The v e l o c i t y e q u a t i o n i s ^-= c o s ( k x - ( j t )-exp(-£^)cos ( kx-a>t-^) ....... (3.7) m where the f a c t o r X must be d e t e r m i n e d e x p e r i m e n t a l l y . I n the l a m i n a r f l o w c o n d i t i o n , X=1, and the s o l u t i o n i s i d e n t i c a l 50 t o t h a t g i v e n by Lamb. In t u r b u l e n t f l o w , X l i e s i n the range 1.0<X<1.8 a c c o r d i n g t o S l e a t h ' s r e s u l t s . E q u a t i o n (3.7) g i v e s t h e v e l o c i t y d i s t r i b u t i o n a d e q u a t e l y , but the phase near the bed i s s i g n i f i c a n t l y d i f f e r e n t because of v o r t e x f o r m a t i o n o c c u r r i n g near bed roughness e l e m e n t s . For c e r t a i n r oughnesses, both the a m p l i t u d e and t h e phase may d e v i a t e s i g n i f i c a n t l y from the f l a t smooth bed s o l u t i o n . More complex t h e o r i e s c o n s i d e r the o s c i l l a t o r y boundary l a y e r t o be composed of d i f f e r e n t l a y e r s , and thus p r e s e n t the p o s s i b i l i t y of b e i n g improved e s t i m a t e s because they t a k e the s t r u c t u r e of the boundary l a y e r i n t o c o n s i d e r a t i o n . K a j i u r a (1968) p r e s e n t e d a t h e o r y i n which an eddy v i s c o s i t y a s s u m p t i o n i s based on an a n a l o g y t o s t e a d y s t a t e f l o w . The o s c i l l a t o r y boundary l a y e r he c o n s i d e r e d t o be composed of t h r e e p a r t s - a n i n n e r l a y e r a t the w a l l , an o v e r l a p l a y e r , and an o u t e r l a y e r near i t s o u t e r boundary. T h i s i s s i m i l a r t o the steady t u r b u l e n t boundary l a y e r s t r u c t u r e , but C l a u s e r (1956) i n d i c a t e d t h a t t u r b u l e n t e d d i e s i n steady t u r b u l e n t f l o w i n t r o d u c e s h e a r i n g s t r e s s e s f o r which no r e l i a b l e method of c a l c u l a t i o n e x i s t s . T e l e k i and Anderson (1972) i n q u o t i n g C l a u s e r added t h a t t h i s e d dying phenomenon s h o u l d o n l y be more complex i n time p e r i o d i c f l o w s . K a j i u r a mentioned t h a t the i n s t a n t a n e o u s s t a t e of t u r b u l e n c e i s l e f t e n t i r e l y out of the q u e s t i o n , as t h e eddy v i s c o s i t y v a l u e i s t e m p o r a l l y and s p a t i a l l y averaged f o r each l a y e r , and t h a t t h e r e i s no c e r t a i n t y as t o how the 51 i n s t a n t a n e o u s s t a t e of t u r b u l e n c e and the average s t a t e of t u r b u l e n c e can be r e l a t e d . One problem encountered w i t h the t u r b u l e n t p r o f i l e s i s t h a t the v e l o c i t y a t a h e i g h t y above the w a l l i s not r e l a t e d e a s i l y t o a t y p i c a l v e l o c i t y a t a n o t h e r p o i n t such as the v e l o c i t y above the boundary l a y e r , but r a t h e r i s r e l a t e d t o the shear s t r e s s e x e r t e d on the w a l l by the moving f l u i d . K a j u i r a (1964) used ' t u r b ^ y l f <3'8> as the r e l a t i o n between the shear s t r e s s i n t h e f l u i d and the v e l o c i t y p r o f i l e s l o p e , where i s the t u r b u l e n t eddy v i s c o s i t y , T T ^ i s the t u r b u l e n t shear s t r e s s i n the f l u i d , and i s the v e l o c i t y g r a d i e n t i n the f l u i d . The boundary l a y e r i s m o d e l l e d by K a j i u r a (1968) f o r smooth and rough b o u n d a r i e s as f o l l o w s : f o r smooth boundary K^= < v 0^y<D, *U*y D,<y<3 k«u$a a<y r v 0<y<D f o r rough boundary K^= < »cU$y D r<y<3 . K U$3 3<y where U* i s the shear v e l o c i t y g i v e n by ( p ) T U§ i s the shear v e l o c i t y a t the bed D, i s the t h i c k n e s s of the l a m i n a r s u b l a y e r over a 52 smooth boundary and i s g i v e n by D,=-^j^; and i s the t h i c k n e s s of the i n n e r l a y e r over a rough boundary and i s g i v e n by D r = l 5 y 0 f where y 0 i s the roughness l e n g t h 3 i s the upper l i m i t h e i g h t of the o v e r l a p l a y e r above the bottom and i s g i v e n by KU 5* a= m com where U m i s the v e l o c i t y a m p l i t u d e o u t s i d e the boundary l a y e r , K i s a c o n s t a n t = .02, k i s a c o n s t a n t = .40, 5* i s the wave d i s p l a c e m e n t t h i c k n e s s , g i v e n by /°(U-u)dz 6*=Amp-0 Um and where U* =-L- i s the a m p l i t u d e of the f r i c t i o n shear com pU* v e l o c i t y a t the bottom. Making use of t h e s e eddy v i s c o s i t y a s s u m p t i o n s , the e q u a t i o n of o s c i l l a t o r y motion i n the boundary l a y e r i s s o l v e d f o r the c a s e of a boundary l a y e r t h a t i s t h i n w i t h r e s p e c t t o the t o t a l d e p t h of the f l u i d . The t h e o r y a l s o assumes t h a t c o n d i t i o n s a r e such t h a t the o v e r l a p l a y e r i s p r e s e n t , r e q u i r i n g t h a t the o r b i t a l a m p l i t u d e of the motion i s l a r g e . Horikawa and Watanabe (1968) i n v e s t i g a t e d many a s p e c t s of K a j i u r a ' s t h e o r y , and a t t e m p t e d t o answer the f o l l o w i n g q u e s t i o n s : (1) whether the a s s u m p t i o n f o r the eddy v i s c o s i t y i s a p p r o p r i a t e t o the t u r b u l e n t o s c i l l a t o r y boundary l a y e r problem (2) whether the b a s i c e q u a t i o n of o s c i l l a t o r y motion i n 53 the boundary l a y e r i s adequate t o e x p r e s s the flow c h a r a c t e r i s t i c s near a r i p p l e d bottom (3) what i s the o r i g i n of the y a x i s f o r a rough bottom (4) whether the assumption f o r the e s t i m a t e of and y 0 i s r e a s o n a b l e . The c o n c l u s i o n s drawn by t h e i r study a r e as f o l l o w s : (1) the eddy v i s c o s i t y i s a time dependent f u n c t i o n and a c o n t i n u o u s f u n c t i o n of y, such t h a t K a j i u r a ' s a s s u m p t i o n i s not always s a t i s f a c t o r y . The eddy v i s c o s i t y i s d e t e r m i n e d u s i n g the e q u a t i o n (2) the f l o w i s v e r y c o m p l i c a t e d near r i p p l e s because of v o r t e x f o r m a t i o n and the b a s i c e q u a t i o n (3.5) i s i n a d e q u a t e t o e x p r e s s the f l o w c h a r a c t e r i s t i c s near a r i p p l e . (3) t h e r e i s no r u l e d e f i n i n g the o r i g i n f o r the y a x i s f o r a rough bottom. (4) the r e a s o n a b i l i t y of the a ssumptions of the e s t i m a t e of D r and y 0 i s s t i l l u n c e r t a i n . The d a t a was o b t a i n e d by the a u t h o r s u s i n g the hydrogen bubble t e c h n i q u e , and a d d i t i o n a l d a t a e x p l o y e d f o r c o n f i r m a t i o n was t h a t of Jonsson (1963). G e n e r a l l y , agreement between t h e o r y and e x p e r i m e n t a l d a t a was good. B r e v i k (1981) q u e s t i o n e d whether i t i s p o s s i b l e t o reduce t h e m a t h e m a t i c a l c o m p l e x i t y of K a j i u r a ' s t h e o r y and y e t m a i n t a i n good agreement w i t h the o b s e r v a t i o n s made by 3y 54 Jonsson (1963), Jonsson and C a r l s e n (1976), K a l k a n i s (1964), and Horikawa and Watanabe (1968). The m a t h e m a t i c a l c o m p l e x i t y a r i s e s because of the requirement f o r matching of the a n a l y t i c s o l u t i o n s of the b a s i c d i f f e r e n t i a l e q u a t i o n s a t the two b o u n d a r i e s between the o u t e r and o v e r l a p l a y e r s and the o v e r l a p and i n n e r l a y e r s . B r e v i k assumed a two l a y e r model by o m i t t i n g the i n n e r p a r t of the boundary l a y e r . T h i s causes the a n a l y t i c s o l u t i o n t o be t e r m i n a t e d a t a f i x e d l e v e l above the bed, but such a s i t u a t i o n i s w e l l known from the rough t u r b u l e n t u n i d i r e c t i o n a l f l o w t h e o r y w i t h a l o g a r i t h m i c v e l o c i t y f l o w p r o f i l e . Assuming t h a t the c o n d i t i o n s a r e such t h a t the o v e r l a p l a y e r i s p r e s e n t , then two a s s u m p t i o n s are made. The f i r s t i s t o choose a form of the mean t u r b u l e n t v i s c o s i t y i n t h i s l a y e r , and the second i s t o e s t i m a t e the p o s i t i o n of the b o u n d a r i e s of t h i s l a y e r . The form of the v i s c o s i t y f o l l o w s and t h e b o u n d a r i e s a r e chosen as y = y 0 f and y=A, where A has two d i f f e r e n t proposed v a l u e s which a r e compared e x p e r i m e n t a l l y . In t h e o u t e r l a y e r , where y>A, efc i s assumed as c o n s t a n t , and t o g i v e c o n t i n u i t y , i n the o u t e r l a y e r . Reasonable agreement w i t h the o b s e r v a t i o n s was found, p a r t i c u l a r l y f o r a c h o i c e of A=-=-, where 8 i s d e f i n e d by e . (y ) = /cU* y t J amr (3.9) e t ( y ) = K U £ m A (3.10) 55 J o n s son and C a r l s e n (1976) as 6=0.072(a O m 3k)' J' (3.11) where a 0 m i s the h o r i z o n t a l p a r t i c l e a m p l i t u d e o u t s i d e the wave boundary l a y e r . The r e s u l t s a r e not v e r y c r i t i c a l w i t h r e s p e c t t o the d i f f e r e n t i n p u t v a l u e s f o r A. ( I t s h o u l d be noted here the problem i n v o l v e d i n d e f i n i n g the v e l o c i t y and the v e l o c i t y a m p l i t u d e a t the bed. Under waves, the "bed" u s u a l l y means " j u s t above the boundary l a y e r " because the boundary l a y e r t h i c k n e s s i s v e r y s m a l l r e l a t i v e t o the t o t a l water d e p t h . In boundary l a y e r s t u d i e s , t h e r e i s a d i s t i n c t i o n made between the v e l o c i t y a t the t o p of the boundary l a y e r and t h a t a t the p h y s i c a l bed. In t h i s r e p o r t , the boundary l a y e r d e s c r i p t i o n s use the s u b s c r i p t " 0" f o r bed v a l u e s (eg. U 0 m ) , and drop i t f o r v a l u e s o u t s i d e the boundary l a y e r (eg. U^). The more g e n e r a l d e s c r i p t i o n s of the near bed p a r t i c u l a r s use t h e s u b s c r i p t " 0 " f o r v a l u e s o u t s i d e the boundary l a y e r . In g e n e r a l , l a b o r a t o r y c o n d i t i o n s where waves t r a v e l over sand beds are such t h a t the boundary l a y e r i s e x t r e m e l y t h i n , and the v e l o c i t y o u t s i d e the boundary l a y e r i s then synonymous w i t h t h a t a t the bed.) The r e s u l t s show the two l a y e r model t o be s a t i s f a c t o r y , and a l s o s u p p o r t the i d e a of the time independent t u r b u l e n t v i s c o s i t y , a t l e a s t i n the near-bed r e g i o n . 56 However, Fredsoe (1984) mentioned t h r e e s h o r t c o m i n g s i n K a j i u r a ' s and B r e v i k ' s t h e o r y : (1) the p r e v i o u s l y mentioned p o i n t t h a t the eddy v i s c o s i t y ' s time dependency was n e g l e c t e d , (2) the wave boundary l a y e r t h i c k n e s s i s a l s o time dependent, and t h a t t h i s was n e g l e c t e d , and f i n a l l y (3) the bed shear s t r e s s v a r i a t i o n was assumed t o be s i n u s o i d a l . In view of the e x p e r i m e n t a l r e s u l t s , the mention of these s h o r t c o m i n g s i s i n c l u d e d o n l y f o r c o m p l e t e n e s s . Myrhaug (1982) used a model s i m i l a r t o t h a t of B r e v i k i n t h a t h i s model d e s c r i b e d the motion i n a t u r b u l e n t wave boundary l a y e r near a rough bed by u s i n g a time i n v a r i a n t two l a y e r eddy v i s c o s i t y model. In the i n n e r l a y e r , the eddy v i s c o s i t y i n c r e a s e s q u a d r a d i c a l l y w i t h h e i g h t , and i n the o u t e r l a y e r , i t i s ta k e n t o be c o n s t a n t . The mean v e l o c i t y and shear s t r e s s as f u n c t i o n s of h e i g h t a re p r e s e n t e d w i t h the bottom shear s t r e s s as a f u n c t i o n of the phase a n g l e . The d a t a from Jonsson and C a r l s e n (1976) and Jdnsson (1980) were used f o r c o m p a r i s o n , and the r e s u l t s match the t h e o r e t i c a l p r e d i c t i o n r e a s o n a b l y w e l l . 3. 3 TRANSITION 3.3.1 INTRODUCTION The p o i n t of t r a n s i t i o n from l a m i n a r t o t u r b u l e n t motion i n o s c i l l a t o r y boundary l a y e r s i s d i f f i c u l t t o 57 d e t e r m i n e . E x p e r i m e n t a l d i f f e r e n c e s and s u b j e c t i v e d e f i n i t i o n s of t u r b u l e n c e account f o r the d i f f i c u l t i e s e x p e r i e n c e d . A s h o r t review of the more w e l l known r e s e a r c h e f f o r t s a r e i n c l u d e d h e r e . 3.3.2 SELECTED THEORETICAL AND EXPERIMENTAL ANALYSES The t r a n s i t i o n from l a m i n a r t o t u r b u l e n t motion i n o s c i l l a t o r y boundary l a y e r s has been e x p e r i m e n t a l l y i n v e s t i g a t e d u s i n g d i f f e r e n t o s c i l l a t i o n methods and o b s e r v a t i o n t e c h n i q u e s . The measurements of L i (1954) and Manohar (1955) were made u s i n g an o s c i l l a t i n g p l a t e i n s t i l l w a t e r , and v a r i o u s t y p e s of roughness e l e m e n t s . V i n c e n t (1958) o b s e r v e d t r a n s i t i o n c l o s e t o sand beds under waves i n a wave flume by examining dye d i s p e r s i o n . L ' H e r m i t t e (1958) a l s o used waves i n a wave fl u m e , and o b s e r v e d what he c a l l e d a p a r t i a l l y t u r b u l e n t / f u l l y t u r b u l e n t l i m i t . C o l l i n s (1963) measured mass t r a n s p o r t i n the o s c i l l a t o r y boundary l a y e r under waves and d e t e r m i n e d the t r a n s i t i o n t o be the p o i n t where the measured t r a n s p o r t began t o d i f f e r from the t h e o r e t i c a l p r e d i c t i o n of L o n g u e t - H i g g i n s (1953). Jonsson (1966,1980) performed a n a l y s e s based on the f u n c t i o n a l r e l a t i o n s h i p s between f a ° m and RE and between f and — r — , d e r i v e d from t h e water W s p a r t i c l e v e l o c i t y p r o f i l e s under waves over rough beds i n a wave flume. Kamphuis (1975) a l s o performed t h e s e a n a l y s e s but used the d a t a of R i e d e l , Kamphuis, and Brebner (1972), who performed e x p e r i m e n t s w i t h a shear p l a t e i n an 58 o s c i l l a t o r y water t u n n e l . Ramaprian and M u e l l e r (1980) used L a s e r D o p p l e r Anemometry t o measure wave p a r t i c l e v e l o c i t i e s near the bed under waves i n a wave flume. For the c o n d i t i o n of f u l l y d e v e l o p e d m i x i n g over a rough bed, the r e s u l t s of L i and Manohar f o r the c r i t i c a l t r a n s i t i o n have been p l o t t e d by S l e a t h (1974a), and f a l l a l o n g the c u r v e g i v e n by — — = i 00 (-£—)" 1 - 2 9 (3.12) w k 1 i n the range 0.03<-|—<5.0, i n which k,=^, where X i s the w a v e l e n g t h of the bed roughness. The r e s u l t s of V i n c e n t and L ' H e r m i t t e have a l s o been p l o t t e d by S l e a t h , and f a l l v e r y c l o s e t o the c u r v e g i v e n by e q u a t i o n ( 3 . 1 2 ) . K n i g h t (1978) has t e n t a t i v e l y s u g g e sted t h a t the p a r t i c l e d i a m e t e r , D, may be the same as the wavelength of the bed roughness, such t h a t e q u a t i o n (3.12) can be m o d i f i e d t o U 0 m 241 7^ ( 0 D ) O - 2 9 T h i s can a l s o be e x p r e s s e d as RE=5760(-i^l) •« 5  K s U ° m a ° m by u s i n g RE= and U 0 =ua 0 , where k i s the same as D. _J 3 v " m m s L i p roposed f o r a smooth f l a t bed the r e l a t i o n s h i p Do. 'm \Ju>v = 400 (3.13) or RE=1.6-10 5. I t i s noted t h a t the e q u a t i o n (3.12) f o r t r a n s i t i o n over a rough bed j o i n s smoothly o n t o the curve 59 f o r the smooth bed case as the roughness term approaches K1 z e r o . L i o b s e r v e d t h a t the t r a n s i t i o n between the l a m i n a r and t u r b u l e n t m otion i s not s h a r p l y d e f i n e d . V i n c e n t (1958) used a c t u a l wave boundary l a y e r s and used the dye d i s p e r s i o n t e c h n i q u e f o r o b s e r v i n g the t r a n s i t i o n t o t u r b u l e n c e . He found t h a t the t r a n s i t i o n t o t u r b u l e n c e o c c u r r e d much e a r l i e r than the l i m i t suggested by L i , a t a p p r o x i m a t e l y RE=6200. The l a r g e d i f f e r e n c e between h i s r e s u l t s and those of L i , he f e l t t o depend on e i t h e r the c r i t e r i o n of t r a n s i t i o n or more l i k e l y on the d i f f e r e n t e x p e r i m e n t a l methods used. V i n c e n t s p e c u l a t e d t h a t i f the bed has been p h y s i c a l l y "smooth", then t h e l i m i t of t r a n s i t i o n would have been a t a Reynolds number a t l e a s t as l a r g e as t h a t g i v e n by L i . S l e a t h (1974a) p o i n t e d out t h a t over the n a t u r a l bed used by V i n c e n t , the i n i t i a l f o r m a t i o n of tongues of dye r e p r e s e n t s t r a n s i t i o n around the l a r g e s t g r a i n s r a t h e r than over the bed as a whole, hence t r a n s i t i o n i s p e r c e i v e d as h a v i n g a lower l i m i t o v e r a rough bed. F u r t h e r , S l e a t h (1970) mentioned t h a t i n dye t e s t s most o b s e r v e r s f e e l t h a t the f l o w i s t u r b u l e n t w h i l e the r e a l c o n d i t i o n i s not r e a l t u r b u l e n c e , but l a m i n a r v o r t e x i n g around i n d i v i d u a l roughness e l e m e n t s . C o l l i n s (1963) proposed a v e r y d i f f e r e n t c u r v e from L i f o r the smooth bed c a s e , g i v i n g RE=1.28«10°. By measuring the mean l o n g i t u d i n a l mass t r a n s p o r t i n t h e o s c i l l a t o r y boundary l a y e r under l a b o r a t o r y waves, C o l l i n s d e t e r m i n e d 60 the t r a n s i t i o n t o be a t t h a t p o i n t where the measured t r a n s p o r t began t o d i f f e r from the t h e o r e t i c a l p r e d i c t i o n of L o n g u e t - H i g g i n s (1953). However, the e m p i r i c a l r e l a t i o n C o l l i n s p r e s e n t e d does not g i v e the c o r r e c t l i m i t o u t s i d e the range of e x p e r i m e n t a l c o n d i t i o n s f o r which i t was o b t a i n e d . S l e a t h (1974b) has shown the d i s c r e p a n c y observed t o be p u r e l y a l a m i n a r e f f e c t , and not an i n d i c a t i o n of t u r b u l e n c e . C o l l i n s ' r e s u l t s l i e between those g i v e n by L i and V i n c e n t , but V i n c e n t f e l t t h a t h i s r e s u l t s would have been at l e a s t as l a r g e as those g i v e n by L i , had V i n c e n t ' s bed been smooth. S l e a t h (1974a) a n a l y z e d the r e l a t i o n s of t h e s e r e s e a r c h e r s , and c o n c l u d e d t h a t t h e r e i s no reason not t o a c c e p t the c u r v e s as p r e s e n t e d by L i . More r e c e n t work by Ramaprian and M u e l l e r (1980) i n an o s c i l l a t o r y water t u n n e l u s i n g L a s e r Doppler Anemometry t o measure the v e l o c i t y p r o f i l e a c r o s s the o s c i l l a t o r y boundary l a y e r g i v e s another r e s u l t . T h e i r experiment i n d i c a t e d t h a t p u r e l y o s c i l l a t o r y f l o w over a smooth s u r f a c e can be t u r b u l e n t a t a Reynolds number of l e s s than RE=6.85•10 f t. However, the presence of t u r b u l e n c e was i n f e r r e d m a i n l y from the o b s e r v a t i o n of the i n s t a n t a n e o u s v e l o c i t y s i g n a l s from the f l o w , and the v e l o c i t y d i s t r i b u t i o n s were o n l y m i l d l y d i f f e r e n t from those p r e d i c t e d by l a m i n a r t h e o r y . T h i s type of f l o w i s s i m i l a r ' t o the weakly t u r b u l e n t c a t e g o r y of f l o w s seen i n o s c i l l a t o r y f l o w i n p i p e s . Hino e t . a l . (1976) o b s e r v e d two t r a n s i t i o n s i n o s c i l l a t o r y p i p e f l o w — one i n w hich the f l o w became weakly t u r b u l e n t , and one i n which the 61 f l o w i n t e r m i t t e n t l y took on f u l l y d e v e l o p e d t u r b u l e n c e c h a r a c t e r i s t i c s . Ramaprian and M u e l l e r f e l t t h a t they had p o s s i b l y observed the f i r s t t r a n s i t i o n , and a l s o t h a t C o l l i n s and V i n c e n t may have done the same. Jonsson (1966) performed an a n a l y s i s based on the a°m f u n c t i o n a l r e l a t i o n s of f vs RE and f vs —«—. These W W K S r e l a t i o n s were d e r i v e d u s i n g waves over s t a t i o n a r y rough beds i n a wave fl u m e , where the f r i c t i o n f a c t o r s were d e r i v e d from the v e l o c i t y p r o f i l e s over the bed. Jonsson d e f i n e d t h r e e f l o w c a t e g o r i e s , namely; l a m i n a r , smooth t u r b u l e n t , and rough t u r b u l e n t f l o w s . He proposed t h r e e t r a n s i t i o n s : from l a m i n a r t o smooth t u r b u l e n t , l a m i n a r t o rough t u r b u l e n t , and from smooth t u r b u l e n t t o rough t u r b u l e n t . For the l a m i n a r t o smooth t u r b u l e n t t r a n s i t i o n , he gave RE=1.26'10" f o r the b e g i n n i n g of t r a n s i t i o n , which i s v e r y c l o s e t o C o l l i n s ' r e s u l t , and RE=3'10" f o r the end of t r a n s i t i o n . For the t r a n s i t i o n from l a m i n a r t o rough '32 l k 2 ^ 0 t u r b u l e n t , Jonsson proposed RE=4^» (—j-^) 2 f o r the b e g i n n i n g of a°m k s t r a n s i t i o n and RE=500 (—r—) (-?) f o r the end of t r a n s i t i o n . For s 0 the t r a n s i t i o n from smooth t u r b u l e n t t o rough t u r b u l e n t , he gave the r e l a t i o n -£=0.287 f o r the onset of t r a n s i t i o n , and 77 2 . . RE= >\ f o r the end of t r a n s i t i o n . * w Jonsson (1980) has s i n c e changed th e s e v a l u e s t o b r i n g them more i n l i n e w i t h the r e s u l t s of o t h e r r e s e a r c h e r s . For the t r a n s i t i o n from l a m i n a r t o smooth t u r b u l e n t , he proposed RE=10 5. For the l a m i n a r t o rough t u r b u l e n t t r a n s i t i o n , he s u g g e s t e d use of the r e s u l t s of S l e a t h (1974a) g i v e n by 62 e q u a t i o n (3.12) t o be used over a rough bed. Jonsson e s t i m a t e d the bed roughness k g t o be about h a l f the 3 w a v e l e n g t h of the bed, so t h a t k , and he gave S K ^ RE=4130(-j^)°' 4 t s (3.14) s T h i s c u r v e f a l l s v e r y c l o s e t o the upper bound f o r t h i s t r a n s i t i o n as he had proposed e a r l i e r ( J o n s s o n , 1966). F i n a l l y , f o r the smooth t u r b u l e n t t o rough t u r b u l e n t t r a n s i t i o n , he suggested the use of the r e s u l t s o b t a i n e d by K a j i u r a (1968) f o r a l e s s rough" bed. K a j i u r a (1968) had g i v e n a t r a n s i t i o n l i m i t of RE=2000-J2 (3.14') K s f o r f u l l y d e v e l o p e d rough t u r b u l e n c e . T h i s r e s u l t i s s i m i l a r t o t h a t g i v e n by Kamphuis (1975), and i t g i v e s t r a n s i t i o n a t a f a r g r e a t e r Reynolds number than t h a t which Jonsson had p r e v i o u s l y chosen. Kamphuis (1975) performed the same s o r t af a n a l y s e s as J o n s s o n , except h i s f r i c t i o n v a l u e s were d e r i v e d from the shear s t r e s s e s measured w i t h a shear p l a t e i n an o s c i l l a t o r y water t u n n e l . These measurements were conducted by R i e d e l , Kamphuis, and Brebner ( 1 9 7 2 ) , and t h e i r f u n c t i o n a l r e l a t i o n s were r e a s o n a b l y s i m i l a r t o t h o s e found by J o n s s o n . They d e f i n e d the same t y p e s of f l o w as Jonsson had d e f i n e d , but o n l y two t r a n s i t i o n s were g i v e n - one from l a m i n a r t o smooth t u r b u l e n t f l o w , and the o t h e r from laminar/smooth t u r b u l e n t t o rough t u r b u l e n t . 63 Kamphuis proposed RE=10" f o r the t r a n s i t i o n from l a m i n a r t o smooth t u r b u l e n t . However, K n i g h t (1978), i n r e v i e w i n g the data of Kamphuis, f e l t t h a t the l a m i n a r range extended much f u r t h e r , and t h a t the c h o i c e s h o u l d have been a p p r o x i m a t e l y RE=1.6«10 5. Kamphuis found the lower l i m i t of the smooth t u r b u l e n t regime t o be RE=6*10 5, which c o r r e s p o n d s w e l l t o t h a t d e r i v e d by K a j i u r a (1968). For the t r a n s i t i o n from laminar/smooth t u r b u l e n t t o rough t u r b u l e n t , k U* he found t h a t the lower l i m i t was a p p r o x i m a t e l y —|—=15, or a°m 9 1 R E = 1 5 - T ^ ( T 2 - ) 7 s w k U* The upper l i m i t was g i v e n by —-—=70-200, or a°m 9 1 RE=(70-200)- jp( I^)' 2" s w k U* where the v a l u e of RE or tends towards the lower v a l u e f o r l a r g e —^—, which i s the v a l u e f o r u n i d i r e c t i o n a l f l o w , s The c u r v e i s s i m i l a r t o t h a t g i v e n by K a j i u r a as s uggested by Jonsson (1980). V o n g v i s e s s o m j a i (1985) summarized the c r i t i c a l v a l u e s of R e y n o l d s number f o r the i n c e p t i o n of t u r b u l e n c e on smooth beds. He noted t h a t the v a l u e s d e r i v e d from o s c i l l a t i n g bed e x p e r i m e n t s a r e g e n e r a l l y about t e n t i m e s h i g h e r than those d e r i v e d from o s c i l l a t i n g f l u i d e x p e r i m e n t s . Other i n v e s t i g a t o r s made s i m i l a r c l a i m s as w e l l . C o l l i n s (1963) and L ' H e r m i t t e (1958) f e l t t h a t t h e r e are b a s i c d i f f e r e n c e s i n t r a n s i t i o n between f l o w s under r e a l waves and f l o w s over o s c i l l a t i n g "beds. V i n c e n t (1958) f e l t t h a t the o s c i l l a t i n g 64 p l a t e work of L i may have caused the d i s c r e p a n c i e s i n t h e i r r e s u l t s , though he b e l i e v e d t h a t the d i f f e r e n c e i n bed roughness was the s i g n i f i c a n t r e a s o n . However, V o n g v i s e s s o m j a i f e l t t h a t the d i f f e r e n c e s between o s c i l l a t i n g bed t r a n s i t i o n and wave boundary l a y e r t r a n s i t i o n would be s l i g h t a t the low v e l o c i t i e s observed i n the d a t a he r e v i e w e d . 