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Maximum scour around cylinders induced by wave and current action Abusbeaa, Abubaker Mohamed 1986

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MAXIMUM SCOUR AROUND CYLINDERS INDUCED BY WAVE AND CURRENT ACTION by ABUBAKER MOHAMED ABUSBEAA A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES Depa r tmen t o f 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 c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA J U N E , 1986 © ABUBAKER MOHAMED ABUSBEAA, 1986 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 requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h 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 study. 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 copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted 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 understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date June, 1986 ABSTRACT T h e maximum p o s s i b l e s c o u r a r o u n d c y l i n d r i c a l s t r u c t u r e s u n d e r t h e a c t i o n o f c o m b i n e d wave and c u r r e n t was i n v e s t i g a t e d i n t h i s s t u d y . A r e v i e w o f t h e l i t e r a t u r e showed t h a t t h e r e a r e no a d e q u a t e t h e o r i e s o r methods f o r p r e d i c t i n g maximum p o s s i b l e s c o u r d e p t h s a r o u n d s t r u c t u r e s u n d e r t h e a c t i o n o f wave p l u s c u r r e n t . D e v e l o p m e n t o f s u c h a s c o u r p r e d i c t i o n i s o f c o n s i d e r a b l e e c o n o m i c i m p o r t a n c e f o r t h e d e s i g n o f o f f -s h o r e s t r u c t u r e s . The s t u d y s t a r t e d w i t h e x p e r i m e n t a l t e s t s o f t h e s c o u r a r o u n d c y l i n d r i c a l s t r u c t u r e s and a c o m p a r i s o n was made be tween s c o u r u n d e r s t e a d y c u r r e n t s a l o n e , waves a l o n e and combined wave and c u r r e n t c o n d i t i o n s . E x i s t i n g t h e o r i e s f o r s c o u r and f l o w v e l o c i t i e s u n d e r waves a l o n e and c u r r e n t s a l o n e were a n a l y z e d and s e t s o f e x p e r i m e n t s were p e r f o r m e d f o r t h e t h r e e f l o w c o n d i t i o n s o f waves a l o n e , c u r r e n t s and waves p l u s c u r r e n t s u s i n g t h r e e s e d i m e n t s i z e r a n g e s and f i v e c y l i n d e r s i z e s . The dependence o f maximum s c o u r o n b o t h c y l i n d e r s i z e and s e d i m e n t s i z e f o r t h e t h r e e f l o w c a s e s was s t u d i e d and g r a p h i c a l r e l a -t i o n s h i p s were e s t a b l i s h e d . The maximum s c o u r u n d e r combined wave and c u r r e n t a t t h r e s h o l d c o n d i t i o n s i n t h e a p p r o a c h i n g f l o w was i n v e s t i g a t e d i n d e t a i l and t h e dependence o f maximum s c o u r on t h i s c r i t i c a l t h r e s h o l d f l o w c r i t e r i o n was shown. The maximum s c o u r d e p t h c a n be r o u g h l y e s t i m a t e d u s i n g t h i s s t u d y p r o v i d e d t h a t t h e f l o w c o n d i t i o n s , s e d i m e n t p r o p e r t i e s and s t r u c t u r e d i m e n s i o n s a r e d e f i n e d . - i i -TABLE OF CONTENTS Page ABSTRACT i i L I S T OF TABLES v L I S T OF FIGURES v i NOMENCLATURE i x ACKNOWLEDGEMENT x i i 1 . INTRODUCTION 1 1.1 GENERAL 1 1.2 LITERATURE REVIEW 2 1 . 2 . 1 SCOUR 2 1 . 2 . 2 UNIDIRECTIONAL FLOWS - STEADY CURRENTS - SCOUR 2 1 . 2 . 3 O S C I L L A T I N G FLOW - WAVES - SCOUR 8 1 . 2 . 4 COMBINED WAVES AND CURRENTS FLOW SCOUR 8 1 . 2 . 5 SCOUR DEPTH PREDICTION FORMULAS 14 2 . THRESHOLD OF MOTION 17 2 . 1 HYDRODYNAMIC FORCES 18 2 . 1 . 1 DRAG 19 2 . 1 . 2 L I F T 20 2 . 1 . 3 GRAVITY 21 2 . 2 A N A L Y S I S OF THE SHIELDS CRITERION 24 2 . 3 THRESHOLD MEASUREMENTS 26 3 . THEORETICAL BACKGROUND 31 3 . 1 WAVE AND CURRENT INTERACTION 31 - i i i -TABLE OF CONTENTS (Continued) 3.2 NEAR-BOTTOM SHEAR STRESSES 34 3.2.1 UNIDIRECTIONAL FLOW SHEAR STRESSES 34 3.2.2 OSCILLATORY FLOW SHEAR STRESSES 35 3.2.3 COMBINED SHEAR STRESSES AT BED 36 3.2.4 SHEAR STRESSES COMPARISON 39 3.3 WAVE THEORIES 41 3.4 COEFFICIENT OF REFLECTION 42 3.5 SCALE EFFECTS 45 3.6 SOME IMPORTANT NUMBERS 45 3.6.1 FROUDE NUMBER 45 3.6.2 REYNOLDS NUMBER 46 3.6.3 KEULEGAN-CARPENTER NUMBER 47 4. EXPERIMENTAL SET UP AND PROCEDURE 48 4.1 EXPERIMENTAL APPARATUS AND EQUIPMENTS 48 4.2 EXPERIMENTAL PROCEDURE 48 5. PRESENTATION AND DISCUSSION OF RESULTS 55 6. CONCLUSIONS AND RECOMMENDATIONS 85 6.1 CONCLUSIONS 85 6.2 RECOMMENDATION FOR FURTHER STUDY 87 BIBLIOGRAPHY 89 - i v -LIST OF TABLES Page Table 2.1 Shear v e l o c i t i e s and maximum near-bed v e l o c i t i e s a f t e r Quick et a l . (1985) 29 3.1 Combined wave and current shear stresses due to d i f f e r e n t methods f o r three sediment s i z e s at threshold conditions 39 3.2 (Modified a f t e r Sarpkaya and Isaacson, 1981). Results of Stokes second order theory 44 5.1 Experimental r e s u l t s under steady currents alone 56 5.2 Measured and estimated maximum scour depths i n (cm) at threshold conditions i n the approaching flow f o r steady currents alone; sediment siz e range = 0.85-1.16 mm 60 5.3 Measured and estimated maximum r e l a t i v e scour at threshold conditions i n the approaching flow f o r steady currents alone; sediment siz e range = 0.85-1.16 mm 60 5.4 Measured and estimated maximum scour depths i n (cm) at threshold conditions i n the approaching flow f o r steady currents alone; sediment s i z e range = 1.16-1.70 mm 61 5.5 Measured and estimated maximum r e l a t i v e scour at threshold conditions i n the approaching flow f o r steady currents alone; sediment siz e range = 1.16-1.70 mm 61 5.6 Experimental r e s u l t s of o s c i l l a t o r y waves 62 5.7 Experimental r e s u l t s of combined waves and currents 67 5.8 Contribution of waves and currents i n threshold v e l o c i t y (%) 72 - v -LIST OF FIGURES Page Figure 1.1 T y p i c a l scour v e l o c i t y r e l a t i o n s h i p , Shen et a l . (1969) 5 1.2 Graph of scour versus Pier Reynold's number, Shen et a l . (1969) 6 1.3 Graph of scour versus Pier Reynold's number for three d i f f e r e n t p i l e s i z e s , Shen et a l . (1969) 7 1.4 Graph of scour versus Pier Reynold's number for two d i f f e r e n t sand s i z e s , Shen et a l . (1969) 7 1.5 Relative scour versus r e l a t i v e depth, Wells and Sorensen (1970) 9 1.6 Relative scour versus wave steepness, Wells and Sorensen (1970) 10 1.7 Relative scour versus sediment number, Wells and Sorensen (1970) 11 1.8 Relative scour versus Pier Reynold's number, Wells and Sorensen (1970) 12 2.1 Primary forces acting on an i n d i v i d u a l sand p a r t i c l e , Wells and Sorensen (1970) 22 2.2 The Shield's entrainment function 25 2.3 Modified Shield's diagram, Bagnold (1966) 27 2.4 Measured threshold v e l o c i t y p r o f i l e s , Quick et a l . (1985) 30 3.1 D e f i n i t i o n sketch for a progressive wave t r a i n on a steady current 31 3.2 Wave p r o f i l e s with and without current, (a) Rough boundary layer; (b) Smooth boundary layer 32 3.3 Flow chart diagram for c a l c u l a t i o n of f ^ , Tanaka and Shuto (1984) 38 3.4 D e f i n i t i o n sketch f o r a progressive wave t r a i n 41 3.5 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 , Quick et a l . (1983) 43 - v i -LIST OF FIGURES (Continued) 4.1 Experimental equipment, schematic diagram, Quick (1983) 49 4.2 OTT current meter 50 4.3 Wave recorder 51 4.4 C y l i n d r i c a l p i l e s 52 5.1 Relative scour versus time under steady currents alone 57 5.2 Relative scour versus time under steady currents alone 58 5.3 Relative scour versus time under pure waves 63 5.4 Relative scour versus time under pure waves 64 5.5 Maximum r e l a t i v e scour versus cylinder diameter f o r o s c i l l a t o r y flow at threshold conditions; sediment siz e range = 0.85-1.16 mm 65 5.6 Relative scour versus time under combined waves and currents 70 5.7 Relative scour versus time under combined waves and currents 71 5.8 Maximum scour versus combined waves and currents threshold v e l o c i t y ; sediment s i z e range = 0.3-0.85 mm 73 5.9 Maximum r e l a t i v e scour versus combined waves and currents threshold v e l o c i t y ; sediment si z e range = 0.3-0.85 mm 74 5.10 Maximum scour versus c y l i n d e r diameter at threshold conditions of combined waves and currents; sediment siz e range = 0.3-0.85 mm 76 5.11 Maximum r e l a t i v e scour versus cylinder diameter at threshold conditions of combined waves and currents; sediment size range = 0.3-0.85 mm 77 5.12 Maximum scour depth versus cylinder diameter f o r three sediment s i z e s under threshold conditions of combined waves and currents; 25% waves and 75% currents 78 - v i i -LIST OF FIGURES (Continued) 5.13 Maximum r e l a t i v e scour versus c y l i n d e r diameter f o r three sediment s i z e s at threshold of combined waves and currents; 25% wave and 75% currents 79 5.14 T y p i c a l scour hole pattern for combined waves and currents 80 5.15 Typ i c a l scour hole pattern for combined waves and currents 81 5.16 T y p i c a l r i p p l e pattern for waves plus currents tests 83 5.17 Typical r i p p l e pattern f o r waves plus currents tests 83 - v i i i -NOMENCLATURE maximum d i m e n s i o n o f t h e p a r t i c l e a m p l i t u d e o f h o r i z o n t a l p a r t i c l e m o t i o n i n t h e o s c i l l a t o r y f l o w p r o j e c t e d a r e a o f o b j e c t n o r m a l t o f l o w d i r e c t i o n i n t e r m e d i a t e d i m e n s i o n o f t h e p a r t i c l e c y l i n d e r d i a m e t e r minimum d i m e n s i o n o f t h e p a r t i c l e wave c e l e r i t y r e l a t r i v e t o t h e f i x e d c o o r d i n a t e s y s t e m f o r m c o e f f i c i e n t f o r m c o e f f i c i e n t r e l a t e d t o t h e e f f e c t i v e s u r f a c e a r e a o f t h e p a r t i c l e i n t h e d i r e c t i o n o f t h e d r a g f o r c e f o r m c o e f f i c i e n t r e l a t e d t o t h e e f f e c t i v e s u r f a c e a r e a o f t h e p a r t i c l e i n t h e d i r e c t i o n o f t h e l i f t f o r c e d r a g 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 wave c e l e r i t y r e l a t i v e t o t h e c o o r d i n a t e f r o m m o v i n g w i t h c u r r e n t w a t e r d e p t h i n t h e c h a n n e l maximum s c o u r d e p t h e q u i l i b r i u m s c o u r d e p t h d i a m e t e r o f a s p h e r e h a v i n g t h e same vo lume a s t h e p a r t i c l e d i a m e t e r o f a s p h e r e h a v i n g t h e same s u r f a c e a r e a a s t h e p a r t i c l e mean p a r t i c l e s i z e o f bed r o u g h n e s s e l e m e n t s c h a r a c t e r i s t i c d i a m e t e r o f t h e p a r t i c l e f r i c t i o n f a c t o r f o r u n i d i r e c t i o n a l f l o w 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 f = f r i c t i o n f a c t o r f o r combined wave and c u r r e n t wc » d r a g f o r c e F = g r a v i t y f o r c e F T = l i f t f o r c e la F = p i l e F r o u d e number P F = F r o u d e number r F,j, = t o t a l f o r c e F * = e n t r a i n m e n t f u n c t i o n s g = g r a v i t a t i o n a l c o n s t a n t H = wave h e i g h t IL^ = h e i g h t o f r e f l e c t e d wave H , H . = maximum and minimum wave h e i g h t s measu red i n t h e f l u m e max m i n k. = wave number = 2TT/L k = h e i g h t o f b e d r o u g h n e s s e l e m e n t s K c = K e u l e g a n - C a r p e n t e r number U ^ T / b K r = r e f l e c t i o n c o e f f i c i e n t f o r f l u m e waves = H ^ / H n = M a n n i n g ' s r o u g h n e s s c o e f f i c i e n t P = P a s c a l a R = R e y n o l d ' s number e R* = p a r t i c l e R e y n o l d ' s number = U * D / v Rp = p i l e R e y n o l d ' s number = U b / v = h y d r a u l i c r a d i u s o f c h a n n e l S = s p e c i f i c g r a v i t y o f p a r t i c l e s s S = c h a n n e l b e d s l o p e o S = maximum s c o u r d e p t h S = shape f a c t o r F T = wave p e r i o d T = p e r i o d o f wave o n c u r r e n t a s s e e n f r o m m o v i n g c o o r d i n a t e a * f rame - x -U = f l o w v e l o c i t y TJ, = maximum o r b i t a l v e l o c i t y a t t h e b e d a s p r e d i c t e d by f i r s t 1 ro o r d e r t h e o r y U = s t e a d y c u r r e n t a v e r a g e v e l o c i t y c U = f l o w c r i t i c a l v e l o c i t y c r U = maximum o s c i l l a t o r y w a t e r p a r t i c l e v e l o c i t y w U * = f r i c t i o n v e l o c i t y = ( t / p ) 1 / 2 U * = f r i c t i o n v e l o c i t y a t t h r e s h o l d c o n d i t i o n s c r U * = e q u i v a l e n t s h e a r v e l o c i t y f o r c o m b i n e d wave and c u r r e n t f l o w wc v = f a l l v e l o c i t y a = c o n s t a n t Y = s p e c i f i c w e i g h t o f f l u i d •y = s p e c i f i c w e i g h t o f s e d i m e n t y = d y n a m i c v i s c o s i t y v = k i n e m a t i c v i s c o s i t y = u / p p = f l u i d d e n s i t y p = s e d i m e n t d e n s i t y s T = s h e a r s t r e s s x = s h e a r s t r e s s a t b e d o x = s t e a d y c u r r e n t s h e a r s t r e s s c T = c r i t i c a l s h e a r s t r e s s c r T. = u n s t e a d y f l o w s h e a r s t r e s s w T = combined wave and c u r r e n t s h e a r s t r e s s wc cj> = a n g l e o f r e p o s e o f t h e submerged s e d i m e n t tj) = a n g l e made by u n i d i r e c t i o n a l c u r r e n t w i t h t h e d i r e c t i o n o f wave p r o p a g a t i o n - x i -ACKNOWLEDGEMENTS T h e a u t h o r i s v e r y g r a t e f u l f o r t h e g u i d a n c e and encouragemen t g i v e n by h i s s u p e r v i s o r , D r . M . C . Q u i c k t h r o u g h o u t t h e p r e p a r a t i o n o f t h i s t h e s i s . F i n a n c i a l s u p p o r t i n t h e f o r m o f a s c h o l a r s h i p f r o m A l - F a t c h U n i v e r s i t y , T r i p o l i i s g r a t e f u l l y a c k n o w l e d g e d . The a u t h o r w i s h e s t o t h a n k M r . 