3.3.3 CONCLUSIONS The t r a n s i t i o n zones have been g i v e n by v a r i o u s i n v e s t i g a t o r s u s i n g d i f f e r e n t o b s e r v a t i o n and c a l c u l a t i o n t e c h n i q u e s . The c r i t i c a l v a l u e s f o r the t r a n s i t i o n s a r e d i f f i c u l t t o d e s c r i b e because the c r i t e r i o n f o r t r a n s i t i o n i s not d e f i n e d , and because of the d i f f e r i n g e x p e r i m e n t a l c o n d i t i o n s used. Over a smooth bed i t i s recommended t h a t the r e s u l t s of L i ( 1 9 5 4 ) , as g i v e n by e q u a t i o n ( 3 . 1 3 ) , a r e used t o d e s c r i b e the l a m i n a r t o smooth t r a n s i t i o n . For a rough bed the r e s u l t s of S l e a t h (1974a), as used by Jonsson (1980) and as g i v e n by e q u a t i o n ( 3 . 1 4 ) , are p r e f e r r e d f o r d e s c r i b i n g t r a n s i t i o n from l a m i n a r t o rough t u r b u l e n t f l o w . For a l e s s rough bed, t h o s e of K a j i u r a (1968), g i v e n by e q u a t i o n (3.14') a r e used t o d e s c r i b e t h e t r a n s i t i o n from smooth t u r b u l e n t t o rough t u r b u l e n t f l o w . In the w e l l d e f i n e d f l o w a r e a s , the shear s t r e s s e s are g i v e n by v a r i o u s e q u a t i o n s , and w i l l be d e s c r i b e d i m m e d i a t e l y . In the t r a n s i t i o n a r e a s , d i s c r e t i o n must be 65 used when a p p l y i n g any of the shear s t r e s s f ormulae t o be d e v e l o p e d . The f l o w p a t t e r n i n the t r a n s i t i o n r e g i o n s h o u l d be c l o s e l y m o n i t o r e d b e f o r e any g e n e r a l s t a t e m e n t s about the f l o w type can be made. There appears t o be a d i f f e r e n c e i n the t r a n s i t i o n R e y n o l d s number between o s c i l l a t i n g p l a t e and wave boundary l a y e r e x p e r i m e n t s , though t h i s d i f f e r e n c e cannot p r e s e n t l y be c o n f i r m e d or e x p l a i n e d i n more than a s p e c u l a t i v e manner. 3. 4 SHEAR STRESS EQUATIONS From the l i n e a r i z e d N a v i e r - S t o k e s e q u a t i o n f o r f l o w i n a t h i n o s c i l l a t o r y boundary l a y e r where the c o n v e c t i v e terms a r e n e g l e c t e d , i t i s a s i m p l e m a t t e r t o o b t a i n the l i n e a r i z e d e q u a t i o n of motion i n the o s c i l l a t o r y boundary l a y e r i n terms of shear s t r e s s : - | u + | U + l | r ( 3 < 1 5 ) 9t 9t p9y E q u a t i o n (3.15) i s o b t a i n e d by c o m b i n i n g the l i n e a r i z e d e q u a t i o n of motion (3.2) w i t h the e q u a t i o n f o r the l a m i n a r shear s t r e s s 1=4"- (3.16) P 9y An i n t e g r a t i o n of e q u a t i o n (3.15) y i e l d s I=-/|-(U-u)dy (3.17) P y 9t where the h e i g h t d c o r r e s p o n d s t o the d e p t h where the shear s t r e s s i n the f l o w i s z e r o . I f the v e l o c i t y f i e l d i s 66 measured i n the boundary l a y e r , then the shear s t r e s s i n the l a y e r can be c a l c u l a t e d u s i n g e q u a t i o n ( 3 . 1 7 ) . 3.4.1 LAMINAR SHEAR STRESS EQUATIONS For l a m i n a r f l o w , the s o l u t i o n t o the l i n e a r i z e d N a v i e r - S t o k e s e q u a t i o n (3.2) has a l r e a d y been found f o r the v e l o c i t y p r o f i l e u=U ( c o s ( k x - w t ) - e x p ( - 7 j ) c o s ( k x - c j t - 7 j ) ) (3.3) m Combining e q u a t i o n (3.3) w i t h the e q u a t i o n f o r the l a m i n a r shear s t r e s s ( 3 . 1 6 ) , Jonsson (1963) o b t a i n e d ^=i/2j/j3U mexp(-0y)cos(kx-a;t-j3y+!) (3.18) The maximum shear s t r e s s a t the w a l l i s then g i v e n by —^•=^2vPUm (3.19) where the s u b s c r i p t " 0 " s i g n i f i e s a q u a n t i t y a t the w a l l . U s i n g the d e f i n i t i o n of the f r i c t i o n f a c t o r , f , as i n t r o d u c e d by Jonsson (1966), we have — = ^ U m 2 (3.20) p 2 m Combining (3.19) and ( 3 . 2 0 ) , we o b t a i n V T R E • • • • • ( 3 - 2 1 ) . which i s t h e f r i c t i o n . f a c t o r f o r a l a m i n a r boundary l a y e r i n o s c i l l a t o r y f l o w , where RE i s an a m p l i t u d e Reynolds number, U ° m a ° m RE= . Here i t has been assumed by t h i s w r i t e r t h a t the 67 boundary l a y e r i s v e r y t h i n , and as l a b o r a t o r y waves are used t o ge n e r a t e t o g e n e r a t e the o s c i l l a t o r y f l o w , the U 0 m and U*m a r e assumed t o be i n t e r c h a n g e a b l e . Kamphuis (1975) made measurements on the shear s t r e s s on the w a l l i n an o s c i l l a t o r y f l o w tank by use of a shear p l a t e , and found t h a t w i t h i n the l a m i n a r range, the agreement between the t h e o r y ( e q u a t i o n 3.21) and the e x p e r i m e n t a l r e s u l t s was v e r y good. 3.4.2 ROUGH TURBULENT SHEAR STRESS EQUATIONS F u l l y analogous t o l a m i n a r f l o w , the l i n e a r i z e d N a v i e r - S t o k e s e q u a t i o n i n a t h i n o s c i l l a t o r y boundary l a y e r i s au.au,. a 2u at + at  eW - £ H + ™ + e £ _ £ = 0 (3.22) U s i n g K a j i u r a ' s e q u a t i o n f o r rough t u r b u l e n t shear s t r e s s and r e a l i z i n g t h a t e and K y a r e both terms r e f l e c t i n g the k i n e m a t i c eddy v i s c o s i t y and a r e e q u i v a l e n t , we a g a i n o b t a i n - | ^ + | U + l | l = o (3.15) at at pay So f o r rough t u r b u l e n t f l o w , we o b t a i n by i n t e g r a t i o n of e q u a t i o n (3.11) an e x p r e s s i o n f o r the shear s t r e s s •5--/|t(0-u)dy (3.17) which i s t h e same as f o r l a m i n a r f l o w . 68 K a j i u r a (1968) i n s e r t e d e q u a t i o n (3.17) i n t o e q u a t i o n (3.8) t o get 1 y By s u b s t i t u t i n g h i s v a l u e of Ky i n t o e q u a t i o n ( 3 . 2 3 ) , K a j i u r a was a b l e t o s o l v e i t f o r u and U*. Jonsson (1963) s i m p l y measured the v e l o c i t y p r o f i l e and c a l c u l a t e d the shear s t r e s s everywhere d i r e c t l y from e q u a t i o n ( 3 . 1 7 ) . He assumed t h a t t h e steady s t a t e e x p r e s s i o n f o r the t u r b u l e n t v e l o c i t y p r o f i l e over a rough bottom i s v a l i d near the bottom, such t h a t ^ = 5 . 7 5 1 o g ^ ..(3.24) where y i s the h e i g h t above the t h e o r e t i c a l bed l e v e l , which i s yo=-krQf where k i s the roughness p a r a m e t e r . Jonsson o b s e r v e d t h a t e q u a t i o n (3.17) y i e l d s the bottom shear s t r e s s as ^ = - j 4 r ( U - u ) d y (3.25) P (j^dt where g y = 5 . 7 5 1 o g ^ (3.26) As U-u=5.75U*log^, then a f t e r assuming t h a t k<<306 and 6=0 f o r t=0, and n e g l e c t i n g s m a l l i t e m s , Jonsson and C a r l s e n (1976) o b t a i n 69 U * 6 = U ^ U * 2 d t (3.27) Xb. /b I n s e r t i n g (3.26) i n t o (3.27) and u s i n g U=U 0 mcos (kx-a>t) U * 5 = 5 - 7 5 ^ U ° i n 2 ^ C O S 2 ( k 3 o f ) d t ( 3 ' 2 8 ) b. /b m 0 l o g 2 ( £ U ^ ) and u * 6 = T r 4 = 4 u 0 2 1 ( t . s i n 2 ( k x - c t ) ) b . . ( 3 b 2 9 ) log (-y> U s i n g 6, f o r the h e i g h t where U=U 0 m, then by combining (3.26) and (3. 2 9 ) , and u t i l i z i n g U 0 =wa 0 , they o b t a i n 3 um m •* 30&T n 306, a ° m _ n x s Jonsson (1980) a p p r o x i m a t e d e q u a t i o n (3.30) w i t h e q u a t i o n ( 3 . 1 1 ) , and s t a t e d t h a t f o r rough t u r b u l e n t f l o w , 6 i s a p p r o x i m a t e l y 2 t o 4% of the wave p a r t i c l e a m p l i t u d e a t the bed. As the f r i c t i o n f a c t o r has been d e f i n e d as < 3 - 3 1 > m then u s i n g e q u a t i o n (3.31) and e q u a t i o n (3.26) t - 2 & ) * - °-0*0* (3.32) w Uo m l o g2 (306_ L ) Combining (3.30) and (3.32) y i e l d s a p p r o x i m a t e l y ^ r - + l o g ( I ^ - ) = m f + l o g ( ^ ) (3.33) where m^  i s a f a c t o r d e t e r m i n e d from measurement, and t u r n s out t o be -0.08 as e a r l i e r d e t e r m i n e d by Jonsson and 70 Lundgren (1964). K a j i u r a (1968) deduced a s i m i l a r e x p r e s s i o n over a rough bed 4.05v/f + l o g ( ^ r - ) = - 0 . 2 5 4 + l o g ( ^ ) (3.34) w w S Measurements made by Kamphuis (1975) y i e l d e d the e m p i r i c a l r e l a t i o n g i v e n by ^ T - + l o g ( ^ ) = - 0 . 3 5 + | l o g ( ^ 1 ) (3.35) a°m which i s q u i t e c l o s e t o the e q u a t i o n (3.33) f o r —^—<50. s 3.4.3 SMOOTH TURBULENT SHEAR STRESS EQUATIONS Over a smooth bed i n t u r b u l e n t f l o w , K a j i u r a (1968) o b t a i n e d j-Q^ 7 ?-+21og(^|-)=-0.270 + 21og(RE)' z" (3.36) w h i l e Jonsson (1966) o b t a i n e d 47f- + 2 l°g(47f->=- 1 -55 + l o g ( R E ) (3.37) Measurements by Kamphuis (1975) y i e l d e d r e s u l t s which p l o t 25 t o 30% below the c u r v e p r e d i c t e d by K a j i u r a i n e q u a t i o n ( 3 . 3 6 ) . 3.4.4 CONCLUSIONS-C o n s i s t e n c y between the s e m i - e m p i r i c a l r e l a t i o n s o b t a i n e d by v a r i o u s i n v e s t i g a t o r s s u g g e s t s t h a t the f r i c t i o n f a c t o r s g i v e n by the f o r e g o i n g r e l a t i o n s a r e c o r r e c t t o a 7 1 f a i r degree of a c c u r a c y . The shear s t r e s s a t the w a l l i n o s c i l l a t o r y boundary l a y e r s i s then g i v e n by e q u a t i o n ( 3 . 2 0 ) u s i n g the a p p r o p r i a t e f r i c t i o n f a c t o r s . The ranges of v a l i d i t y of the s e f r i c t i o n f a c t o r s have been d e t e r m i n e d , as the y a r e d e l i n e a t e d by the t r a n s i t i o n r e g i o n b o u n d a r i e s . 3. 5 CONCLUSIONS T h i s c h a p t e r has d e s c r i b e d the v e l o c i t y p r o f i l e s i n o s c i l l a t o r y f l o w f o r both the l a m i n a r and the t u r b u l e n t c a s e s . Some d e r i v a t i o n of the shear s t r e s s r e l a t i o n s have been p r o v i d e d t o show how t h i s i s done. S e v e r a l d i f f e r e n t r e s u l t s have a l s o been p r e s e n t e d , showing the c l o s e agreement between the works of v a r i o u s i n v e s t i g a t o r s . I t i s seen t h a t the d i f f e r e n t e q u a t i o n s p r e s e n t e d g i v e shear s t r e s s e s t h a t a r e i n r e a s o n a b l e agreement. A l s o , some r e l a t i o n s a r e g i v e n which d e s c r i b e the t r a n s i t i o n under o s c i l l a t o r y f l o w . These f o r m u l a e a r e i n c l o s e agreement among i n v e s t i g a t o r s f o r l a m i n a r f l o w , but t h e r e i s some q u e s t i o n as t o the t r a n s i t i o n i n t u r b u l e n t f l o w over 'a rough bed. One must be c a r e f u l t o observe the t r a n s i t i o n , and not c o n f u s e i t w i t h l a m i n a r v o r t e x i n g around the rough bed f e a t u r e s . 4. WAVE THEORIES 4. I GOVERNING EQUATIONS 4.1.1 INTRODUCTION A C a r t e s i a n c o o r d i n a t e system ( x , y , z ) i s d e f i n e d w i t h x measured i n the d i r e c t i o n of wave p r o p a g a t i o n , y measured upwards from the s t i l l water l e v e l , and z o r t h o g o n a l t o x and y. The waves a r e assumed t o be t w o - d i m e n s i o n a l i n the x-y p l a n e and t o t r a v e l i n the p o s i t i v e x d i r e c t i o n over a smooth h o r i z o n t a l bed. The f l u i d i s taken t o be i n c o m p r e s s i b l e and i n v i s c i d , and the f l o w t o be i r r o t a t i o n a l . F i g u r e 4.1 shows . a wave t r a i n of permanent form w i t h wave h e i g h t H and w a v e l e n g t h L i n a f l u i d of u n d i s t u r b e d depth d. The wave p e r i o d T i s the time i n t e r v a l between s u c c e s s i v e c r e s t s p a s s i n g a s t a t i o n a r y p o i n t and the wave speed c i s t h e speed of the wave t r a v e l l i n g t h r o u g h the f l u i d . The v e r t i c a l d i s t a n c e above the s t i l l water l e v e l , Wove speed, c , y Sur fac shown L d e e levat ion at t = 0 F i g u r e 4.1. D e f i n i t i o n s k e t c h f o r a p r o g r e s s i v e wave t r a i n . 72 73 y=0, t o the f l u i d s u r f a c e i s g i v e n by r ? ( x , t ) . The a n g u l a r 2 7T 2 fr e q u e n c y of the wave, CJ=-=;, and the wave number, k = — a r e two c o n v e n i e n t terms used f o r wave d e s c r i p t i o n , and a r e such t h a t oj=kc. The v e l o c i t y p o t e n t i a l , <t>, f o r the f l u i d r e g i o n must be d e t e r m i n e d . T h i s p o t e n t i a l s a t i s f i e s the L a p l a c e e q u a t i o n ¥4+^4=0 (4 1) s u b j e c t t o the boundary c o n d i t i o n s §|=0 a t y=-d (4.2) || + ^ [ ( | | ) 2 + ( | | ) 2 ] + g 7 ? = f { t ) a f c y = 7 ? ( 4 > 4 ) <^(x,y , t ) = ^ ( x - c t , y ) (4.5) E q u a t i o n (4.1) i s v a l i d and the p o t e n t i a l <j> e x i s t s because of the ass u m p t i o n s of i n c o m p r e s s i b i l i t y and i r r o t a t i o n a l i t y of the f l u i d . E q u a t i o n (4.2) r e f l e c t s the c o n d i t i o n a t the seabed t h a t t h e r e can be no v e r t i c a l v e l o c i t y component. The k i n e m a t i c s a t the f r e e s u r f a c e a r e d e s c r i b e d by e q u a t i o n (4.3) i n t h a t the f l u i d v e l o c i t y normal t o the f r e e s u r f a c e must be e q u a l t o the v e l o c i t y of the f r e e s u r f a c e i t s e l f i n t h a t d i r e c t i o n . The dynamic c o n d i t i o n s a t the f r e e s u r f a c e a r e d e s c r i b e d by e q u a t i o n (4.4) which s t a t e s t h a t the f r e e s u r f a c e p r e s s u r e 74 i s c o n s t a n t . F i n a l l y , the p e r i o d i c n a t u r e of the wave t r a i n i s d e s c r i b e d by e q u a t i o n ( 4 . 5 ) . 4.1.2 SMALL AMPLITUDE WAVE THEORY An e x a c t s o l u t i o n f o r the wave t r a i n i s d i f f i c u l t t o o b t a i n because of the n o n l i n e a r i t y of the f r e e s u r f a c e boundary c o n d i t i o n s , and because the f r e e s u r f a c e p o s i t i o n i s u s u a l l y unknown. W i t h t h i s a s s u m p t i o n , the n o n l i n e a r terms i n e q u a t i o n s (4.3) and (4.4) a r e then v e r y s m a l l i n comparison w i t h the r e m a i n i n g l i n e a r terms, and they can be n e g l e c t e d . The second problem i s a l s o s o l v e d , because the f r e e s u r f a c e i s now assumed t o be a t the s t i l l water l e v e l , y=0, and t h e r e the f r e e s u r f a c e boundary c o n d i t i o n s can be a p p l i e d . The f r e e s u r f a c e boundary c o n d i t i o n s as g i v e n i n e q u a t i o n s (4.3) and (4.4) a r e now d<j>_ 3r? By 3t = 0 a t y=0 (4.6) and 30 at +gT?=0 at y=0 (4.7) which combine t o g i v e a t y=0 (4.8) and (4.9) 75 The s o l u t i o n t o the L a p l a c e e q u a t i o n ( 4 . 1 ) , s u b j e c t t o the boundary c o n d i t i o n e q u a t i o n s ( 4 . 2 ) , ( 4 . 5 ) , ( 4 . 8 ) , and ( 4 . 9 ) , i s g i v e n by * - < S > c o c ' ^ h { K ) ) ) « 1 " ' ^ > ••• <4-10> such t h a t c o m b ining e q u a t i o n s (4.8) and (4.10) r e s u l t i n the l i n e a r d i s p e r s i o n r e l a t i o n c 2 = f e a n h ( k d ) (4.11) £. IT such t h a t e q u a t i o n (4.10) can a l s o be e x p r e s s e d as • • ' C T ' ^ h l K ) " ' 1 " ' 1 " - ^ ' u - ' 2 > A complete s o l u t i o n has now been o b t a i n e d f o r the v e l o c i t y p o t e n t i a l , and the r e s u l t s can be e a s i l y o b t a i n e d f o r the p a r t i c l e d i s p l a c e m e n t s , v e l o c i t i e s , and a c c e l e r a t i o n s , and f o r the f l u i d p r e s s u r e , energy f l u x and d e n s i t y , and the r a d i a t i o n s t r e s s e s . R e s u l t s of l i n e a r wave t h e o r y a r e p r e s e n t e d i n T a b l e 4.1. 4.1.3 STOKES FINITE AMPLITUDE WAVE THEORY Stokes (1847,1880) c o n s i d e r e d a p e r t u r b a t i o n p r o c e d u r e t o d e v e l o p s u c c e s s i v e a p p r o x i m a t i o n s such t h a t the v a r i a b l e s d e s c r i b i n g the f l o w a r e expanded as a power s e r i e s . I t i s assumed t h a t the v e l o c i t y p o t e n t i a l <j> and the a s s o c i a t e d v a r i a b l e s (7j, u, v, ...) may be w r i t t e n as <l>=€<t>^ + e2<j>2 + (4.13) 76 V e l o c i t y P o t e n t i a l D i s p e r s i o n R e l a t i o n S u r f a c e E l e v a t i o n H o r i z o n t a l P a r t i c l e D i s p l a c e m e n t V e r t i c a l P a r t i c l e D i s p l a c e m e n t H o r i z o n t a l P a r t i c l e V e l o c i t y V e r t i c a l P a r t i c l e V e l o c i t y H o r i z o n t a l P a r t i c l e A c c e l e r a t i o n V e r t i c a l P a r t i c l e A c c e l e r a t i o n P r e s s u r e T a b l e 4.1. ( M o d i f i e d a f t e r Sarpkaya and I s a a c s o n , 1981). R e s u l t s of l i n e a r wave t h e o r y . where e i s the p e r t u r b a t i o n parameter. Each v a r i a b l e <t>-i ,4>i > • - •<t>n> i s assumed t o be of the same o r d e r of magnitude such t h a t each a d d i t i o n a l term i n the s e r i e s i s s m a l l e r than the p r e v i o u s one by the f a c t o r e. P e r e g r i n e (1972) has shown t h a t t h i s method of e x p a n s i o n i s v a l i d f o r L < < 1 ' A N D ^<<(kd) 2 f o r kd<1, and the s e c o n d i t i o n s impose a wave h e i g h t r e s t r i c t i o n i n s h a l l o w water. Sarpkaya and I s a a c s o n (1981) have shown the way i n which the r e s u l t i n g e q u a t i o n s c r e a t e d by i n s e r t i n g e q u a t i o n (4.13) i n t o e q u a t i o n s (4.1) th r o u g h (4.5) a r e s o l v e d f o r the f i r s t and second o r d e r s . The f i r s t o r d e r TTH c o s h ( k s ) . m *-kT s i n h ( k d ) s i n ( 6 ) c 2=^4=|tanh(kd) 7?=jCOS ( 6 ) { _ _ H c o s h ( k s ) . m *~ 2 s i n h ( k d ) s i n ( 0 ) ,_H s i n h ( k s ) , a^ S"2 s i n h ( k d ) c o s ( g ) 7rH c o s h ( k s ) / a x u = " f s i n h ( k d ) c o s ( g ) fi_7rH s i n M k s ) . , .. u " s i n h ( k d ) s i n ( e ) 9u 27r 2 H cosh( ks) „ • , „ n Tt"~W s i n h ( k d ) S i n m dv 27r 2 H s i n h ( k s ) , D^ dt = —T*~ s i n h ( k d ) c o s ( 6 ° .1 ..cosh(ks) /„ x P = - P 9 y ^ g H C o s h ( k d ) c o s ( e ) s=y+d 77 s o l u t i o n i s i d e n t i c a l t o the s o l u t i o n f o r l i n e a r t h e o r y , and the r e s u l t s of the second o r d e r t h e o r y a r e g i v e n i n T a b l e 4.2. For s t e e p e r waves i n s h a l l o w water, the second o r d e r terms become l a r g e and the second o r d e r t h e o r y becomes i n v a l i d . F i f t h o r d e r Stokes t h e o r y as p r e s e n t e d by S k j e l b r e i a and H e n d r i c k s o n (1960) i s a p p l i c a b l e t o e n g i n e e r i n g p r a c t i c e , though the v a l u e s of kd and a parameter X must be e v a l u a t e d n u m e r i c a l l y by an i t e r a t i v e p r o c e d u r e . Sarpkaya and I s a a c s o n (1981) r e p o r t e d t h a t the development of a computer program t o de t e r m i n e the c h a r a c t e r i s t i c s of the f i f t h o r d e r t h e o r y i s not d i f f i c u l t . As f o r the second o r d e r S t o k e s t h e o r y , the h i g h e r o r d e r terms become l a r g e and the t h e o r y becomes i n v a l i d f o r s t e e p e r waves i n s h a l l o w w a t e r . 4.1.4 NONLINEAR SHALLOW WATER THEORIES N o n l i n e a r s h a l l o w water t h e o r i e s have been d e v e l o p e d f o r l o n g s h a l l o w water waves because the St o k e s t h e o r i e s a r e ina d e q u a t e f o r s t e e p waves i n s h a l l o w water. Korteweg and de V r i e s (1895) d e v e l o p e d the c n o i d a l wave t h e o r y i n which the wave c h a r a c t e r i s t i c s a r e e x p r e s s e d i n terms of the J a c o b i a n e l l i p t i c f u n c t i o n , c . Second and t h i r d n a p p r o x i m a t i o n s t o c n o i d a l t h e o r y have been made by L a i t o n e (1961) and Chappelear (1962) r e s p e c t i v e l y , and Fenton (1979) developed a t h e o r y c a p a b l e of use t o any o r d e r . 78 V e l o c i t y P o t e n t i a l D i s p e r s i o n R e l a t i o n S u r f a c e E l e v a t i o n H o r i z o n t a l P a r t i c l e D i splacement V e r t i c a l P a r t i c l e D i s p l a c e m e n t H o r i z o n t a l P a r t i c l e V e l o c i t y V e r t i c a l P a r t i c l e V e l o c i t y H o r i z o n t a l P a r t i c l e A c c e l e r a t i o n V e r t i c a l P a r t i c l e A c c e l e r a t i o n P r e s s u r e , 7TH c o s h ( k s ) „ - i a \ * = k T s i n h ( k d ) s i n ( e ) .3 7rH/7rHvCOsh(2ks)^.. , 0 / 3 v + 8 T c T ( - L ) s i n h * ( k d ) s i n ( 2 g ) _ 2 _ Cl)' H =£tanh(kd) 7j=gcos(0) + g ( ^ ) s W ( k S ) [ 2 + C 0 S h ( 2 k d ) 3cos(2e) , H cosh(ks)„ . / 0x * = _ 2 s i n h ( k d ) s i n ( g ) • H ,.. / 7rH \ 1 f 1 3 c o s h ( 2 k s ) / o f lx + 8 ( ^ ) s i n h ^ ( k d ) [ 1 ' 2 s i n h M k d ) ] s i n ( 2 g ) u=-.H,*rHxC0sh(2ks) / . x + 4 ( ~ ) s i n h M k d ) ( u ) t ) t_H s i n h ( k s ) , f l )  S _ 2 s i n h ( k d ) c o s m 3H / irH x sijTh{2ks) i o a\ 7rH c o s M k s ) ,,x s i n h ( k d ) c o s ( e ) TTH , wH x c o s h ( 2 k s ) „ , 0 fl x - T ( - L ) s i n h " ( k d ) C 0 5 ( 2 e ) s i n h ( k s ) • , p x f s i n h ( k d ) s i n ( e ) ,3 7rH,7rHxSinh(2ks). . , 0 0 x 9u 27r2H c o s h ( k s ) • , f l x . 3 7r2 H / 7rH x c ^ s h ( 2 k s ) _ . , - Q x + 1 F - ( ^ ; ) S i n h M k d ) S i n ( 2 g ) y=-4 7TH 3t" 27r2H s i n h ( k s ) /flx, -T»- s i n h ( k d ) c o s ( g ) 37r 2H,7rHxSinh(2ks) L ' s i n h M k d ) c o s ( 2 0 ) .3 _ t W f f H x 1 r c o s h ( 2 k s ) 1 V g H ( T ) s i n h ( 2 k d ) [ s i n h ^ ( k d ) 3 ] c ° 5 ( 2 g ) - ^ H ( I £ ) S i n h ( 2 k d ) [ c 0 s h ( 2 k 5 ) - 1 3 s=y+d T a b l e 4.2. ( M o d i f i e d a f t e r Sarpkaya and I s a a c s o n , 1981). R e s u l t s of S t o k e s second o r d e r t h e o r y . 79 The s o l i t a r y wave i s a s p e c i a l case of the c n o i d a l wave i n which the wavelength and wave p e r i o d a r e i n f i n i t e . B o u s s i n e s q (1871) f i r s t a n a l y z e d the s o l i t a r y wave, and Munk (1949) gave a d e s c r i p t i o n of the t h e o r y w i t h the i n t e n t i o n of making the t h e o r y a p p l i c a b l e t o e n g i n e e r i n g use. 4.1.5 STREAM FUNCTION THEORY Stream f u n c t i o n t h e o r y uses a n u m e r i c a l method t o p r e d i c t t w o - d i m e n s i o n a l wave c h a r a c t e r i s t i c s , and i s based on a stream f u n c t i o n r e p r e s e n t a t i o n of the f l o w . T h i s method was p r e s e n t e d by Dean (1965), who t r i e d t o s o l v e how t o n u m e r i c a l l y o b t a i n stream f u n c t i o n r e p r e s e n t a t i o n s of a wave w i t h a g i v e n p r o f i l e , and of a p e r i o d i c wave t r a i n of permanent form. The stream f u n c t i o n , \jj=\p(x ,y) , must s a t i s f y the L a p l a c e e q u a t i o n w i t h i n the f l u i d o2i>. o24> (4.14) 3T2" ay"2" s u b j e c t t o the boundary c o n d i t i o n s a t y=-d (4.15) u = u a t y=7] (4.16) ^_(u2 + l )2) + J ? = Q a t y = 7 ? (4.17) where Q i s a B e r n o u l l i c o n s t a n t . The stream f u n c t i o n i s 80 assumed t o be of the form i//(x ,y )=cy+ZX nsinh(nk (y+d) ) c o s ( n k x ) (4.18) E q u a t i o n (4.18) s a t i s f i e s e q u a t i o n s (4.14) t h r o u g h ( 4 . 1 6 ) , the l a t t e r p r o v i d i n g t h a t i//(x,y) i s a c o n s t a n t a t the s u r f a c e , g i v e n by such t h a t the s u r f a c e v a l u e i s g i v e n by \p =cT?+IXnsinh(nk(r?+d) ) c o s ( n k x ) (4.19) We must determine the c o e f f i c i e n t s X , the wavenumber k, and n the s u r f a c e v a l u e of the stream f u n c t i o n , by u s i n g the dynamic f r e e s u r f a c e boundary c o n d i t i o n g i v e n by e q u a t i o n ( 4 . 1 7 ) . Guesses a r e made f o r X, and k, and s u c c e s s i v e a p p r o x i m a t i o n s f o r X n a r e o b t a i n e d u s i n g a s u i t a b l e n u m e r i c a l p r o c e d u r e . An i t e r a t i o n p r o c e s s i s used whereby an e r r o r of f i t i s c a l c u l a t e d f o r the B e r n o u l l i c o n s t a n t f o r each s t e p and i s m i n i m i z e d over the c o u r s e of i t e r a t i o n , such t h a t the e s t i m a t e s f o r X a r e s u c c e s s i v e l y n J improved. 4.1.6 CONCLUSIONS Wave t h e o r y r e q u i r e s an i n c o m p r e s s i b l e , i n v i s c i d f l u i d h a v i n g i r r o t a t i o n a l f l o w . The wave t r a i n must be p r o g r e s s i v e and t w o - d i m e n s i o n a l , t r a v e l l i n g i n water of c o n s t a n t d e p t h . There e x i s t s " a v e l o c i t y p o t e n t i a l , <j>, t h a t s a t i s f i e s t he L a p l a c e e q u a t i o n , s u b j e c t t o s e v e r a l boundary c o n d i t i o n s . 8 1 S e v e r a l wave t h e o r i e s have been d e s c r i b e d h e r e . For a more i n - d e p t h d e s c r i p t i o n of the s e t h e o r i e s , the reader i s r e f e r r e d t o Sarpkaya and I s a a c s o n (1981), who a l s o d e s c r i b e d l e s s w e l l known t h e o r i e s and some c o m p u t a t i o n a l methods. 4. 2 COMPARISON OF WAVE THEORIES 4.2.1 INTRODUCTION The c h o i c e of the most s u i t a b l e wave t h e o r y i s d i f f i c u l t t o make. The f i r s t problem i s t h a t f o r a s p e c i f i c wave t r a i n , d i f f e r e n t wave t h e o r i e s w i l l a d e q u a t e l y reproduce d i f f e r e n t c h a r a c t e r i s t i c s of i n t e r e s t . For example, one t h e o r y may d e s c r i b e the h o r i z o n t a l water p a r t i c l e v e l o c i t i e s v e r y w e l l but f a i l t o p r o v i d e a proper p r e s s u r e d i s t r i b u t i o n , whereas a second t h e o r y w i l l a c c u r a t e l y p r e d i c t the p r e s s u r e s but i n a d e q u a t e l y p r e d i c t the h o r i z o n t a l v e l o c i t i e s . A comparison between t h e o r i e s must be made f o r a p a r t i c u l a r c h a r a c t e r i s t i c , and no g e n e r a l i z a t i o n s can be made r e g a r d i n g the comparison of th e s e t h e o r i e s f o r o t h e r c h a r a c t e r i s t i c s . A nother p o i n t t o c o n s i d e r i s t h a t the most s u i t a b l e wave t h e o r y may not be the one t h a t i s s i m p l y the most a c c u r a t e . The g o v e r n i n g c r i t e r i o n i n a g i v e n e n g i n e e r i n g a p p l i c a t i o n may be t o choose a t h e o r y t h a t i s s i m p l e and c o n v e n i e n t t o use, a t the c o s t of some a c c u r a c y . 82 4.2.2 COMPARISONS BASED ON THEORY Dean (1970) t h e o r e t i c a l l y compared s e v e r a l t h e o r i e s u s i n g the c r i t e r i o n of t h e c l o s e n e s s of f i t of the p r e d i c t e d m o tion t o the two n o n l i n e a r f r e e s u r f a c e boundary c o n d i t i o n s . The c l o s e n e s s of f i t c r i t e r i o n does not need t o be a p p l i e d t o the bottom boundary c o n d i t i o n or t o the L a p l a c e e q u a t i o n , as t hey a r e s a t i s f i e d e x a c t l y i n the t h e o r i e s Dean c o n s i d e r e d : l i n e a r wave t h e o r y , S t o k es t h i r d and f i f t h o r d e r t h e o r i e s , the f i r s t and second a p p r o x i m a t i o n s t o b oth c n o i d a l and s o l i t a r y wave t h e o r y , and the s t r e a m f u n c t i o n t h e o r i e s . He found t h a t the most s u i t a b l e t h e o r i e s are l i n e a r wave t h e o r y , S t o k e s f i f t h o r d e r t h e o r y , f i r s t o r d e r c n o i d a l t h e o r y , and the stream f u n c t i o n t h e o r i e s , and the'ir ranges of u s e f u l n e s s a r e i n d i c a t e d i n F i g u r e 4.2, from Sarpkaya and I s a a c s o n . Le Mehaute (1976) based a p l o t of the ranges of v a l i d i t y of s e v e r a l t h e o r i e s not on any q u a l i t a t i v e assessment but a r b i t r a r i l y . The p l o t i s p r e s e n t e d i n F i g u r e 4.3 from Sarpkaya and I s a a c s o n and c l o s e l y resembles the p l o t g i v e n by Dean i n F i g u r e 4.2. Both f i g u r e s i n d i c a t e t h a t f o r l a r g e r waves, c n o i d a l t h e o r y i s s u i t a b l e f o r s h a l l o w water and h i g h e r o r d e r S t o k e s t h e o r i e s f o r deep w a t e r . Based on the t h e o r e t i c a l c omparisons of many i n v e s t i g a t o r s , Sarpkaya and I s a a c s o n c o n c l u d e d t h a t the S t o k e s and c n o i d a l f i f t h o r d e r t h e o r i e s a r e s u f f i c i e n t l y a c c u r a t e f o r most e n g i n e e r i n g p u r p o s e s , and y e t a r e 83 0.04 H g T 2 0 .00003 Q 0 0 0 3 . 0.4 d F i g u r e 4.2. ( A f t e r Sarpkaya and I s a a c s o n , 1981). Ranges of wave t h e o r i e s g i v i n g the best f i t t o the dynamic f r e e s u r f a c e boundary c o n d i t i o n , a f t e r Dean (1970). r e l a t i v e l y s i m p l e t o use. Fenton (1979) recommended t h a t c n o i d a l t h e o r y be used f o r ^>8 and Stokes t h e o r y o t h e r w i s e . 4.2.3 COMPARISON BASED ON EXPERIMENT Le Mehaute, D i v o k y , and L i n (1968) measured the maximum p a r t i c l e v e l o c i t y f o r r e l a t i v e l y s h a l l o w water c o n d i t i o n s and i n comparing the r e s u l t s w i t h v a r i o u s t h e o r i e s , d e t e r m i n e d t h a t no one t h e o r y was p a r t i c u l a r l y o u t s t a n d i n g . T s u c h i y a and Yamaguchi (1972) compared the v e r t i c a l d i s t r i b u t i o n s of the h o r i z o n t a l p a r t i c l e v e l o c i t y i n phase w i t h the wave c r e s t , and found t h a t the f i n i t e a m p l i t u d e t h e o r i e s p r e d i c t the v e l o c i t i e s w e l l . However, they made no recommendation as t o which t h e o r y i s best over any g i v e n 84 0.05 H g r 0.0002 0.0001 1 r i — r Linear Theory Intermediate Depth Waves Deep Water Waves 0.00005 o.OOt 0.002 0.005 0.01 0.02 Q05 0.1 0.2 d gT 2 F i g u r e 4.3. ( A f t e r Sarpkaya and I s a a c s o n , 1981). Ranges of s u i t a b i l i t y f o r v a r i o u s wave t h e o r i e s as suggested by Le Mehaute (1976). range. Hansen and Svendsen (1974) measured waves g e n e r a t e d by a s i n u s o i d a l l y moving p i s t o n t y pe wave g e n e r a t o r , and de t e r m i n e d t h a t the .waves produced can be d e s c r i b e d as second o r d e r Stokes waves superimposed by a f r e e second harmonic wave. T h i s f r e e second harmonic wave i s p r e s e n t because a r i g i d p a d d l e cannot produce the p a r t i c l e motion c o r r e s p o n d i n g t o p r o g r e s s i v e waves. Fontanet (1961) showed T t h e o r e t i c a l l y t h a t t he p e r i o d of t h i s harmonic wave i s ^, and t h a t i t s h e i g h t i s of the o r d e r of H 2. Hansen and 85 Svendsen checked on no c h a r a c t e r i s t i c o t h e r than the wave p r o f i l e . Grace (1976) found t h a t the l i n e a r wave t h e o r y p r e d i c t e d the maximum h o r i z o n t a l v e l o c i t y w e l l i n the ocean environment. Ohmart and G r a t z (1978) compared water p a r t i c l e v e l o c i t i e s and a c c e l e r a t i o n s as w e l l as water s u r f a c e p r o f i l e s i n the n a t u r a l environment w i t h l i n e a r , Stokes f i f t h o r d e r , and Stream f u n c t i o n p r e d i c t i o n s . They found t h a t the l i n e a r t h e o r y p r e d i c t i o n s were c l o s e t o those of Stokes f i f t h o r d e r except f o r the h i g h e s t waves, and t h a t the Stokes f i f t h o r d e r and the r e g u l a r stream f u n c t i o n t h e o r i e s gave i n d i s t i n g u i s h a b l e p r e d i c t i o n s . C h a k r a b a r t i (1980) measured water s u r f a c e e l e v a t i o n s , p a r t i c l e k i n e m a t i c s , and dynamic p r e s s u r e s i n a l a b o r a t o r y flume and compared the r e s u l t s w i t h Stokes f i r s t , t h i r d , and f i f t h o r d e r t h e o r i e s , f i r s t and second o r d e r c n o i d a l t h e o r i e s , K e u l e g a n - P a t t e r s o n t h e o r y , and stream f u n c t i o n t h e o r y . The h o r i z o n t a l water p a r t i c l e v e l o c i t y was found t o be w e l l a p p r o x i m a t e d by l i n e a r t h e o r y , which p o o r l y r e p r e s e n t e d the p r e s s u r e d i s t r i b u t i o n . The wavelength was best e s t i m a t e d by Stokes t h i r d o r d e r t h e o r y , and the i r r e g u l a r stream f u n c t i o n t h e o r y was the b e s t f o r matching p r o f i l e s and dynamic p r e s s u r e s . However, f o r l o n g e r p e r i o d waves, the p r o f i l e became non-symmetric, and the stream f u n c t i o n t h e o r y c o u l d no l o n g e r p r e d i c t the p a r t i c l e k i n e m a t i c s . C h a k r a b a r t i c o n c l u d e d t h a t f o r low p e r i o d waves l i n e a r t h e o r y works w e l l , and f o r l a r g e p e r i o d waves no 86 t h e o r y i s p a r t i c u l a r l y good because of p r o f i l e asymmetry. 4.2.4 CONCLUSIONS In a sediment movement s t u d y , i t i s of p r a c t i c a l i mportance t o have an a c c u r a t e knowledge of the near-bed water p a r t i c l e v e l o c i t i e s . In l i g h t of the e x p e r i m e n t a l comparisons as p r e v i o u s l y mentioned, i t seems t h a t i f an a c c u r a t e r e p r e s e n t a t i o n of a near-bed h o r i z o n t a l v e l o c i t y i s n e c e s s a r y , i t may be best t o not r e l y on a p a r t i c u l a r t h e o r y at a l l , i n s t e a d c h o o s i n g t o c a r e f u l l y measure the v e l o c i t y u s i n g a proven i n s t r u m e n t a t i o n t e c h n i q u e . 4.3 MASS TRANSPORT UNDER WAVES 4.3.1 INTRODUCTION In a t w o - d i m e n s i o n a l wave t r a i n , the f l u i d p a r t i c l e motions i n c l u d e a stead y d r i f t , which has a speed termed the mass transport v e l o c i t y . S e v e r a l t h e o r i e s have been s e t f o r t h , of which the most w e l l known a r e t o be p r e s e n t e d h e r e . 4.3.2 STOKES THEORY Stokes (1880) f i r s t t h e o r e t i c a l l y p r e d i c t e d t h i s d r i f t f o r s i n u s o i d a l l y t r a v e l l i n g i r r o t a t i o n a l waves. Stokes s o l u t i o n i s g i v e n by "M°8sinh"( kd) [c°sh(2k(y+d) ) - ' l n ^ M > ] ...(4.20) 87 where U M i s the mass t r a n s p o r t v e l o c i t y . The p r o f i l e s g i v e n by e q u a t i o n (4.20) g i v e a n e g a t i v e mass t r a n s p o r t a t the seabed, which i s not u s u a l l y o b s e r v e d . 4.3.3 LONGUET-HIGGINS THEORY L o n g u e t - H i g g i n s (1953) d e v e l o p e d a g e n e r a l mass t r a n s p o r t t h e o r y which i n c l u d e d the f l u i d v i s c o s i t y e f f e c t s and p r e d i c t e d the f o r w a r d mass t r a n s p o r t near the bed. He found t h a t the mass tra.nsport v e l o c i t y j u s t o u t s i d e the l a m i n a r boundary l a y e r a t the bed i s g i v e n by U M = l 6 s i n h " u d ) a t * = " d ( 4 ' 2 1 ) and t h a t the v e r t i c a l g r a d i e n t of the mass t r a n s p o r t v e l o c i t y j u s t below the f r e e s u r f a c e boundary l a y e r i s g i v e n by = H 2cjk 2coth(kd) a t y = 0 (4.22) 3z L o n g u e t - H i g g i n s o b t a i n e d two e q u a t i o n s which d e s c r i b e the mass t r a n s p o r t i n the f l u i d . u n d e r d i f f e r e n t wave c o n d i t i o n s . I f the wave a m p l i t u d e i s much s m a l l e r than t h e boundary 2 v 1 l a y e r t h i c k n e s s , a 0 <<8, where 6 = ( — ) T , then the mass t r a n s p o r t i s g i v e n by the "conduct i on" e q u a t i o n V^7=0 (4.23) where i s t h e stream f u n c t i o n c o r r e s p o n d i n g t o (u7, t>T) . The e q u a t i o n i s s o l v e d u s i n g the e q u a t i o n s (4.21) and (4.22) f o r the two boundary c o n d i t i o n s , and by making the depth 88 averaged t o t a l h o r i z o n t a l mass t r a n s p o r t e q u a l t o z e r o . The s o l u t i o n g i v e s p r o f i l e s p l o t t e d i n F i g u r e 4.4 f o r some d i f f e r e n t wave c o n d i t i o n s . For com p a r i s o n , some p r o f i l e s g i v e n by the Stokes s o l u t i o n , e q u a t i o n ( 4 . 2 0 ) , a re a l s o p l o t t e d i n F i g u r e 4.4. The second e q u a t i o n , c a l l e d the "convection" e q u a t i o n , i s v a l i d when the wave a m p l i t u d e i s much l a r g e r than the boundary l a y e r t h i c k n e s s , a 0 m>>6, and i s g i v e n as ( uM 2 f e + v M 2 l y ) v 2 ^ = 0 ( 4 - 2 4 ) where and V M 2 a r e m a s s t r a n s p o r t v e l o c i t y components. T h i s e q u a t i o n g i v e s r e s u l t s which depend on a d d i t i o n a l boundary c o n d i t i o n s a t the ends of the c h a n n e l , and Sarpkaya and I s a a c s o n r e p o r t e d t h a t t h e s e boundary c o n d i t i o n s make the g i v e n r e s u l t s d i f f i c u l t t o i n t e r p r e t . 4.3.4 EXPERIMENTAL ANALYSES R u s s e l l and O s o r i o (1958) made mass t r a n s p o r t measurements which i n d i c a t e d s i m i l a r i t y t o the " c o n d u c t i o n " s o l u t i o n i n many c a s e s , even though the c o n d i t i o n a 0 m<<6 was v i o l a t e d . L i u and D a v i s (1977) d e v e l o p e d a m o d i f i c a t i o n t o the " c o n d u c t i o n " s o l u t i o n which does not r e q u i r e the c o n d i t i o n t h a t a 0 m<<6f and as such, a g r e e s w i t h the r e s u l t s of R u s s e l l , and O s o r i o . In c o n t r a s t , S l e a t h (1973) a c q u i r e d mass t r a n s p o r t v e l o c i t y p r o f i l e s d i f f e r e n t from the L o n g u e t - H i g g i n s ' s " c o n d u c t i o n " s o l u t i o n and f o r s m a l l a m p l i t u d e waves, the 89 F i g u r e 4.4. ( M o d i f i e d a f t e r Sarpkaya and I s a a c s o n , 1981). N o n - d i m e n s i o n a l mass t r a n s p o r t v e l o c i t y p r o f i l e s w i t h d e p t h f o r v a r i o u s v a l u e s of the depth parameter kd. 90 mass t r a n s p o r t was even i n the o p p o s i t e d i r e c t i o n t o the waves a t the bed. Most work, however, shows bottom d r i f t i n the d i r e c t i o n of wave p r o p a g a t i o n . R e c e n t l y , I s a a c s o n (1978) compared the mass t r a n s p o r t v e l o c i t y o u t s i d e the l a m i n a r boundary l a y e r measured by experiment u s i n g s k e i n s of dye w i t h the mass t r a n s o r t v e l o c i t y as p r e d i c t e d by two t h e o r i e s . The t h e o r i e s used were the second a p p r o x i m a t i o n t o c n o i d a l t h e o r y and the Stokes second o r d e r t h e o r y , where s h a l l o w water c o n d i t i o n s c o r r e s p o n d w e l l t o the c n o i d a l t h e o r y and deep water c o n d i t i o n s t o the Stokes t h e o r y . F i g u r e 4.5 shows the c u r v e s proposed by the e x p e r i m e n t s , and the v a r i a t i o n of the t h e o r i e s from t h e s e c u r v e s . I s a a c s o n b e l i e v e s . t h a t the c u r v e s •remain v a l i d when the boundary l a y e r becomes t u r b u l e n t . 04 03 X 0 1 "0 0.05 0.10 015 020 F i g u r e 4.5. ( A f t e r I s a a c s o n , 1978). No n - d i m e n s i o n a l mass t r a n s p o r t v e l o c i t y o u t s i d e t h e seabed boundary l a y e r . , d e p a r t u r e of the c n o i d a l and Stokes wave t h e o r i e s from proposed c u r v e s . 9 1 In an appendix t o the paper of R u s s e l l and O s o r i o (1958), L o n g u e t - H i g g i n s showed t h a t the mass t r a n s p o r t v e l o c i t y j u s t o u t s i d e the bottom boundary l a y e r i s v a l i d f o r a t u r b u l e n t boundary l a y e r . T h i s i s an imp o r t a n t r e v e l a t i o n t o the e n g i n e e r w o r k i n g w i t h a bed of sediment where the boundary . l a y e r i s changing w i t h v a r y i n g wave p r o p e r t i e s . 4.3.5 CONCLUSIONS The work of I s a a c s o n (1978) seems b e s t t o use f o r the p r e d i c t i o n of mass t r a n s p o r t . The maximum v e l o c i t y i s then g i v e n by " U m a x = U ° m + V where U m i s the maximum v e l o c i t y a t the bed, U 0 i s the max •* um maximum o r b i t a l v e l o c i t y a t the bed as p r e d i c t e d by f i r s t o r d e r t h e o r i e s , and U M i s the mass t r a n s p o r t v e l o c i t y . U n f o r t u n a t e l y , the t h e o r i e s p r e d i c t i n g U 0 m a r e not a c c e p t a b l e f o r use by t h e e n g i n e e r who r e q u i r e s an a c c u r a t e knowledge of the near-bed v e l o c i t y . I t i s then n e c e s s a r y t o measure a c c u r a t e l y t h e near-bed v e l o c i t i e s by use of an i n s t r u m e n t a t i o n t e c h n i q u e . 92 4. 4 COMBINED WAVES AND CURRENTS 4.4.1 INTRODUCTION In d e v e l o p i n g a t h e o r y f o r waves and c u r r e n t s t h a t i s s i m i l a r t o the l i n e a r wave t h e o r y p r e s e n t e d by S t o k e s , i t i s prope r t o c o n s i d e r a r e f e r e n c e frame (x',y) moving w i t h the same v e l o c i t y as the c u r r e n t . The wave h e i g h t and l e n g t h a re u n a l t e r e d , but the p e r i o d of the wave i s now T r as p e r c e i v e d by the s t a t i o n a r y frame. From the moving c o o r d i n a t e frame, the p e r i o d r e v e r t s t o i t s i n i t i a l p e r i o d , T , and the wave moves on a s t a t i o n a r y water body, so t h a t the u s u a l wave t h e o r y i s a g a i n a p p l i c a b l e . 4.4.2 APPLICABLE EQUATIONS " R e f e r r i n g t o F i g u r e 4.6, the f o l l o w i n g e q u a t i o n s h o l d : where L i s the w a v e l e n g t h , T i s the wave p e r i o d , c i s the wave c e l e r i t y , and V i s the stea d y c u r r e n t v e l o c i t y . The s u b s c r i p t s "a" and " r " r e f e r t o the o r i g i n a l f i x e d c o o r d i n a t e frame and the r e l a t i v e c o o r d i n a t e frame moving w i t h the c u r r e n t , r e s p e c t i v e l y . By r e o r g a n i z i n g e q u a t i o n s (4.26) and ( 4 . 2 7 ) , we o b t a i n L=c T = a a c T r r (4.26) c =c +V a r (4.27) (4.28) 93 Wive Spe«<ttCr*Ca-V Wova 5pee«l, c a F i g u r e 4 . 6 . D e f i n i t i o n s k e t c h f o r a p r o g r e s s i v e wave t r a i n on a s t e a d y c u r r e n t . The r e l a t i v e wave p e r i o d , T r, can now be c a l c u l a t e d , and the u s u a l wave formulae a re a p p l i c a b l e , such as c 2 = 2 L t a n h ( 2 ^ d ) ( 4 > 2 g ) Z £.11 Li U o - ( ^ ) - 1 -cos-27Tt ( 4.30) r s i n h ( k d ) r where U 0 i s the l i n e a r wave t h e o r y water p a r t i c l e v e l o c i t y j u s t above the bed. Note t h a t the p a r t i c l e v e l o c i t y as seen by the bed i s c y c l o i d i c i n the f i r s t o r d e r t h e o r y , as the v e l o c i t y V must be t a k e n i n t o a c c o u n t . H i g h e r o r d e r t h e o r i e s i n c l u d e the mass t r a n s p o r t v e l o c i t y , which i s not time dependent and t h i s would i n c r e a s e the c y c l o i d i c d e p a r t u r e as seen by the bed f o r t h e case of waves moving w i t h the c u r r e n t , but would 94 d e c r e a s e t h i s d e p a r t u r e f o r the case of the a d v e r s e c u r r e n t . 4.4.3 WAVES TO BE GENERATED As the wave moves from a r e g i o n of no c u r r e n t t o a r e g i o n where t h e r e i s a s t e a d y c u r r e n t , the wave form i s m o d i f i e d such t h a t the wave h e i g h t and the wav e l e n g t h and speed change, but the wave p e r i o d r e l a t i v e t o the s t a t i o n a r y c o o r d i n a t e s i s unchanged. L o n g u e t - H i g g i n s and Ste w a r t (1960,1961) w i t h t h e i r p r i n c i p l e of r a d i a t i o n s t r e s s , d e s c r i b e d t h i s m o d i f i c a t i o n m a t h e m a t i c a l l y , but i t seems b e s t t o use a t r i a l method t o o b t a i n the d e s i r e d wave c h a r a c t e r i s t i c s a t the t e s t s e c t i o n . The p e r i o d s remain the same, hence L L -2=-*-=—^ (4.31) c g c a C r + V where L i s the w a v e l e n g t h , d i s the water d e p t h , V i s the c u r r e n t v e l o c i t y , and c i s the wave c e l e r i t y . The s u b s c r i p t s "g" and "a" r e f e r t o the wave on the water h a v i n g no c u r r e n t ( i . e . t he g e n e r a t e d wave), and t o the wave i n the r e g i o n h a v i n g a s t e a d y c u r r e n t , r e s p e c t i v e l y . 4. 5 COEFFICIENT OF REFLECTION The near bed water p a r t i c l e v e l o c i t y due s o l e l y t o wave a c t i o n i s g i v e n by u o = ^ ) i T h T T k d T c o s ( k x - w t ) ( 4 ' 3 3 ) where H i s t h e wave h e i g h t and T i s the wave p e r i o d . I n the 95 wave flu m e , r e f l e c t e d waves from the energy a b s o r b i n g end w i l l have an i n f l u e n c e on the water p a r t i c l e v e l o c i t i e s . I t i s i m p o r t a n t t o dete r m i n e the a m p l i t u d e of t h e i r c o n t r i b u t i o n t o the water p a r t i c l e v e l o c i t i e s near the bed. The r a t i o of r e f l e c t e d wave h e i g h t t o i n c i d e n t wave h e i g h t i s d e s i g n a t e d by the r e f l e c t i o n c o e f f i c i e n t , K r, where K r = ^ (4.34) where H R i s the h e i g h t of the r e f l e c t e d wave. R e f l e c t e d waves t r a v e l i n the o p p o s i t e d i r e c t i o n of the i n c i d e n t waves, so the r e s u l t i n g wave form i s g i v e n by r?=HA(kx)cos( 6 ( k x ) - w t ) (4.35) where A(kx) = [ 1+K r 2 + 2 K r c o s ( 2 k x ) (4.36) 1 _ K r 6 ( k x)=tan" 1 [ ( — - I ) t a n ( k x ) ] (4.37) 1+Kr As shown by Sarpkaya and I s a a c s o n (1981), the c o e f f i c i e n t of r e f l e c t i o n can be d e t e r m i n e d s i m p l y by t r a v e r s i n g t he flume i n the d i r e c t i o n of wave p r o p a g a t i o n w i t h a wave probe t o measure the maximum and minimum wave h e i g h t s H and H . , r e s p e c t i v e l y . Then, 3 max mm' v J K = " m a x ^ m i n (4.38) max min and the i n c i d e n t wave h e i g h t H i s g i v e n by 96 H=i(H m +H . ) (4.39) 2. max min In the wave flume used, the r e f l e c t i o n c o e f f i c i e n t was found t o be q u i t e s m a l l . 4. 6 CONCLUSIONS A s h o r t r e v i e w of s e v e r a l wave t h e o r i e s has been p r e s e n t e d . Some i n - d e p t h d e s c r i p t i o n s of t h e o r i e s h a v i n g more wi d e s p r e a d a p p l i c a t i o n have been i n c l u d e d , and o n l y a s h o r t mention of l e s s a d a p t a b l e t h e o r i e s , such as c n o i d a l t h e o r y , has been made. The r e s u l t s of comparisons based on t h e o r y and experiment have been d e s c r i b e d , and i t has been shown t h a t the r e s e a r c h e r ' s d e s i r e d v a l u e s s h o u l d be measured e x p e r i m e n t a l l y , w i t h the t h e o r y used o n l y t o c o n f i r m the r e s u l t s . The mass t r a n s p o r t t h e o r i e s have been d e s c r i b e d , as i t i s a t t e m p t e d t o u n d e r s t a n d the phenomenon r e s p o n s i b l e f o r wave d i r e c t i o n d r i f t near the sea bed. The c u r v e s proposed by I s a a c s o n a r e recommended f o r e s t i m a t i n g the near-bed d r i f t . The t h e o r y has been p r e s e n t e d f o r the case of a wave superimposed by a c u r r e n t . A moving c o o r d i n a t e frame i s adopted, so t h a t the u s u a l wave formulae are a p p l i c a b l e . A f o r m u l a i s g i v e n f o r a p p r o x i m a t i n g the wave g e n e r a t o r p e r i o d f o r waves of d e s i r e d wavelength on a known c u r r e n t . 97 R e f l e c t e d waves are a b l e t o d i s r u p t the water s u r f a c e p r o f i l e such t h a t the near-bed water p a r t i c l e v e l o c i t i e s may be d i f f i c u l t t o d e t e r m i n e . The c o e f f i c i e n t of r e f l e c t i o n i s d e f i n e d , and the p r o c e d u r e i s g i v e n f o r i t s d e t e r m i n a t i o n . 5. INITIATION OF SEDIMENT MOTION 5. I INTRODUCTION The s u b j e c t of p o s s i b l y the g r e a t e s t i n t e r e s t i n the i n v e s t i g a t i o n of near-bed shear s t r e s s e s c o n c e r n s the movement of sediment. I t i s g e n e r a l l y a c c e p t e d t h a t a f l u i d f l o w i n g over a sediment bed e x e r t s a shear s t r e s s on the p a r t i c l e s which causes them to.move i f i t i s s u f f i c i e n t l y l a r g e . T h i s s t r e s s a t which the p a r t i c l e s b e g i n t o move i s known as the c r i t i c a l shear stress, and i s a s s o c i a t e d w i t h a f l u i d v e l o c i t y known as the c r i t i c a l v e l o c i t y . Q u e s t i o n s have been r a i s e d as t o whether t h e shear s t r e s s e x p l a i n s the whole s t o r y . O b s e r v a t i o n s of sediment movement t h r e s h o l d seem t o i n d i c a t e t h a t t h e r e i s a n o t h e r mechanism i n v o l v e d i n the i n i t i a t i o n of p a r t i c l e movement. That mechanism i s the f l o w t u r b u l e n c e near the p a r t i c l e s , and though a q u a n t i t a t i v e d e s c r i p t i o n of the importance of f l o w t u r b u l e n c e i s u n a v a i l a b l e , i t i s apparent t h a t the t u r b u l e n c e p l a y s a r o l e i n the onset of t h r e s h o l d . U s u a l l y , the S h i e l d s e n t r a i n m e n t f u n c t i o n ( S h i e l d s , 1936) i s used t o d e f i n e the shear s t r e s s f o r i n i t i a l p a r t i c l e movement, but e x p e r i m e n t a l r e s u l t s p l o t t e d on t h e S h i e l d s diagram show c o n s i d e r a b l e s c a t t e r . T h i s s c a t t e r may be a t t r i b u t a b l e t o such f a c t o r s as the random shear s t r e s s e x e r t e d by the moving f l u i d , t he random shear s t r e s s n e c e s s a r y t o move bed p a r t i c l e s , and each o b s e r v e r ' s QR 9 9 d e f i n i t i o n of c r i t i c a l movement ( W i l l i a m s and Kemp, 1 9 7 1 ) . D e s p i t e the s c a t t e r , the S h i e l d s c u r v e remains about the b e s t i n d i c a t o r of c r i t i c a l m o t i o n . Bagnold ( 1 9 4 6 ) has done a famous experiment u s i n g a sand bed on an o s c i l l a t i n g p l a t e t o f i n d c r i t i c a l m o t i o n . Komar and M i l l e r ( 1 9 7 4 ) have p l a c e d h i s o s c i l l a t o r y d a t a on the u n i d i r e c t i o n a l f l o w S h i e l d s diagram and have c o n c l u d e d t h a t the S h i e l d s c u r v e works w e l l f o r o s c i l l a t o r y f l o w . T h i s f i n d i n g has been c o n f i r m e d by Madsen and Grant ( 1 9 7 5 ) and i s an i m p o r t a n t advancement i n the study of the onset of motion under waves. 