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 t h e l a b o r a t o r y . 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 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 i i -1 1. INTRODUCTION 1.1 GENERAL I n t h e a c c e l e r a t e d e x p l o r a t i o n o f o f f s h o r e r e s o u r c e s t h e r e i s a g r o w i n g need t o p l a c e o b j e c t s o r s t r u c t u r e s on t h e s e a b e d . D e s i g n c o n s i d e r a t i o n s must i n c l u d e a n a n a l y s i s o f t h e i r s t a b i l i t y , p a r t i c u l a r l y i f t h e s e a bed i s composed o f n o n - c o h e s i v e s e d i m e n t s . A s a n d - b o t t o m m a t e r i a l i s g e n e r a l l y i n a c o n d i t i o n o f dynamic s t a b i l i t y u n d e r t h e p r e v a i l i n g c u r r e n t s and w a v e - i n d u c e d c u r r e n t s , when a n o b j e c t i s p l a c e d i n o r on t h e s e a - b e d , t h e e q u i l i b r i u m may be d i s t u r b e d and l o c a l v e l o c i t i e s a r o u n d t h e o b j e c t s a r e i n c r e a s e d and t h e r a t e o f t r a n s p o r t e d m a t e r i a l i s i n c r e a s e d . T h i s i n c r e a s e i n t h e t r a n s p o r t e d m a t e r i a l may c a u s e s c o u r i n g ( e r o s i o n ) a r o u n d t h e s t r u c t u r e s . C u r r e n t and wave c o n d i t i o n s i n t h e v i c i n i t y o f t h e s t r u c t u r e w i l l c h a n g e . T h i s change may c a u s e l a r g e changes i n t h e b o t t o m t o p o g r a p h y i n t h e v i c i n i t y o f t h e s t r u c t u r e and t h u s c a u s e e r o s i o n i n some and d e p o s i t i o n I n o t h e r a r e a s . The d e p t h o f s c o u r a r o u n d s t r u c t u r e s i s i m p o r t a n t i n c o m p u t i n g t h e minimum p e n e t r a t i o n d e p t h o f p i l e s f o r f i x e d s t r u c t u r e s . F o r o b j e c t s p l a c e d o n t h e b o t t o m , o r f o r " s i t - o n - b o t t o m " p l a t f o r m s t h e s c o u r may cause s e t t l e m e n t o f t he s u p p o r t i n g members . The m a i n p u r p o s e o f t h i s i n v e s t i g a t i o n i s t o d e v e l o p p r e d i c t i o n s o f s c o u r d e p t h a r o u n d o f f s h o r e s t r u c t u r e s u n d e r t h e combined a c t i o n o f c u r r e n t s and w a v e s . S c o u r d e p t h e x p e r i m e n t s f o r c y l i n d r i c a l p i l e s were c o n d u c t e d i n t h i s s t u d y and c o m p a r i s o n was made be tween t h e s c o u r p r o d u c e d by c u r r e n t s a l o n e , waves a l o n e and waves p l u s c u r r e n t s . The s t u d y w i l l be r e s t r i c t e d t o d e t e r m i n i n g s c o u r d e p t h s a r o u n d c y l i n d r i c a l 2. s t r u c t u r e s p l a c e d i n n o n - c o h e s i v e g r a n u l a r bed m a t e r i a l and w i t h f l o w s i n t h e d i r e c t i o n o f wave p r o p a g a t i o n . The m a i n e m p h a s i s w i l l be c o n c e r n e d w i t h e s t i m a t i n g t h e maximum p o s s i b l e s c o u r . 1.2 LITERATURE REVIEW 1 . 2 . 1 SCOUR Many t y p e s o f s c o u r e x i s t , f o r example g e n e r a l s c o u r , where bed l e v e l d e g r a d e s b e c a u s e s e d i m e n t r e m o v a l e x c e e d s s e d i m e n t s u p p l y , l o c a l s c o u r where t h e bed l e v e l l o c a l l y d r o p s , u s u a l l y c a u s e d by l o c a l a c c e l e r a t i o n o f t h e f l o w a r o u n d a s t r u c t u r e . Th es e t y p e s a r e s u b - d i v i d e d i n t o l i v e bed and c l e a r w a t e r s c o u r . L i v e bed s c o u r o c c u r s where s e d i m e n t s u p p l y e q u a l s s e d i m e n t r e m o v a l w h i l e c l e a r w a t e r s c o u r o c c u r s when t h e r e i s no s e d i m e n t s u p p l y . I n g e n e r a l , c l e a r w a t e r s c o u r d e p t h s a r e g r e a t e r t h a n g e n e r a l s c o u r d e p t h s as p o i n t e d o u t by many i n v e s t i g a t o r s , e . g . M e l v i l l e (1975) and J a i n and F i s h e r ( 1 9 8 0 ) , hence o n l y c l e a r w a t e r s c o u r c a s e w i l l be d e a l t w i t h i n t h i s s t u d y . The s c o u r due t o s t e a d y c u r r e n t s a l o n e , waves a l o n e , a n d c o m b i n e d c u r r e n t p l u s waves i s d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n . 1 . 2 . 2 UNIDIRECTIONAL FLOWS - STEADY CURRENTS - SCOUR S c o u r a r o u n d c y l i n d r i c a l p i l e s has l o n g been o f i m p o r t a n c e i n t h e d e s i g n o f b r i d g e p i e r s and o c e a n j e t t i e s f o r more t h a n a c e n t u r y and i t h a s b e e n t h e s u b j e c t o f many i n v e s t i g a t i o n s s u c h as A n d e r s o n ( 1 9 7 4 ) , B r e u s e r s e t a l . ( 1 9 7 7 ) . J a i n and F i s h e r ( 1 9 7 9 ) , N e i l l (1964a) and Shen e t a l . ( 1 9 6 6 ) . Numerous s c o u r p r e d i c t i o n s f o r m u l a s have been p u b l i s h e d a s a r e s u l t o f t h e s e s t u d i e s , bu t p r e d i c t e d s c o u r d e p t h s v a r y w i d e l y , 3. e s p e c i a l l y when t h e f o r m u l a s a r e a p p l i e d t o f l o w c o n d i t i o n s o u t s i d e t h e r a n g e o f c o n d i t i o n s f o r w h i c h t h e y were d e v e l o p e d . However i n s p i t e o f t h e v a r i a t i o n s , t h e r e does a p p e a r t o be c o n s e n s u s . Once t h e c o n d i t i o n s o f g e n e r a l s e d i m e n t m o t i o n h a v e b e e n e s t a b l i s h e d i n a s t r e a m , t h e r e w i l l b e no f u r t h e r i n c r e a s e i n s c o u r d e p t h w i t h v e l o c i t y , b e c a u s e t h e r a t e a t w h i c h s e d i m e n t e n t e r s t h e r e g i o n o f s c o u r i s e q u a l t o t h e r a t e o f r e m o v a l by t h e s c o u r i n g p r o c e s s . Few i n v e s t i g a t i o n s have been made f o r f l o w s a t h i g h e r v e l o c i t i e s , i . e . , f o r f l o w r e g i m e s w e l l b e y o n d t h e o n s e t o f g e n e r a l s e d i m e n t t r a n s p o r t . T h i s l i m i t e d number o f i n v e s t i g a t i o n s i s b e c a u s e p e r f o r m i n g e x p e r i m e n t s a t h i g h v e l o c i t i e s beyond t h e o n s e t o f m o t i o n ( T h r e s h o l d v e l o c i t y ) needs a c o n t i n u o u s s u p p l y o r r e c i r c u l a t i o n o f s e d i m e n t m a t e r i a l w h i c h i s a d i f f i c u l t p r o c e s s . W a t e r f l o w i n g p a s t a s t r u c t u r e f o u n d e d on a n e r o d i b l e s e d i m e n t bed i s f r e q u e n t l y o b s e r v e d t o p r o d u c e l o c a l s c o u r i n g . T h i s s c o u r i n g r e s u l t s f r o m t h e a c c e l e r a t i o n and d e c e l e r a t i o n o f t h e w a t e r f l o w f i e l d a s i t f l o w s p a s t t h e s t r u c t u r e and t h e s e a r e t h e r e g i o n s o f l o c a l e r o s i o n and d e p o s i t i o n o f s e d i m e n t . The d o m i n a n t f e a t u r e s o f t h e f l o w n e a r a p i e r i s t h e l a r g e - s c a l e eddy s t r u c t u r e , o r t h e s y s t e m o f v o r t i c e s w h i c h d e v e l o p a r o u n d t h e p i e r . T h e s e v o r t e x s y s t e m s a r e t h e b a s i c mechanism o f l o c a l s c o u r , w h i c h has l o n g b e e n r e c o g n i z e d by many i n v e s t i g a t o r s i n c l u d i n g M e l v i l l e ( 1 9 7 5 ) , N e i l l ( 1 9 6 4 b ) , R o p e r ( 1 9 6 7 ) , I m b e r g e r e t a l . ( 1 9 8 2 ) . Due t o t h e p l a c i n g o f a n o b s t r u c t i o n i n t h e f l o w f i e l d , l o c a l l y h i g h v e l o c i t i e s a r e p r o d u c e d a r o u n d t h e p i e r w h i c h i n c r e a s e t h e t r a c t i v e f o r c e o n t h e s e d i m e n t a t t h e p i e r b a s e . Due t o t h e n o n - u n i f o r m v e l o c i t y d i s t r i b u t i o n i n a s t r e a m f l o w , t h e p r e s s u r e f i e l d I n d u c e d by t h e p i e r : (1 ) c r e a t e s a downward 4. v e l o c i t y along the lower leading face of the pi e r ; and (2) produces a three-dimensional separation of the boundary layer leading to the forma-t i o n of a horseshoe vortex. According to M e l v i l l e (1975), once the process has begun, i t i s the downward flow impinging on the bed, and the horseshoe vortex transporting dislodged p a r t i c l e s away which causes the scour. In the case of c l e a r water scour as the scour depth Increases, the strength of the downward flow decreases near the bottom u n t i l f i n a l l y , i t can no longer dislodge p a r t i c l e s . This condition represents the maximum scour to be attained by the p r e v a i l i n g flow conditions. Experimental observations by Roper et a l . (1967) show that the side slopes of the scour hole i n the c l e a r water regime are approximately equal to the nat u r a l angle of repose of the bed material. This i n d i c a t e s that the scour mechanism i s i n the immediate v i c i n i t y of the structure base, and the sediment s l i d e s i n towards the base before i t i s removed. In the case i n which there i s a net sediment transport i n the stream, the strength of the flow near the bottom s i m i l a r l y decreases with increasing scour depth, but maximum scour i s attained when the rate of sediment removal i s equal to the rate of sediment transported i n t o the scour hole by the stream. The v i s u a l observation of the scour process i n the general scour regime at high flow v e l o c i t i e s show that a dynamic equi-l i b r i u m e x i s t s between scour hole and stream flow which i s not apparent when the flow i s stopped, the flow f i e l d near the base of the p i e r i s strong enough to support the sides of the scour hole at angles greater than the angle of repose of the sediment. The sides of the scour hole p e r i o d i c a l l y collapse and dump sediment i n t o the hole, e i t h e r as a dune encroaches upon the pi e r , or as the f l u i d forces supporting these sides become unstable, see J a i n and Fisher (1980). 5. I n te rms o f s t r e a m v e l o c i t y , t h e maximum s c o u r d e p t h a r o u n d a p i e r i n c r e a s e s w i t h i n c r e a s i n g v e l o c i t y u n t i l i n c i p i e n t s e d i m e n t c o n d i t i o n s i n t h e s t r e a m a r e r e a c h e d , F i g u r e 1 . 1 . F i g u r e 1.1 T y p i c a l s c o u r - v e l o c i t y r e l a t i o n s h i p , Shen e t a l . ( 1 9 6 9 ) . E x p e r i m e n t a l o b s e r v a t i o n s by I m b e r g e r e t a l . (1982) and J a i n and F i s h e r ( 1 9 8 0 ) , t h e n show t h a t as t h e s t r e a m v e l o c i t y i s f u r t h e r i n c r e a s e d , s c o u r d e p t h no l o n g e r i n c r e a s e s b u t , i n f a c t , d e c r e a s e s s l i g h t l y , b e c a u s e t h e a p p r o a c h i n g f l o w i s now p r o d u c i n g a s e d i m e n t d i s c h a r g e i n t o t h e s c o u r h o l e . I t s h o u l d be c l e a r t h a t t h e d e f i n i t i o n o f o n s e t o f m o t i o n I s n o t u n i q u e f o r a l l t he s t u d i e s . Some e x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d by Shen e t a l . (1969) as shown i n F i g u r e s 1 .1 t h r o u g h 1 . 4 . T h e s e t e s t s e m p l o y e d a s t e a d y c u r r e n t . F i g u r e 1.1 shows t h e g e n e r a l t r e n d o f s c o u r f o r a g i v e n p i l e d i a m e t e r a n d s e d i m e n t s i z e . The 10% r e d u c t i o n i n s c o u r d e p t h r e s u l t e d f r o m u p s t r e a m l o c a l s u p p l y t o t h e s c o u r h o l e . F i g u r e 1.2 i n d i c a t e s t h e s c o u r d e p t h a s a f u n c t i o n o f t h e R e y n o l d ' s number . F i g u r e s 1.3 and 1.4 i m p l y t h a t t h e s c o u r d e p t h a l s o depends on p i l e d i a m e t e r and s e d i m e n t s i z e . Many f o r m u l a s a r e a v a i l a b l e i n t h e l i t e r a t u r e f o r s c o u r d e p t h p r e d i c t i o n i n c a s e o f u n i d i r e c t i o n a l f l o w s u c h a s t h o s e g i v e n by R o p e r e t a l . ( 1 9 6 7 ) , I m b e r g e r e t a l . (1982) and B r e u s e r s e t a l . ( 1 9 7 7 ) . An K> 90 «0 4 '0.00073*t TOT I 1 ro w o» oe o* !0 FOOT riUUC. 3 FOOT Pit* •7 Pltr Silt.n 01 OOB OO* 004 m Ctobtrt 6 C*attfi*ftrl3) to if* to its o Chabtn a CnotlflnairtSI \03!3 • SI**, r> alKSI 030 * Situ f al 00833 Shrn, tr al 0130 V Tiiax l&l Oi • Tenpartliei oisr Toroport IPS) OIST nam am SaocimOSI °"~\\ • Otto* (61 OS a Cfitlt IS) OS 1 » CDiat IS) OS J* h 0!S OS! 014 imi 04S fm) 04s rmi oso 030 0130 OIT Oft OIS OSS 10' »• Reynolds Number, It 10' Figure 1.2 Graph of scour versus Pier Reynold's number, Shen et a l . (1969). Figure 1.3 Graph of scour versus Pier Reynold' number for three d i f f e r e n t p i l e s i z e s , Shen et a l . (1969). I0r f 1 * ; • t • i • 1 / J 1 4 • f 1 Al : 1 | Saint • ShK. Hal an » oiobti a EnoHfinf (3) <£o«« t./l oftnm as JO OUST nt m - tfi ' ' • • • • • 10* Pilr RtynoM* Numbir, #7 Figure 1.4 Graph of scour versus P i e r Reynold's number f o r two d i f f e r e n t sand s i z e s , Shen et a l . (1969). 