5. 2 ANALYSIS OF THE SHIELDS CRITERION 5 . 2 . 1 INTRODUCTION The S h i e l d s c u r v e ( F i g u r e 5 . 1 ) has been based on i i i i i i i i i — i l l i i i i i i *-Q o> Q. II * in 0.1 0.01 The threshold of movement I I 0.056 10 R * = ^ ° e v 100 1000 F i g u r e 5 . 1 . The S h i e l d s e n t r a i n m e n t f u n c t i o n . 100 e x p e r i m e n t s i n l a b o r a t o r y flumes w i t h f u l l y d e v e l o p e d t w o - d i m e n s i o n a l f l o w s over f l a t sediment beds. For the t u r b u l e n t boundary l a y e r , a l o g a r i t h m i c v e l o c i t y p r o f i l e had been assumed, and i n d e f i n i n g t h e c r i t i c a l shear s t r e s s v a l u e s on the bed f o r i n i t i a l p a r t i c l e movement, S h i e l d s used the te m p o r a l mean shear s t r e s s . The c r i t i c a l shear s t r e s s was o b t a i n e d by e x t r a p o l a t i n g a graph of obser v e d sediment d i s c h a r g e v e r s u s shear s t r e s s and i t does not depend on a q u a l i t a t i v e c r i t e r i o n (Task Committee, 1966). The Task Committee Report r e v e a l e d t h a t one of the main reasons f o r the d a t a s c a t t e r on t h e S h i e l d s diagram stems from the d i f f i c u l t y e n c o u n t e r e d i n c o n s i s t e n t l y d e f i n i n g -c r i t i c a l f l o w , c o n d i t i o n s . C o n s i s t e n c y i s d i f f i c u l t t o a c h i e v e because of the random shear s t r e s s e x e r t e d by the moving f l u i d , and . because of the random p a r t i c l e s u s c e p t i b i l i t y t o movement under a l o c a l i n s t a n t a n e o u s shear s t r e s s . The e x e r t e d shear s t r e s s i s random because of the t u r b u l e n c e i n the f l u i d f l o w , and any i n s t a n t a n e o u s shear s t r e s s i s a f u n c t i o n of the t e m p o r a l mean shear s t r e s s , the f l u i d d e n s i t y and v i s c o s i t y , and the f l o w boundary c o n d i t i o n s i n c l u d i n g the p a r t i c l e geometry. The p a r t i c l e s u s c e p t i b i l i t y depends on the shape, w e i g h t , and placement of any p a r t i c l e , and the o v e r a l l p a r t i c l e s u s c e p t i b i l i t y can be d e s c r i b e d by a p r o b a b i l i t y d i s t r i b u t i o n ( G r a s s , 1970). 101 There i s the problem of d i f f e r e n t o b s e r v e r s h a v i n g d i f f e r e n t p e r c e p t i o n s as t o the onset of sediment moti o n . Some may p r e d i c t motion when the v e r y f i r s t g r a i n s a re i n movement, and o t h e r s not u n t i l a s u b s t a n t i a l f r a c t i o n of the bed p a r t i c l e s a r e i n mo t i o n . A f u r t h e r problem i s caused by the use of d i f f e r e n t wave fl u m e s . Grass e x p l a i n e d t h a t because d i f f e r e n t wave flumes have d i f f e r e n t boundary c o n d i t i o n s , the boundary r e g i o n t u r b u l e n c e s a r e n e c e s s a r i l y d i f f e r e n t and they no l o n g e r show s i m i l a r i t y w i t h r e s p e c t t o the average c r i t i c a l shear s t r e s s v a l u e s d e r i v e d from the S h i e l d s c u r v e . 5.2.2 EXERTED SHEAR STRESS DISTRIBUTION S h i e l d s used the t e m p o r a l mean shear s t r e s s as the c r i t i c a l shear s t r e s s t h a t w i l l cause p a r t i c l e movement. I t has been p r e d i c t e d by K a l i n s k e (1947) t h a t the i n s t a n t a n e o u s bed shear s t r e s s can be up t o t h r e e t i m e s the t e m p o r a l mean shear s t r e s s . So i t i s p o s s i b l e f o r p a r t i c l e s t o move when the mean shear s t r e s s i s c o n s i d e r a b l y l e s s than the r e a l c r i t i c a l movement v a l u e . As t h e c r i t i c a l shear s t r e s s i s d e f i n e d as a mean v a l u e , i t s h o u l d be n e c e s s a r y t o s p e c i f y a mean amount of p a r t i c l e movement but t o do t h i s poses d i f f i c u l t y . The n a t u r e of the boundary l a y e r has been d e s c r i b e d as h a v i n g a t u r b u l e n t l a y e r and a v i s c o u s s u b l a y e r . For f i n e sands l y i n g c o m p l e t e l y w i t h i n t h e v i s c o u s s u b l a y e r , motion i n a s m a l l a r e a of the bed o c c u r s i n g u s t s . T h i s o c c u r s 1 02 because i n the v i s c o u s s u b l a y e r , t h e r e a r e unsteady v e l o c i t i e s p r e s e n t , and thes e a r e caused by h i g h v e l o c i t y b u r s t s i n the t u r b u l e n t l a y e r . The f l o w p a t t e r n i n the v i s c o u s s u b l a y e r then t a k e s the form of h i g h and low v e l o c i t y s t r e a k s , and has a d e f i n i t e and r e g u l a r s t r u c t u r e e x t e n d i n g t o the bed, a c c o r d i n g t o W i l l i a m s and Kemp. K l i n e e t . a l . (1967) d e s c r i b e d how the h i g h v e l o c i t y b u r s t i n t he t u r b u l e n t r e g i o n reaches i n t o the v i s c o u s s u b l a y e r and touches down t o the bed. Low momentum f l u i d i s e j e c t e d from e i t h e r s i d e of the s t r e a k and i n t o the o u t e r r e g i o n so momentum t r a n s f e r o c c u r s between the v i s c o u s s u b l a y e r and the t u r b u l e n t l a y e r , c a u s i n g a s e v e r e l y r e t a r d e d l o c a l v e l o c i t y p r o f i l e , and the c y c l e i s r e p e a t e d . For sands p r o t r u d i n g above the v i s c o u s s u b l a y e r , the p a r t i c l e motion i n a s m a l l a r e a of the bed a l s o o c c u r s i n g u s t s , t h i s time because of the t u r b u l e n c e i n the f l u i d f l o w and not because of any s t r e a k i n g phenomenon. The i n s t a n t a n e o u s shear s t r e s s can be d e s c r i b e d by a p r o b a b i l i t y d i s t r i b u t i o n because i t i s dependent on the i n s t a n t a n e o u s f l u i d v e l o c i t y which has random t u r b u l e n t f l u c t u a t i o n s about a t e m p o r a l mean v a l u e . The i n s t a n t a n e o u s shear s t r e s s d i s t r i b u t i o n w i l l have a mean te m p o r a l shear s t r e s s , and w i l l depend on the f l u i d d e n s i t y and v i s c o s i t y , and on the f l o w boundary c o n d i t i o n s i n c l u d i n g the p a r t i c l e geometry. A t y p i c a l d i s t r i b u t i o n i s shown i n F i g u r e 5.2, where 77 i s the mean e x e r t e d shear s t r e s s a t the bed. 103 Bed shear stress distribution Value of r F i g u r e 5.2. E x e r t e d shear s t r e s s p r o b a b i l i t y d i s t r i b u t i o n . 5.2.3 CRITICAL SHEAR STRESS DISTRIBUTION Sediment n o n u n i f o r m i t y i n s i z e and shape f u r t h e r c o m p l i c a t e s m a t t e r s as some p a r t i c l e s a r e more prone t o movement. Some p a r t i c l e s a r e a l s o more exposed as they l i e i n a random manner on the s u r f a c e of the bed. The p a r t i c l e s u s c e p t i b i l i t y t o the l o c a l i n s t a n t a n e o u s shear s t r e s s can be d e s c r i b e d by a p r o b a b i l i t y d i s t r i b u t i o n where the most s u s c e p t i b l e p a r t i c l e s w i l l be moved by the lowe s t shear s t r e s s e s . F i g u r e 5.3 shows a t y p i c a l p r o b a b i l i t y d i s t r i b u t i o n , where i s the mean c r i t i c a l shear s t r e s s n e c e s s a r y t o move the p a r t i c l e s . Value of r F i g u r e 5.3. C r i t i c a l shear s t r e s s p r o b a b i l i t y d i s t r i b u t i o n . 104 One q u e s t i o n s the v a l i d i t y of the assumption t h a t the shear s t r e s s i s r e s p o n s i b l e f o r p a r t i c l e movement. The l i f t f o r c e may be r e s p o n s i b l e f o r e j e c t i n g the p a r t i c l e from the bed, or the moments of the dr a g and l i f t f o r c e s a g a i n s t the immersed weight may be r e s p o n s i b l e f o r p a r t i c l e r o l l i n g . However, no g e n e r a l l y a c c e p t e d a n a l y t i c a l t h e o r y or e x p e r i m e n t a l r e s u l t can d e s c r i b e t h e i n i t i a t i o n of p a r t i c l e m o t i o n . R a u d k i v i (1963) made measurements a l o n g a r i p p l e f a c e and found t h a t sediment t r a n s p o r t o c c u r r e d a l l a l o n g the r i p p l e f a c e even where the shear s t r e s s was l e s s than the S h i e l d s c r i t i c a l v a l u e . The t u r b u l e n c e i n the f l o w was s u s p e c t e d , and i t was s u g g e s t e d by R a u d k i v i (1967) t h a t h y d r o s t a t i c p r e s s u r e c o u l d e j e c t p a r t i c l e s from the bed when a p a s s i n g eddy lo w e r e d the l o c a l p r e s s u r e , such t h a t the shear s t r e s s would be r e s p o n s i b l e o n l y f o r t r a n s p o r t i n g the p a r t i c l e , not f o r e n t r a i n i n g i t . The q u e s t i o n as t o what f o r c e or what c o m b i n a t i o n of f o r c e s moves a p a r t i c l e remains unanswered, but i n the S h i e l d s diagram, t h e r e i s a t l e a s t a c o n s i s t e n t f i n d i n g t h a t each p a r t i c l e moves under some s p e c i f i e d s t r e s s . A nother d i f f i c u l t y e x p e r i e n c e d by o b s e r v e r s of f r e s h l y groomed beds of sediment i s t h a t of bed a r m o r i n g . I f the f l u i d v e l o c i t y i s s l o w l y i n c r e a s e d beyond the c r i t i c a l shear s t r e s s v e l o c i t y , some exposed p a r t i c l e s b e g i n t o move but soon have f a l l e n t o some p r o t e c t e d p o s i t i o n and movement c e a s e s . A f u r t h e r i n c r e a s e d shear s t r e s s has the same 105 r e s u l t . Only when the shear s t r e s s i s i n c r e a s e d beyond some minimum v a l u e w i l l some movement remain. The a r m o r i n g p r o c e s s can be d e s c r i b e d by a s h i f t i n g t o the r i g h t of the p r o b a b i l i t y d i s t r i b u t i o n as a r m o r i n g t a k e s p l a c e as shown i n F i g u r e 5.4. 5.2.4 THRESHOLD OF MOTION There have been p r e s e n t e d two p r o b a b i l i t y d i s t r i b u t i o n s : one g i v i n g the i n s t a n t a n e o u s shear s t r e s s e x e r t e d on the bed by the f l u i d f l o w , and the o t h e r g i v i n g the i n s t a n t a n e o u s shear s t r e s s n e c e s s a r y t o move a c e r t a i n number of bed s u r f a c e p a r t i c l e s . When thes e two d i s t r i b u t i o n s b e g i n t o o v e r l a p w i t h i n c r e a s i n g f l u i d v e l o c i t y , t h e most s u s c e p t i b l e g r a i n s b e g i n t o move d u r i n g the peak v e l o c i t y g u s t s , and t h e f l o w would then have the t h e o r e t i c a l mean shear s t r e s s , as shown i n F i g u r e 5.5. T h i s d e f i n i t i o n of the i n i t i a l movement c o n d i t i o n s i s i m p r a c t i c a l because of t h e d i f f i c u l t i e s i n e x p e r i m e n t a l l y e s t a b l i s h i n g the f l o w t h a t would produce the n e c e s s a r y VahMofT F i g u r e 5.4. P r o b a b i l i t y d i s t r i b u t i o n f o r bed a r m o r i n g . 106 Value of r F i g u r e 5.5. ( A f t e r W i l l i a m s and Kemp, 1971). P r o b a b i l i t y d i s t r i b u t i o n f o r c r i t i c a l movement. t h e o r e t i c a l mean shear s t r e s s . N o r m a l l y , the flo w i s i n c r e a s e d u n t i l movement appears t o be g e n e r a l as t h i s i s more f e a s i b l e e x p e r i m e n t a l l y . H o w e v e r , t h i s has the drawback t h a t each o b s e r v e r ' s p e r s o n a l judgment becomes i m p o r t a n t . Each must make a judgment as t o what p e r c e n t a g e of p a r t i c l e s i n motion c o n s t i t u t e s i n i t i a l movement. I t i s easy t o u n d e r s t a n d the e x p e r i m e n t a l d i f f i c u l t y p r e s e n t i n e s t i m a t i n g i n i t i a l movement i n t h i s manner, as the p e r c e n t a g e of bed p a r t i c l e s i n motion i s c o n s t a n t l y c h a n g i n g because of the f l o w t u r b u l e n c e . At t i m e s , o n l y a s m a l l f r a c t i o n of t h e p a r t i c l e s , or no p a r t i c l e s a t a l l , may be i n m o t i o n , when a sudden b u r s t of t u r b u l e n c e may sweep the m a j o r i t y of the p a r t i c l e s i n t o the f l o w . G r a s s q u a n t i t a t i v e l y d e f i n e d i n i t i a l movement i n terms of the o v e r l a p of the two d i s t r i b u t i o n s . T h i s o v e r l a p i s d e f i n e d as the m u l t i p l e , n, of the sum of the s t a n d a r d d e v i a t i o n s of the two d i s t r i b u t i o n s t h a t s e p a r a t e s the two mean v a l u e s . G r a s s d e t e r m i n e d t h a t n=0.625 c o r r e s p o n d s t o the c a s e of S h i e l d s ' s i n i t i a l p a r t i c l e movement c o n d i t i o n s . 107 There i s a p e r c e n t a g e of p a r t i c l e s t h a t w i l l never move under t h e s e c o n d i t i o n s , and a l a r g e p e r c e n t a g e t h a t move o n l y i n f r e q u e n t l y . A f t e r a s h o r t t i m e , many p a r t i c l e s w i l l have become armored so t h a t o n l y a few p a r t i c l e s w i l l remain i n m o t i o n . W i l l i a m s and Kemp d e t e r m i n e d t h a t n=0.2 f o r no ar m o r i n g t o occur i n f i n e sands. T h i s c o r r e s p o n d s t o the c r i t i c a l c o n d i t i o n t h a t all the s u r f a c e sand p a r t i c l e s can be put i n t o motion by the h i g h e s t e x e r t e d shear s t r e s s e s . T h i s d e f i n i t i o n of the c r i t i c a l c o n d i t i o n i s one r a r e l y e n c o u n t e r e d , as most o b s e r v e r s use a c r i t i c a l c o n d i t i o n w i t h much l e s s movement. D i f f e r e n t o b s e r v e r s use d i f f e r e n t wave flumes i f waves are used t o ge n e r a t e o s c i l l a t o r y m o t i o n . O t h e r s use o s c i l l a t i n g p l a t e s (Bagnold, 1946; Manohar, 1955), and o s c i l l a t o r y water t u n n e l s (Ranee and Warren, 1969). In the s e d i f f e r e n t e x p e r i m e n t s , the f l o w boundary c o n d i t i o n s a re not s i m i l a r , and the near-bed t u r b u l e n t s t r u c t u r e i s not s i m i l a r . I t has been d i s c u s s e d t h a t because of the t u r b u l e n c e or because of i n c r e a s e d shear s t r e s s e s on the p a r t i c l e , i t s h o u l d be expected t h a t f l o w c o n d i t i o n s h a v i n g d i f f e r e n t t u r b u l e n c e l e v e l s w i l l have d i f f e r e n t e x p e r i m e n t a l t e m p o r a l mean shear s t r e s s e s . 5.2.5 CONCLUSIONS The f o r c e or c o m b i n a t i o n of f o r c e s r e s p o n s i b l e f o r sediment e n t r a i n m e n t i s unknown. S h i e l d s has devel o p e d an e n t r a i n m e n t f u n c t i o n e x p l a n a t i o n w h i c h p l o t s as a curve 108 a g a i n s t a p a r t i c l e Reynolds number. Other o b s e r v e r s ' d a t a f a l l near the c u r v e w i t h some s c a t t e r which may be e x p l a i n e d by d i f f e r e n c e s i n p e r s o n a l judgment and d i f f e r i n g t u r b u l e n t c o n d i t i o n s near the boundary. A l l appearances a r e t h a t the S h i e l d s c u r v e g i v e s a r e a s o n a b l e i n d i c a t i o n as t o the onset of m o t i o n , but t h a t d i f f e r e n t boundary c o n d i t i o n s u s i n g the same sand may y i e l d s l i g h t l y d i f f e r e n t mean c r i t i c a l shear s t r e s s e s n e c e s s a r y f o r movement. 5.3 BAGNOLD'S OSCILLATING PLATE EXPERIMENT Bagnold (1946) performed e x p e r i m e n t s on sand beds u s i n g an o s c i l l a t i n g p l a t e i n what were c o n s i d e r e d the major advancements i n r i p p l e f o r m a t i o n d e t e r m i n a t i o n a t t h a t t i m e . The o b j e c t of the e x p e r i m e n t s was t o o b t a i n q u a n t i t a t i v e d a ta on - t h e s i z e and c h a r a c t e r of sand r i p p l e s made by waves - t h e d r a g t o which t h e s e r i p p l e s g i v e r i s e - t h e minimum o s c i l l a t i n g water motion r e q u i r e d t o d i s t u r b the sand i n the f i r s t i n s t a n c e . In . t h i s s t u d y , the l a s t c o n s i d e r a t i o n i s of p r i m a r y i n t e r e s t , but Bagnold's o b s e r v a t i o n of r i p p l e s sheds some l i g h t on the i n i t i a t i o n of p a r t i c l e motion on a f l a t sediment bed. 5.3.1 DESCRIPTION OF EXPERIMENTAL PROCEDURE The experiment was c a r r i e d out i n a narrow wave tank by means of an o s c i l l a t i n g p l a t e on which sand was s p r e a d . The 109 sand was c o n t a i n e d i n a c r a d l e suspended from a p i v o t i n the r o o f so t h a t i t c o u l d be o s c i l l a t e d t h r o u g h a c i r c u l a r a r c i n s t i l l water i n the tank. The o s c i l l a t i o n was m a i n t a i n e d by a motor d r i v e n c r a n k . The sand was spre a d over the c r a d l e and the s u r f a c e was c a r e f u l l y smoothed t o a f l a t bed. The c r a d l e was o s c i l l a t e d and w h i l e the o r b i t a l a m p l i t u d e was kept c o n s t a n t , the o s c i l l a t i o n speed was i n c r e a s e d u n t i l movement of the p a r t i c l e s began. C o n c l u s i o n s s t i l l have not been drawn as t o the v a l i d i t y of bed o s c i l l a t i o n r a t h e r than f l u i d osc i l l a t i o n . The i n e r t i a l f o r c e s on the p a r t i c l e s a r e d i f f e r e n t f o r both systems, but t h e i r s m a l l s i z e means t h a t t h e s e can be n e g l e c t e d f o r a l l ca s e s except f o r t h o s e h a v i n g l a r g e a c c e l e r a t i o n s . N i e l s e n (1979) d e t e r m i n e d t h a t the r a t i o of the maximum volume f o r c e s t o the maximum drag f o r c e s on a p a r t i c l e o s c i l l a t e d on a bed i s s m a l l f o r a l l c a s e s except f o r sediments of h i g h s p e c i f i c w e i g h t . Brebner and R i e d e l (1972) c o n c l u d e d t h a t the boundary l a y e r above an o s c i l l a t i n g p l a t e remains l a m i n a r t o a l a r g e r a m p l i t u d e Reynolds number than does the boundary l a y e r above :a f i x e d bed under o s c i l l a t o r y f l o w . S p e c u l a t i o n i s t h a t the o s c i l l a t i n g p l a t e t u r b u l e n c e i s damped by the s t i l l w a t e r , whereas a f i x e d bed l a m i n a r boundary l a y e r may be s e t i n t o t u r b u l e n c e by an i r r e g u l a r i t y i n the o s c i l l a t i n g f l o w . Bagnold noted the absence of any s i g n s of t u r b u l e n c e i n the water as an o u t s t a n d i n g f e a t u r e of the e x p e r i m e n t . 110 O b s e r v a t i o n of the t u r b u l e n c e was made u s i n g a p a r t i c l e of dye, which l e f t an u n d i s t u r b e d s k e i n even a f t e r many bed o s c i l l a t i o n s . I t may be e r r o n e o u s t o make c o n c l u s i o n s c o n c e r n i n g the c r i t i c a l o s c i l l a t o r y motion f o r the i n i t i a t i o n of movement under an o s c i l l a t o r y f l o w i f those c r i t i c a l c o n d i t i o n s were d e t e r m i n e d u s i n g an o s c i l l a t i n g bed. U n t i l the e f f e c t of t u r b u l e n c e on the c r i t i c a l motion of a p a r t i c l e on which wave a c t i o n i s e x e r t e d has been a d e q u a t e l y a s s e s s e d , d i s c r e t i o n s h o u l d be used when e x t r a p o l a t i n g o s c i l l a t o r y bed measurements t o o s c i l l a t i n g f l o w c o n d i t i o n s . 5.3.2 DESCRIPTION OF EXPERIMENTAL RESULTS At the c r i t i c a l speed of the water motion a t which the p a r t i c l e s j u s t b e g i n t o move, the p a r t i c l e s were observed t o r o l l back and f o r t h over the s u r f a c e but not be l i f t e d o f f of i t . The movement o c c u r s i n phase w i t h the v e l o c i t y , and the l e n g t h of t h e p a r t i c l e p a t h i s obser v e d t o be s h o r t . As the v e l o c i t y i s i n c r e a s e d , the p a r t i c l e p a t h i s a l s o i n c r e a s e d . I f the p a r t i c l e motion i s a l l o w e d t o c o n t i n u e f o r a t i m e , a r i p p l e i s formed, and i s known as a r o l l i n g g r a i n r i p p l e . A f e a t u r e noted by Bagnold i s t h a t a v o r t e x r i p p l e forms i f t h e r o l l i n g g r a i n r i p p l e h e i g h t i n c r e a s e s t o a v a l u e l a r g e r than some c r i t i c a l h e i g h t . A v o r t e x motion forms on t h e l e e s i d e of the d i s t u r b a n c e , and sand p a r t i c l e s a r e s c o u r e d from the t r o u g h and the sand i s suspended i n the 111 f l u i d . The v o r t e x r i p p l e mechanism w i l l o p e r a t e and p a r t i c l e s w i l l be put i n t o movement over any s u f f i c i e n t l y l a r g e sand bed d i s t u r b a n c e even a t speeds below t h a t needed t o i n i t i a t e p a r t i c l e movement on a smoothed out s u r f a c e . I t i s i m p o r t a n t t o c a r e f u l l y smooth the bed when d o i n g onset of motion s t u d i e s , but as making the bed p e r f e c t l y smooth i s not p o s s i b l e , the o b s e r v e r must be p r e p a r e d t o d i s c o u n t motion t h a t i s o c c u r r i n g near o u t s t a n d i n g bed f e a t u r e s . F u r t h e r s o u r c e s of p o s s i b l e e r r o r common t o a l l of Bagnold's measurements of the f l u i d o s c i l l a t i o n f o r i n i t i a l p a r t i c l e movement were -the d i f f i c u l t y of m a i n t a i n i n g a c o n s i s t e n t s t a n d a r d as t o the p r o p o r t i o n of moving p a r t i c l e s on a s u r f a c e which s h o u l d c o n s t i t u t e i n i t i a l movement. T h i s d i f f i c u l t y was enhanced by the use of d i f f e r e n t o s c i l l a t o r y a m p l i t u d e s and d i f f e r e n t p a r t i c l e s i z e s , - the i m p o s s i b i l i t y of u s i n g a c o m p l e t e l y u n i f o r m m a t e r i a l . The f i r s t s u r f a c e p a r t i c l e s t o move might have been t h o s e of e x c e p t i o n a l s i z e or shape, -th e p o s s i b i l i t y of t i n y a i r b u b b l e s a t t a c h e d t o the p a r t i c l e s o f low d e n s i t y making these ones the f i r s t t o move. Bagn o l d watched f o r t h i s e r r o r , but a d m i t t e d t h a t i t may not have been e n t i r e l y e x c l u d e d . These f i n a l t h r e e s o u r c e s of e r r o r can be e x p l a i n e d i n terms of the p r o b a b i l i t y d i s t r i b u t i o n s as seen e a r l i e r . The d i f f e r e n c e i s t h a t here Bagnold had o b s e r v e d p a r t i c l e movement under o s c i l l a t o r y f l u i d f l o w , and the p r e v i o u s 1 12 d i s c u s s i o n c o n c e r n s f l o w t h a t i s u n i d i r e c t i o n a l . I t i s i m p o r t a n t t h a t the e s t i m a t i o n of the i n i t i a t i o n of the onset of motion i n v o l v e s the same problems i n the two f l o w t y p e s . I t s u g g e s t s t h a t the f l o w s may be of the same order f o r i n i t i a t i o n of m o t i o n . S t u d i e s have been made comparing the shear s t r e s s e s n e c e s s a r y t o g i v e the onset of motion f o r b o th u n i d i r e c t i o n a l and o s c i l l a t o r y f l o w . 5. 4 COMPARISON OF UNIDIRECTIONAL AND OSCILLATORY CRITICAL CONDITIONS 5.4.1 INTRODUCTION Much may be r e v e a l e d about the t h e o r i e s f o r p a r t i c l e movement by comparing the t h r e s h o l d c o n d i t i o n s f o r u n i d i r e c t i o n a l and o s c i l l a t o r y f l u i d f l o w s . An a n a l y s i s of the f o r c e s a c t i n g on the p a r t i c l e g i v e s a comparison of the s i z e of the volume f o r c e s and the drag f o r c e s i n both f l o w s . A c t u a l measurement of the f l o w c o n d i t i o n s a t t h r e s h o l d g i v e s d a t a p o i n t s t o compare w i t h p r e v i o u s l y p r e s c r i b e d c u r v e s . Both methods i n d i c a t e t h a t t h e r e s h o u l d be v e r y l i t t l e d i f f e r e n c e between the f l o w c o n d i t i o n s n e c e s s a r y t o i n i t i a t e m otion f o r both t y p e s of f l o w . 5.4.2 ANALYSIS OF FORCES For a l l p r a c t i c a l p u r p o s e s , t h e r e i s no reason why the mechanism of the i n i t i a t i o n of sediment motion under waves s h o u l d d i f f e r from t h a t under s t e a d y f l o w ( N i e l s e n , 1979). A 1 13 q u a l i t a t i v e a n a l y s i s of the f o r c e s a c t i n g on a p a r t i c l e a t r e s t on a sediment bed r e v e a l s t h a t the f o r c e s under waves a r e v e r y much l i k e t h o s e under s t e a d y f l o w . R a u d k i v i (1967) mentioned the f o l l o w i n g f o r c e s a c t i n g on the p a r t i c l e shown i n F i g u r e 5.6: -th e g r a v i t y f o r c e 1 F G = p ( s - D ( | D 3 ) -th e drag f o r c e c o n s i s t i n g of i ) the form drag F D F = C F I ( ! D 2 ) U I U I F i g u r e 5.6. D e f i n i t i o n s k e t c h of f o r c e s a c t i n g on the p a r t i c l e . 1 1 4 i i ) the s u r f a c e d r ag F D S = C s p ( D 2 ) u | u | which combine t o g i v e F D = C D I ( ! d 2 ) u m - t h e l i f t f o r c e F =c — (— D 2 ) u 2 - t h e volume f o r c e F =F +F V P H where F p i s due t o the p r e s s u r e g r a d i e n t of the f l o w where F H i s due t o the added hydrodynamic mass of the pa r t i c 1 e FH= CM'<! D'>fi For o s c i l l a t o r y f l o w , U 0 i s the water v e l o c i t y o u t s i d e the boundary l a y e r U 0=a 0 OJCOS ( kx-wt) m and u i s the water v e l o c i t y a t the l e v e l of the p a r t i c l e u=aa 0 r nwcos (kx-wt + 0) 1 15 So, F v=p(^D 3) [ a o m w 2 s i n ( k x - a ) t ) + C M a a O n i w 2 s i n ( k x - c j t + 0) ] For u n i d i r e c t i o n a l f l o w , F v i s assumed t o be i n the o r d e r of z e r o . In a n a l y z i n g t h e s e f o r c e s , N i e l s e n assumed t h a t the i n f l u e n c e of the l i f t f o r c e on the i n i t i a t i o n of motion i s the same under waves as i n s t e a d y f l o w . T h i s assumption i g n o r e s the t u r b u l e n t p r e s s u r e and v e l o c i t y f l u c t u a t i o n s found i n u n i d i r e c t i o n a l f l o w as w e l l as i g n o r i n g the tendency f o r the s t r o n g e r v e l o c i t y g r a d i e n t i n o s c i l l a t o r y f l o w t o e n t r a i n the p a r t i c l e . As the s e c o n t r i b u t i o n s t o the p a r t i c l e movement a r e not q u a n t i t a t i v e l y d e t e r m i n a b l e , ' the assumption i s a c c e p t a b l e . The drag f o r c e as g i v e n i s d i f f i c u l t t o d e t e r m i n e , so i t i s d e t e r m i n e d from the shear s t r e s s , u s i n g F D = l D 2 r ( t ) The f a c t o r £ r e f l e c t s the uneven p a r t i c l e p o s i t i o n i n g on the bed, and g i v e s those p a r t i c l e s p r o t r u d i n g above s u r r o u n d i n g p a r t i c l e s a g r e a t e r p o r t i o n of the shear s t r e s s . Fenton and Abbot (1977) d e t e r m i n e d £=10, w h i l e White (1940) proposed £=7. White suggested t h a t i n l a m i n a r f l o w the s u r f a c e drag would be l a r g e r and a c t h i g h e r on the p a r t i c l e than i n t u r b u l e n t f l o w . T h i s would lower the c r i t i c a l v a l u e of T 0 , e x p l a i n i n g the d i p i n the S h i e l d s c u r v e . 116 The shear s t r e s s i s g i v e n by r(t)=£ wfu|u| such t h a t F D = i D ' f K f u | U | F V . The r a t i o ^r- i s g i v e n by D F„ [sin(kx-wt)+C„asin(kx-ut+0)] V _ L IT — M F D ~ 3 a 0 m f w c o s ( kx-wt) |cos(kx-o>t) | and so F Vmax 7rDr FDmax 3 a ° m f w In l a m i n a r f l o w : f..=2(r-A7T)^ w a 0 m ' u m so Vmax_7rrD,o)x4-FDmax = 6 v T h i s r a t i o i s s m a l l so t h e e f f e c t of the volume f o r c e on the i n i t i a t i o n of sediment motion f o r l a m i n a r f l o w under waves can be n e g l e c t e d . I f the f l o w i s t u r b u l e n t , Jonsson (1963) has proposed a 47f- + l o g47f- =-°- 0 8 + l o g (-r n ) w w s where a 0 i s Che a m p l i t u d e of the h o r i z o n t a l o r b i t a l m d i s p l a c e m e n t and k i s the sand roughness and i s here e q u a l t o the p a r t i c l e d i a m e t e r . T h i s can be a p p r o x i m a t e d as 1 17 £ „ = ° - ' < T r > T i n a r e l e v a n t range w i t h i n a s m a l l margin of e r r o r a c c o r d i n g t o N i e l s e n . So t h e r a t i o f o r the maximum volume f o r c e s t o the maximum dra g f o r c e s becomes FVmax 7r / a°nu "-§-a°m The r e l e v a n t range of (—=j—) i s r e p o r t e d by N i e l s e n as <TT> € [70;2000] such t h a t F. Vmax ^ [.009;.09] F Dmax Then the e f f e c t of the volume f o r c e on the i n i t i a t i o n of sediment motion under waves can a l s o be n e g l e c t e d i n the t u r b u l e n t c a s e . I f the o s c i l l a t o r y motion i s made u s i n g an o s c i l l a t i n g p l a t e r a t h e r than an o s c i l l a t i n g f l u i d , t hen the p r e s s u r e f o r c e F p w i l l not e x i s t , but t h e r e w i l l be a f i c t i t i o u s f o r c e F =ps (-5D3 )a 0„tJ 2cos(kx-cJt) A 6 m due t o the p l a t e a c c e l e r a t i o n s . Then F Vmax 7rDrs FDmax 3 a ° m f w The i n i t i a t i o n of motion i s u n a f f e c t e d by the volume f o r c e s 118 except f o r measurements u s i n g v e r y l a r g e d e n s i t y m a t e r i a l s . I t i s r e c o g n i z e d t h a t the i n e r t i a l terms f o r the o s c i l l a t i n g p l a t e and the o s c i l l a t i n g f l o w a r e d i f f e r e n t . However, f o r sand p a r t i c l e s of s m a l l s i z e , t h i s d i f f e r e n c e i s so s m a l l when compared w i t h the drag f o r c e magnitudes t h a t i t i s of no consequence. These f i n d i n g s , f o r l a m i n a r and t u r b u l e n t f l o w and f o r o s c i l l a t i n g p l a t e s , i n d i c a t e t h a t the drag f o r c e s , here synonymous w i t h the shear f o r c e s , dominate f o r both u n i d i r e c t i o n a l and o s c i l l a t o r y f l o w s . The shear f o r c e f o r the u n i d i r e c t i o n a l case i s d e t e r m i n e d from the u n i v e r s a l l o g law, w h i l e t h a t f o r the o s c i l l a t o r y case, i s c a l c u l a t e d u s i n g f r i . c t i o n f a c t o r f o r m u l a e , where the v e l o c i t y used i s t h a t j u s t o u t s i d e the boundary l a y e r . 5.4.3 ANALYSIS OF THRESHOLD MEASUREMENTS Much work has been done on the t h r e s h o l d under u n i d i r e c t i o n a l s t e a d y c u r r e n t c o n d i t i o n s . The f i n d i n g of S h i e l d s (1936) i s u n i v e r s a l l y a c c e p t e d as the f o r e r u n n e r i n sediment e n t r a i n m e n t s t u d i e s . Bagnold (1963) has p r e s e n t e d a cu r v e s i m i l a r t o t h a t of S h i e l d s , but p r e s e n t e d i t i n a more c o n v e n i e n t form h a v i n g r e p l a c e d the p a r t i c l e Reynolds number w i t h the p a r t i c l e d i a m e t e r . Much work has been done i n d e t e r m i n i n g the sediment movement t h r e s h o l d under o s c i l l a t o r y f l o w c o n d i t i o n s . Komar and M i l l e r (1974) found from the da t a of p r e v i o u s r e s e a r c h e r s t h a t the sediment e n t r a i n m e n t c o n d i t i o n s a r e 119 s i m i l a r f o r a l l t y p e s of o s c i l l a t i o n i n v e s t i g a t e d . These c o n d i t i o n s have been a t t a i n e d by use of waves i n flumes, o s c i l l a t i n g water t u n n e l s , o s c i l l a t i n g water b l o c k s , and o s c i l l a t i n g p l a t e s . P r o t o t y p e p e r i o d s , o r b i t a l d i a m e t e r s , and o r b i t a l v e l o c i t i e s cannot be reproduced i n wave t e s t s i n o r d i n a r y t a n k s because the p e r i o d i s so r e s t r i c t e d . The o t h e r methods a r e a b l e t o g e nerate t h e s e p r o t o t y p e c o n d i t i o n s , but may not be a b l e t o r e - c r e a t e the e x p e c t e d p r o t o t y p e t u r b u l e n c e and cannot r e - c r e a t e the p r o t o t y p e p r e s s u r e s and c o n v e c t i v e a c c e l e r a t i o n s as e x p e r i e n c e d under waves. Komar and M i l l e r r e p o r t e d t h a t t h e s e methods of g e n e r a t i n g an o s c i l l a t i n g f l o w over a sediment bed l e a d t o the s i m i l a r c o n c l u s i o n t h a t the shear s t r e s s r e q u i r e d t o move sediment under waves i s the same as t h a t r e q u i r e d t o move sediment under a u n i d i r e c t i o n a l c u r r e n t . Komar and M i l l e r used f i v e s e t s of p u b l i s h e d d a t a i n a n a l y s i s of the t h r e s h o l d . Bagnold (1946) and Manohar (1955)' used o s c i l l a t i n g p l a t e s , Ranee and Warren (1969) used an o s c i l l a t i n g water t u n n e l , w h i l e Horikawa and Watanabe (1967) and E a g l e s o n , Dean and P e r a l t a (1958) used waves i n a wave flume. The l a s t two s e t s of d a t a were not used much except t o s u p p o r t the c o n c l u s i o n s as d e t e r m i n e d u s i n g the f i r s t t h r e e d a t a s e t s . The sediment p a r t i c l e d i a m e t e r s range from .009 t o 4.8 cm, and t h e i r d e n s i t i e s from 1.052 t o 7.90 §^T. The data ' cm J p o i n t s o b t a i n e d range over the e n t i r e spectrum of Reynolds 1 20 number used i n the S h i e l d s d i a g r a m , so a complete comparison of the u n i d i r e c t i o n a l and o s c i l l a t o r y c r i t i c a l shear s t r e s s e s i s p o s s i b l e . S e v e r a l f o r m u l a t i o n s were i n v e s t i g a t e d , as p r e s e n t e d by S i l v e s t e r and Mogridge (1970), but t h o s e of Bagnold (1946) and Ranee and Warren (1969) were found t o work the b e s t . Bagnold d e t e r m i n e d an e m p i r i c a l r e l a t i o n s h i p f o r the t h r e s h o l d of p a r t i c l e s from the o s c i l l a t i n g p l a t e experiment as o_ P c P i n t n r " T - a [ ( — ) g F - j u 0 where d 0 = 2 a 0 i s the t o t a l o r b i t a l e x c u r s i o n a t the bed. For wm s m a l l sand p a r t i c l e s , the d r a g f o r c e s are much l a r g e r than the i n e r t i a l f o r c e s f o r both o s c i l l a t i n g f l o w and f o r o s c i l l a t i n g p l a t e s , so t h i s r e l a t i o n i s e x p e c t e d t o be v a l i d f o r p a r t i c l e e n t r a i n m e n t under waves. The v a l i d i t y of t h i s r e l a t i o n under waves appears t o have been c o n f i r m e d by the r e s e a r c h of Romar and M i l l e r . T h i s r e l a t i o n has been r e a r r a n g e d by Komar and M i l l e r u s i n g U 0 m = ^ t o P U o J A 1 3 ( p s - p ) g D " a (D- ) 3 The f a c t o r D7^ makes the e q u a t i o n d i m e n s i o n a l , and o n l y s e r v e s t o reduce s c a t t e r , so i t i s dropped and we f i n a l l y have 121 where a'' i s d e t e r m i n e d as a''=.21 from the e x p e r i m e n t a l p l o t . Bagnold p o i n t e d out t h a t the c o n d i t i o n s were always l a m i n a r i n h i s e x p e r i m e n t . Manohar had much the same r e s u l t f o r the l a m i n a r boundary l a y e r , but had a''=.39. Komar and M i l l e r j u d i c i o u s l y chose a"'=.30, b e l i e v i n g the d i f f e r e n c e t o be caused by the p e r s o n a l d i f f e r e n c e between the o b s e r v e r s ' judgments. For the t u r b u l e n t boundary l a y e r , Manohar d e t e r m i n e d t h a t p U ° m _ .463 ( P s - p ) g T . { d o i ) b e s t f i t the d a t a . A g a i n s u b s t i t u t i n g u o m = _ i f r - r Komar and M i l l e r o b t a i n e d The c o n v e n i e n c e of e q u a t i o n s (5.1) and (5.2) i s t h a t the l e f t hand s i d e of each i s found t o be the same as the S h i e l d s e n t r a i n m e n t f u n c t i o n , e x c e p t f o r a c o n s t a n t and an a p p l i c a b l e f r i c t i o n f a c t o r . Then, F*=.30( dr2-)" 2'^ f o r l a m i n a r f l o w S D Z and , ,f F * = . 4 6 3 f f ( S £ ) T - ^ f o r t u r b u l e n t f l o w S D 2 where F* denotes the e n t r a i n m e n t f u n c t i o n , s 122 For g i v e n wave and sediment c o n d i t i o n s , i t i s p o s s i b l e t o d e t e r m i n e whether the near-bed boundary l a y e r i s l a m i n a r or t u r b u l e n t . The approach used by Komar and M i l l e r (1973) was t o examine the Reynolds numbers f o r the boundary l a y e r , and t o d e t e r m i n e the c r i t i c a l v a l u e a t which t r a n s i t i o n o c c u r r e d . Three d i f f e r e n t forms of the Reynolds number were used, ——, _ _ ^ u 0 m ( - F , ( g - J — o c - ^ , as w e l l as - y and were examined t o dete r m i n e which worked the b e s t f o r d e t e r m i n i n g t h e boundary l a y e r t y p e . The Reynolds number, U ° m D — — , was found t o be the most s u c c e s s f u l a t p r e d i c t i n g the boundary l a y e r t y p e , as a p l o t shows t h a t f o r D<0.5 mm, the sediment t h r e s h o l d i s reached b e f o r e t r a n s i t i o n , and f o r D>0.5 mm, t h e boundary l a y e r t y pe a t t h r e s h o l d i s more complex t o . d e t e r m i n e . Data used c o n s i s t s of t h a t of Bagnold (1946) and Manohar (1955). Madsen and Grant (1975) commented t h a t the wave f r i c t i o n f a c t o r , f , can be e v a l u a t e d a n a l y t i c a l l y o n l y f o r l a m i n a r boundary l a y e r s . U s i n g J o n s s o n ' s f r i c t i o n f a c t o r i n c a l c u l a t i n g t he shear s t r e s s f o r Bagnold's o s c i l l a t i n g p l a t e d a t a , i t i s seen t h a t t h e d a t a f o r the i n i t i a t i o n of sediment movement under o s c i l l a t o r y f l o w c o n d i t i o n s p l o t r emarkably w e l l w i t h the S h i e l d s c u r v e . Manohar's d a t a p l o t s somewhat above the S h i e l d s c u r v e , and t h i s d e v i a t i o n i s a t most a f a c t o r of two which i s the d e v i a t i o n found even f o r e x p e r i m e n t s w i t h s t e a d y u n i d i r e c t i o n a l f l o w s . 1 23 5.4.4 OTHER THRESHOLD MEASUREMENTS S i l v e s t e r and Mogridge (1970) prese.nted t h i r t e e n e m p i r i c a l r e l a t i o n s d e r i v e d f o r the i n c i p i e n t motion of sand p a r t i c l e s on a f l a t bed under o s c i l l a t o r y f l o w . These r e l a t i o n s g i v e the maximum near-bed f l u i d v e l o c i t y i n terms of the s p e c i f i c bouyant d e n s i t y , the g r a v i t a t i o n a l c o n s t a n t , the p a r t i c l e d i a m e t e r , the f l u i d v i s c o s i t y , and the o s c i l l a t o r y p e r i o d . These terms a re a l l r a i s e d t o a m u l t i t u d e of powers, depending on each i n v e s t i g a t o r ' s r e s u l t s . P l o t s of o r b i t a l a m p l i t u d e a g a i n s t o s c i l l a t o r y p e r i o d f o r i n c i p i e n t motion r e v e a l a wide d i v e r s i t y i n the r e s u l t s ( S i l v e s t e r and Mogridge, 1970). These p l o t s a r e p r o b a b l y extended beyond the range of v e r i f i c a t i o n however, so any c o n c l u s i o n t h a t a p a r t i c u l a r e m p i r i c a l r e l a t i o n does not work s h o u l d not be made. I t does appear t h a t some e m p i r i c a l r e l a t i o n s do work b e t t e r than o t h e r s over a wide range. In the l a m i n a r range, Komar and M i l l e r (1974) have found t h a t the e q u a t i o n p r e s e n t e d by Bagnold (1946) works b e s t , and t h a t the e m p i r i c a l graph of Ranee and Warren (1969) best f i t s t he t u r b u l e n t range. The work of V i n c e n t (1958) seems t o work w e l l f o r both t y p e s of boundary l a y e r . V i n c e n t has done work w i t h d i f f e r e n t sands i n t o the t u r b u l e n t regime, and found t h a t l a r g e r p a r t i c l e s ease the t r a n s i t i o n t o t u r b u l e n c e . The onset of t u r b u l e n c e seems t o be f a c i l i t a t e d by an i n c r e a s e d p o r o s i t y such as t h a t brought 1 24 about by an i n c r e a s e d sand bed t h i c k n e s s on the flume f l o o r . I t i s shown t h a t f o r c e r t a i n m a t e r i a l s , one o b s e r v e s f i r s t the development under waves of the l a m i n a r o s c i l l a t o r y boundary l a y e r , then the development of t u r b u l e n c e i n the boundary l a y e r , and f i n a l l y the movement of the f i r s t p a r t i c l e s . However, f o r some f i n e m a t e r i a l s ' exposed t o o s c i l l a t o r y p e r i o d s of about two seconds, the f i r s t p a r t i c l e s may b e g i n t o move b e f o r e the development of t u r b u l e n c e i n the boundary l a y e r . T h i s would seem t o agree w e l l w i t h the f i n d i n g s of Bagnold (1946) and Komar and M i l l e r (1973). V i n c e n t showed t h a t f o r a p a r t i c u l a r sediment s i z e , the sediment b e g i n s t o move when the v e l o c i t y j u s t o u t s i d e the near-bed boundary l a y e r , U =U 0 + U „ , rea c h e s a c e r t a i n J J max m M v a l u e . F i g u r e 5.7 shows the v a l u e s of U vs L f o r seven 3 max d i f f e r e n t m a t e r i a l s . The c o n s i s t e n c y of th e s e r e s u l t s f o r each p a r t i c l e d i a m e t e r i n d i c a t e s t h a t , f o r t h i s e xperiment at l e a s t , the p a r t i c l e e n t r a i n m e n t can be p r e d i c t e d s a t i s f a c t o r i l y . I t s h o u l d a g a i n be emphasized t h a t the r e l a t i o n s o b t a i n e d a r e u s u a l l y l i m i t e d by the range of e x p e r i m e n t a l c o n d i t i o n s from which they were d e r i v e d and a r e not of the g e n e r a l n a t u r e of the S h i e l d s c r i t e r i o n f o r u n i d i r e c t i o n a l s t e a d y f l o w . The number of r e l a t i o n s r e v i e w e d by S i l v e s t e r and Mogridge show t h i s c l e a r l y . Madsen and Grant (1976) p r e f e r r e d the use of the S h i e l d s c r i t e r i o n t o a l l r e l a t i o n s , and s t a t e d t h a t the 125 10 cm/sac n S a n d n*l -S a n d n* 2 — O fl o A Sand n* Pumlc* i f l . i . * 0 - f - ^ 1 " o D • • 8 u h om. <o Clrclt Sond n* 1 dgo. o,oei Sand n°2 d go.0,04< h c m . to Full cltcl. s,,,,, B o S d » o . o , O M h cm.20 Croti Po.Pumict n° l dgo.o.it hcm.K> A.t .H.k P u m l M ^ • • o - o . i l Po,- Pollopot n° l dgo.O.H Pot-Pollopot n ° 2 d » 0,OBt & — « -Pumlc* "'2, , f - ^ " ^ a a • PoMopaan* I — a • + t i -*• +- ^ p o l l a p a t n*2 1 0 100 200 ioo 400 L cm BOO F i g u r e 5.7. ( M o d i f i e d a f t e r V i n c e n t , 1958). Near-bed c r i t i c a l v e l o c i t y for' sands -of d i f f e r e n t s i z e under waves of v a r y i n g w a v e l e n g t h . e n t r a i n i n g f o r c e i s a d e q u a t e l y r e p r e s e n t e d by the bottom shear s t r e s s , which can be c a l c u l a t e d u s i n g J o n s s o n ' s f r i c t i o n f a c t o r r e s u l t s . They used a parameter s*=47[<s-l)gD]* i n p l a c e of the c o n v e n t i o n a l p a r t i c l e Reynold's number. The s c a t t e r i s found t o be a t most 30% from the mean v a l u e which i s r e a s o n a b l e when one c o n s i d e r s how s u b j e c t i v e i s the d e f i n i t i o n of the onset of mot i o n . No t r e n d i s found i n the e r r o r , and i t i s c o n c l u d e d t h a t the v a r i a t i o n i s p r i m a r i l y due t o e x p e r i m e n t a l s c a t t e r . They c o n c l u d e d t h a t the S h i e l d s c r i t e r i o n f o r the i n i t i a t i o n of sediment movement as d e r i v e d from s t e a d y u n i d i r e c t i o n a l f l o w c o n d i t i o n s s e r v e s as a q u i t e 126 a c c u r a t e and g e n e r a l c r i t e r i o n i n o s c i l l a t o r y f l o w , p r o v i d e d the boundary shear s t r e s s i s p r o p e r l y e v a l u a t e d . 5. 5 CONCLUSIONS A n a l y t i c a l l y , the f o r c e s e x e r t e d on a sediment p a r t i c l e w h ich cause i t t o move a r e unknown. The l i f t , d r a g , and the r e s u l t i n g o v e r t u r n i n g moments may a c t i n some c o m b i n a t i o n t o move a p a r t i c l e , or the p r e s s u r e f l u c t u a t i o n s i n t u r b u l e n t f l o w may cause p a r t i c l e e n t r a i n m e n t under some c o n d i t i o n s . The volume f o r c e s a r e regard e d as s m a l l , and the drag f o r c e i s d e t e r m i n e d t o be predominant. The shear f o r c e i s assumed t o be d i r e c t l y r e l a t e d t o the drag f d r c e , and t h i s shear f o r c e can be d e t e r m i n e d e x p e r i m e n t a l l y . The S h i e l d s c r i t e r i o n e x p r e s s e s the c r i t i c a l v a l u e of the r a t i o of e n t r a i n i n g t o s t a b i l i z i n g f o r c e s a c t i n g on a sediment p a r t i c l e on a sediment bed. The e n t r a i n i n g f o r c e i s r e l a t e d t o the shear s t r e s s e x e r t e d on the bed by the f l u i d , and the s t a b i l i z i n g f o r c e t o the submerged weight of the sediment p a r t i c l e . When the r a t i o of the two f o r c e s exceeds a c r i t i c a l v a l u e , sediment movement i s i n i t i a t e d . The S h i e l d s c r i t e r i o n i s an e m p i r i c a l r e l a t i o n s h i p which a p p l i e s t o any f l u i d , f l o w , and sediment c h a r a c t e r i s t i c s as l o n g as the sediment i s c o h e s i o n l e s s . There i s a good d e a l of s c a t t e r on the S h i e l d s diagram, but i f one c o n s i d e r s the s u b j e c t i v i t y i n v o l v e d i n i n t e r p r e t i n g the i n i t i a t i o n of sediment m o t i o n , as w e l l as the d i f f e r e n t e x p e r i m e n t a l s e t u p s and the wide range of 1 27 m a t e r i a l s and o r b i t a l p e r i o d s used, t h i s s c a t t e r i s a c c e p t a b l e . The o r d e r of s c a t t e r f o r u n i d i r e c t i o n a l f l o w and o s c i l l a t o r y f l o w t u r n out t o be s i m i l a r , so t h a t i t may be c o n c l u d e d t h a t the S h i e l d s c r i t e r i o n f o r the i n i t i a t i o n of sediment movement as d e r i v e d from u n i d i r e c t i o n a l f l o w c o n d i t i o n s s e r v e s as a good c r i t e r i o n f o r the i n i t i a t i o n of sediment movement under o s c i l l a t o r y f l o w . T u r b u l e n c e seems t o be i n v o l v e d t o a l a r g e degree i n the i n i t i a t i o n of movement. Whether the reason i s because of an i n c r e a s e d v e l o c i t y d u r i n g a t u r b u l e n t b u r s t or because of some p r e s s u r e d i f f e r e n c e d u r i n g the passage of an eddy has been a p o i n t of q u e s t i o n . The d i f f e r e n c e i n the i n i t i a t i o n of sediment motion f o r o s c i l l a t i n g and u n i d i r e c t i o n a l f l o w seem t o i n d i c a t e t h a t the t u r b u l e n c e and the boundary l a y e r s t r u c t u r e need t o be c o n s i d e r e d a l o n g w i t h the S h i e l d s c r i t e r i o n . Other e m p i r i c a l r e l a t i o n s appear t o work w e l l , but i n g e n e r a l , each works o n l y f o r a l i m i t e d range as d e t e r m i n e d by the e x p e r i m e n t a l bounds. Model c o n d i t i o n s s h o u l d not be e x t r a p o l a t e d t o p r o t o t y p e c o n d i t i o n s w i t h o u t c a r e f u l e v a l u a t i o n of the f l o w t u r b u l e n c e and the boundary l a y e r c o n d i t i o n s . 6. SHEAR PLATE EXPERIMENT 6.1 INTRODUCTION The purpose of the e x p e r i m e n t s was t o measure shear s t r e s s e s a t the bed e x e r t e d by waves and s t e a d y c u r r e n t s and by the c o m b i n a t i o n of waves and c u r r e n t s . There were s e v e r a l w avelengths used i n the s t u d y c o r r e s p o n d i n g t o the n o - c u r r e n t p e r i o d s of T=1.6, 1.8, 2.0, 2.2, and 2.4 seconds. There were f o u r c u r r e n t v e l o c i t i e s used, which were V=0, 0.20, 0.30, and 0.50 m/s i n the d i r e c t i o n of wave p r o p a g a t i o n . The wave h e i g h t s were i n c r e a s e d from v e r y s m a l l t o near b r e a k i n g f o r e v e r y c u r r e n t c o n d i t i o n , and the d i s p l a c e m e n t of a f l e x u r a l l y s u p p o r t e d shear p l a t e was measured f o r each wave h e i g h t . I n t h i s way was i t d e t e r m i n e d t o compare the shear s t r e s s under a wave w i t h the shear s t r e s s under an e q u i v a l e n t h e i g h t and shape wave on a c u r r e n t . 6. 2 GENERAL DESCRIPTION OF SHEAR PLATE In F i g u r e 6.1, the square shear s e n s i t i v e p l a t e (1) has a s u r f a c e a r e a of 0.360 square m e t r e s , and i s mounted on f l e x u r e l e g s . The s t r u c t u r e i s p o s i t i o n e d i n the c h a n n e l so t h a t i n i t s e q u i l i b r i u m p o s i t i o n t h e r e i s a 1 mm gap a t each end t h r o u g h which the p l a t e can d e f l e c t . At the s i d e s of the p l a t e , c l e a r a n c e gaps of 0.4 mm a r e p r o v i d e d . The v e r t i c a l p o s i t i o n of the shear p l a t e can be a d j u s t e d by t h r e e a d j u s t i n g screws (2) l o c a t e d on the base p l a t e ( 3 ) . The 128 129 F i g u r e 6.1. D e f i n i t i o n s k e t c h of shear p l a t e . l a t e r a l p o s i t i o n i n g of the shear p l a t e i s done by hand. S e v e r a l r e p e t i t i o n s a r e n e c e s s a r y t o a d j u s t the shear p l a t e v e r t i c a l l y and h o r i z o n t a l l y so- t h a t i t s s u r f a c e i s f l u s h w i t h the a d j a c e n t s u r f a c e s . I t i s p o s s i b l e t o a d j u s t the shear p l a t e s u r f a c e t o w i t h i n 0.5 mm. The shear p l a t e i s s u p p o r t e d by 5 s t a i n l e s s s t e e l l e g s (4) which a r e 1 mm t h i c k and 6 mm wide. These l e g s are one p i e c e w i t h s u p p o r t i n g f e e t which are f a s t e n e d t o the base p l a t e and t o the shear p l a t e by c o u n t e r s u n k screws. F u r t h e r s u p p o r t i s p r o v i d e d by 2 s i m i l a r l y f a s t e n e d l e g s (5) of 10 mm d i a m e t e r , h a v i n g s l i d i n g c o v e r s which l o c k the shear p l a t e t o p r e v e n t s t r a i n gauge damage d u r i n g h a n d l i n g . The shear p l a t e i s 1.5 mm t h i c k and i s l o n g i t u d i n a l l y and t r a n s v e r s e l y s t i f f e n e d by f o u r 3 mm a n g l e s (6) b o l t e d onto i t s bottom. These a n g l e s f a i l e d t o p r e v e n t some bowing of the p l a t e edges under v e r t i c a l " l o a d i n g . T h i s bowing o c c u r r e d upwards under the t r o u g h s of the l a r g e r waves but l i t t l e bowing o c c u r r e d downwards under the wave c r e s t s . 