8. e x t e n s i v e r e v i e w o f l o c a l s c o u r a r o u n d b r i d g e p i e r s i s g i v e n by B r e u s e r s e t a l . ( 1 9 7 7 ) . 1 . 2 . 3 OSCILLATING FLOW - WAVES - SCOUR V e r y l i t t l e e x p e r i m e n t a l work has been done on s c o u r due t o o s c i l l a t i n g wave c o n d i t i o n s , whe reas t h e r e e x i s t s a w e a l t h o f k n o w l e d g e o n s c o u r i n open c h a n n e l f l o w . S i n c e t h e f o r c e s t h a t c a u s e s c o u r v i r t u a l l y a r e t h e same f o r o s c i l l a t i n g f l o w as f o r open c h a n n e l ( s t e a d y s t a t e ) f l o w , t h a t i s , h y d r o d y n a m i c i n n a t u r e , t h e k n o w l e d g e g a i n e d f r o m e x p e r i m e n t s i n o p e n c h a n n e l f l o w was a p p l i e d , w i t h c e r t a i n r e s e r v a t i o n s , by some i n v e s t i g a t o r s t o o s c i l l a t o r y m o t i o n . The m a j o r i t y o f t h e work done o n s c o u r i n o s c i l l a t o r y m o t i o n s has been c o n c e r n e d p r i m a r i l y w i t h s c o u r o f beaches and l i t t o r a l s e d i m e n t t r a n s p o r t . Ko ( 1 9 6 7 ) , M a r p h y ( 1 9 6 4 ) , and V a n W e l l s (1965) s t u d i e d s c o u r i n f r o n t o f s e a w a l l s o f v a r i o u s a n g l e s , and t h e i r r e s u l t s a r e summar i zed by H e r b i c h e t a l . ( 1 9 6 5 ) . A s t u d y o f t h e s c o u r due t o o s c i l l a t o r y wave m o t i o n was c o n d u c t e d by W e l l s and S o r e n s e n ( 1 9 8 0 ) . F i g u r e s 1.5 t h r o u g h 1 .8 a r e g r a p h s w h i c h r e s u l t e d f r o m t h e i r i n v e s t i g a t i o n . F i g u r e 1 .5 shows a d e f i n i t e r e l a t i o n s h i p b e t w e e n t h e r e l a t i v e s c o u r and t h e r e l a t i v e d e p t h f o r v a r i o u s s e d i m e n t s . F i g u r e s 1.5 and 1.6 show a n i n c r e a s i n g s c o u r d e p t h w i t h a n i n c r e a s i n g wave p e r i o d . F i g u r e s 1.7 and 1.8 show t h e r e l a -t i v e s c o u r as a f u n c t i o n o f s e d i m e n t number , and p i e r R e y n o l d ' s number r e s p e c t i v e l y . 1 . 2 . 4 COMBINED WAVES AND CURRENTS FLOW SCOUR S c o u r due t o waves and c u r r e n t s a r o u n d o f f s h o r e f a c i l i t i e s i s becom-i n g a n e v e r - i n c r e a s i n g p r o b l e m a s t h e m a r i n e c o n s t r u c t i o n i n d u s t r y o -cS>- A A _0L SYMBOL H / h 0 0128 ± 0.000 • 0.198 ± 0.002 A 0.146 O O.203 ± 0.002 0 0.212 ± 0 . 0 0 4 • 0.330 A 0.240 9 O.I83 ± 0 0 3 t Q ° A 10" 10 10" F i g u r e 1 . 6 . R e l a t i v e s c o u r v e r s u s wave s t e e p n e s s , W e l l s and S o r e n s e n (1970) 5 N S INC IP IENT o SAND NO. I 0.737 • S A N D N O 2 0 . 9 0 6 A S A N D NO. 3 0 .935 / 1 1 ii 1 ll 1 ** ^ "™ JQS? 2 5 I 0 5 2 5 10 Sediment Number NS = /( S s - l ) g D Figure 1.7. Relative scour versus sediment number, Wells and Sorensen (1970). . 6MCIPIENT N„ p O SAND NO. 1 3.26 X to' • SAND NO. 2 3.33 X 10* A SAND NO. 3 2.90 X 10* /~\ / / / o / i 1 i / oj o i ru-• • 1 1 I 1 1 1 1 t IC? 2 5 \0* 2 5 I S * P i l e R e y n o l d ' s Number R = — P v F i g u r e 1 . 8 . R e l a t i v e s c o u r v e r s u s P i e r R e y n o l d ' s number , W e l l s and S o r e n s e n ( 1 9 7 0 ) . 1 3 . continues to grow. The s t a b i l i t y of a structure placed i n a non-cohesive seabed may be heavily dependent upon the magnitude of scour experienced a t the structure foundation. There have been many f i e l d observations des c r i b i n g the amount of scour that can occur at a structure's base. A few of these observations have been noted by Johnes (1970). He reported scour depths of 11 feet around the foundation p i l i n g of Diamond Shoal Lighthouse o f f the Coast of North Carolina. Also i n North Carolina, the State Port Authority at Morehead C i t y experienced the collapse of a waterfront warehouse. D i r e c t exposure to waves and t i d a l currents was the cause of the f a i l u r e i n June 1962. For steady flow the f l o o d of the Beaver River i n Alberta, Canada caused excessive scour around the La Corey Bridge and the Beaver Crossing Bridge. N e i l l (1964b) reported a scour depth of 10 f e e t at the La Corey Bridge and 17 f e e t at the Beaver Crossing Bridge. Abad and Machemehi (1974) studied scour due to combined waves and currents. They concluded that scour increases with an increase of wave length, but t h e i r t e s t s were l i m i t e d i n scope and t h i s increase would not be expected to continue when general sediment motion occurred. Armbrust (1982) i n h i s experimental study described l o c a l scour produced by wave and current motion and the dependency of scour around structures on flow c h a r a c t e r i s t i c s . Wang and Herbich (1983) introdued a complex parameter and for t h e i r range of t e s t s the scour depth i s r e l a t e d to t h i s parameter by a c e r t a i n formula. The parameter i s formed by m u l t i p l y i n g together f i v e dimensional groups which are derived from t h e i r dimensional a n a l y s i s , but no j u s t i f i c a t i o n was made f o r making such an assumption. This parameter i s a f u n c t i o n of stream v e l o c i t y to the power four and wave height to the 14. same power. It can be seen that scour depth according to t h e i r assump-t i o n i s not l i m i t e d , i . e . the scour depth w i l l increase with current v e l o c i t y and wave height increase which i s not true once the threshold conditions have been exceeded. This formula i s l i m i t e d to the t e s t conditions and r e l i a b l e i n that range only. The findings of the present study i n d i c a t e that t h e i r formula should be used with great care. In conclusion, the majority of scour studies conducted i n the past dealt with scour due to steady currents. In the l a s t two decades some work has been done on the scour r e s u l t i n g from o s c i l l a t o r y wave motion. To date, only l i m i t e d work, as c i t e d above, has been conducted i n v o l v i n g both waves and currents. 1.2.5 SCOUR DEPTH PREDICTION FORMULAS In the case of pure waves and waves plus currents, no t h e o r e t i c a l formula e x i s t s f o r estimating the scour, but some empirical r e l a t i o n s e x i s t which r e l a t e depth of scour to various flow parameters. For u n i -d i r e c t i o n a l flow - steady currents -, numerous references on l o c a l scour experiments on p i e r s can be found i n l i t e r a t u r e . Few of them, however, cover a s u f f i c i e n t l y general range of conditions with independent v a r i a -t i o n of parameters. In most cases v e l o c i t i e s were below or at the c r i t i c a l v e l o c i t y f o r i n i t i a t i o n of motion. Some of the i n t e r e s t i n g references are summarized below. 1) Larras (1963, 1960) analyzed the data given by Chabert and Engeldinger (1956). He concentrated on the maximum scour depth near the threshold v e l o c i t y of the undisturbed material and gave a r e l a t i o n expressing scour depth -as a function of p i e r diameter, with water depth and grain size neglected: 15. d = 1.05 b ° * 7 5 (1.1) sm where d = maximum possible scour depth sm b = p i e r diameter 2) Hincu (1965) gave experimental r e s u l t s f o r c i r c u l a r p i e r s (b = 3, 4.7, 6, 13, and 20 cm) i n coarse material (Dgg = 0.5, 2 and 5 mm). The scour depth was c o n s t a n t (d = d ) above a c e r t a i n c r i t i c a l v e l o c i t y r s sm J (u ). At lower v e l o c i t i e s a l i n e a r r e l a t i o n with v e l o c i t y obtained IT-IT—1) (1-2) sm cr The influence of water depth was n e g l i g i b l e f o r d/b > 1. The r e s u l t s were correlated with the expression: gb with a r e l a t i o n given f o r u c r i o / ^ / P s ~ p N /oxO.2 . _,0.3 . 0.2 0.5 U c r = 1.2 / g D ( — — ) (—) = 1.54 D d Q g (1.4) for natural sands, the r e l a t i o n may be converted into: 3.3 < £ ) 0 - 2 ( i ) 0 * 1 3 (1.5) b b b 16. where d Q = water depth U = steady flow v e l o c i t y D = sediment s i z e 3) Shen, Schneider, Karaki (1966a, 1969), Roper, Schneider, Shen (1967), Shen (1971) concluded that the maximum l o c a l scour near c r i t i c a l v e l o c i t y for i n i t i a t i o n of sediment w i l l be: dsm = 0.00022 R e ° * 6 1 9 (m - units) (1.6) d . _ 0.43 , do.0.355 sm = 2 Fr ( r — ) (1.7) b where Re = Reynold's number Fr = Froude number 4) Laursen and Toch (1956) suggested t h a t d g m changes with time f o r U > U c r and h i s formula may be presented as: dsm d o 0.3 — — = 1.35 (r— ) * f o r c i r c u l a r p i e r s (1.8) b b Most of the above equations give maximum possible scour as a c l e a r water scour near or at the threshold of motion i n the approaching flow. Scour depth does not i n c r e a s e with steady flow v e l o c i t y U f o r U > ^ CJ.) as given by Chabert and Engeldinger (1965) but, i n f a c t , decreases s l i g h t l y , as reported by Chabert and Shen (1956, 1965). 17. 2. THRESHOLD OF MOTION The term threshold defines a l i m i t i n g condition which forms the boundary between one state of a f f a i r s and another. Like many threshold conditions the threshold of sediment motion cannot be defined with absolute p r e c i s i o n . Eagleson and Dean (1961) described the threshold of movement as, "a state of flow reached when the resultant of a l l a c t i v e forces of the p a r t i c l e i n t e r s e c t the l i n e connecting the bed p a r t i c l e contact points". Some of these active forces w i l l be discussed l a t e r . In t h i s study threshold was assumed to correspond to a moderate number of p a r t i c l e s moving. If threshold i s assumed to correspond to very few p a r t i c l e s moving, t h i s motion i s usually just a few small or l i g h t p a r t i c l e s , which are soon removed, and the motion then ceases. Therefore a s l i g h t l y more a c t i v e p a r t i c l e movement i s a more true threshold which w i l l be maintained, and w i l l not cease. It i s generally accepted that a f l u i d flowing over a sediment bed exerts 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 large. This shear stress at which the p a r t i c l e s begin to move i s known as c r i t i c a l shear stress, and i s associated with 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 . Usually, the Shields entrainment function (Shields, 1936) i s used to define 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 experimental r e s u l t s p l o t t e d on the Sheilds diagram show considerable s c a t t e r . This s c a t t e r may be a t t r i b u t a b l e to such f a c t o r s as the random shear stress exerted by the moving f l u i d , the random shear stress necessary to move bed p a r t i c l e s , and each observer's d e f i n i t i o n of c r i t i c a l movement (Williams and Kemp, 1971). Despite the sc a t t e r , the Shields curve 18. remains about the best i n d i c a t o r of c r i t i c a l motion; d e t a i l s of the Shields c r i t e r i o n w i l l be given l a t e r . Bagnold (1946) has done a famous experiment using a sand bed on an o c i l l a t i n g p l a t e to f i n d c r i t i c a l motion. Komar and M i l l e r (1974) have placed h i s o s c i l l a t o r y data on the u n i d i r e c t i o n a l flow Shields diagram and have concluded that the Shields curve works w e l l f o r o s c i l l a t o r y flow. This f i n d i n g has been confirmed by Madsen and Grant (1975) and i s an important advancement i n the study of the onset of motion under waves. Observations of sediment movement threshold seems to i n d i c a t e that the shear s t r e s s i s not the only f a c t o r i n the mechanism, but there i s another mechanism involved i n the i n i t i a t i o n of p a r t i c l e movement. The mechanism i s the flow turbulence near the p a r t i c l e s , though a quanti-t a t i v e d e s c r i p t i o n of the importance of flow turbulence i s unavailable. I t i s apparent that the turbulence plays a r o l e i n the onset of thresh-o l d . Three primary a c t i v e forces involved and r e l a t i o n s h i p to Incipi e n t motion s h a l l be discussed. 2.1 HYDRODYNAMIC FORCES The forces i n f l u e n c i n g bed p a r t i c l e motion are hydrodynamic and co n s i s t of the forces of steady flow drag, l i f t and the so- c a l l e d i n e r t i a f orces of ac c e l e r a t i n g flow as proposed by Morison et a l . (1950). However since the force due to i n e r t i a i s a function of body volume ( p a r t i c l e diameter D 3 i n t h i s case) which i s very small hence i t w i l l never predominate and therefore i t w i l l be neglected. The t o t a l hydro-dynamic force w i l l be the combination of the l i f t force and the drag f o r c e . The hydrodynamic forces are opposed by the force of gra v i t y and by f r i c t i o n a l forces. !9. 