1 30 Bed s u p p o r t s (7) a c t as f l o w i n t e r r u p t o r s t o impede f l o w under the shear p l a t e from any f l o w s below the f a l s e bed (8) i n the flume. The s m a l l c l e a r a n c e between the shear p l a t e and the end f l a n g e s of the f a l s e bottom r e s u l t s i n h i g h f l o w r e s i s t a n c e and s m a l l f l o w s . A s t r a i n gauge (9) mounted on the downstream support l e g i s used to measure any shear p l a t e d e f l e c t i o n . The s t r a i n gauge i s p r o t e c t e d w i t h i n an i n v e r t e d p l a s t i c c a p s u l e (10) which p r e v e n t s water from c o n t a c t i n g i t . A f u l l b r i d g e c i r c u i t i s used g i v i n g a l i n e a r o u tput over the f u l l range of shear f o r c e . S t a t i c c a l i b r a t i o n s were performed u s i n g the c a n t i l e v e r system shown i n F i g u r e 6.2. Both p o s i t i v e and n e g a t i v e l o a d i n g was performed, y i e l d i n g the c a l i b r a t i o n c u r v e shown i n F i g u r e 6.3. The s t r a i n gauge c i r c u i t c o n s i s t s of two s t r a i n gauges mounted on both s i d e s of one of the downstream f l e x u r e l e g s , and an e x c i t a t i o n v o l t a g e source d e l i v e r i n g 6.5 v o l t s . The t r a n s d u c e r w i r i n g runs t h r u g h a copper tube (11) t o the • 2 IS cm 4crr, F i g u r e 6.2. D e f i n i t i o n s k e t c h of c a n t i l e v e r system. 131 Pl0+e —i 1 1 1 1 — -loo Zoo Soo 4a> SCO to 6 4 F i g u r e 6.3. C a l i b r a t i o n c u r v e f o r shear p l a t e o utput s i g n a l d e t ermined from e x p e r i m e n t . t r a n s v e r s e edge of the p l a t e , and t h r o u g h a s m a l l gap i n the p l a t e t o t h e water s u r f a c e . The tube a l s o a c t s as a s t o p t o p r e v e n t l a r g e d i s p l a c e m e n t s of the shear p l a t e so t h e r e w i l l not be any s t r a i n gauge damage. A f o u r c h a n n e l c h a r t r e c o r d e r w i t h s e n s i t i v i t y of 10 mV/chart d i v i s i o n i s used t o r e c o r d t h e t r a n s d u c e r o u t p u t . The s e n s i t i v i t y of the shear p l a t e was measured by u s i n g a v e r y s m a l l l o a d which c o u l d be r e a p p l i e d a t any l o a d s i t u a t i o n t o g i v e the same d e t e c t a b l e added o u t p u t , and was d e t e r m i n e d t o be 0.006 Pa. The shear p l a t e i s c o n s i d e r e d t o be a s i m p l e m e c h a n i c a l s t r u c t u r e r e p r e s e n t a b l e by a spr i n g - m a s s - d a s h p o t system. The e q u a t i o n of mo t i o n i s then F(t)=mx+cx+kx (6.1) 1 32 where m i s the shear p l a t e mass c i s the damping c o e f f i c i e n t r e s u l t i n g from the water r e s i s t a n c e k i s the s p r i n g c o n s t a n t x i s the d i s p l a c e m e n t i n the d i r e c t i o n of motion x i s the v e l o c i t y i n the d i r e c t i o n of motion x i s the a c c e l e r a t i o n i n the d i r e c t i o n of motion F ( t ) i s the f o r c i n g f u n c t i o n e x e r t e d on the system. The dynamic response of t h i s system i s d e s c r i b e d by Thomson (1981), and the damping r a t i o and n a t u r a l f r e q u e n c y a r e d e s c r i b e d as The n a t u r a l f r e q u e n c y was measured i n a i r , and i n water the damping r a t i o was found t o be s m a l l . The resonant f r e q u e n c y i s then almost i d e n t i c a l t o the n a t u r a l f r e q u e n c y . F u r t h e r , the f r e q u e n c y of the f o r c i n g f u n c t i o n under waves i s so s m a l l when compared w i t h the resonant f r e q u e n c y t h a t the phase between the output response and the f o r c i n g f u n c t i o n i s z e r o , and the g a i n f a c t o r f o r the response i s u n i t y . 6. 3 ASSUMPTIONS FOR USE OF SHEAR PLATE I t was assumed t h a t the drag on the u n d e r s i d e of the p l a t e would be n e g l i g i b l e , as the water v e l o c i t y t h e r e would be s m a l l . I t was f u r t h e r assumed t h a t the p r e s s u r e f o r c e s m (6.2) (6.3) would be t r a n s m i t t e d t h r o u g h the gaps of the p l a t e such t h a t the v e r t i c a l l o a d i n g on the p l a t e would be m i n i m a l . I t was a l s o assumed t h a t the p r e s s u r e f o r c e s on the edges of the p l a t e c o u l d be c a l c u l a t e d from the a p p l i c a b l e t h e o r y ( l i n e a r or second o r d e r S t o k e s ) and t h a t t h e s e f o r c e s c o u l d be d e t e r m i n e d and s u b t r a c t e d from the shear p l a t e o utput at c e r t a i n phase i n t e r v a l s . The method of P.S. E a g l e s o n (1962) of u s i n g p l a t e s of d i f f e r e n t t h i c k n e s s was employed t o de t e r m i n e the p r e s s u r e f o r c e s . 6. 4 EXPERIMENTAL SETUP AND PROCEDURE The wave flume used i s d e s c r i b e d i n an e a r l i e r c h a p t e r . For t h i s experiment a f a l s e bed was p l a c e d on the flume bottom. I t co m p r i s e d f o u r 2.5 m l o n g s e c t i o n s of 7.5 cm d e p t h , made of aluminum and p l a s t i c . S u p p o r t s were of p l a s t i c and a l s o s e r v e d t o b l o c k u n d e r f l o w . The p i e c e s were c a r e f u l l y p l a c e d i n t o the flume such t h a t a l l a i r was a l l o w e d t o escape, such t h a t the p i e c e s were p r o p e r l y i n t e r l o c k e d t o p r o v i d e a c o n t i n u o u s f l a t bed. T h i n metal s t r i p s had t o be i n s t a l l e d a t i n t e r v a l s a l o n g the s i d e s of the flume t o h o l d the bed down under the p r e s s u r e f o r c e s caused by the wave a c t i o n . The shear p l a t e was p o s i t i o n e d i n the r e c e s s e d bed and w e i g h t s were p l a c e d on the base p l a t e t o p r e v e n t the a p p a r a t u s from moving about. The r e c e s s was then c l o s e d up such t h a t the gaps around the shear p l a t e edges were about 1 mm, and the bed was then p r o p e r l y s e c u r e d . 1 34 The water depth was s e t t o 0.40 m, and the waves were g e n e r a t e d . A R o b e r t s o n L e v e l - T e l Model 157-B2 wave probe r e q u i r i n g 26.5VDC Supply was p o s i t i o n e d above the c e n t e r p o i n t of the p l a t e and the o u t p u t s from the probe and from the shear p l a t e were r e c o r d e d s i m u l t a n e o u s l y on the c h a r t r e c o r d e r . The wave h e i g h t t r a c e and the shear p l a t e o utput t r a c e were then r e c o r d e d , and a r e shown i n F i g u r e 6.4. The waves used were of p e r i o d T = 1.6, 1.8, 2.0, 2.2, and 2.4 seconds on no c u r r e n t , and were of many wave h e i g h t s , from near z e r o t o near b r e a k i n g . Waves of i d e n t i c a l w avelengths were g e n e r a t e d f o r c o - c u r r e n t c o n d i t i o n s of 0.20, 0.30, and 0.50 m/s, where a g a i n t h e r e were wave h e i g h t s from near z e r o t o near b r e a k i n g g e n e r a t e d . F i g u r e 6.4. T y p i c a l t r a c e of wave p r o f i l e and c o r r e s p o n d i n g shear p l a t e o u t p u t s i g n a l . 135 6. 5 EXPERIMENTAL RESULTS 6.5.1 OBSERVATION I t was obser v e d t h a t the p l a t e bowed t o a c e r t a i n degree under the t r o u g h s of the l a r g e r waves. T h i s was not e x p e c t e d t o cause any d i f f i c u l t y i n measurement because t h i s l o a d i n g was deemed t o be v e r t i c a l o n l y . I t was thought t h a t t h e r e would be no acknowledgement of t h i s f o r c e i n the o u t p u t , as the s t r a i n gauges would measure o n l y the d i f f e r e n t i a l s t r a i n caused by the bending of the f l e x u r e l e g s under a h o r i z o n t a l l o a d . The f l o w under the p l a t e was found t o be q u i t e s m a l l as e x p e c t e d , but t h e r e was g r e a t v o r t e x motion seen around the p l a t e edges, and t h i s agrees w i t h the observed p l a t e bowing. 6.5.2 NUMERICAL RESULTS The measured f o r c e s on the p l a t e were d e t e r m i n e d by a p p l y i n g the n e c e s s a r y f a c t o r s t o the measured t r a n s d u c e r o u t p u t . The f i r s t o r d e r p r e s s u r e end terms were c a l c u l a t e d u s i n g the t o t a l f r o n t a l a r e a , and f o r each phase a n g l e of -g, thes e were s u b t r a c t e d from the t o t a l measured f o r c e . The r e s u l t a n t f o r c e was b e l i e v e d t o be the measured shear f o r c e , but i t was much l a r g e r than t h a t e x p e c t e d . Then i t was d e c i d e d t o use E a g l e s o n ' s s i m p l e approach of u s i n g a p l a t e of a n o t h e r t h i c k n e s s and comparing the two r e s u l t s . The d i f f e r e n c e i n measured f o r c e would s i m p l y be the d i f f e r e n c e as caused by the d i f f e r e n c e i n t h e end p r e s s u r e f o r c e s , so 136 the end p r e s s u r e c o u l d be c a l c u l a t e d i f the end a r e a s were both known. T h i s approach f a i l e d as the p r e s s u r e was c a l c u l a t e d t o have many d i f f e r e n t v a l u e s , both p o s i t i v e and n e g a t i v e . I t was then r e a l i z e d t h a t something was i n h e r e n t l y wrong w i t h the d a t a procurement. A v e r t i c a l l o a d was then a p p l i e d t o the p l a t e and an output was o b s e r v e d . Then, the p l a t e was c a l i b r a t e d f o r a v e r t i c a l l o a d and i t was d e t e r m i n e d t h a t a s m a l l v e r t i c a l l o a d a c r o s s the p l a t e would c r e a t e a l a r g e t r a n s d u c e r o u t p u t . The problems here a r e t h a t the o u t p u t i s l a r g e , and t h a t the output depends not o n l y on the l o a d but a l s o on the l o c a t i o n of the p l a t e where the l o a d i s a p p l i e d . A . u n i f o r m p r e s s u r e d i f f e r e n c e a c r o s s the p l a t e would have t o be measured t o 1 mm of water head a c c u r a c y t o g i v e a response the s i z e of the e x p e c t e d measured shear f o r c e . F u r t h e r , g i v e n the n o n l i n e a r n a t u r e of the above p l a t e p r e s s u r e f i e l d , and the l o a d p o s i t i o n dependent response, t h i s p r e s s u r e d i f f e r e n c e would have t o be measured v i r t u a l l y a t e v e r y l o c a t i o n a c r o s s the shear p l a t e . The s i t u a t i o n i s f u r t h e r c o m p l i c a t e d by a n o n l i n e a r i t y of the v e r t i c a l l o a d o u t p u t response which a l s o depends on the p o s i t i o n of l o a d i n g . I t must be c o n s i d e r e d i f i t i s a t a l l p o s s i b l e t o measure the shear s t r e s s under a wave by use of a shear p l a t e of t h i s t y p e . An i n t e r e s t i n g f i n d i n g from t h e s e e x p e r i m e n t s was t h a t of the importance of the gap w i d t h between t h e t r a n s v e r s e edges of the shear p l a t e and the bed. I t was found t h a t by 137 i n c r e a s i n g the gap by a s m a l l amount i t was p o s s i b l e t o change the measured t o t a l f o r c e by over two and o n e - h a l f t i m e s . I t was t h e o r e t i c a l l y d e r i v e d by Brown and J o u b e r t (1969) t h a t the gap w i d t h s h o u l d p l a y an im p o r t a n t r o l e i n d e t e r m i n i n g the s i z e of the measured f o r c e . Then the p r e s s u r e measurement a t the gaps under o s c i l l a t o r y f l o w would appear t o be c r i t i c a l as t h i s gap r e l a t i o n c e r t a i n l y p r e c l u d e s any p o s s i b l e t h e o r e t i c a l p r e s s u r e d e t e r m i n a t i o n . 6.6 CONCLUSIONS The shear s t r e s s e s t o be measured a r e v e r y s m a l l . Use of a shear p l a t e t o measure t h e s e s t r e s s e s r e q u i r e s c a r e f u l c a l i b r a t i o n and measurement of p r e s s u r e f o r c e s . C a r e f u l d e s i g n of the shear p l a t e must ensure t h a t i n e r t i a l terms a r e n e g l i g i b l e , and t h a t p r e s s u r e f o r c e s can be p r o p e r l y measured. Gap w i d t h between p l a t e and a d j o i n i n g s u r f a c e s may p l a y an i m p o r t a n t r o l e i n c h a n g i n g the p r e s s u r e f o r c e s around the p l a t e , n e c e s s i t a t i n g t h e i r measurement, r a t h e r than s i m p l y t h e i r e s t i m a t i o n . Because of the n o n - l i n e a r p r e s s u r e d i s t r i b u t i o n under waves, i t i s q u e s t i o n a b l e whether a shear p l a t e can even be used t o measure the shear f o r c e s a t the bed l e v e l . 7. LASER DOPPLER ANEMOMETRY EXPERIMENT 7.I INTRODUCTION When a n a l y z i n g the combined e f f e c t of waves and c u r r e n t s on sediment t r a n s p o r t , i t i s n e c e s s a r y t o know how t o combine a near-bed wave shear s t r e s s w i t h a u n i d i r e c t i o n a l c u r r e n t shear s t r e s s . At the p r e s e n t t i m e , t h e r e a r e no t h e o r i e s t o p r e d i c t the shear s t r e s s under a combined o s c i l l a t o r y and u n i d i r e c t i o n a l f l o w motion. T h e o r i e s do e x i s t f o r c a l c u l a t i n g the wave shear s t r e s s f o r wave motion a l o n e as a f u n c t i o n of the maximum near-bed v e l o c i t y and the near-bed water p a r t i c l e a m p l i t u d e . These t h e o r i e s make use of a wave f r i c t i o n f a c t o r , w hich has a v a l u e u s u a l l y dependent on the near-bed water p a r t i c l e a m p l i t u d e and the bed roughness v a l u e . There a r e a l s o w e l l known t h e o r i e s f o r c a l c u l a t i n g the shear s t r e s s under steady c u r r e n t s a l o n e . The p r e s e n t work i s concerned w i t h examining how t h e s e s e p a r a t e s t r e s s components compare w i t h the combined s t r e s s under both waves and c u r r e n t s . Measurements were t h e r e f o r e c a r r i e d out t o determine the near-bed v e l o c i t y p r o f i l e s over f l a t sand beds at v e l o c i t i e s near the c r i t i c a l f o r p a r t i c l e m o t i o n . The measurements were made i n a wave flume w i t h use of a L a s e r D o p p l e r Anemometer (LDA). The measurements were made f o r waves of two c h a r a c t e r i s t i c wavelengths and f o r c o m b i n a t i o n s of waves and c u r r e n t s h a v i n g i d e n t i c a l c h a r a c t e r i s t i c w a v e l e n g t h s . Measures were a l s o made of the c r i t i c a l 1 139 v e l o c i t i e s f o r c u r r e n t s a l o n e . P r o f i l e s were c o n s t r u c t e d by o b t a i n i n g v e l o c i t y measurements a t d i f f e r e n t h e i g h t s above the bed. The t h r e s h o l d v e l o c i t i e s a t the bed a r e o b t a i n e d f o r t h e v a r i o u s c o n d i t i o n s . These a c q u i r e d p r o f i l e s can then be compared w i t h t h e o r e t i c a l p r o f i l e s of water p a r t i c l e v e l o c i t y . A l s o , p r e d i c t i o n s can be made of the bed shear s t r e s s e s from v a r i o u s f r i c t i o n f a c t o r v a l u e s , and from v e l o c i t y p r o f i l e l a w s . The f r i c t i o n f a c t o r r e f e r r e d t o above f o l l o w s a c e r t a i n t r e n d , d e c r e a s i n g as the water p a r t i c l e a m p l i t u d e i n c r e a s e s f o r a c o n s t a n t bed roughness, and i s g i v e n by a number of e q u a t i o n s by d i f f e r e n t a u t h o r s ( J o n s s o n , 1966; K a j i u r a , 1968; and Kamphuis, 1975). The v a l u e of the f r i c t i o n f a c t o r f o r smooth t u r b u l e n t f l o w a c c o r d i n g t o th e s e formulae r a r e l y d i f f e r s by more than 25 p e r c e n t . For rough t u r b u l e n t f l o w , the v a l u e s as p r e d i c t e d by Jonsson and Kamphuis a r e w i t h i n 20%, but K a j i u r a ' s v a l u e s seem v e r y h i g h . Most of the bed shear s t r e s s t h e o r i e s f o r a u n i d i r e c t i o n a l c u r r e n t g i v e the s t r e s s as a f u n c t i o n of the v e l o c i t y p r o f i l e . Examples of t h i s a r e the u n i v e r s a l l o g law, the 1/7th power law, and the l a m i n a r law. S c h l i c h t i n g (1979) gave the shear s t r e s s as a f u n c t i o n of the d i s t a n c e x from the edge of a f l a t p l a t e , but over a c o n t i n u o u s sand bed t h i s d i s t a n c e cannot be d e f i n e d and so the f l o w i s assumed t o be f u l l y d e v e l o p e d rough t u r b u l e n t w i t h t h e boundary l a y e r t h i c k n e s s e q u a l t o the depth of 1 40 f l o w . For t he combined o s c i l l a t o r y and u n i d i r e c t i o n a l f l o w m o t i o n , i t has been w i d e l y assumed t h a t the v e l o c i t y under t h i s combined c o n d i t i o n i s s i m p l y the v e c t o r i a l sum of the i n d i v i d u a l components. P r e s u m a b l y , i f the v e l o c i t i e s can be added v e c t o r i a l l y , then t h e shear s t r e s s e s can be added p r o p o r t i o n a t e l y as f u n c t i o n s of the v e l o c i t y s q u a r e d . The assumption of the v e c t o r i a l a d d i t i o n of v e l o c i t i e s has been q u e s t i o n e d by George and S l e a t h (1976), whose measurements r e v e a l e d t h a t the near-bed d r i f t was l e s s than the sum of the i n d i v i d u a l components. The measurements p e r f o r m e d appear t o v e r i f y the r e s u l t s of Kamphuis f o r o s c i l l a t o r y f l o w and the l o g v e l o c i t y p r o f i l e law f o r st e a d y u n i d i r e c t i o n a l f l o w . They a l s o p r o v i d e s t r o n g e v i d e n c e f o r a g e n e r a l c r i t e r i o n f o r the onset of sediment m o t i o n , and e s t a b l i s h a r u l e f o r the d e s c r i p t i o n of the v e l o c i t y under combined waves and c u r r e n t s i n terms of t h e component wave and c u r r e n t v e l o c i t i e s . 7.2 EXPERIMENTAL EQUIPMENT AND PROCEDURE 7.2.1 FACILITY The d e s c r i p t i o n of t h e d i m e n s i o n s of the wave flume and i t s boundary c o n d i t i o n s has been g i v e n i n an e a r l i e r c h a p t e r . 141 The t h i c k n e s s of the g l a s s w a l l s i s 16 mm. They were t h o r o u g h l y c l e a n e d b e f o r e the experiment so t h a t t h e i r o p t i c a l q u a l i t y was f a i r l y good. Sand of a p a r t i c u l a r s i z e range was s p r e a d i n t h e w o r k i n g s e c t i o n t o a f l a t bed c o n d i t i o n . The roughness of a sand bed v a r i e s a l o n g i t s l e n g t h and a l o n g i t s b r e a d t h , so the f l o w p a t t e r n i s not as t w o - d i m e n s i o n a l as one would l i k e . However, the measurements were taken a t a s i n g l e p o i n t i n the flume so d i f f e r e n c e s caused by the w a l l s and by the bed roughness v a r i a t i o n a c r o s s the flume b r e a d t h a r e n o n - e x i s t e n t . R e f l a t t e n i n g of the bed a f t e r each measurement would however be e x p e c t e d t o cause a d i f f e r e n c e i n the l o n g i t u d i n a l roughness, and presumably i n the near-bed v e l o c i t y as w e l l . There was a s e r i o u s problem w i t h the water t u r b i d i t y . In the w i n t e r of 1983, heavy r a i n s caused the Vancouver, B.C., water s u p p l y t o become so c l o u d y t h a t i t was i m p o s s i b l e t o see t h r o u g h the water t o the o t h e r s i d e of the flume. I t was n e c e s s a r y t o use alum c r y s t a l s and f i l t r a t i o n t o c l e a r up the w a t e r , and a f t e r t r e a t m e n t was c o m p l e t e d , the s i g n a l - t o - n o i s e r a t i o of the Doppler s i g n a l was e x c e l l e n t . 7.2.2 INSTRUMENTATION The v e l o c i t y measurements were made a t a p o i n t some 4.9 m downstream from the downstream end of the c o n v e r g i n g s e c t i o n w i t h a L a s e r Doppler Anemometer c o m p r i s i n g the f o l l o w i n g u n i t s : 142 - S p e c t r a P h y s i c s S t a b i l i t e , Model 120 He/Ne L a s e r - O p t i k o n O p t i c a l U n i t w i t h phase s h i f t e r - O p t i k o n R e c e i v i n g O p t i c s -EMI Gencom I n c . P h o t o m u l t i p l i e r -Model 3000R EMI Gencom I n c . High V o l t a g e Power Supply -DISA Type 55 L97 High V o l t a g e Power Supply -DISA Type 55 N21 T r a c k e r Main U n i t -DISA Type 55 N24 D i s p l a y Module - H e w l e t t P a c k a r d 3582A Spectrum A n a l y z e r -Type 561A O s c i l l o s c o p e - S p e c t r a P h y s i c s L a s e r E x c i t e r - H e w l e t t - P a c k a r d Model 7100B Ch a r t R e c o r d e r 'A b l o c k d i a g r a m of the u n i t s i s p r e s e n t e d i n F i g u r e 7.1. A c c u r a t e v e r t i c a l l o c a t i o n of the i n t e r s e c t i o n volume was f a c i l i t a t e d by use of p r e c u t m e t a l s t r i p s p l a c e d under the l a s e r and o p t i c s . A l s o , the d epth below a known water l e v e l was measured by use of a meter s t i c k and hook and p o i n t e r guage. The p r e s ence of a r e t u r n f l o w p i p e b e s i d e the flume made i t i m p o s s i b l e t o j o i n the p r o j e c t i n g and r e c e i v i n g o p t i c s i n t o an assembly, so they had t o be r a i s e d and l o w e r e d i n d e p e n d e n t l y and r e a l i g n e d . T h i s p r a c t i c e was t e d i o u s and as t h e r e was a time c o n s t r a i n t , i t was d e c i d e d t o run the s e p a r a t e wave and c u r r e n t c o n d i t i o n s f o r each h e i g h t above the bed. However, t h e r e were drawbacks i n " t h i s p r o c e d u r e as w e l l , as the wave h e i g h t c o u l d not be r e - c r e a t e d w i t h g r e a t a c c u r a c y f o r e v e r y t e s t r u n , though 143 3oooR EMI Analyzer lisplc^ Velocity Signals Velocity Dt&pl b^ s RMS VeUtiiy Evcifet-F i g u r e 7.1. B l o c k diagram of the LDA setup, the p e r i o d c o u l d be r e s e t a c c u r a t e l y . The LDA system was o p e r a t e d i n the d i f f e r e n t i a l D o ppler mode w i t h f o r w a r d s c a t t e r . The s i g n a l p r o c e s s i n g was a c c o m p l i s h e d by u s i n g the DISA Type 55 N21 Frequency T r a c k e r . The Doppler s i g n a l s were u s u a l l y f u r t h e r enhanced by c u t t i n g out the f r e q u e n c i e s i n ex c e s s of 59 Hz w i t h lowpass f i l t e r i n g . The Doppler s i g n a l was d i s p l a y e d on the Type 561A o s c i l l o s c o p e i n o r d e r t o f i n e tune the s i g n a l by 144 a c c u r a t e l a t e r a l and v e r t i c a l adjustment of the r e c e i v i n g o p t i c s , such t h a t the i n t e r s e c t i o n volume was f o c u s s e d on the p h o t o m u l t i p l i e r . S p e c t r a l a n a l y s i s was done u s i n g F a s t F o u r i e r T r a n s f o r m s and the t ime averaged v e l o c i t y was d i s p l a y e d by the DISA Type 55 N24 D i s p l a y Module. S p e c t r a l a n a l y s i s was performed on the v e l o c i t y t r a c e g i v e n by the HP 3582A Spectrum A n a l y z e r , which was found t o be u n c a l i b r a t e d . I t became n e c e s s a r y t o get a c a l i b r a t e d v e l o c i t y t r a c e because the v e l o c i t y spectrum c o u l d not be used f o r c o m p u t a t i o n of the maximum v e l o c i t y . There was no use made of the f r e q u e n c y s h i f t e r because of a problem w i t h a l a r g e s i g n a l - t o - n o i s e r a t i o when i t was employed. S p e c t r a l a n a l y s i s becomes q u e s t i o n a b l e when the v e l o c i t y f a l l s out of the f r e q u e n c y range chosen f o r the t r a c k i n g p r o c e s s . As the f r e q u e n c y s h i f t i n g was i m p o s s i b l e , f l o w r e v e r s a l was o b s e r v e d t o o c c u r and t r a c k i n g was broken as the v e l o c i t y c r o s s e d z e r o . F u r t h e r , as the spectrum does not r e c o g n i z e n e g a t i v e v e l o c i t i e s as d i f f e r e n t from p o s i t i v e ones, they b o t h were summed f o r the p e r i o d c o r r e s p o n d i n g t o o n e - h a l f of the a c t u a l p e r i o d . As seen i n the v e l o c i t y t r a c e i n F i g u r e 7.2, the maximum and minimum v e l o c i t i e s d i f f e r , so the s p e c t r a l a n a l y s i s c o u l d not be used f o r v e l o c i t y d e t e r m i n a t i o n . F u r t h e r , f o r t h e c a s e s of added o s c i l l a t o r y and u n i d i r e c t i o n a l c u r r e n t , the f r e q u e n c y t r a c k i n g range was not wide enough i n most c a s e s t o c a t c h both the maximum and minimum v e l o c i t i e s . S p e c t r a l a n a l y s i s was abandoned and the 1 45 F i g u r e 7.2. T y p i c a l t r a c e of near-bed water p a r t i c l e v e l o c i t y . c h a r t r e c o r d e r was used t o r e c o r d the v e l o c i t i e s . However, v e l o c i t i e s near z e r o were immeasurable because of a v e r y s m a l l f r e q u e n c y range b e i n g n e c e s s a r y t o r e c o r d t h e s e . T h i s caused c l i p p i n g of the output s i g n a l , as the v e l o c i t y d i p p e d i n t o the s m a l l f r e q u e n c y range f o r such a s h o r t time t h a t the s i g n a l c o u l d not be t r a c k e d . Q u e s t i o n s have been r a i s e d as t o the performance of the LDA i n the boundary l a y e r . Beech (1977) a c h i e v e d good r e s u l t s i n a l a m i n a r boundary l a y e r down t o 0.3 mm above the boundary. In the p r e s e n t s t u d y , the minimum h e i g h t above the bed was 0.3 mm and the t r a c k e r e x p e r i e n c e d no d i f f i c u l t y i n f o l l o w i n g the s i g n a l f o r o t h e r than near z e r o v e l o c i t i e s . No sediment p a r t i c l e s were observed t o pass through the l a s e r r a y s so i t i s assumed t h a t the r e s u l t s a c h i e v e d a r e good. 7. 