2.1.1 DRAG Due to v i s c o s i t y and boundary layer e f f e c t s a separation of flow occurs on the boundary of the object and a wake i s formed. The point of separation i s a function of the shape of the object and the l o c a l Reynold's number. The drag force i s the combination of form drag due to pressure d i f f e r e n t i a l and the viscous drag due to s k i n f r i c t i o n . For d i f f e r e n t bodies, one type of drag may dominate the other, as i n the case of a pe r f e c t sphere when the t o t a l drag i s primarily pressure drag. The point through which the drag force acts depends on the r e l a t i v e magnitude of the l i f t and drag force components which are functions of bed p a r t i c l e s geometry, lo c a t i o n , and l o c a l Reynold's number. The steady f o r c e due to drag as developed i n any elementary f l u i d mechanics text can be shown to be equal FD = r - p A V ( 2 * 1 } where A i s the p r o j e c t e d area of object normal to flow d i r e c t i o n , C Q i s the drag c o e f f i c i e n t , p i s the f l u i d density and U c i s the fr e e stream v e l o c i t y . Since drag i s a viscous phenomena, the c o e f f i c i e n t of drag i s pr i m a r i l y influenced by the Reynold's number. The drag c o e f f i c i e n t f o r spheres has been studied by various authors. Drag i s measured using f a l l v e l o c i t y data. Under steady state conditions the f a l l v e l o c i t y i s c a l l e d the terminal v e l o c i t y and the drag on the p a r t i c l e i s equal to the submerged weight. Therefore for a sphere, C D = ^ U 1 ( ^ ) (2-2) 20. where V i s the f a l l v e l o c i t y , D i s the p a r t i c l e s i z e diameter, g i s the grav i t y and p i s the density of sediment, s Alger and Simons (1968) adjusted the sphere drag c o e f f i c i e n t f o r i r r e g u l a r shaped p a r t i c l e s through the use of shape f a c t o r , dA SF -p (2.3) n where SF i s Corey shape f a c t o r = c /ab Q, a i s the maximum dimension of the p a r t i c l e , b Q i s the intermediate dimension of the p a t i c l e , c i s the minimum dimension of the p a r t i c l e , d^ i s the diameter of a sphere having the same surface area as the p a r t i c l e and d Q i s the diameter of a sphere having the same volume as the p a r t i c l e . The drag c o e f f i c i e n t increases as the shape f a c t o r increases f o r the same Reynold's number. It i s not only a fu n c t i o n of Reynold's number, but also p a r t i c l e geometry, and i s also influenced by adjacent p a r t i c l e s as pointed out by White (1940). 2.1.2 LIFT The r e l a t i o n s h i p f o r the force due to l i f t i s s i m i l a r to that f o r form drag and i s given by F L = ^ p A U c (2.4) where F i s the l i f t f o r c e and C i s the l i f t c o e f f i c i e n t . Since the Li Li f l u i d i s passing above a p a r t i c l e , there i s a decrease i n pressure above the p a r t i c l e , whereas below the p a r t i c l e the pressure remains f a i r l y h y d r o s t a t i c . Although the l i f t i s very d i f f i c u l t to measure, but i t can be taken i n consideration among the drag force since both of them are functions of U 2 . c 2.1.3 GRAVITY The hydrodynamic forces are opposed by the weight of the p a r t i c l e s and f r i c t i o n . The f r i c t i o n i s usually expressed i n terms of f r i c t i o n angle - angle of natural repose -, and the g r a v i t y force may be simply expressed as the p a r t i c l e ' s submerged weight. The submerged weight of a perfect sphere i s given as where i s g r a v i t a t i o n a l force, D i s the sphere diameter, y g i s the u n i t weight of the sphere and y i s the unit weight of the f l u i d . Figure 2.1 shows the three primary forces on a hypothetical sand p a r t i c l e . When these hydrodynamic forces a c t i n g on a gr a i n of sediment reached a value that, i f increased even s l i g h t l y the grain w i l l move, c r i t i c a l or threshold conditions are said to have been reached. Under these c r i t i c a l conditions the hydrodynamic forces are j u s t balanced by the r e s i s t i n g force of the p a r t i c l e , i . e . , the sum of the moments about the contact point of F^ and F^ , equals zero. White (1940) studied the equilibrium of a p a r t i c l e i n laminar flow and defined a c r i t i c a l shear stress as: F = g i 0 3 <* 8 - y> (2.5) T c r = 0.18 (y ~Y) D tan<t> s (2.6) for turbulent flow, the drag force i s (2.7) 22. Figure 2.1. Primary forces acting on an i n d i v i d u a l sand p a r t i c l e , Wells and Sornesen (1970). where x i s the bed shear s t r e s s , D i s a c h a r a c t e r i s t i c diameter of the o s p a r t i c l e , i s the e f f e c t i v e surface area of the p a r t i c l e exposed to the shear s t r e s s x , x denotes the c r i t i c a l shear st r e s s and C 0 i s a o c r 2 form c o e f f i c i e n t d e f i n i n g the e f f e c t i v e surface area of the p a r t i c l e , that Is the area of the pr o j e c t i o n of the p a r t i c l e on a plane perpendicular to the d i r e c t i o n of the f l u i d flow. For the c r i t i c a l conditions x = x o cr For f u l l y turbulent flow considering the drag and the l i f t force <YS-Y) c r (2.8) — + — cot o> C l C l or cr ^ s ^ ^ s k l + k 2 c o t • (2.9) C L C 4 C2 w h e r e k i = _ _ , ^ = C C C4 2 \ > i s a form c o e f f i c i e n t , and are form c o e f f i c i e n t s r e l a t e d to the e f f e c t i v e surface area of the p a r t i c l e i n the d i r e c t i o n of the drag and the l i f t f o r c e r e s p e c t i v e l y and $ i s the angle of repose of the submerged sediment. If CL =0, equation (2.9) reduces to 24. T The parameter r r — i s the r a t i o of the drag force to the g r a v i t a -c r •Y)I s s t i o n a l force, often referred as Sheilds parameter. Hence, t h i s dimensionless number i s a type of Froude number that i s related to the grain s i z e and the shear v e l o c i t y . A dimensional analysis y i e l d s x p U* " - " (2.11) < V Y > D s ( V Y ) D s hence x - p U* 2 (2.12) c r cr where U* i s the shear v e l o c i t y at the threshold condition, c r ' 2.2 ANALYSIS OF THE SHIELDS CRITERION The Shields curve (Figure 2.2) has been based on experiments i n laboratory flumes with f u l l y developed two-dimensional flows over f l a t sediment bed. For the turbulent boundary layer, a logarithmic v e l o c i t y p r o f i l e had been assumed, and i n defining the c r i t i c a l shear stress values on the bed f o r i n i t i a l p a r t i c l e movement, Sheilds used the temporal mean shear s t r e s s . The c r i t i c a l shear stress was obtained by extrapolating a graph of observed sediment discharge versus shear stress 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 revealed that one of the main reasons f o r the data s c a t t e r on the Shields diagram stems from the d i f f i c u l t y 25. encountered In consistently d e f i n i n g c r i t i c a l flow conditions. Consist-ency i s d i f f i c u l t to achieve because of the random shear s t r e s s exerted 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 to movement under a l o c a l instantaneous shear s t r e s s . I The exerted shear stress i s random because of the turbulence i n the f l u i d flow, and any instantanteous shear s t r e s s i s a function of the temporal mean shear s t r e s s , the f l u i d density and v i s c o s i t y and the flow boundary conditions including 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, weight, and place-ment 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 described 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 (Grass, 1970). There i s the problem of d i f f e r e n t observers having d i f f e r e n t percep-t i o n s as to the onset of sediment motion. Some may predict motion when the very f i r s t grains are i n movement, and others 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 are i n motion. 26. A fur t h e r problem i s caused by the use of d i f f e r e n t wave flumes. Grass explained that because d i f f e r e n t wave flumes have d i f f e r e n t bound-ary conditions, the boundary region turbulence are necessarily d i f f e r e n t and they no longer show s i m i l a r i t y with respect to the average c r i t i c a l shear stress values derived from the Shields curve. 2.3 THRESHOLD MEASUREMENTS Much work has been done on the threshold under u n i d i r e c t i o n a l steady current conditions. The fi n d i n g of Shields (1936) i s u n i v e r s a l l y accepted as the fore-runner i n sediment entrainment studies. Bagnold (1966) has presented a curve s i m i l a r to that of Shields, but presented i n a more convenient form having replaced the p a r t i c l e Reynold's number with the p a r t i c l e diameter (Figure 2.3). Much work has been done i n determining the sediment movement thresh-o l d under o s c i l l a t o r y flow conditions. Komar and M i l l e r (1974) found from the data of previous researchers that the sediment entrainment conditions are s i m i l a r f o r a l l types of o s c i l l a t i o n investigated. These conditions have been attained by use of waves i n flumes, o s c i l l a t o r y water tunnels, and o s c i l l a t o r y p l a t e s . Prototype periods, o r b i t a l diameters, and o r b i t a l v e l o c i t i e s cannot be reproduced i n wave tests i n ordinary tanks because the period i s so r e s t r i c t e d . The other methods are able to generate these prototype conditions, but may not be able to recreate the prototype pressures and convective accelerations as experienced under waves. Komar and M i l l e r (1974) reported that these methods of generating an o s c i l l a t i n g flow over a sediment bed lead to the s i m i l a r conclusion that the shear s t r e s s required to move sediment under waves i s the same as 27. Grain Size, i n Millimeters. Figure 2.3. Modified Shield's diagram after Bagnold (1966). 28. that required to move sediment under a u n i d i r e c t i o n a l current. Komar and M i l l e r used f i v e sets of published data i n analysis of the threshold. Bagnold (1946) and Monohar (1955) used o s c i l l a t o r y plates, Ranee and Warren (1969) used an o s c i l l a t i n g water tunnel, while Horikawa and Watanabe (1967) nad Eagleson, Dean and Peralata (1958) used waves i n a wave flume. The l a s t two sets of data were not used much except to support the conclusions as determined using the f i r s t three data sets. The sediment p a r t i c l e diameters range from 0.009 to 4.8 cm, and t h e i r d e n s i t i e s from 1.52 to 7.9 gm/cm3. The data points obtained range over the e n t i r e spectrum of Reynold's number used i n the Shields diagram, 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 stresses i s possible. Madsen and Grant (1975, 1976) demonstrated that Shields 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 derived from steady u n i d i r e c -t i o n a l flow conditions serves as quite an accurate and general 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 i n o s c i l l a t o r y flow provided that the boundary shear stress i s properly evaluated. Quick et a l . (1985) have studied the onset of sediment motion under combined waves and steady currents. They assumed that the threshold of sediment motion represents a c r i t i c a l l e v e l of shear st r e s s at the bed and they concluded that a s i m i l a r maximum v e l o c i t y condition, measured very close to the bed, causes onset of motion f o r a l l the conditions tested, currents, waves, and combined waves plus currents. The study assumed that the near bed maximum wave and average current v e l o c i t y at one roughness height above the bed combine l i n e a r l y and show to be i n reasonable agreement with Shields threshold c r i t e r i o n f o r steady current; the r e s u l t of that study for the maximum v e l o c i t y c r i t e r i o n i s shown i n Table 2.1 and Figure 2.4. 29. Hs»»»T*4 * Ot/»K UlaAntt Mauaaa 'alKltlu fl«i.t)l i ».«. C u r a t • )m 1 C i r m i F r o •*ss la t I M . 1 4 . t l l far tel. t (fcUMatll l*r M». 1 Maaolaf •UtMtO) 0.3 - 0.»S H . T M.I r . « i . n 1.12 1.11 It.7 }4.4 13.1 t . » 5 - l . « f«.T J t . t s .u 1.J1 3.12 14.1 2I.« 24.4 1.1* - 1.T0 12.4 11.> M . l 2.44 1.14" 2.S1 10.4 n . > 21.1 i . n - s.w 11.4 %4.» S».» I.M 1.41 1.14 11.1 4} . J 1T.J S.eo - S . » M . « O J . i l W.J M . l l.« 4.04 l.*J 24.1 44.3 41.5 Table 2.1. Shear v e l o c i t i e s and maximum near-bed v e l o c i t i e s a f t e r Quick et a l . (1985). I t should be emphasized that some of the previously mentioned r e s u l t s obtained are usually l i m i t e d by the range of experimental condi-t i o n s from which they were derived and are not of the general nature of the Shields 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 steady flow. It i s therefore concluded that the Shields C r i t e r i o n f o r onset of sediment motion i s a good t o o l to determine the l e v e l of c r i t i c a l shear stresses at which p a r t i c l e s s t a r t to move, bearing i n mind a representa-t i v e s i z e of the sediment siz e range. * As shown by Quick et a l . (1985), the Shields C r i t e r i o n can be extended f o r any kind of flow, since the c r i t i c a l shear stresses at threshold conditions w i l l be almost the same whether i t was attained by steady currents alone or waves alone or combined waves and currents. 3 0 . • l e V I . VO0CITT. C" 'S (a) Rang* 0.30 Ie O.BBmm (c) Rang* 1.16 Ie 1.70mm I %.% » t ».t «t v a o c i i i . c n / i g.e •( * t (b)'Range 0.6S to 1.16mm o w • 5. ya • • IS.8 ID.O T.f> « C V I H I 1 1 WlCUIi . Cn-5 CtJ) Range 1.70 to 2.00mm I. i ; / v u o t i u . cx i (a) Range t .00 Ie t.96mm >r n. »c *•«• •>• mnr in. M tO Range 1.16 Ie I.TOmm - Ad»ef»e Current Figure 2.4 Measured threshold v e l o c i t y p r o f i l e s after Quick et a l . (1985). 31. 3. THEORETICAL BACKGROUND 3.1 WAVE AND CURRENT INTERACTION A basic understanding of wave current interation i s fundamental for studying scour under combined waves and currents. Waves and currents interact in two ways. The f i r s t i s the modification of a wave which travels from a zero region to a region where currents exist. The second i s the behaviour of a wave which i s already superimposed on a current; for this situation the resultant i s a simple combination of the wave and the current flow f i e l d s . As discussed by Quick (1983), the major features of the wave and current combination problem can be defined by transforming the situation of a wave on a uniform current into an exactly similar wave on water at rest. This transformation i s achieved simply by subtracting the uniform current as shown i n Figure 3.1. The wave height and length are unaltered, but the wave period i s now T r > on zero current. The velocity f i e l d of the equivalent wave can then be analyzed using standard wave v. o ^///////////////////7/f/////////////////////. Figure 3.1. Definition sketch for a progressive wave train on a steady current. 3 2 . theory. This argument i s confirmed by Kemp and Simons (1982); they concluded that comparison with t h e o r e t i c a l p r o f i l e s suggests that the ad d i t i o n of a current on a wave has l i t t l e e f f e c t on the c l o s e agreement between the measured waves and those of both second- and third-order theory, provided that the wave period i s reasonably adjusted. Figure 3.2 allows further comparison, by p l o t t i n g equivalent p r o f i l e s of the same waves with and without currents. Referring to Figure 3.1, the following equations hold: L C T a a = C T r r (3.1) C a C + U r c (3.2) 10 I M.W.L. H 10 s Figure 3.2. Wave p r o f i l e s with and without current, (a) Rough boundary l a y e r ; (b) smooth boundary layer. , wave with current; , wave alone, Kemp and Simons (1982) 33. L T c a where L i s the wave length, T i s the wave period, C i s the wave c e l e r i t y , and " c i s the steady current v e l o c i t y . The subscripts 'a' and 'r' r e f e r to the absolute and r e l a t i v e coordinate with the current, r e s p e c t i v e l y . Many i n v e s t i g a t o r s have attempted to give a f u l l explanation of how the presence of a wave can a f f e c t the steady current boundary layer which i s important when modelling problems. Most of these studies r e l y on mathematical or t h e o r e t i c a l models f o r the wave current i n t e r a c t i o n and attempt to f i n d t h e o r e t i c a l r e l a t i o n s f o r near bottom v e l o c i t i e s and consequently for shear stress estimation. Grant and Madsen (1979) presented an a n a l y t i c a l theory to describe the combined motion of waves and currents i n the v i c i n i t y of a rough bottom and the associated boundary shear s t r e s s . C h a r a c t e r i s t i c shear v e l o c i t i e s were defined f o r wave and current boundary layer regions by u s i n g a combined wave-current f r i c t i o n f a c t o r 'f ' which was given i n a very complex form. The maximum bottom shear stress | T, | i s g i D y HI 3.X ven as, where |T. | - 7 f p a |U. | 2 (3.4) 1 b.max1 2 cw 1 b 1 oc - 1 + (|U |/|U. | ) 2 + 2(|U |/|U,|) cos 4 (3.5) where |u I i s the magnitude of the steady current v e l o c i t y vector at a 1 a 1h e i g h t a above the bottom; |U^| i s the maximum near-bottom o r b i t a l v e l o c i t y from l i n e a r wave theory, and $ i s the angle made by U with the c a d i r e c t i o n of wave propagation. 34. Brevik and Aas (1980) studied three aspects of wave-current behaviour. F i r s t l y they studied the wave amplitude v a r i a t i o n when a p e r i o d i c wave, i n i t i a l l y on s t i l l water, propagates i n t o a following or an opposing current, fed from below. Secondly they investigated wave attenuation on homogeneous currents or on s t i l l water and the r e s u l t s were used to determine the appropriate bed f r i c t i o n f a c t o r . T h i r d l y , the h o r i z o n t a l f l u i d v e l o c i t y components i n a wave-current system were measured. They gave the wave-current f r i c t i o n f a c t o r f as a function wc of wave a t t e n u a t i o n f a c t o r ~^(H) along the wave flume; i n the l i m i t i n g case f o r pure waves, t h i s f r i c t i o n f a c t o r i s clos e to Jonsson's f a c t o r . Kemp and Simons (1982,1983) concluded that u n i d i r e c t i o n a l turbulent boundary l a y e r i s reduced i n thickness by the superposition of waves propagating with current on both rough and smooth bed sand within 2 roughness heights of the roughbed. The turbulence c h a r a c t e r i s t i c s are dominated by the p e r i o d i c formation of v o r t i c e s at the bed. According to them the shear st r e s s measurements under waves alone are i n a good agree-ment with values estimated using wave f r i c t i o n f a c t o r s fw (Jonsson) c a l c u l a t e d from I f "1/2 + l o g i n (^ f _ 1 / 2 ) = l o g . n (a k " 1 ) and f 4 w ° 10 ^ 4 w "ID o i s w (Kajiura) calculated from fw = 0.37 (a /k )~2/3 w h e r e a i s the maximum om s om wave displaement at bottom estimated using p o t e n t i a l wave theory and k s i s the roughness s i z e . Fredsoe (1984) calculated the mean current v e l o c i t y i n the combined wave-current motion. 3.2 NEAR-BOTTOM SHEAR STRESSES 3.2.1 UNIDIRECTIONAL FLOW SHEAR STRESSES The shear stress at bed under u n i d i r e c t i o n a l flow i s given by the well known formula 3 5 . T = p U* 2 (3.6) o c where T i s the shear stress at bed, p i s the f l u i d density and U* i s the o c shear v e l o c i t y of the free stream. This formula has been i n use f o r long time and i t has been checked against laboratory and f i e l d measurements. The mean current v e l o c i t y near the bed can be calculated using the Manning-Strickler r e l a t i o n s h i p which i s based on a wealth of rough boundary steady flow open channel measurements. This equation can be r e - w r i t t e n i n terms of shear v e l o c i t y U* and the sediment or roughness size D, by using the re l a t i o n s h i p s (Henderson, 1966) n = 0.038 D 1' 3 (d i n meters) (3.7) U* 2 = g R h S o (3.8) Then the Manning equation becomes, \ 1/6 U = 8.4 (-"-) U* (3.9) c D c where U c i s the mean current v e l o c i t y , i s the hydraulic radius, D i s the sediment s i z e , n i s the Manning's roughness c o e f f i c i e n t and S q i s the bed slope. For a given mean v e l o c i t y , flow cross section and sediment s i z e , the equation y i e l d s quite robust estimates of U*. 3.2.2 OSCILLATORY FLOW SHEAR STRESSES The near-bottom shear stress due to o s c i l l a t o r y flow i s expressed as: w \ fw 36. (3.10) T i s the shear s t r e s s a t bottomn due to pure wave motion, f i s a w r » w f r i c t i o n f a c t o r f o r x w, and U"lm i s wave p a r t i c l e v e l o c i t y at the bottom. The f r i c t i o n f a c t o r f i s a f u n c t i o n of the type of flow, laminar or smooth or rough turbulent regimes, and also function of the bed rough-ness. There are many formulas i n the l i t e r a t u r e f o r p r e d i c t i n g the f r i c -t i o n f a c t o r , see f o r instance K a j i u r a (1968), Jonsson (1966) and Kamphius (1975). Under natural conditions, the bed boundary layer below sea waves i s often rough turbulent, see Brevik (1981) and Fredsoe (1984). 3.2.3 COMBINED SHEAR STRESSES AT BED The shear stresses under combined waves and currents can be estimated i n d i f f e r e n t ways, assuming a simple combination of the flow f i e l d s , that i s a l i n e a r a d d i t i o n of the v e l o c i t y f i e l d s , i ncluding both the mean and turbulent components. If the flow i s turbulent, such a l i n e a r a d d i t i o n of flow f i e l d s would r e s u l t i n a nonlinear combination of shear stresses x = T + T + y-2 T T (3.11) wc w c w c where x i s t h e n e a r bed shear s t r e s s due to combined waves and wc c u r r e n t s , T i s t h e s t e a d y c u r r e n t s h e a r s t r e s s , and x i s the c w o s c i l l a t o r y flow shear s t r e s s . This agrees with Quick et a l . (1985) r e s u l t s . The near-bottom shear st r e s s can be estimated using what w i l l be c a l l e d e q u i v a l e n t or c h a r a c t e r i s t i c shear v e l o c i t y U* . The shear n wc stress i n t h i s case w i l l be defined as: 3 7 . T = p (U* )2 (3.12) wc wc Tanaka and Shuta (1984) In t h e i r paper found out that shear stress for combined wave and current i s given as: T = T p f U 2 (3.13) wc 2 K wc lm where f i s a f r i c t i o n f a c t o r under combined wave and current and IL i s wc lm the amplitude of h o r i z o n t a l v e l o c i t y of the o s c i l l a t i n g component just outside the boundary layer. It may at f i r s t appear somewhat curious that Equation (3.13) e x p l i c i t l y contains only the unsteady v e l o c i t y component. However because f c a l c u l a t i o n includes the steady current component, the d e f i n i t i o n i n E q u a t i o n (3.13) i s not unreasonable. The f can be ^ wc ca l c u l a t e d using the flow chart diagram as shown i n Figure 3.3. The terms i n the flow chart are: U i s the maximum value of h o r i z o n t a l v e l o c i t y of the unsteady component, w a i s the h o r i z o n t a l excursion length of a water p a r t i c l e i n o s c i l l a t o r y m moton g i v e n by p o t e n t i a l t h e o r y , z^ i s the depth of flow, Z q i s the roughness length, a i s the angular v e l o c i t y , v i s the kinematic v i s c o -s i t y , T i s the maximum bottom shear s t r e s s , p i s the f l u i d density, R J ' om a i s the Reynold's number and equal U a / v , R i s the Reynold's number and J w m c equal U z,/v, U i s the h o r i z o n t a l v e l o c i t y of the steady component, and c n c f i s the f r i c t i o n c o e f f i c i e n t f o r a wave-current coe x i s t i n g system, wc noting that U i s given by the following formula w Uw = sinh (2, z u/L) ( 3 ' 1 4 ) 3 8 . j hyeriulic cr.aracceris cicf ind bea for= property E" 0.738 f ( -S- J " 0 - 4 0 8 2 Eq.CH) - a • B «s. + c i 2 u V 02 C - - (0.25 + 0.101 ( In 1) V ) J r 1 / 2 - 0.5 ln £ +2 cw U a In -Zrr1 + 0.568 )' }" Ea.(5) 0 " z T • ^ f U OB 2 cw w Figure 3.3 Flow chart diagram for c a l c u l a t i o n of f w c a f t e r Tanaka and Shuto (1984). 3 9 . where H i s the wave height, L i s the wave length, and T i s wave period. The compound s i g n i s p o s i t i v e f o r opposing flow and negative f o r follow-i n g f l o w . For U c/U w = 0.0, i . e . , f o r pure waves, the f r i c t i o n c o e f f i -c i e n t obtained by t h i s method i s almost i d e n t i c a l with that of Jonsson (1966). 3.2.4 SHEAR STRESSES COMPARISON The shear stress c a l c u l a t i o n s due to d i f f e r e n t methods are given f o r three sediment siz e ranges f o r waves plus currents at threshold condition, see Table 3.1, noting that the roughness height i s the upper l i m i t of the sediment siz e range. Sediment Size Range Me thod (mm) No. 0.3 - 0.85 0.85 - 1.16 1.16 - 1.7 1 0.576 0.658 0.814 2 0.702 0.838 1.061 3 0.450 0.630 1.050 1.370 4 0.550 0.720 5 0.380 0.520 1.000 1.570 6 0.620 0.820 Table 3.1 Combined wave and current shear stresses i n (Pa) due to d i f f e r e n t methods for three sediment sizes at threshold conditions. The d i f f e r e n t methods are summarized below: Method 1: using the equivalent shear v e l o c i t y concept as given by Equation (3.12) 40. Method 2: using the f r i c t i o n f a c t o r f given i n Figure 3.3. Method 3: using Shields c r i t e r i o n for threshold conditions. Method 4: shear stresses using Jonsson's shear f a c t o r f given as 10g ( J L _ ) = -0.08 + l o g . . (^2L) (3.15) 4/f- 1 0 4 / r - 1 0 k s w w f o r the unsteady component. Method 5: using Kamphius' shear f a c t o r f given as w - 1 — + lOg ( - i - ) - -0.35 l o g . . ( ^ ) (3.16) 4/F" 1 0 4/f" 3 1 0 k s w w Method 6: using K a j i u r a shear f a c t o r f given as — + l o g 1 Q ( — ) = "0.254 + 10g 1 Q (^) (3.17) 4 / f - 4/F" w w where a, i s the amplitude of h o r i z o n t a l p a r t i c l e motion j u s t outside the lm boundary l a y e r i n o s c i l l a t i n g flow, f i s a 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 flow, and k i s the height of p a r t i c l e f o r equivalent s sand roughness. Refe r r i n g to Table 3.1 i t can be seen that most of the above mentioned methods give shear s t r e s s values close to each other e s p e c i a l l y f o r the small sediment siz e ranges. It i s also seen that Method 1 using the e q u i v a l e n t shear v e l o c i t y U* c gives shear stresses estimation which i s i n good agreement with most of the other methods. 41. 