3 MEASUREMENTS TAKEN Two s e t s of measurements were o b t a i n e d : the f i r s t s e t by t h e w r i t e r , and the second by Quick e t . a l . (1985). The p r e l i m i n a r y d a t a s e t was the f i r s t t o be o b t a i n e d u s i n g the 1 46 d e s c r i b e d p r o c e d u r e , and was found t o be more v a l u a b l e f o r d i s c o v e r i n g what s p e c i f i c problems a r i s e and what d e t a i l e d measurements s h o u l d be made than f o r use as a sound d a t a base. The poor r e p e a t a b i l i t y of the r e s u l t s and the s h o r t a g e of d a t a meant t h a t s u b s t a n t i v e c o n c l u s i o n s c o u l d not be drawn. The second, more comprehensive s e t of d a t a was then o b t a i n e d u s i n g the r e f i n e d e x p e r i m e n t a l p r o c e d u r e s , w i t h the purpose of o b t a i n i n g a broad d a t a base near the bed i n o r d e r t h a t the . s p e c u l a t i v e c o n c l u s i o n s drawn from the f i r s t s e t c o u l d be c o n f i r m e d . 7.3.1 FIRST SET OF EXPERIMENTS S t i l l water depth was v e r y near t o 0.40 m f o r t h i s s e t of e x p e r i m e n t s , and two wavelengths were used, L=3.73 m f o r the 2.0 second p e r i o d wave, and L=2.83 m f o r the 1.6 second p e r i o d wave. T h i s f i r s t s e t of e x p e r i m e n t s was d i v i d e d i n t o two s e c t i o n s . The bed. was sand of average d i a m e t e r 0.5 mm f o r the f i r s t group of e x p e r i m e n t s and was pea g r a v e l of average d i a m e t e r 6.0 mm f o r the second group. Over the sand bed, the c u r r e n t used was 0.20 m/s. Over the g r a v e l bed, two c u r r e n t s were used, one . of 0.20 m/s and the second of 0.50 m/s. W i t h the superimposed c u r r e n t s , t h e wave p e r i o d s were m o d i f i e d a c c o r d i n g l y t o p r e s e r v e the same w a v e l e n g t h s . For t h e sand bed, the wave h e i g h t was i n c r e a s e d u n t i l sand p a r t i c l e motion was o b s e r v e d f o r both the c u r r e n t and n o - c u r r e n t c a s e s , f o r both w a v e l e n g t h s . A l s o , an e q u i v a l e n t h e i g h t of wave, H , was made f o r the n o - c u r r e n t case h a v i n g 147 the same wave h e i g h t as f o r the case of imposed c u r r e n t . U n f o r t u n a t e l y , t h e s e wave h e i g h t s were not i d e n t i c a l because of a mixup i n r e f e r e n c e water s u r f a c e l e v e l s f o r the c u r r e n t and n o - c u r r e n t c a s e s . F i g u r e 7.3 shows how the s e wave h e i g h t s and wavelengths a r e d e f i n e d . For the g r a v e l bed, the wave h e i g h t c o u l d not be i n c r e a s e d u n t i l movement o c c u r r e d because the l a r g e wave s i z e r e s u l t e d i n l a r g e amounts of water b e i n g thrown over the flume w a l l s near the wave p a d d l e . For the i n s t a n c e of the 0.50 m/s c u r r e n t , the wave was i n c r e a s e d u n t i l the g r a v e l was ve r y n e a r l y i n m o t i o n , but o v e r a l l c r i t i c a l m otion was not o b s e r v e d . R a t h e r , t h e r e would be " b u r s t s " when some p a r t i c l e s would move, s e p a r a t e d by i n t e r v a l s when » O F i g u r e 7.3. D e f i n i t i o n s k e t c h f o r wave c o n d i t i o n s . 1 48 t h e r e would be no movement. The wave used f o r t h i s c u r r e n t was re p r o d u c e d i n h e i g h t and wav e l e n g t h f o r the 0.20 m/s c u r r e n t and the n o - c u r r e n t c a s e s such t h a t the a d d i t i v e n a t u r e of the wave and c u r r e n t v e l o c i t i e s c o u l d be i n v e s t i g a t e d . The t r a n s m i t t i n g and r e c e i v i n g o p t i c s were s e t up and the c o n t r o l volume was p o s i t i o n e d near the c e n t r e of the flume a t a c e r t a i n h e i g h t above the sand bed. A l l wave and c u r r e n t c o n d i t i o n s were run and the i n s t a n t a n e o u s v e l o c i t y t r a c e s were r e c o r d e d on the c h a r t r e c o r d e r . The wave p e r i o d was checked u s i n g the spectrum a n a l y z e r and the c h a r t r e c o r d e r . The wave h e i g h t was measured by use of a hook and p o i n t e r gauge, as e a r l i e r measurements made by use of the wave probe p r o v e d t o be i n c o n s i s t e n t . The wave probe c o u l d not be c o r r e c t l y c a l i b r a t e d though s e v e r a l a t t e m p t s were made. A f t e r the measurements of t h e h o r i z o n t a l v e l o c i t y at t h i s h e i g h t above the bed were made, the l a s e r and o p t i c s were moved t o a new h e i g h t and the c o n d i t i o n s r e p e a t e d , such t h a t a v e r t i c a l t r a v e r s e was made. A time c o n s t r a i n t d i s a l l o w e d the making of measurements a t many h e i g h t s , but they were made a t 0.5 cm, 0.75 cm, 1.00 cm, 2.10 cm, and at a h e i g h t w e l l above the bed, a t 28.7 cm. 7.3.1.1 E x p e r i m e n t a l R e s u l t s From the v e l o c i t y t r a c e , f o u r or f i v e peaks were averaged f o r bo t h t h e f o r w a r d and r e v e r s e maximum v e l o c i t i e s , and t h e v a l u e s r e c o r d e d i n Ta b l e 7.1. These L=3.73 Height Wave above Height bed (cm) (cm) 28.7 2.1 2.1 1.0 0.75 0.50 2.1 1.0 0.75 0.50 11.24 11.24 11.24 10.75 10.75 10.75 5.70 5.70 5.70 5.70 Current Measured Velocity Velocity (cm/s) (cm/s) Umax -Umax 0.0 24.6 -20.7 0.0 23.2 -18.0 0.0 21.0 -15.9 0.0 22.5 -18.4 0.0 25.6 -18.8 0.0 24.8 -17.1 0.0 12.9 -10.6 0.0 13.6-11.2 0.0 13.5 -10.9 0.0 14.2 -11.2 Fi r s t Order Velocity (cm/s) Umax -Umax 26.6 24.0 23 22 22 22 12 12 12.1 12.1 -26.6 -24.0 -23.9 -22.9 -22.9 -22.9 -12.1 -12.1 -12.1 -12.1 First Order plus Mass Transport (cm/s) Umax -Umax 27.0 26.9 25.8 25.8 25.8 -21.0 -20.9 -20.0 -20.0 -20.0 13.0 -11.2 13.0 -11.2 13.0 -11.2 13.0 -11.2 Second Order Velocity (cm/s) Umax -Umax 32.8 -20.4 28.3 -19.7 28.2 -19.6 26.8 -19.0 26.8 -19.0 26.8 -19.0 13.2 -11.0 13.2 13.2 13.2 -11.0 -11.0 -11.0 28.7 7 .64 20.5 41.2 4.6 38. 2 2.8 39.8 4.4 2.1 7 .64 16.9 30.8 5.6 31. 6 2.2 33. 1 3 .7 32.5 3.1 2.1 7 .64 15.0 31.2 3.7 29. 8 0.2 31. 3 1 .7 30.7 1.1 1.0 6 .85 16.2 32.2 3.4 29. 4 3.0 30. 6 4 .2 30.1 3.7 0.75 6 .85 15.5 31.4 4.6 28. 7 2.3 29. 9 3 .5 29.4 3.0 0.50 6 .85 14.4 29.6 3.0 27. 6 1.2 28. 8 2 .4 28.3 1.9 Height Wave Current Measured above Height Velocity Velocity bed (cm) (cm) (cm/s) (cm/s) Umax -Umax 28.7 2.1 2.1 1.0 0.75 0.50 2.1 1.0 0.75 0.50 L=2.83 m Firs t Order Velocity (cm/s) Umax -Umax 10.45 0.0 26.2 -24.0 24.2 -24.2 10.45 0.0 22.2 -19.5 20.1 -20.1 10.45 0.0 17.5 -16.4 20.2 -20.2 11.09 0.0 22.1 -18.6 21.3 -21.3 11.09 0.0 23.2 -19.2 21.3 -21.3 11.09 0.0 21.8 -18.5 21.3 -21.3 64 0.0 15.0 -12.5 14.7 -14.7 64 0.0 14.9 -12.6 14.7 -14.7 64 0.0 14.9 -12.4 14.7 -14.7 64 0.0 16.1 -13.1 14.7 -14.7 First Order plus Mass Transport (cm/s) Umax -Umax 22.7 22.7 24.2 24.2 24.2 16.2 16.2 16.2 16.2 -17.5 -17.5 -18.4 -18.4 -18.4 -13.2 -13.2 -13.2 -13.2 Second Order Velocity (cm/s) Umax -Umax 25.8 21.7 21.8 23. 23. 23.2 15.6 15.6 15.6 15.6 -22.6 -18.5 -18. -19. -19. -19. -13.8 -13.8 -13.8 -13.8 28.7 5.70 20 .5 38. 3 8.2 34.0 7.0 - - 35.6 8. 0 2.1 5.70 16 .9 31. 7 6.7 29.1 4.7 30.0 5.6 30.1 5. 8 2.1 5.70 15 .0 38. 8 3.0 27.1 2.9 28.0 3.8 28.2 4. 0 1.0 5.03 16 .2 30. 8 5.1 26.9 5.5 27.6 6.2 27.8 6. 3 0.75 5.03 15 .5 29. 7 5.0 26.2 4.8 26.9 5.5 27.1 5. 6 0.50 5.03 14 .4 31. 7 8.7 25.1 3.7 25.8 3.4 26.0 4. 5 T a b l e 7.1. Measured and t h e o r e t i c a l f o r w a r d and r e v e r s e v e l o c i t i e s . 1 50 v a l u e s were compared w i t h the l i n e a r wave t h e o r y maximum v e l o c i t i e s , the second o r d e r S t o k e s wave t h e o r y v e l o c i t i e s , and the v e l o c i t i e s c a l c u l a t e d from the a d d i t i o n of the l i n e a r wave t h e o r y v e l o c i t y and the mass t r a n s p o r t v e l o c i t y above the boundary l a y e r over a smooth bed as p r e d i c t e d by I s a a c s o n (1978). These v a l u e s a r e a l s o g i v e n i n T a b l e 7.1. I t i s i n t e r e s t i n g t o note the agreement between the a c t u a l v e l o c i t i e s and the t h e o r e t i c a l ones. I t seems t h a t f o r the l o n g e r p e r i o d wave t h a t the s m a l l e r wave h e i g h t s g i v e l a r g e r v e l o c i t i e s than the t h e o r e t i c a l , w h i l e the l a r g e r h e i g h t s g i v e s m a l l e r v e l o c i t i e s than the t h e o r e t i c a l . T h i s t r e n d was noted by R u s s e l l and O s o r i o (1958). A l s o , one t h e o r y may p r e d i c t the maximum f o r w a r d v e l o c i t y v e r y c l o s e l y , w h i l e p r e d i c t i n g the maximum r e v e r s e v e l o c i t y not as w e l l , but t h a t another t h e o r y w i l l p r e d i c t the l a t t e r w e l l but the former q u i t e p o o r l y . O v e r a l l , i t appears t h a t the l i n e a r t h e o r y i s q u i t e i n a d e q u a t e when compared t o the l i n e a r t h e o r y w i t h added mass t r a n s p o r t and t o the Stokes second o r d e r t h e o r y . 7.3.2 SECOND SET OF EXPERIMENTS The p r o c e d u r e s used here a r e s i m i l a r t o those e s t a b l i s h e d i n the f i r s t s e t of e x p e r i m e n t s . S t i l l water depth was chosen as 0.40 m f o r t h i s , s e t of e x p e r i m e n t s , and one wave l e n g t h was used, L=2.83 m, c o r r e s p o n d i n g t o a r e l a t i v e p e r i o d of 1.6 seconds. The sediments were s i e v e d i n t o f i v e narrow s i z e r a n g e s , 0.3 t o 151 0.85 mm, 0.85 t o 1.16 mm, 1.16 t o 1.70 mm, 1.70 t o 2.00 mm, and 2.00 t o 2.35 mm. Most ranges were f a i r l y narrow, and so f o r a p a r t i c u l a r t e s t , the onset of motion c o u l d be a s s o c i a t e d w i t h a r e a s o n a b l y s p e c i f i c sediment s i z e . F o r each s e r i e s of t e s t s , sediment of one s i z e range was s p r e a d i n t o a f l a t bed c o n d i t i o n . The onset of sediment motion under a steady c u r r e n t was then o b s e r v e d , and the bed was a l l o w e d t o armor f o r about 15 minutes under f l o w s s l i g h t l y i n excess of c r i t i c a l . Quick (1985) found t h a t the measurements of the t h r e s h o l d c o n d i t i o n were more r e p e a t a b l e a f t e r t h i s a g i n g p r o c e s s . A t e s t was then made f o r waves a l o n e , where the wave h e i g h t was i n c r e a s e d u n t i l sediment motion o c c u r r e d . A t h i r d t e s t was made f o r a s m a l l e r s t e a d y c u r r e n t and a d i f f e r e n t s i z e " e q u i v a l e n t " wave h a v i n g the r e q u i r e d p e r i o d , where the wave h e i g h t was i n c r e a s e d u n t i l sediment motion o c c u r r e d . A f i n a l t e s t was made f o r one s i z e range where the stea d y c u r r e n t was adverse t o the wave d i r e c t i o n . Measurements u s i n g the LDA were made of the v e l o c i t y f i e l d i m m e d i a t e l y above t h e bed f o r each t e s t , and a l s o f o r the s m a l l e r steady c u r r e n t a l o n e and f o r t h e " e q u i v a l e n t " wave a l o n e . The measurements were made f o r a few c e n t i m e t e r s above t h e bed and down t o w i t h i n a p p r o x i m a t e l y 0.3 mm of the bed. An OTT meter was a l s o used t o de t e r m i n e the mean stea d y c u r r e n t v e l o c i t i e s . 1 52 7.3.2.1 E x p e r i m e n t a l R e s u l t s The v e l o c i t y d a t a has been a r r a n g e d i n T a b l e s 7.2 t o 7.7. I m m e d i a t e l y i t i s n o t i c e d t h a t f o r a g i v e n sediment s i z e , onset of sediment motion o c c u r r e d at the same maximum v e l o c i t y v e r y c l o s e t o the bed f o r each of the t e s t s p e r formed. These t h r e s h o l d p r o f i l e s a re shown i n F i g u r e 7.4. I t i s shown t h a t f o r the same maximum i n s t a n t a n e o u s c r i t i c a l v e l o c i t y w i t h i n 1 mm of the bed, t h r e s h o l d of sediment motion was produced, independent of the f l o w type and i t s a s s o c i a t e d upper p o r t i o n of the v e l o c i t y p r o f i l e . The comparisons of the measured t h r e s h o l d wave v e l o c i t y p r o f i l e s w i t h f i r s t and second o r d e r Stokes t h e o r y a r e shown i n F i g u r e 7.5. The comparisons of the measured t h r e s h o l d c u r r e n t v e l o c i t y p r o f i l e s w i t h the range of p r e d i c t e d v e l o c i t y p r o f i l e s from the l o g v e l o c i t y law are shown i n F i g u r e 7.6. T h i s range of p r o f i l e s i s produced by u s i n g a minimum and a maximum U*, where the U* have been c a l c u l a t e d u s i n g the Mannings f o r m u l a , where D has been chosen as D . and D 3 min max The comparisons of the measured t h r e s h o l d wave and c u r r e n t v e l o c i t y p r o f i l e s w i t h the v e c t o r i a l sums of the " e q u i v a l e n t " wave v e l o c i t y p r o f i l e s and the added c u r r e n t v e l o c i t y p r o f i l e s a r e shown i n F i g u r e 7.7. These v e l o c i t y p l o t s e n a b l e one t o draw immediate c o n c l u s i o n s c o n c e r n i n g s e v e r a l of the q u e s t i o n s t h a t have been r a i s e d : 1 53 Height Measured Threshold V e l o c i t i e s Measured Added V e l o c i t i e s Above (cm/s) (cm/s) The Bed Wave and Wave Current Alone Wave Current Alone (mm) Current Alone Max. Avg. Alone Max. Avg. 0.30 27.6 26.7 26.7 15.0 21.5 12.4 6.3 0.76 28.7 26.6 26.7 15.6 22.3 12.3 6.3 1.22 31.3 26.6 25.6 15.7 22.9 13.2 7.4 1 .57 32.3 26.6 27.2 17.4 22.4 14.7 8.8 2.03 32.8 26.6 28.7 17.6 22.2 14.5 9.5 2.49 32.3 25.6 28.7 18.9 22.0 15.4 9.7 2.84 . 32.4 25.1 28.2 19.9 21.8 15.4 10.1 3.53 33.8 25.1 30.8 19.8 21.6 16.4 11.3 4.80 34.0 24.8 29.9 20.9 21.5 16.9 11.6 6.07 34.3 24.8 32.3 22.6 21.5 16.9 12.0 6.76 34.9 24.2 32.3 21 .5 21.3 17.9 12.2 9.99 35.4 24.2 33.8 ' 24.8 20.8 18.5 13.0 13.22 35.6 24. 1 3511 24.7 • 20.6 19.5 13.5 16.45 35.9 23.6 35.6 25.9 20.5 19.5 14.7 35.35 38.5 22.2 . 37.6 30.5 20.0 20.5 16.1 54.25 38.0 22.4 36.9 30.4 20.0 20.7 16.4 T = 1.60 seconds r e l a t i v e to current. H = 8.58 cm wave height of equivalent wave for threshold where wave and current are combined. H = 11.22 cm wave height for threshold where there i s a wave alone. V e l o c i t y measured with OTT meter = 16.80 cm/s for threshold where wave and current are combined. = 35.00 cm/s for threshold where there i s a current alone. T a b l e 7.2. Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 0.30 t o 0.85 mm. 154 r Height Measured Threshold V e l o c i t i e s Measured'Added V e l o c i t i e s Above (cm/s) (cm/s) The Bed Wave and Wave Current Alone Wave Current Alone (mm) Current Alone Max. Avg. Alone Max. Avg. 0.30 29.5 28.7 29.7 20.0 22.9 13.8 7.5 0.76 33.8 29.2 30.5 20.0 25.6 15.1 9.0 1.22 34.3 28.7 31.7 22.0 25.4 15.3 9.7 1 .57 35.8 30.2 35.4 22.3 25.3 15.8 10.2 2.03 35.8 29.0 35.4 24.0 24.6 15.9 10.6 2.49 36.7 28.9 35.0 24.0 22.9 16.4 1 1 .2 2.84 36.0 28.0 36.9 26.0 22.9 16.9 1 1 .7 3.53 36.9 27.6 36. 9 26. 1 22.7 16.4 11.8 4.80 36.5 27.6 37.7 27.0 22.5 16.4 12.0 6.07 37.0 26.8 . 37.9 27.3 22.5 17.6 12.7 6.76 37.0 26.7 38.4 27.5 22.4 18.0 12.7 9.99 38.1 27.6 > 3 9 ' 5 30.0 22.3 18.5 13.2 13.22 38.0 27.6 41.2 31.0 22.3 18.5 13.7 16.45 39.6 27.5 43. 1 33. 1 21.4 18.9 14 .1 35.35 40.0 26. 1 43.7 35.0 21.2 19.7 15.9 54.25 40.2 25.7 45.6 37.6 21 .1 20.2 16.4 T = 1.60 seconds r e l a t i v e to current. H = 9.40 cm wave height of equivalent wave for threshold where wave and current are combined. H = 12.30 cm wave height for threshold where there i s a wave alone. V e l o c i t y measured with OTT meter = 17.30 cm/s for threshold where wave and current are combined. = 41.00 cm/s for threshold where there is a current alone. T a b l e 7.3. Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 0.85 t o 1.16 mm. 1 55 Height Measured Threshold V e l o c i t i e s Measured Added V e l o c i t i e s Above (cm/s) (cm/s) The Bed Wave and Wave Current Alone Wave Current Alone (mm) Current Alone Max. Avg. Alone Max. Avg. 0.30 32.8 31.3 32.4 14.5 27.7 15.9 9.2 0.76 35.4 33.3 32.2 14.0 27.8 19.4 10.2 1 .22 34.5 33.3 32.8 17.0 31 .9 17.9 9.6 1 .57 36.7 33.3 34.5 17.4 33.9 20.4 11.8 2.03 40.6 32.6 40.1 20.4 34.1 20.6 11.9 2.49 39.5 36.4 37.7 20.7 34. 1 21.0 12.5 2.84 41 . 1 37.6 41 .7 23.6 34.3 19.6 12.3 3.53 41.4 35.2 43.2 23.9 33.9 22.2 14.0 4.80 41.8 34.0 37.8 25.3 31.3 21.6 14.5 6.07 42.0 34.0 43.0 26.8 30.6 23.9 16.2 6.76 42.2 34.9 43.0 28.6 30.5 23.8 16 .1 9.99 44. 1 27.0 46.0 30.7 29.5 23.6 16.4 13.22 44.3 27.5 46. 1 . , 3 1 . 7 29.2 23.6 17.5 16.45 44.0 26.9 47.2 34.7 . 29.6 24.2 18.0 35.35 46.5 26. 1 50.4 37.3 28.0 25.9 20.2 54.25 48.9 27.7 5 1 .7 40. 1 28.0 26. 1 20.9 T = 1.60 seconds r e l a t i v e to current. H = 12.32 cm wave height of equivalent wave for threshold where wave and current are combined. H = 13.86 cm wave height for threshold where there i s a wave alone. V e l o c i t y measured with OTT meter = 20.60 cm/s for threshold where wave and current are combined. - = 48.40 cm/s for threshold where there i s a current alone. T a b l e 7.4. Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 1.16 t o 1.70 mm. 156 Height Measured Threshold V e l o c i t i e s Measured Added V e l o c i t i e s Above (cm/s) (cm/s) The Bed Wave and Wave Current Alone Wave Current Alone (mm) Current Alone Max. Avg. Alone Max. Avg. 0.30 35.9 34.9 35.4 18.6 27.2 19.0 9.6 0.76 36.9 35.7 33.3 19.4 28.7 21.5 10.1 1 .22 38.7 36.9 34.0 20.4 30.0 20.0 12.0 1 .57 38.9 37.4 35.5 21.7 30.5 20.5 12.7 2.03 41.0 38.0 37.0 23.6 31.0 21 .8 13.7 2.49 42.0 38.1 39.9 26.6 3 1.7 24.6 14.2 2.84 42.1 37.4 41.3 26.8 32.3 24.0 14.3 3.53 44.1 36.4 41.7 26.5 30.8 23.8 15.3 4.80 44. 1 35.4 44.3 27.8 30.5 25.6 17.1 6.07 45.0 35.4 46.1 29.8 28.9 26.0 17.8 6.76 42.5 33.3 • 46.6 30.0 28.9 27. 1 17.7 9.99 46.6 •31 .9 50.2 34.0 • 26.7 27.0 18.9 13.22 48.2 31 .8 52.0 38.9 26.7 28.2 18.9 16.45 49.0 31.9 53.3 40.9 26.7 28.2 21.3 35.35 55.4 31.8 57.9 46.9 26.3 30.9 24.8 54.25 54.0 31.1 58. 1 - 50.8 26.9 32.4 26.4 T = 1.60 seconds r e l a t i v e to current. H = 12.50 cm wave height of equivalent wave for threshold where wave and current are' combined. H = 15.00 cm wave height for threshold where there i s a wave alone. V e l o c i t y measured with OTT meter = 25.40 cm/s for threshold where wave and current are combined. = 54.60 cm/s for threshold where there i s a current alone. T a b l e 7.5. Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 1.70 t o 2.00 mm. 157 Height Measured Threshold V e l o c i t i e s Measured Added V e l o c i t i e s Above (cm/s) (cm/s) The Bed Wave and Wave Current Alone Wave Current Alone (mm) Current Alone Max. Avg. Alone Max. Avg. 0.30 38.2 39.9 33.6 18.1 26.7 20.5 11.6 0.76 41.8 39.8 36.9 21.6 31 .8 21 .4 11.7 1.22 45.3 41.0 37.9 23. 1 35.7 22.8 11.9 1 .57 47.7 41.6 39.1 24.3 36.9 23.1 12.7 2.03 47.2 41.5 40.8 25.5 37.9 23. 1 13.3 2.49 47.6 41.5 42.7 26.8 39.0 24.6 14.7 2.84 47. 1 41.8 42.2 27.4 40.7 24.4 14.4 3.53 49.8 42.1 - - 42.3 24.6 15.1 4.11 - - 48.2 30.7 - - -4.80 50.3 41.2 47.0 29.2 42.4 25.6 16.3 6.07 51.6 40.8 - - 41.0 27.2 17.2 6.76 51.6 41.0 - - 39.0 28.3 18.4 8.03 - - 52.7 35.0 - -9.99 ' 51.8 37.6 - - 37.5 29.5 19.9 1 1 .26 - - 53.3 37.5 - - -13.22 55.1 37.3 - - 36.2 29.9 20.7 14.49 - - 52.7 38. 1 - - -16.45 55.3 37. 1 - - 36.2 32.8 23.3 17.72 - - 56.3 42.2 - - -20.95 - - 57.3 42.2 - - _ 35.35 59.9 36.6 - - 36.1 34.5 27.2 39.85 - - . 62.5 49.7 - - -54.25 62.7 36.4 - - 35.9 35.4 29. 1 T = 1.60 seconds r e l a t i v e to current. H = 14.74 cm wave height of equivalent wave for threshold where wave and current are combined. H = 16.70 cm wave height for threshold where there i s a wave alone. V e l o c i t y measured with OTT meter = 24.80 cm/s for threshold where wave and current are combined. = 55.40 cm/s for threshold where there i s a current alone. T a b l e 7.6. Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 2.00 t o 2.35 mm. 158 Height Measure'd Threshold V e l o c i t i e s Above (cm/s) Measured Added V e l o c i t i e s (cm/s) The Bed Wave and Wave Current Alone Wave Current Alone (mm) . Current Alone Max. Avg. Alone Max. Avg. 0.30 32.8 31.3 32.4 14.5 21 .8 17.6 10.1 0.76 32.8 33.3 32.2 14.0 20.3 17.3 1 1.0 1.22 35.0 33.3 32.8 17.0 20.7 18.2 12.8 1.57 37.2 33.3 34.5 17.4 22.0 18.8 13.6 2.03 37.4 32.6 40. 1 20.4 20.8 19.6 13.3 2.49 38.1 36.4 37.7 20.7 21.3 20.6 15.5 2.84 37.6 37.6 41.7 23.6 21.9 20.5 15.1 3.53 39.0 35.2 43.2 23.9 20.3 20.9 15.4 4.80 39.5 34.0 37.8 25.3 19.9 21.1 15.5 6.07 38.2 34.0 43.0 26.8 19.9 22.2 16.8 6.76 40.0 34.9 . 43.0 • 28.6 19.5 21.3 16.9 13.22 40.2 27.5 46. 1 31 .7 • 19.5 24.6 19.4 35.35 46.1 26. 1 50.4 37.3 ' 18.5 25.9 21.0 54.25 45.6 27.7 51 .7 40. 1 18.5 26.9 22.0 T = 1.60 seconds r e l a t i v e to current. H = 8.80 cm wave height of equivalent wave for threshold where wave and current are combined. H = 13.86 cm wave height for threshold where there i s a wave alone. V e l o c i t y measured with OTT meter = 20.60 cm/s for threshold where wave and- current are combined. = 48.40 cm/s for threshold where there i s a current alone. T a b l e 7.7. Measured near-bed v e l o c i t y d a t a f o r sediment s i z e range 1.16 t o 1.70 mm. Adverse c u r r e n t . 159 mtosured wove and curr«nl m c o t u r c d Current meoturcd wove 0.0 10.0 20.0 n .O 40.0 VELOCITY, CM/S SO.O 80.0 0.0 1D.0 20.0 3O.0 10.0 50.0 60.0 70.0 vaocnr, CM/S (a) Range 0.30 to 0.65mm (b) Range 0.85 to 1.16mm 0.0 10.0 ZO.O 30.0 40.0 50.0 VCL0C1TT, CM/5 0.0 10.0 20.0 30.0 10.0 50.0 60.0 vaocnr, CM/S (c) Range 1.16 to 1.70mm (d) Range 1.70 to 2.00mm XI.0 10.0 VCLOCIlr, CM/5 o.o 10.0 30.0 10.0 SD.O v i x n r i T r , c n/s (e) Range 2.00 to 2.35mm (t) Range 1.16 to 1.70mm - A d v e r s e Current F i g u r e 7.4. Measured t h r e s h o l d v e l o c i t y p r o f i l e s . 160 8 6c £8] O o mtotur.d Itif.thold v o d ttt.ot.llcol linaot woe* th.ot.tKOI Sloh.i lacond ord.t 20. 0 30.0 TO.O W O VELOCITY, CM/5 (a) Range 0.30 to 0.85mm 0.0 10.0 20.(1 W O <0.0 SO-0 60.0 VCLOCITf, Cll/S (b) Range 0.85 to 1.16mm 20.0 V.o «n.n ^".0 v n L o e n r , o u s (c) Range 1.16 to 1.70mm 30.0 tO.O v n . o c m . CM-s (d) Range 1.70 to 2.00mm v ru i r . i 10. n I t . CM/5 (e) Range 2.00 to 2.35mm F i g u r e 7 . 5 . Measured and t h e o r e t i c a l wave v e l o c i t y p r o f i l e s . 161 30.0 . 10.0 vciocnv, CM/S io.n V C L H u n t , CM'5 (c) Range 1.16 to 1.70mm (d) Range 1.70 to 2.00mm (e) Range 2.00 to 2.35mm F i g u r e 7.6. Measured and t h e o r e t i c a l c u r r e n t v e l o c i t y p r o f i l e s . 162 81 0.0 10.0 20.0 30.0 10.0 50.0 80.0 VELOCITY, Ctt/5 (a) Range 0.30 to 0.85mm I oi H ial i tO.O 20.0 30,0 10.0 50.0 vaocm, CM/S 60.0 10. 0 <b) Range 0.B5 to 1.16mm mvuturtd wot! ontJ curffnt lhr*«hn!d •n.tisurad oddad vrova pHi« *i«o*ur«d o*«rao« odd«d co'tent f, m«Qiur*d oddtd "iovt plut mtosMiad ma it mum odd*d e u f t n l 10.0 20.0 30.0 10.0 VELOCITY, CM/S M.O 60.0 (c) Range 1.16 to 1.70mm 0.0 10.0 w.o in.n VCLOCIlt, CM/5 W.O 60.0 (d) Range 1.70 to 2.00mm 0.0 10.0 30.0 10.0 VCLOCIIY, CM/5 60.0 '0.0 <e) Range 2.00 to 2.35mm 0.0 10.0 20.0 30.0 40.0 V T L X I T T , CM/S SO.O 60.0 (f) Range 1.16 to 1.70mm - Adverse Current F i g u r e 7.7. Measured wave and c u r r e n t v e l o c i t y p r o f i l e s . 163 — F o r a g i v e n sediment s i z e , the v e l o c i t i e s a t the bed are the same f o r the onset of t h r e s h o l d r e g a r d l e s s of the f l o w t y p e , be i t under waves, c u r r e n t s , or c o m b i n a t i o n s of waves and c u r r e n t s . —The second o r d e r Stokes t h e o r i e s g i v e good p r e d i c t i o n s as t o the maximum v e l o c i t i e s under waves above the boundary l a y e r f o r the wave p e r i o d t e s t e d . Very near the bed, the measured v e l o c i t y i s i n ex c e s s of the t h e o r e t i c a l by about 10%. —The average c u r r e n t v e l o c i t y p r o f i l e s f o r t h r e s h o l d c o n d i t i o n s f a l l w i t h i n the ranges g i v e n by l o g v e l o c i t y law p r o f i l e s , where the shear v e l o c i t i e s f o r these p r o f i l e s have been e s t i m a t e d from Manning's f o r m u l a , u s i n g p a r t i c l e s i z e ranges from D .„ t o D_ 3 c 3 min max —The measured maximum wave and c u r r e n t v e l o c i t i e s seem t o be s i m p l e v e c t o r i a l a d d i t i o n s of the measured " e q u i v a l e n t " wave maximum v e l o c i t i e s and of the measured average added c u r r e n t v e l o c i t i e s . 