3.3 WAVE THEORIES Wave theory requires an incompressible, i n v i s c i d f l u i d having i r r o -t a t i o n a l flow. The wave t r a i n must be progressive and two-dimensional, t r a v e l l i n g i n water of constant depth as shown i n Figure (3.4). Wove speed, c y Surfoc shown L d e elevotion of t * 0 Figure 3.4 D e f i n i t i o n sketch f o r a progressive wave t r a i n . A choice of the most s u i t a b l e wave theory i s d i f f i c u l t to make. The f i r s t problem i s that 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 theories w i l l adequately reproduce d i f f e r e n t c h a r a t e r i s t i c s of i n t e r e s t . A comparison between theories 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 ge n e r a l i z a t i o n can be made regarding the comparison of these theories f o r other c h a r a c t e r i s t i c s . Another point to consider i s that the most s u i t a b l e wave theory may not be the one that i s simply the most accurate. The governing c r i t e r i o n i n a given engineering a p p l i c a t i o n may be to choose a theory that i s simple and convenient to use, at the cost of some accuracy. Based on the t h e o r e t i c a l comparison of many i n v e s t i g a t o r s , Sarpkaya and Isaacson (1981) concluded that the stokes and cn o i d a l f i f t h order theories are s u f f i c i e n t l y accurate f o r most engineering purposes, and yet are r e l a t i v e l y simple to use. Fenton (1979) recommended that cnoidal 42. theory can be used f o r L/d > 8 and Stokes theory otherwise. For a d e t a i l e d d e s c r i p t i o n of d i f f e r e n t wave theories, the reader i s referred to Sarpkaya and Isaacson (1981), who described the w e l l known and the less well known theories and some computation methods. Quick et a l . (1985) found out that, f o r the range of experimental t e s t s , the near bottom v e l o c i t i e s calculated by Stokes second order theory i s i n a reasonable agreement with the measured v e l o c i t i e s , see Figure 3.5. The r e s u l t s of Stokes second order theory are presented i n Table 3.2. 3.4 COEFFICIENT OF REFLECTION In the wave flume, r e f l e c t e d waves from the energy absorbing end of the flume w i l l have an infl 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 . It i s important to determine the amplitude of t h e i r contribution to the 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 height to incident wave height i s designated by the r e f l e c t i o n c o e f f i c i e n t , K , where where H i s t h e h e i g h t of the r e f l e c t e d wave, which t r a v e l i n the opposite d i r e c t i o n of the incident waves. As shown by Sarpkaya and Isaacson (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 determined simply by trav e r s i n g the flume i n the d i r e c t i o n of wave propagation with a wave probe to measure the maximum and the minimum wave height H and H , r e s p e c t i v e l y . Then K r - HR/H (3.18) max H - H min K = r max (3.19) H + H min max 43. 6 S 5" s I : a.i *>• vCLOCltr, cn/S (a) Rang* 0.30 le 0.66mm •.t »• »« Ktior.itr. oi'S e too tot » t (b) Range 0.86 le 1.16mm s f5 § 5* • 0 Ml VtLOCIlf. £"'S m.e » o wo vUOCIIf. Cn-5 u r ro.f> (e) Rang* 1.16 to 1.70mm (d) Range 1.70 to 2.00mm o P s. J> yflU in. m i <e) Range «.00 1o t.36mm Figure 3.5 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 a f t e r Quick et a l . (1985). 44. 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 Surface 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 Displacement V e r t i c a l P a r t i c l e Displacement 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 Pressure . ffH coshUs) . , „» ***T s i n h ( k d ) s i n ( e ) 3 ffH,iTH,cosh{2ks) / +B" F f ^ s i n h ^ k d ) 5 1 " * 2 * * °j£«j|tanh(kd) 7j=|cos(f3) + 0( ^ i l n T W l 2 + C O S h ( 2 k d n c O S { 2 e ) . H cosh ( k s ) . , - r H f * H l 1 f 1 - 3 c 0 s h ( 2 k s ) 1 - i n f ? f t , V n C ' s i n h ' U d i V 2 s i n h ' ( k d ) 3 s i n l 2 e ) .H,jTH,cosh(2ks) , ^ 4 l ^ ) ? T n T r T k Q T ( c J t ) , H sinh(ks) .» 3H/TTH» sinh(2ks) f 1 cosh(ks)_^, / a N  u — f s i n h ( k d ) c 0 5 ( e ) .3 ffH,irH»cosM2ks) + 4 " ? ( T > s i n h M k d ) C O B ( 2 g ) TTH sinh(ks) . , „ v y = ^ f sTnTTTkcTT .3 ^H,7THvSjj}h(2ks) e._ f, ax 3u 2ir 2H cosh ( k s ) . , a x • S t - T T - s i n h ( k d ) 5 i n ( g ) .3jr^ frH,cosh(2ks) ._ ot> 2ir 2H sinh(ks) ^/^x • o t = ' - T ^ " s i n h ( k d ) C 0 S m 37T2H,7THxsinh(2ks) , . — f 7 " (T )iTHPlW c o s ( 2 B ] ^1 „cosh(ks) ^„, ., P ° - p g y ^ p g h c o s h ( k d ) c o s ( e ) ^3 , f W ^ H x 1 rCQsh(2ks)_1 T ^ ^ P f 0 f l ^ V g H ("L ) s i n h(2kd ) I s i n h ^ k d ) 3 ] c O S ( 2 g ) I / s i n H j I H y [ c o s h ( 2 k s ) - l ] s«y*d Table 3.2 (Modified a f t e r Sarpkaya and Isaacson, 1981). Results Stokes Second Order Theory. and the incident wave height H i s given by 45. H = 1/2 (H max + H min ) (3.20) 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 to be l e s s than 0.07. 3.5 SCALE EFFECTS The improper s c a l i n g of the sediment s i z e i s usually accepted i n a study of t h i s type. Proper sediment s i z e s c a l i n g would lead to a cohe-s i v e model sediment, and a cohesive sediment possesses properties vastly d i f f e r e n t from a non-cohesive sediment. Wave damping due to the side walls w i l l not pose a problem i n t h i s study since the wave properties w i l l be measured at the test section. According to B i j k e r (1967), the boundary layer r e s u l t i n g from a combination of current and waves i s d i r e c t l y proportional to the boundary roughness, and thus the boundary f r i c t i o n f a c t o r . This implies that the boundary layer thickness f o r the model and the prototype w i l l be approximately the same. 3.6 SOME IMPORTANT NUMBERS 3.6.1 FROUDE NUMBER In the case of steady f l o w s , Froude number F^ i s defined by the following equation, U F c (3.21) r •gd 46. where d i s the flow depth, and i s the steady current v e l o c i t y . The P i e r or P i l e Froude number F i s given as: P U F = — (3.22) P / l b where b i s the pier diameter. 3.6.2 REYNOLD'S NUMBER Under u n i d i r e c t i o n a l flow, Reynolds number Re i s given as U r Re = — — (3.23) where r i s a c h a r a c t e r i s t i c length dimension; i n case of open channels r Is equal to the hydraulics radius R^, and v i s the kinematic v i s c o s i t y . For p i e r s or p i l e s Reynold's number R^ i s given i n the form of: U b R = — — (3.24) p v v ' some references give i t as U b (p - p ) R = — - (3.25) P U where ( p g - p ) i s the s e d i m e n t - f l u i d densty d i f f e r e n c e , and \i i s the dynamic v i s c o s i t y 47. v = 2- (3.26) i n case of pure waves U c i n a l l the above equations i s replaced by Uffl. 3.6.3 KEULEGAN-CARPENTER NUMBER For pure waves Keulegan-Carpenter number K c i s given by: U T K = (3.27) c b where T i s the wave period. 48. 4. EXPERIMENTAL SETUP AND PROCEDURE 4.1 EXPERIMENTAL APPARATUS AND EQUIPMENTS The experiments were conducted i n a p l e x i g l a s s flume. The flume i s approximately 23 m long, 0.6 m wide, and 0.7 m deep. A wave paddle i s mounted i n a deeper section 1 m deep at one end of the flume. There i s a l s o an entry tank approximately 4 m long, 2 m wide, and 1.3 m deep which i s used as a s t i l l i n g basin to dampen waves. Currents can be generated using a pump and c i r c u l a t i n g system. The t e s t section consists of a 4 m long bed of sediment material and 21 cm deep. It was convenient to use concrete blocks f o r most of the upstream part of the flume to save material as seen i n Figure 4.1. V e l o c i t y measurements were taken with a p r o p e l l e r type OTT current meter. This current measuring device i s shown i n Figure 4.2. The wave generator i s an o s c i l l a t i n g pendulum type whose stroke and consequently wave height can be varied by adjusting the e c c e n t r i c i t y of the paddle arm on the flywheel. The period i s c o n t r o l l e d by a v a r i a b l e speed d r i v e . The wave height was measured by a capacitance wave gauge connected to a Hewlett Packard dual channel recorder (Model No. 17 501A). Figure 4.3 shows the wave recorder. 4.2 EXPERIMENTAL PROCEUDRE A t o t a l of seventy experiments were conducted f o r t h i s study. The experiments u t i l i z e d f i v e c y l i n d r i c a l p i l e s of 1.27, 2.54, 5.08, 8.26, and 11.43 cm i n diameter as shown i n Figure 4.4. Figure 4.1 Experimental equipment, schematic diagram a f t e r Quick (1983). 53. Sand was the sediment material f o r t h i s study. These sediments were sieved i n t o a narrow siz e range so that f o r a p a r t i c u l a r experiment onset of sediment motion could be associated with a f a i r l y s p e c i f i c sediment s i z e , these sediment s i z e ranges are 0.3-0.85 mm, 0.85-1.16 mm, and 1.16-1.70 mm. For a p a r t i c u l a r t e s t s e r i e s , sediment of c e r t a i n s i z e was spread uniformly i n t o a f l a t bed condition. A l l te s t s were c a r r i e d out with the same average water depth of 35 cm. The experiments can be c l a s s i f i e d i n t o three groups, 15 runs under currents alone, 12 runs with waves alone, and 43 runs under combined waves and currents. For each group of tes t s the d i f f e r e n t p i l e s i z e s and sediment size ranges were u t i l i z e d at d i f f e r e n t flow states, i . e . below, at, and above threshold conditions of each sediment s i z e . For the case of waves alone, the wave period was kept constant at 1.6 seconds and hence the wave length was 2.69 meters. For the case of combined waves and currents, the current v e l o c i t y i s f i x e d at a c e r t a i n value and the wave period i s adjusted so that the r e l a t i v e wave period i s kept the same (1.6 seconds) so that the wave length i s unaltered. The wave height and the current v e l o c i t y were adjusted for each test to the required value. At the onset of each t e s t , the tank was slowly f i l l e d to avoid d i s t u r b i n g the sediment. A f t e r a s u f f i c i e n t amount of water had accumu-l a t e d i n the tank, the flow rate through the tank was adjusted to the appropriate s e t t i n g . The wave generator then started and the period s e t t i n g and stroke length were adjusted u n t i l the desired wave length and wave height were attained. Once t h i s was completed, everything was turned o f f and the sand bed was r e l e v e l l e d , before the t e s t was started. Scour depth was recorded at various time increments a f t e r the t e s t was under way, the increments becoming longer as the tes t continued. Each 54. t e s t was conducted u n t i l there appeared to be no further increase i n scour depth. For the wave-current t e s t s , the experiments ran from 18-26 hours, whereas f o r the wave only and current alone t e s t s , the running time was l e s s . The maximum scour depth i s the maximum depth reached during the test run. 55. 5. PRESENTATION AND DISCUSSION OF RESULTS A s e r i e s of tests were performed using d i f f e r e n t c y l i n d e r and sediment s i z e s , and under d i f f e r e n t flow conditions i n the approaching flow, i . e . , below, at, and above threshold of motion. These experiments can be grouped i n t o three sets of t e s t s , currents alone, waves alone, and waves p l u s c u r r e n t s , and are g i v e n the l e t t e r s C, W, and WC, r e s p e c t i v e l y . Each run i n a set was given a number following the set l e t t e r , f o r example, WC12 r e f e r s to the run number 12 i n the set of combined waves and currents. The reader should note that both water depth and wave period (hence wave length) are kept constant throughout a l l the experiments. The aim of the t e s t i n g was to in v e s t i g a t e the ultimate maximum scour which could occur and t h i s was found to be when the approach flow conditions reached c r i t i c a l s tress f o r onset of motion of sediment p a r t i c l e s , i r r e g a r d l e s s of water depth or wave length. The f i r s t set of tests was under the a c t i o n of currents alone, runs C l through C15, Table 5.1 from which i t can be seen that the maximum scour occurs at threshold conditions i n the approaching flow which agrees with the r e s u l t s of Chabert and Engeldinger (1956). The scour development with time f o r some of these runs i s shown i n Figures 5.1 and 5.2 i n the form of r e l a t i v e scour (scour depth over c y l i n d e r diameter) versus time. These two fi g u r e s show a decrease i n scour depth as the flow conditions i n the approaching flow exceed the threshold state which confirms the findings of Chabert and Shen (1966). In order to check the t e s t r e s u l t s f o r steady current alone, comparison was made with some w e l l known formulas which already e x i s t . Run No. Sediment Size D Roughness k s Cylinder Dia. b Water Depth d St eady Current V e l o c i t y u c (cm/sec) Max. Scour S Relative Scour S/b F r State of Approaching Flow u * Bed Shear R e x l 0 5 (mm) (mm) (cm) (cm) (cm) (cm/sec) T o (Pa) C l 1.16-1.70 1.70 11.43 35 27.48 4.50 0.39 0.148 Below Threshold 0.0153 0.235 0.444 C2 1.16-1.70 1.70 11.43 35 30.28 6.50 0.57 0.163 Below Threshold 0.0169 0.285 0.489 C3 1.16-1.70 1.70 11.43 35 36.44 14.00 1.22 0.197 Below Threshold 0.0203 0.412 0.589 C4 1.16-1.70 1.70 11.43 35 44.60 17.00 1.49 0.241 Below Threshold 0.0249 0.618 0.720 C5 1.16-1.70 1.70 11.43 35 47.73 19.50 1.71 0.258 At Threshold 0.0266 0.708 0.771 C6 1.16-1.70 1.70 11.43 35 53.80 19.00 1.66 0.290 Above Threshold 0.0330 0.899 0.856 C7 1.16-1.70 1.70 11.43 35 59.20 18.50 1.62 0.320 Above Threshold 0.0300 1.100 0.956 C8 1.16-1.70 1.70 2.54 35 41.38 4.25 1.67 0.223 Below Threshold 0.0231 0.532 0.662 C9 1.16-1.70 1.70 2.54 35 47.73 8.10 3.18 0.258 At Threshold 0.0266 0.708 0.771 CIO 1.16-1.70 1.70 2.54 35 53.80 6.80 2.68 0.290 Above Threshold 0.0300 0.899 0.869 C l l 0.85-1.16 1.16 11.43 35 41.00 22.20 1.94 0.221 At Threshold 0.021 0.460 0.662 C12 0.85-1.16 1.16 8.26 35 41.00 18.10 2.19 0.221 At Threshold 0.021 0.460 0.662 C13 0.85-1.16 1.16 5.08 35 41.00 13.10 2.57 0.221 At Threshold 0.021 0.460 0.662 C14 0.85-1.16 1.16 2.54 35 41.00 9.30 3.66 0.221 At Threshold 0.021 0.460 0.662 C15 0.85-1.16 1.16 1.27 35 41.00 4.80 3.78 0.221 At Threshold 0.021 0.460 0.662 Table 5.1 Experimental results of steady currents alone. Haxiwun r e l a t i v e scour S/b 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 ' '. t i J ' , 1 , ' | , ' ' I I I I I I I I I I I I l _ X \ \ \ \ ) JO JO \ / 3 3 i ; i \ JO JO JO JO JO w \ \ 3 3 o r> o o o —i co cn -t* co o o \ | 1 i i i I I I I I I I I I I I I I—I I I I I | i I J I I I I 11II J I I I 1111 ' • I I 1111 I I—I I I 111 Run C1 Run C2 Run C3 — Run C4 Run C5 \ ~~ to -- - - Run C6 Run C7 y y T 3 M l " 5 7 104 10' I i I • I • 'I 3 5 7 10* I ' I ' I "I 3 5 7 10' — i 1 i | i | 11| 3 S 7 10* Time (min.) i ' i > i " i 3 s 7 io» Figure 5.2. Relative scour versus time under steady currents alone. 