7.3.3 COMPARISON OF STEADY CURRENT RESULTS WITH OTHER  VELOCITY PROFILES The u n i d i r e c t i o n a l c u r r e n t c r i t i c a l v e l o c i t y p r o f i l e s have been shown i n the p r e c e d i n g s e c t i o n . Manning's form u l a has been used t o c a l c u l a t e a range of U* v a l u e s , and these have been used i n the u n i v e r s a l l o g law t o g i v e v e l o c i t y p r o f i l e s comparing w i t h the measured p r o f i l e s . I t was seen t h a t t h e s e compared v e r y w e l l , and i t was c o n c l u d e d t h a t 164 Manning's f o r m u l a and the u n i v e r s a l l o g law c o u l d be used t o p r e d i c t the c r i t i c a l v e l o c i t y a t the bed f o r a g i v e n sediment s i z e . I t i s i n t e r e s t i n g t o compare these measured p r o f i l e s w i t h t h o s e p r e d i c t e d from o t h e r w e l l known p r o f i l e e q u a t i o n s . The f o r m u l a f o r the 1/7th power law and the n u m e r i c a l r e s u l t s of Howarth (1938) f o r the l a m i n a r law are used t o c a l c u l a t e v e l o c i t y p r o f i l e s . These p r o f i l e s a r e not i n c l u d e d i n t h i s r e p o r t , but the r e s u l t s a r e d i s c u s s e d here. The p r o f i l e s as g i v e n by the 1/7th power law g i v e v e l o c i t i e s much l a r g e r than those a c t u a l l y o b s e r v e d . V a l u e s of minimum U* were c a l c u l a t e d from both Manning's f o r m u l a and the S h i e l d s c r i t e r i o n and i t was found f o r most sediment s i z e ranges t h a t the p r e d i c t e d v a l u e s exceeded the maximum measured c u r r e n t v a l u e s by a t l e a s t 10 t o 20%. Only f o r the case of the s m a l l e s t sediment s i z e range was t h e r e any s i m i l a r i t y between the t h e o r e t i c a l and measured v a l u e s , and then o n l y f o r the near-bed v a l u e s . T h i s f i n d i n g i s q u i t e d i f f e r e n t from those d i s c u s s e d e a r l i e r by Hansen (1928) and B u r g e r s ( 1 9 2 4 ) , but i t i s r e a l i z e d t h a t the l e n g t h Reynolds numbers a r e l a r g e r than 10 6 and a r e beyond the range t e s t e d by them. As t h e l e n g t h Reynolds numbers a r e so l a r g e , the l a m i n a r f l o w p r o f i l e s h o u l d not be e x p e c t e d t o h o l d . In f a c t , the roughness Reynolds numbers a r e so l a r g e t h a t the sediment s i z e a l o n e p u t s the near-bed f l o w i n t o a t l e a s t the t r a n s i t i o n regime even i f the e x t e r n a l f l o w was not 1 65 k U* t u r b u l e n t . The v a l u e s of — ^ — a r e g i v e n f o r the sediment s i z e ranges i n T a b l e 7.8. I t was f e l t by Quick (1985) t h a t the e x t e r n a l f l o w was f u l l y d e v e l o p e d rough t u r b u l e n t and t h a t t h i s t u r b u l e n c e extended down t o the bed. The good r e s u l t s g i v e n by the Manning r e l a t i o n and the u n i v e r s a l l o g v e l o c i t y law i n d i c a t e rough t u r b u l e n t f l o w by t h e i r good agreement w i t h the measured p r o f i l e . 7.3.4 COMPARISON OF THE MEASURED AND THEORETICAL  OSCILLATORY FLOWS The o s c i l l a t o r y f l o w c r i t i c a l v e l o c i t y p r o f i l e s have been shown a g a i n s t the f i r s t and second o r d e r Stokes t h e o r i e s . I t i s seen f o r the upper p o r t i o n of the f l o w t h a t the measured v a l u e s f a l l q u i t e c l o s e t o second o r d e r t h e o r y and a r e between f i r s t and second o r d e r p r e d i c t i o n s . C l o s e r t o t he bed, the c h a r a c t e r i s t i c " b u l g e " i s seen t o occur such t h a t the maximum v e l o c i t y i s l a r g e r than t h a t w e l l above the bed. Very near the bed, the v a l u e s measured are o n l y s l i g h t l y l a r g e r than the second o r d e r p r e d i c t i o n s , and i t i s f e l t t h a t second o r d e r p r e d i c t i o n f a l l s w i t h i n e n g i n e e r i n g k U* Sediment Size Range Value of U* Value of - 5 — Flow Type 0.30 to 0.85 mm 0.30 8.1 t r a n s i t i o n 0.85 to 1.16 mm 0.45 16.5 t r a n s i t i o n 1.16 to 1.70 mm 0.71 38.2 t r a n s i t i o n 1.70 to 2.00 mm 0.96 60.7 t r a n s i t i o n 2.00 to 2.35 mm 1.04 77.3 rough turbulent k 0* T a b l e 7.8. V a l u e s of - J — and p r e d i c t e d f l o w type f o r a l l "sediment s i z e s . 1 66 a c c u r a c y . When the p o s s i b l e e r r o r s - i n measuring the v e r y s m a l l waves used i n the experiment a r e c o n s i d e r e d , i t i s r e a l i z e d t h a t the agreement i s v e r y good. The b u l g e has been d e s c r i b e d by Lamb (1932) f o r smooth beds and by S l e a t h (1970) f o r rough beds. The smooth bed s o l u t i o n does not a p p r o x i m a t e the measurements even c l o s e l y , except f o r g e n e r a l appearance. For S l e a t h " s t u r b u l e n t f l o w s o l u t i o n , the v a l u e of X has been v a r i e d by the w r i t e r t o f i n d a p r o f i l e t h a t matches the o b s e r v e d p r o f i l e . The measured v a l u e s showed the b u l g e s t o be much c l o s e r t o the bed and of l a r g e r r e l a t i v e magnitudes than d i d t h o s e d e t e r m i n e d by S l e a t h . 7.3.5 RESULTING CALCULATED SHEAR STRESSES : V a l u e s of the shear s t r e s s have been c a l c u l a t e d from v a r i o u s t h e o r i e s . The u n i d i r e c t i o n a l c r i t i c a l shear s t r e s s e s were c a l c u l a t e d from t h e Manning's r e l a t i o n ( e q u a t i o n 2.17). Other u n i d i r e c t i o n a l shear s t r e s s f ormulae as d i s c u s s e d e a r l i e r a r e not used h e r e , because of the f a i l u r e of t h o s e v e l o c i t y p r o f i l e d e s c r i p t i o n s t o match th o s e a c t u a l l y o b s e r v e d . The o s c i l l a t o r y maximum shear s t r e s s e s were c a l c u l a t e d from the r=f -£u0 2 r e l a t i o n s h i p , and the f r i c t i o n w2 um f a c t o r s used were t h o s e d e t e r m i n e d by Jonsson (1963,1966), K a j i u r a (1968), and Kamphuis (1975), as g i v e n by e q u a t i o n s (3.33) t o ( 3 . 3 7 ) . Even though the l a m i n a r o s c i l l a t o r y v e l o c i t y p r o f i l e d e s c r i p t i o n resembled the measured p r o f i l e o n l y i n g e n e r a l form, shear s t r e s s e s u s i n g 1 67 the l a m i n a r f r i c t i o n f o r m u l a ( e q u a t i o n 3.21) have been d e t e r m i n e d f o r the sake of c o m p a r i s o n . The c a l c u l a t i o n s have been made u s i n g the measured near-bed v a l u e s . A l s o , c u r r e n t s t r e s s e s f o r the added c u r r e n t were added v e c t o r i a l l y t o the wave s t r e s s e s f o r the " e q u i v a l e n t " wave. The r e s u l t s of the shear s t r e s s c a l c u l a t i o n s were compared w i t h the r e q u i r e d s t r e s s d e t e r m i n e d from the S h i e l d s diagram. I t s h o u l d be remembered t h a t S h i e l d s d i d not de t e r m i n e e x a c t l y when the sediment began t o move, but e x t r a p o l a t e d when the sediment s h o u l d move from h i s measurements of the sand f l o w a t h i g h e r water v e l o c i t i e s . F u r t h e r , the f l u i d t u r b u l e n c e and the e x p e r i m e n t a l boundary c o n d i t i o n s p r o b a b l y d i f f e r from S h i e l d s ' s e x p e r i m e n t . One might expect the shear s t r e s s p r e d i c t e d , by the S h i e l d s diagram t o be a c c u r a t e o n l y w i t h i n a r e a s o n a b l e u n c e r t a i n t y . The shear s t r e s s e s a r e p r e s e n t e d i n T a b l e 7.9. The u n i d i r e c t i o n a l c u r r e n t s t r e s s e s as c a l c u l a t e d from the Manning's r e l a t i o n g i v e r e s u l t s t h a t a r e v e r y s i m i l a r t o those p r e d i c t e d by S h i e l d s . The s t r e s s e s as c a l c u l a t e d f o r the c r i t i c a l motion of the sand a r e t y p i c a l l y i n the range of s t r e s s e s as p r e d i c t e d from t h e S h i e l d s diagram. For the l a r g e s t s i z e ranges, the measured s t r e s s e s a re up t o 30% l e s s than t h o s e p r e d i c t e d by S h i e l d s . Even t h i s r e s u l t i s r e a s o n a b l e , as Madsen and Grant (1976) mentioned t h a t t h e r e a r e l a r g e d i s c r e p a n c i e s between each o b s e r v e r ' s r e s u l t s f o r u n i d i r e c t i o n a l f l o w . As p r e v i o u s l y mentioned, e x p e r i m e n t a l 168 SEDIMENT SIZE RANGE 0.30 TO 0.65 mm SEDIMENT SIZE RANGE 1.70 TO 2.00 mm THRESHOLD THRESHOLD THRESHOLD SHIELDS THRESHOLD THRESHOLD THRESHOLD SHIELDS WAVE WAVE AND CURRENT CURRENT STRESS WAVE WAVE AND CURRENT CURRENT STRESS STRESS STRESS STRESS STRESS STRESS STRESS (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. 0 . 3 7 (3.21) 0.36 0.38 0 . 5 3 (3.21) 0.63 0.64 0.64 (3.33) 0 . 5 3 0.55 1.60 (3.33) 1 .36 1.37 0 . 7 4 ( 3 . 3 4 ) 0.60 0.62 0 . 2 1 0.30 0 . 1 9 0 . 4 5 1.86 ( 3 . 3 4 ) 1 .55 1.56 0 . 9 1 0.96 1.05 1.30 0.44 ( 3 . 3 5 ) 0.38 0.40 1.13 ( 3 . 3 5 ) 1 .03 1.04 0.42 (3.36) 0 . 3 7 0.39 0.71 (3.36) 0.72 0.73 0.39 ( 3 . 3 7 ) 0 . 3 5 0 . 3 7 0.67 (3.37) 0.69 0.70 SEDIMENT SIZE RANGE 0.85 TO 1.16 mm SEDIMENT SIZE RANGE 2.00 TO 2 . 3 5 mm THRESHOLD THRESHOLD THRESHOLD SHIELDS THRESHOLD THRESHOLD THRESHOLD SHIELDS WAVE WAVE AND CURRENT CURRENT STRESS WAVE WAVE AND CURRENT CURRENT STRESS STRESS STRESS STRESS STRESS STRESS STRESS (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. 0.41 (3.21) 0.42 0.42 0.58 (3.21) 0.73 0.74 0.83 (3.33) 0.71 0.72 1.97 (3.33) 1.90 1 . 91 0.96 ( 3 . 3 4 ) 0.82 0.82 0 . 4 1 0.4S 0 . 4 5 0.63 2.29 ( 3 . 3 4 ) 2 . 1 9 2.20 0.99 1.04 1.30 1.63 0.58 ( 3 . 3 5 ) 0.52 0 . 5 3 1.40 ( 3 . 3 5 ) 1 .41 1 . 4 3 0.48 (3.36) 0 . 4 5 0.46 0 . 8 1 (3.36) 0 . 9 1 0.92 0.45 ( 3 . 3 7 ) 0 . 4 3 0 . 4 3 0.77 (3.37) 0.86 0.87 SEDIMENT SIZE RANGE 1 . 1 6 TO 1.70 ma i SEDIMENT SIZE RANGE 1 . 1 6 TO 1.70 mm Adverse current THRESHOLD THRESHOLD THRESHOLD SHIELDS THRESHOLD THRESHOLD THRESHOLD SHIELDS WAVE WAVE AND CURRENT CURRENT STRESS WAVE WAVE AND CURRENT- CURRENT STRESS STRESS STRESS STRESS STRESS STRESS STRESS (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) Min. Max. Min. Max. Min. Max. Hin. Max. Min. Max. Min. Max. 0.46 (3.21) 0 . 5 9 0.61 0.46 (3.21) 0.41 0.42 1.17 (3.33) 1 . 3 7 1 .39 1.17 (3.33) 0.69 0.71 1.36 ( 3 . 3 4 ) 1 . 5 7 1 . 5 9 0.63 0.71 0.63 1.05 1.36 ( 3 . 3 4 ) 0.79 0.81 0.63 0.71 0.63 1.05 0.83 ( 3 . 3 5 ) 1 .00 1 .02 0.83 ( 3 . 3 5 ) 0.54 0.55 0 . 5 7 (3.36) 0.72 0.74 0.57 (3.36) 0.41 0 . 4 3 0 . 5 3 ( 3 . 3 7 ) 0.69 0.70 0 . 5 3 (3.37) 0.39 0.41 Table 7 .9. E x p e r i m e n t a l shear s t r e s s e s as c a l c u l a t e d from v a r i o u s f o r m u l a e , compared w i t h the S h i e l d s c r i t e r i o n . boundary c o n d i t i o n s d i f f e r between e x p e r i m e n t s , f l o w t u r b u l e n c e d i f f e r s , and p e r s o n a l judgments a r e not s i m i l a r . The c r i t i c a l o s c i l l a t o r y s t r e s s e s as c a l c u l a t e d from the rough t u r b u l e n t r e s u l t s of Kamphuis ( e q u a t i o n 3.35) f a l l i n t he ranges p r e d i c t e d by the S h i e l d s c u r v e . These o s c i l l a t o r y s t r e s s e s a r e g e n e r a l l y l a r g e r than the u n i d i r e c t i o n a l s t r e s s e s , y e t t h i s c o n s i s t e n t l y l a r g e d i f f e r e n c e cannot be caused by a judgmental d i s s i m i l a r i t y or an e x p e r i m e n t a l boundary c o n d i t i o n d i f f e r e n c e , because the same o b s e r v e r i s u s i n g the same flume. T h e r e f o r e , assuming n o t h i n g e l s e has been d i s r e g a r d e d , the f l o w t u r b u l e n c e s o l e l y a c c o u n t s f o r the d i f f e r e n c e i n the c a l c u l a t e d shear 1 69 s t r e s s e s . A p p a r e n t l y , Komar and M i l l e r (1974) d i d not determine the shear s t r e s s f o r a sand under a u n i d i r e c t i o n a l f l o w . Had t h i s been done, perhaps t h e i r r e s u l t s would have been more i n l i n e w i t h the p r e s e n t f i n d i n g s t h a t the u n i d i r e c t i o n a l and o s c i l l a t o r y shear s t r e s s e s a r e not the same. A c l o s e r e x a m i n a t i o n of the shear s t r e s s e s r e v e a l s an i n t e r e s t i n g p o i n t . The shear s t r e s s e s as c a l c u l a t e d by the rough t u r b u l e n t e q u a t i o n g i v e n by Kamphuis. g e n e r a l l y f i t i n the m i d d l e t o upper p o r t i o n of the ranges g i v e n by S h i e l d s . Those c a l c u l a t e d by the u n i v e r s a l l o g law e q u a t i o n f o r c u r r e n t f l o w a r e g e n e r a l l y near the lower end of these same ranges. T h i s d i s c r e p a n c y may be acc o u n t e d f o r i n one or i n a c o m b i n a t i o n of s e v e r a l p o s s i b i l i t i e s : the f r i c t i o n v a l u e as d e t e r m i n e d by Kamphuis may be a l i t t l e l a r g e ; t h e r e may be judgmental d i f f e r e n c e s i n c u r r e d when s e p a r a t e l y o b s e r v i n g the t h r e s h o l d s under waves and those under c u r r e n t s ; f i n a l l y , t he f l o w t u r b u l e n c e s i n the steady f l o w s i n the flume used may be d i f f e r e n t from t h o s e t u r b u l e n c e s measured i n the f i e l d w hich are used i n g i v i n g the Manning r e l a t i o n . I t i s noted t h a t the rough t u r b u l e n t r e l a t i o n of Kamphuis g i v e s s t r e s s v a l u e s a good d e a l lower than those of Jonsson and K a j i u r a , and t h a t the bedform roughnesses used a r e q u i t e s i m i l a r i n s i z e t o tho s e used by t h i s w r i t e r , but are much s m a l l e r than t h a t used by J o n s s o n . I t i s f e l t t h a t Kamphuis's r e s u l t s s h o u l d g i v e a b e t t e r i n d i c a t i o n as t o the 1 70 s t r e s s l e v e l s than the o t h e r r e l a t i o n s , even though l a r g e c o r r e c t i o n s were n e c e s s a r y i n the method he used to de t e r m i n e t h o s e r e s u l t s . The a c c u r a c y of Kamphuis's rough t u r b u l e n t f o r m u l a cannot here be v a l i d a t e d , but i t s g e n e r a l s u i t a b i l i t y t o the w r i t e r ' s r e s u l t s i s here mentioned. Judgmental d i f f e r e n c e s may be r u l e d out because of the c o n s i s t e n c y w i t h which the obser v e d t h r e s h o l d s o c c u r r e d . I t i s f e l t t h a t the t h r e s h o l d s used f o r the sediment ranges are c o r r e c t f o r the c o n d i t i o n s used, and t h a t the same degree of sediment motion was used f o r e v e r y onset c o n d i t i o n . F i n a l l y , the s t r e s s e s g i v e n by the Manning r e l a t i o n are maximum s t r e s s e s based on the average f l o w s . T h i s r e l a t i o n has been e s t a b l i s h e d by a w e a l t h of stream d a t a , but i t r e f l e c t s t he average of the d a t a . For the flume used, i t i s p o s s i b l e t h a t the maximum t u r b u l e n t shear s t r e s s e s a re l a r g e r than those p r e d i c t e d by the Manning r e l a t i o n . Again i t i s m a i n t a i n e d t h a t the Manning r e s u l t s do f a l l i n the range g i v e n by S h i e l d s ' s e n t r a i n m e n t f u n c t i o n , and t h a t t h e s e a r e s a t i s f a c t o r y h e r e . We can a l s o compare the r e s u l t s as g i v e n by smooth t u r b u l e n t f l o w r e l a t i o n s . The c a l c u l a t e d a m p l i t u d e Reynolds numbers and the o r b i t a l a m p l i t u d e t o bed roughness r a t i o s i n d i c a t e t h a t the f l o w f a l l s i n the t r a n s i t i o n regime w e l l below smooth t u r b u l e n t f l o w . U s i n g the form u l a e anyway, i t i s seen t h a t they do g i v e r e s u l t s t h a t a re a c c e p t a b l e o n l y f o r the s m a l l e s t sediment r a n g e s . 171 The l a m i n a r r e s u l t s a r e a l s o compared, though they a re not e x p e c t e d .to work because of the poor agreement between the measured and the t h e o r e t i c a l p r o f i l e s . Much l i k e the smooth t u r b u l e n t p r e d i c t i o n s , the r e s u l t s a r e a c c e p t a b l e o n l y f o r the s m a l l e s t sediment range. The f l o w p a t t e r n i s of some d i s p u t e . Jonsson (1963,1966) m a i n t a i n e d t h a t f l o w over sand beds i n n a t u r e i s always rough t u r b u l e n t whereas l a b o r a t o r y f l o w s a r e always l a m i n a r . A c c o r d i n g t o Bagnold (1963), the p a r t i c l e d i a m e t e r i s the prime c o n s i d e r a t i o n i n the f l o w type d e t e r m i n a t i o n . P a r t i c l e s of l a r g e r than 0.5 mm d i a m e t e r cause t u r b u l e n t f l o w and p a r t i c l e s of l e s s than 0.5 mm diam e t e r g i v e f l o w t h a t i s not t u r b u l e n t . The p a r t i c l e d i a m e t e r s used l i e between 0.30 mm and 2.35 mm, and the q u e s t i o n s t i l l remains as t o the f l o w t y p e . The v e l o c i t y p r o f i l e s suggest rough and t u r b u l e n t i n the range of measurement f o r u n i d i r e c t i o n a l f l o w , y e t the f l o w may have been l a m i n a r c l o s e r t o the bed f o r o s c i l l a t o r y f l o w and s m a l l sediment s i z e . E s t i m a t i o n of f l o w type u s i n g s k e i n s of dye i s both s u b j e c t i v e and d i f f i c u l t because of dye d i s p e r s a l , so t h i s was not r i g o u r o u s l y a t t e m p t e d . 7.4 CONCLUSIONS L a s e r Doppler Anemometry has been used t o measure the maximum and average c r i t i c a l v e l o c i t i e s under u n i d i r e c t i o n a l c u r r e n t s , under waves, and under c o m b i n a t i o n s of c u r r e n t s and waves. Over a sand bed, the v e l o c i t i e s a r e measured t o 172 be v e r y c l o s e t o tho s e g i v e n by second o r d e r Stokes law. When one c o n s i d e r s the s m a l l wave h e i g h t s used i n the expe r i m e n t and the low a c c u r a c y of h e i g h t measurement, i t i s r e a l i z e d t h a t the second o r d e r Stokes t h e o r y p r e d i c t s the f o r w a r d and r e v e r s e maximum v e l o c i t i e s w i t h remarkable agreement. The wave and c u r r e n t v e l o c i t i e s a r e found t o be n e a r l y s i m p l e v e c t o r i a l a d d i t i o n s of the independent c a s e s of the " e q u i v a l e n t " wave v e l o c i t y and the average added c u r r e n t v e l o c i t y . T h i s i s c o n t r a r y t o the f i n d i n g s of George and S l e a t h ( 1 976), who det e r m i n e d t h a t the combined v e l o c i t y came t o be l e s s than the two added i n d e p e n d e n t l y . T h e i r f i n d i n g s were, however, f o r a steady f l o w over an o s c i l l a t i n g bed. The u n i d i r e c t i o n a l shear s t r e s s e s f o r sand movement a r e found t o be a t the low ends of the ranges p r e d i c t e d by S h i e l d s . The o s c i l l a t o r y shear s t r e s s e s f o r sand movement are found t o be a t the m i d d l e t o the upper end of the ranges g i v e n by him. The f l u i d t u r b u l e n c e i s b e l i e v e d t o p l a y a c r i t i c a l r o l e i n t h e onset of p a r t i c l e m o t i o n , because the s t r e s s e s f o r c r i t i c a l motion f o r u n i d i r e c t i o n a l and o s c i l l a t o r y motion a r e c a l c u l a t e d t o be v e r y d i f f e r e n t . The s t r e s s e s as c a l c u l a t e d under waves a r e p r o b a b l y peak s t r e s s e s , as the v e l o c i t i e s used i n the c a l c u l a t i o n s a r e peak v e l o c i t i e s . For u n i d i r e c t i o n a l c u r r e n t s , the s t r e s s e s c a l c u l a t e d a r e a v e r a g e s , based on average v e l o c i t i e s , and the peak s t r e s s e s 1 73 i n t u r b u l e n t v e l o c i t y b u r s t s a r e p r o b a b l y the same s i z e as those under waves. A d e f i n i t i v e c r i t e r i o n has been e s t a b l i s h e d i n t h a t the onset of motion of sediment p a r t i c l e s i s a s s o c i a t e d w i t h near-bed water p a r t i c l e v e l o c i t i e s t h a t a r e the same r e g a r d l e s s of whether the motion i s d r i v e n by waves, c u r r e n t s , or combined waves and c u r r e n t s . 8. CONCLUSIONS The purpose of t h i s s t u d y i s t o determine how waves and c u r r e n t s combine, e s p e c i a l l y i n the r e g i o n j u s t above a f l a t sediment bed. The shear s t r e s s a t the bed has been examined by u s i n g the onset of sediment motion as an i n d i c a t o r of a s p e c i f i c s t r e s s l e v e l . The shear s t r e s s e s a r e det e r m i n e d from the v e l o c i t y p r o f i l e s i n the near-bed r e g i o n , which were measured w i t h a l a s e r d o p p l e r anemometer. These v e l o c i t y p r o f i l e measurements f u r t h e r s e r v e d t o check t h e o r e t i c a l v e l o c i t y f i e l d p r e d i c t i o n s , and t o : answer the q u e s t i o n of how component wave and c u r r e n t v e l o c i t i e s can be used t o d e s c r i b e the combined wave and c u r r e n t v e l o c i t y f i e l d . The measurements a l s o gave r e s u l t s which e s t a b l i s h a d e f i n i t i v e c r i t e r i o n f o r the t h r e s h o l d of sediment m o t i o n . Other measurements of shear s t r e s s u s i n g a shear p l a t e have been a t t e m p t e d , but t h i s approach i s shown to be s u b j e c t t o l a r g e p r e s s u r e induced terms which cannot be s e p a r a t e d from the d e s i r e d shear f o r c e terms. An e x t e n s i v e review has been made of the boundary l a y e r t h e o r y f o r stea d y u n i d i r e c t i o n a l and o s c i l l a t o r y flow c o n d i t i o n s . Wave t h e o r y has a l s o been s t u d i e d , as has the onset of sediment motion under waves and st e a d y c u r r e n t s . T h i s r e s e a r c h r a i s e d a few major p o i n t s f o r c o n s i d e r a t i o n and f o r c o n f i r m a t i o n i n the ex p e r i m e n t s performed. F i r s t , when comparing r e s e a r c h e r s ' r e s u l t s , i t i s e s s e n t i a l t o c a r e f u l l y c o n s i d e r t h e flume d i m e n s i o n s and i n l e t boundary 1 7 A 175 c o n d i t i o n s , a l l bedforms, p a d d l e t y p e , wave p r o f i l e , and water t e m p e r a t u r e . Second, i t i s i m p o r t a n t t h a t f o r any sediment movement s t u d y i n a l a b o r a t o r y s e t t i n g , a c c u r a t e near-bed v e l o c i t y measurements must be made u s i n g a proven i n s t r u m e n t a t i o n t e c h n i q u e . The l a s e r d o p p l e r e x p e r i m e n t s gave the f o l l o w i n g s p e c i f i c r e s u l t s : 1) the near-bed v e l o c i t i e s under the waves i n the flume used can be d e s c r i b e d a c c u r a t e l y > b y S t o k e s second o r d e r wave t h e o r y , 2) the near-bed c u r r e n t v e l o c i t i e s can be d e s c r i b e d a c c u r a t e l y by the u n i v e r s a l l o g v e l o c i t y law, where the shear v e l o c i t y v a l u e i s c a l c u l a t e d from the flume c h a r a c t e r i s t i c s and average- f l o w v e l o c i t y u s i n g the Manning e q u a t i o n , 3) the f l o w i s not always l a m i n a r under l a b o r a t o r y c o n d i t i o n s , but i n t h i s case i s i n a t l e a s t t r a n s i t i o n near the bed s i m p l y because of the sediment roughness. I t i s f e l t from the v e l o c i t y s i g n a l s and from the good agreement between the v e l o c i t y laws and the measured v e l o c i t y p r o f i l e s t h a t the f l o w i s f u l l y rough t u r b u l e n t , 4) the combined wave and c u r r e n t near-bed v e l o c i t y can be d e s c r i b e d by t h e v e c t o r i a l a d d i t i o n of the component wave near-bed v e l o c i t y w i t h the average component c u r r e n t near-bed v e l o c i t y , 5) t h e near-bed maximum v e l o c i t y can be used t o f u l l y 176 e s t a b l i s h the onset of sediment motion f o r a l l c o n d i t i o n s t e s t e d . For a s p e c i f i c sediment s i z e , the maximum v e l o c i t y j u s t above the bed i s the same f o r sediment t h r e s h o l d , independent of whether t h i s t h r e s h o l d i s under waves, c u r r e n t s , or combined waves and c u r r e n t s , 6) t h e boundary l a y e r t h i c k n e s s under waves i s observed t o be v e r y t h i n , 7) the shear s t r e s s d e t e r m i n e d from the near-bed wave p r o f i l e s f o r steady u n i d i r e c t i o n a l c u r r e n t s and c a l c u l a t e d u s i n g the Manning r e l a t i o n f a l l s i n the range s p e c i f i e d by the S h i e l d s e n t r a i n m e n t f u n c t i o n , 8) the shear s t r e s s d e t e r m i n e d from the near-bed v e l o c i t i e s f o r wave c o n d i t i o n s and as c a l c u l a t e d u s i n g Kamphuis's rough t u r b u l e n t f r i c t i o n f a c t o r f a l l s i n the range s p e c i f i e d by the S h i e l d s e n t r a i n m e n t f u n c t i o n , 9) t h e shear s t r e s s under combined waves and c u r r e n t s may be d e s c r i b e d by v e c t o r i a l a d d i t i o n s of the component wave and c u r r e n t shear s t r e s s e s , as t h i s s t r e s s f a l l s v e r y c l o s e t o the range s p e c i f i e d by the S h i e l d s e n t r a i n m e n t f u n c t i o n . 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