5 9 . The comparison i s shown i n Tables 5.2 through 5.5 f o r the three sediment s i z e s tested under threshold conditions and f o r d i f f e r e n t c y l i n d r i c a l s i z e s . It should be noted that the scour depths reported were the maximum possible depths attained during t e s t i n g . The measured values are i n good agreement with most of the estimated values which i s another v e r i f i c a t i o n f o r these formulas. Hence i t can be argued that steady current scour depths could form a base or a reference f o r comparison with scour depths under waves or combined waves and currents. From T a b l e 5.1 i t i s seen t h a t Froude number F i s small and l e s s r than unity, which makes the flow a s u b c r i t i c a l flow. The flow Reynolds number Re i s i n the order of 10 5, accordingly the flow i s f u l l y rough turbulent flow. The second set of tes t s was performed under pure waves, runs Wl through W12. The r s u i t s of these runs are presented i n Table 5.6, as seen from the table, the depth of scour f o r the same p i l e s i z e under the same conditions i n the approaching flow i s much l e s s than the scour depth under the ac t i o n of steady currents. The reason f o r t h i s may be that, although sediment p a r t i c l e s are dislodged due to wave turbulence i n the v i c i n i t y of the cylinder, t h i s turbulence i s i n s u f f i c i e n t to transport the material away because of the r e v e r s a l flow and the o r b i t a l type of h o r i z o n t a l v e l o c i t y of the wave. The scour development with time i s shown i n Figures 5.3 and 5.4. Figure 5.5 i s a p l o t of r e l a t i v e scour depth versus p i l e diameter under the same conditions, from which i t can be seen that the r e l a t i v e scour under bigger diameters i s l e s s than the r e l a t i v e scour produced by the smaller diameters. It follows that the s m a l l e r the K £, Keulegan-Carpenter number, the bigger i s the scour hole depth. Foraula No. Cyl . Size (cm) (1.1) (1.3) (1.6) (1.7) (1.8) Measured Values 1.27 3.97 3.40 4.39 5.90 4.64 4.80 2.54 6.68 5.39 6.75 9.23 7.53 9.30 5.08 11.24 8.56 10.36 14.44 12.24 13.10 8.26 16.17 11.29 14.06 19.65 17.20 18.10 11.43 20.64 14.69 17.14 22.61 21.59 22.20 Table 5.2 Measured and estimated maximum scour depth S (cm) at threshold conditions i n the approaching flow for sediment s i z e range 0.85-1.16mm under steady currents alone. Formula No. Cyl . Size (cm) (1.1) (1-3) (1.6) (1.7) (1.8) Measured Values 1.27 3.13 2.67 3.46 4.65 3.65 3.78 2.54 2.36 2.12 2.66 3.63 2.97 3.66 5.08 2.21 1.69 2.04 2.84 2.41 2.57 8.26 1.96 1.43 1.70 2.38 2.08 2.19 11.43 1.81 1.29 1.50 1.99 1.89 1.94 Table 5.3 Measured and estimated maximum r e l a t i v e scour at threshold conditions i n the approaching flow for sediment siz e range 0.85-1.16mm under steady currents alone. Formula No. C y l . Size (cm) (1.1) (1.3) (1.6) (1.7) (1.8) Measured Values 2.54 6.68 7.63 7.42 7.19 7.53 7.50 11.43 20.64 16.26 18.82 18.98 21.59 19.50 Table 5.4 Measured and estimated maximum scour depth i n (cm) at threshold conditions i n the approaching flow for sediment siz e range 1.16-1.70mm under steady currents alone. Formula No. C y l . Size (cm) (1-1) (1-3) (1.6) (1-7) (1.8) Measured Values 2.54 2.63 3.01 3.00 2.83 2.97 2.95 11.43 1.81 1.42 1.65 1.66 1.89 1.71 Table 5.5 Measured and estimated maximum r e l a t i v e scour at threshold conditions i n the approaching flow for sediment size range 1.16-1.70mm under steady currents alone. Run No. Sediment Size D (mm) (mm) Cylinder Dia. b (cm) State of Approaching Flow Max. Scour S (cm) Relative Maximum Scour S/b H (cm) Uw (cm/s) a l m (cm) R e x 105 K c W3 1.16-1.70 1.70 11.43 At Threshold 2.30 0.20 13.00 31.96 7.13 0.288 4.47 W4 1.16-1.70 1.70 11.43 Above Threshold 1.68 0.51 14.60 36.47 8.01 0.292 5.11 W5 1.16-1.70 1.70 2.54 Below Threshold 1.80 0.71 10.50 25.17 5.76 0.145 15.86 W6 1.16-1.70 1.70 2.54 At Threshold 3.10 1.22 13.00 31.96 7.13 0.288 20.13 W7 1.16-1.70 1.70 2.54 Above Threshold 3.00 1.18 14.00 34.76 7.70 0.267 21.90 W8 0.85-1.16 1.16 11.43 At Threshold 4.50 0.39 11.80 28.70 6.50 0.187 4.02 W9 0.85-1.16 1.16 8.26 At Threshold 4.50 0.53 11.80 28.70 6.50 0.187 5.56 W10 0.85-1.16 1.16 5.08 At Threshold 4.30 0.84 11.80 28.70 6.50 0.187 9.04 Wll 0.85-1.16 1.16 2.54 At Threshold 3.70 1.46 11.80 28.70 6.50 0.187 18.08 W12 0.85-1.16 1.16 1.27 At Threshold 3.50 2.80 11.80 28.70 6.50 0.187 36.16 Table 5.6 Experimental results of o s c i l l a t o r y waves alone. CM eg" I I I I i I i I I t t l l l l l l I I I I I I I I I I I I Run U3 Run U4 Run U5 - - - Run U6 Run U7 cn -to. l_ —* a • 8 3. OJ > to — 3 -J 5 ' CO 1 as ; 1 1 I' 24' CM T—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—r 100 200 300 400 500 600 700 800 900 1000 Time (min.) -1—1—1—1—1—1—1—1—1—1—1—1—1—r 1100 1200 1300 1400 1500 1600 1700 1800 Figure 5.3. Relative scour versus time under pure waves. Run U3 Run U4 Run U5 Run U6 Run U7 t 10 • T 1 I I I I I I | 3 5 7 10* | i | I | I 11 1 1 i I i I "I 3 5 7 10' 3 5 7 10* Time (min.) T — | I i I I I '| 3 5 7 10» T 1—r • | i | 11 3 5 7 104 Figure 5.4. Relative scour versus time under pure waves. 65. o o O Csi' D • 0_| , , r 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Cylinder diameter (cm) Figure 5.5. Maximum r e l a t i v e scour versus cylinder diameter for O s c i l l a t o r y flow at threshold conditions; sediment siz e 0.85-1.16 66. Further s e r i e s of te s t s were c a r r i e d out to inv e s t i g a t e how scour i s produced under the a c t i o n of combined waves and currents. In p a r t i c u l a r , the conditions were investigated which produce the maximum scour depth. Th i s t h i r d set of tests under the combined waves and currents was l a b e l l e d runs WC1 through WC43, and the r e s u l t s are tabulated i n Table 5.7. R e l a t i v e maximum scour versus time p l o t s are shown i n Figures 5.6 and 5.7, both of them show a development with time s i m i l a r to that of u n i d i r e c t i o n a l flows (see Figures 5.1 and 5.2). Referring to Table 5.7, i t can be seen that runs WC3 and WC6 are under s i m i l a r sediment threshold conditions, but the r e s u l t i n g scour depths are d i f f e r e n t , and the same i s true f o r runs WC9 and WC11. This d i f f e r e n c e i n scour depth can be explained because, although a threshold s t a t e condition was attained f o r a l l the runs, the contribution of the waves and currents to the c r i t i c a l v e l o c i t y at bed i n each case i s d i f f e r e n t . For example i n run WC3 the wave height was 10.5 cm and the steady current v e l o c i t y was 23 cm/sec, whereas i n run WC6 the wave height was 9.4 and the steady current v e l o c i t y was 30.28 cm/sec, f o r both of the runs the maximum v e l o c i t y at bed due to combined wave and current was 32.8 cm/sec. From the f i r s t few tests a conclusion was reached, that maximum scour depths under combined waves and currents occurred when threshold sediment conditions existed i n the approaching flow, and therefore most of the r e s t of t h i s set was performed under t h i s threshold condition. An attempt was made to i n v e s t i g a t e the influence on scour of flow parameters, such as c y l i n d e r s i z e , sediment s i z e , and the proportion of wave motion to current v e l o c i t y . Runs WC29 through WC43, C l l through Run No. Sediment Size D (mm) k s (mm) Cylinder Dia. b (cm) State of Approaching Flow Max. Scour S (cm) Maximum Relative Sc our S/b U c (cm/sec) H (cm) Uw (cm/s) ^wc (cm/s) Equiv. U* wc (cm/sec) Twc Pa WC1 1.16-1.70 1.70 11.43 Below Threshold 8.00 0.70 23.00 6.00 13.72 24.00 2.08 0.434 WC2 1.16-1.70 1.70 11.43 Below Threshold 10.70 0.94 23.00 8.50 19.96 30.24 2.61 0.683 WC3 1.16-1.70 1.70 11.43 At Threshold 15.00 1.31 23.00 10.50 25.17 32.80 2.85 0.814 WC4 1.16-1.70 1.70 11.43 Above Threshold 14.00 1.42 23.00 12.00 29.21 40.00 3.48 1.210 WC5 1.16-1.70 1.70 11.43 Below Threshold 16.20 1.51 30.28 7.20 16.68 31.04 2.70 0.729 WC6 1.16-1.70 1.70 11.43 At Threshold 17.30 1.40 30.28 9.40 22.80 32.80 2.85 0.814 WC7 1.16-1.70 1.70 11.43 Above Threshold 16.00 1.22 30.28 11.80 28.66 43.03 3.74 1.400 WC8 1.16-1.70 1.70 2.54 Below Threshold 5.00 1.97 27.76 7.50 17.43 30.60 2.66 0.708 WC9 1.16-1.70 1.70 2.54 At Threshold 5.00 1.97 27.76 9.40 22.30 32.80 2.85 1.040 WC10 1.16-1.70 1.70 2.54 Above Threshold 5.00 1.97 27.76 10.00 23.85 37.00 3.22 0.814 WC11 1.16-1.70 1.70 2.54 At Threshold 3.50 1.38 13.70 10.50 25.17 32.80 2.85 0.814 WC12 1.16-1.70 1.70 2.54 At Threshold 5.00 1.97 22.30 9.10 21.50 32.80 2.85 0.814 WC13 1.16-1.70 1.70 2.54 Below Threshold 7.10 2.80 31.12 6.50 14.95 32.80 2.85 0.814 WC14 1.16-1.70 1.70 11.43 At Threshold 17.40 1.52 31.12 6.40 14.70 32.80 2.85 0.814 WC15 1.16-1.70 1.70 1.27 At Threshold 4.00 3.14 31.12 7.70 17.93 32.80 2.85 0.814 Table 5.7. Experimental results of combined waves and currents. Run No. Sediment Size D (mm) k s (mm) Cylinder Dia. b (cm) State of Approaching Flow Max. Scour S (cm) Maximum Relative Scour S/b U c (cm/sec) H (cm) Uw (cm/s) ^wc (cm/s) Equiv. U* wc (cm/sec) T wc Pa WC16 1.16-1.70 1.70 2.54 At Threshold 7.10 2.80 31.12 7.70 17.93 32.80 2.85 0.814 WC17 1.16-1.70 1.70 5.08 At Threshold 10.50 2.10 31.12 7.70 17.93 32.80 2.85 0.814 WC18 1.16-1.70 1.70 8.26 At Threshold 15.50 1.88 31.12 7.70 17.93 32.80 2.85 0.814 WC19 0.85-1.16 1.16 1.27 At Threshold 4.50 3.54 31.12 6.80 15.68 29.50 2.57 0.658 WC20 0.85-1.16 1.16 2.54 At Threshold 8.00 3.51 31.12 6.80 15.68 29.50 2.57 0.658 WC21 0.85-1.16 1.16 5.08 At Threshold 11.00 2.17 31.12 6.80 15.68 29.50 2.57 0.658 WC22 0.85-1.16 1.16 8.26 At Threshold 16.20 1.96 31.12 6.80 15.68 29.50 2.57 0.658 WC23 0.85-1.16 1.16 11.43 At Threshold 18.30 1.60 31.12 6.80 15.68 29.50 2.57 0.658 WC24 0.30-0.85 0.85 11.43 At Threshold 5.00 3.94 28.60 6.75 15.56 27.60 2.40 0.576 WC25 0.30-0.08 0.85 2.54 At Threshold 8.50 3.35 28.60 6.75 15.56 27.60 2.40 0.576 WC26 0.30-0.08 0.85 5.08 At Threshold 11.80 2.32 28.60 6.75 15.56 27.60 2.40 0.576 WC27 0.30-0.08 0.85 8.26 At Threshold 17.00 2.10 28.60 6.75 15.56 27.60 2.40 0.576 WC28 0.30-0.08 0.85 11.43 At Threshold 19.20 1.66 28.60 6.75 15.56 27.60 2.40 0.576 WC29 0.85-1.16 1.16 11.43 At Threshold 19.30 1.69 34.20 3.50 7.60 29.50 2.57 0.658 WC30 0.85-1.16 1.16 8.26 At Threshold 15.20 1.84 34.20 3.50 7.60 29.50 2.57 0.658 Table 5.7. Continued Run No. Sediment Size D (mm) (mm) Cylinder Dia. b (cm) State of Approaching Flow Max. Scour S (cm) Maximum Relative Scour S/b U c (cm/sec) H (cm) Uw (cm/s) ^wc (cm/s) Equiv. U* wc (cm/sec) T WC Pa WC31 0.85-1.16 1.16 5.08 At Threshold 11.10 2.20 34.20 3.50 7.60 29.50 2.57 0.658 WC32 0.85-1.16 1.16 2.54 At Threshold 7.20 2.83 34.20 3.50 7.60 29.50 2.57 0.658 WC33 0.85-1.16 1.16 1.27 At Threshold 4.10 3.30 34.20 3.50 7.60 29.50 2.57 0.658 WC34 0.85-1.16 1.16 11.43 At Threshold 14.50 1.27 28.60 6.00 13.70 29.50 2.57 0.658 WC35 0.85-1.16 1.16 8.26 At Threshold 10.30 1.25 28.60 6.00 13.70 29.50 2.57 0.658 WC36 0.85-1.16 1.16 5.08 At Threshold 8.50 1.67 28.60 6.00 13.70 29.50 2.57 0.658 WC37 0.85-1.16 1.16 2.54 At Threshold 5.90 2.30 28.60 6.00 13.70 29.50 2.57 0.658 WC38 0.85-1.16 1.16 1.27 At Threshold 4.00 3.15 28.60 6.00 13.70 29.50 2.57 0.658 WC39 0.85-1.16 1.16 11.43 At Threshold 9.80 0.86 17.40 9.00 21.20 29.50 2.57 0.658 WC40 0.85-1.16 1.16 8.26 At Threshold 8.10 0.98 17.40 9.00 21.20 29.50 2.57 0.658 WC41 0.85-1.16 1.16 5.08 At Threshold 5.80 1.14 17.40 9.00 21.20 29.50 2.57 0.658 WC42 0.85-1.16 1.16 2.54 At Threshold 4.70 1.85 17.40 9.00 21.20 29.50 2.57 0.658 WC43 0.85-1.16 1.16 1.27 At Threshold 3.70 2.90 17.40 9.00 21.20 29.50 2.57 0.658 Table 5.7. Continued 1 1 1 1 Run I I I l I UC1 Run UC2 3 " Run UC3 - - - Run UC4 3- Run UC5 - - - Run UC6 \ CO - Run UC7 to_ a J ' ' ' ' ' ' « ' I ' ' I ' I I I I I L I I I I T — i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i— i—r 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Time (min.) Figure 5.6. Relative scour versus time under combined waves and currents. 111 ' I I I I 11 • • • I I I I 111 J I I I 1111 Run UC1 Run UC2 Run UC3 Run UC4 Run UC5 - - - Run UC6 — Run UC7 I I I 11"I 3 5 7 10» I ' I ' M 3 5 7 10' to-T 3 > I i I 111 5 7 10' T 3 I ' I ' M 5 7 10' — i 1 i | I | "I 3 5 7 10* Time (min.) Figure 5.7. Relative scour versus time under combined waves and currents. 72. C15, and W8 through W12 were at threshold conditions f o r a sediment s i z e range of 0.85-1.16 mm using the f i v e c y l i n d e r s i z e s . These runs can be divi d e d i n t o f i v e groups as seen i n Table 5.8 according to the contribu-t i o n of both the wave and current flow components to the t o t a l c r i t i c a l v e l o c i t y at the bed. Percentage of Wave i n Percentage of Current i n Run No. Threshold V e l o c i t y (%) Threshold V e l o c i t y (%) C l l - C15 0 100 WC29 - WC33 25 75 WC34 - WC39 50 50 WC39 - WC43 75 25 W8 - W12 100 0 Table 5.8 Contribution of waves and currents i n threshold v e l o c i t y (%) The r e s u l t s of these tests are presented i n Figures 5.8 through 5.11, each p l o t contains f i v e curves, a curve f o r each c y l i n d e r s i z e . The p l o t of the wave percentage i n the threshold v e l o c i t y of the combined wave and current versus the maximum scour i s shown i n Figure 5.8 and versus the maximum r e l a t i v e scour i s shown i n Figure 5.9. For threshold or c r i t i c a l v e l o c i t y f o r onset of motion i n the approach flow, Figure 5.8 shows that the maximum scour depth Is varying almost l i n e a r l y according to the co n t r i b u t i o n of both waves and currents. I t also shows that the more i s the current percentage i n the c r i t i c a l combined flow v e l o c i t y , the more i s the scour depth. At the l i m i t , scour reaches the steady current value and t h i s i s true f o r a l l the cy l i n d e r s i z e s tested. It should be noted that l a r g e r c y l i n d e r sizes are more s e n s i t i v e to changes i n the c r i t i c a l flow v e l o c i t y than f o r the smaller s i z e s . 73. LEGEND • = Cyl. Diam. 1.27 cm o = Cyl. Diam. 2.54 cm A = Cyl. Diam. 5.08 cm o = Cyl. Diam. 8.26 cm v = Cyl. Diam. 11.43 cm o.. O a • a i i 1 1 0.0 25.0 50.0 75.0 100.0 Wave contribution (%) to the threshold velocity Figure 5.8. Maximum scour versus combined waves and current threshold v e l o c i t y , sediment s i z e 0.30 - 0.85. 74. q o CO O O cn CD > o q CD ^ 3 E X D O 6 _ l • o A O Q TO. "A. . O.. "V.. A. LEGEND Cyl. Diam. 1.27 cm Cyl. Diam. 2.54 cm Cyl. Diam. 5.08 cm Cyl. Diam. 8.26 cm Cyl. Diam. 11.43 cm •-o 0.0 25.0 50.0 75.0 Wave contribution (%) to the threshold velocity 100.0 Figure 5.9. Maximum r e l a t i v e scour versus combined waves and currents threshold v e l o c i t y ; sediment siz e 0.30 - 0.85 mm. 75. Figure 5.10 shows the change of maximum scour depth with respect to the c y l i n d e r s i z e f o r d i f f e r e n t flow combinations. In t h i s f i g u r e the scour depth under the a c t i o n of pure currents increases considerably as the c y l i n d e r s i z e increases, while f o r pure waves the change i n scour depth as the c y l i n d e r s i z e v a r i e s i s not very great. Figure 5.11 shows that maximum r e l a t i v e scour f o r smaller c y l i n d e r s i z e s i s greater than the r e l a t i v e scour of bigger diameters, and t h i s i s v a l i d for the d i f f e r e n t flow cases tested. Figure 5.12 and 5.13 show scour r e s u l t s when contributions to threshold conditions are 25% waves and 75% current. The maximum scour depth i s plotted against c y l i n d e r diameter f o r the three sediment s i z e ranges tested. Although the change i s not so high, the smaller sediment size ranges are scoured more than the bigger sediment s i z e s . From the v i s u a l observations i t was found that the pattern of the scour hole i s very s i m i l a r f o r a l l the flow cases tested; f o r most of the t e s t s the maximum scour depth was at the f r o n t of the p i l e f a c i n g the upstream side of the flow. A t y p i c a l scour hole under combined wave and current i s shown i n Figures 5.14 and 5.15. It was noted that the scour around the c y l i n d e r under the combination of waves and currents takes place because sediment p a r t i c l e s are dislodged or entrained by the wave and the current turbulence and then transported by the flow current. The scour mechanism under the a c t i o n of combined waves and currents seems to be s i m i l a r to the mechanism under currents or waves alone. E r o s i o n i s r e s t r i c t e d to a narrow width adjacent to the p i l e and at the bottom of the scour hole. The r e s t of scour hole i s at the angle of repose and material slumps i n t o the hole as the scour proceeds. For the 76. LEGEND • = 100% Wave o = 75 % Wave A = 50 % Wave o = 25 % Wave v = 0 % Wave v''' .o • o-1 1 1 1 1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Cylinder diameter (cm) Figure 5.10. Maximum scour versus cylinder diameter at threshold conditions of combined waves and currents; sediment s i z e 0.3 -0.85 mm. 77. Q LEGEND •Q. • = 0 % Wave o = 25 % Wave A = 50 % Wave o = 75 %. Wave v = 100 % Wave D. A. G-Cylinder diameter (cm) v V 1 1 1 1 I 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Figure 5.11. Maximum r e l a t i v e scour versus cylinder diameter at threshold conditions of combined waves and currents; sediment siz e 0.3 - 0.85 mm. 78. LEGEND • = Sed. Size 1.16- 1.70 mm . A o = Sed. Size 0 . 8 5 - 1.16 mm = Sed. Size 0 . 3 0 - 0.85 mm . - A " n ' A ' ' ••' a'.--' A . g' —1 1 n 1 i 1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Cylinder diameter (cm) Figure 5.12. Maximum scour depth versus cylinder diameter for three sediment sizes under threshold conditions of combined waves and currents; - 25% wave and 75% current. 79. LEGEND iize 1.16-o = Sed. Size 0 . 8 5 - 1.16 mm Q A. • = Sed. Si 1.16- 1.70 mm Q \ \ \ A = Sed. Size 0 .30 -0 .85 mm O. " Q . " A . 1 1 1 1 1 1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Cylinder diameter (cm) Figure 5.13. Maximum r e l a t i v e scour versus cylinder diameter for three sediment sizes at threshold of combined waves and currents; 25% wave and 75% current. 82. case of waves alone or waves plus currents, the sand bed might r i p p l e as shown i n Figures 5.16 and 5.17. A comparison of the combined wave and current t e s t r e s u l t s with r e s u l t s from other studies has not been possible, p a r t l y because of the lack of such a study, and p a r t l y because even with the few studies that are a v a i l a b l e , the comparison cannot be made because the problem was treated from another viewpoint. For example, previous studies i n v e s t i -gated the dependence of scour on the flow parameters such as wavelength, wave height, water depth, etc., and at flow v e l o c i t i e s l e s s than the c r i t i c a l v e l o c i t y . Also a s i n g l e p i l e s i z e and a s i n g l e sediment s i z e were u t i l i z e d i n these other studies. In the present study the emphasis was to determine the maximum pos s i b l e scour, namely when the sandbed was a t onset of motion, and t h i s condition was studied using several p i l e and sediment s i z e s . It i s worthwhile to note that no d i s t i n c t formulas f o r maximum scour p r e d i c t i o n were reached due to the l i m i t e d number of tests and l i m i t e d studies i n t h i s f i e l d . However a rough estimate of the amount of maximum scour around c y l i n d r i c a l p i l e s can be found using these study r e s u l t s . The bed material c r i t i c a l v e l o c i t y f o r onset of motion can be known i f the material properties are known, then the flow v e l o c i t y at bed can be c a l c u l a t e d provided that flow parameters are defined, hence the contribu-t i o n of each flow component i s determined. Knowing the structure si z e the maximum possible scour around the structure can be estimated using Figures 5.8 through 5.13. 83. Figure 5.17. Ty p i c a l r i p p l e pattern under combined waves and currents. 84. Figure 5.16. T y p i c a l ripple-pattern under combined waves and currents. 6. CONCLUSIONS AND RECOMMENDATIONS 85. 6.1 CONCLUSIONS The purpose of t h i s study i s to compare the maximum possible scour around c y l i n d r i c a l p i l e s under the a c t i o n of combined waves and currents with the scour produced by pure waves and pure currents. The following conclusions were reached: 1. Scour around p i l e s s t a r t s when the approach v e l o c i t y i s as low as 0.3-0.4 of the c r i t i c a l v e l o c i t y . This i s because the flow i s accelerated by the p i l e . 2. The equilbrium scour conditions f o r pure currents are reached i n l e s s time than the equilibrium conditions f o r the combined waves and currents. However maximum scour under pure wave act i o n i s reached i n much l e s s time than f o r currents alone or waves and currents. 3. For a l l the cases tested the maximum possible scour was attained when approach flow conditions upstream of the p i l e reached the c r i t i c a l stress for onset of motion for sediment p a r t i c l e s . 4. If threshold conditions are exceeded, i . e . , the flow v e l o c i t y i s greater than the c r i t i c a l v e l o c i t y f o r onset of motion, the equilibrium scour depth i s l e s s than the maximum scour depths under threshold conditions. The graphs of scour development as a function of time show that scour increases to a maximum and then decreases s l i g h t l y as time increases. This decrease could possibly be caused by s e l e c t i v e armouring of the scour hole with coarser ma t e r i a l . 86. 5. The development of combined wave and current scour with time i s s i m i l a r to that of u n i d i r e c t i o n a l flow, except f o r the rate of development as noted i n (2). 6. The pattern of scour hole i s quite s i m i l a r under the three flow cases tested. 7. For a l l the cases, the maximum scour depth was adjacent to the leading edge of the p i l e . 8. For the case of combined waves and currents at threshold conditions, the maximum scour i s dependent on the r e l a t i v e c o n t r i b u t i o n of both steady and unsteady components of flow to the threshold v e l o c i t y of the sediment material. The higher the steady current percentage, the deeper i s the scour hole, so that i n the l i m i t the pure current produces the deepest scour. On the other hand, the more the c o n t r i b u t i o n of the wave, the l e s s i s the scour depth and i n the l i m i t of pure waves, the scour depth i s minimum, provided that the same size of c y l i n d e r i s used. Therefore, when considering maximum scour depth under combined waves and currents, i t i s necessary to know the amount of each flow component i n the combined threshold v e l o c i t y . 9. The scour depth under waves plus currents i s more s e n s i t i v e to the flow component contributions when the c y l i n d e r size i s large than when the c y l i n d e r Is small. 10. It was observed that the combined wave and current scour depth i s only s l i g h t l y dependent on sediment s i z e , e s p e c i a l l y f or bigger diameter p i l e s . However, scour depths f o r small sediment siz e ranges are s l i g h t l y greater than f o r bigger sediment s i z e ranges. 87. 11. The measured scour depths under currents alone were compared with estimated or calculated values using a v a i l a b l e scour p r e d i c t i o n formula. The measured values are i n a good agreement with most of the estimated values i n s p e c i f i c formulas (1.7) and (1.8). Therefore steady flow estimates of scour provide an upper bound on the scour depth. 12. Knowing the flow parameters and the structure si z e and bed material properties, i t i s pos s i b l e to get at l e a s t a rough estimate of the maximum possible scour. 6.2 RECOMMENDATIONS FOR FURTHER STUDY There are several areas i n which further studies could be made to improve the present study. 1. It would be desi r a b l e to repeat these experiments using d i f f e r e n t s i z e sand ranges to evaluate the e f f e c t of sediment size more p r e c i s e l y . 2. Although t h i s study was r e s t r i c t e d to non-cohesive material, s i m i l a r experiments can be run again using cohesive material. 3. Using l a r g e r p i l e s i z e s and conducting the same experiments i n a la r g e r flume could i n d i c a t e the p i l e s i z e dependence, and to f i n d a c e r t a i n c o r r e l a t i o n or formula for maximum scour. 4. Using an array of p i l e s and varying the spacing could i d e n t i f y a spacing parameter i n d i c a t i n g at what distance the p i l e s must be separated i n order for them to scour independently of one another. 88. 5. I n s t a l l a t i o n of a pro t e c t i v e c o l l a r on p i l e s and repeating the same experiments could i d e n t i f y the spacing and the s i z e of these c o l l a r s to reduce scour. 6. Throughout the experiments the wave used was uniform, two-dimensional of the s i n u s o i d a l type. This study could be extended to in v e s t i g a t e the e f f e c t of random waves on scour depth around structures. 7. Other shapes of p i l e s - rather than the c y l i n d r i c a l shape - could be tested. 89. BIBLIOGRAPHY Abad, G.N. and Machemehl, J.L. 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Wang, R.K. and Herbich, J.B. (1983), "Combined Current and Wave Produced Scour Around a Single P i l e , " Texas Engineering Experiment Station, Report No. COE269, Texas A&M University, College Station, Texas. Wells, D.R. and Sorensen, R.M. (1970), "Scour Around a C i r c u l a r P i l e Due to O s c i l l a t o r y Wave Motion," Sea Grant P u b l i c a t i o n No. 208, COE Report, No. 113, Texas A&M U n i v e r s i t y . White, CM. (1940), "The Equilibrium of Grains on the Bed of a Stream," Proceedings of the Royal Society (A), Vol. 174, pp. 322-338. Williams, P.B. and Kemp, P.H. (1971), " I n i t i a t i o n of Ripples on F l a t Sediment Beds," J. Hydr. Div. ASCE, Vol. 97, No. HY4, pp. 505-522. 

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