Stability of bottom armoring under the attack of solitary waves

U N D E R T H E A T T A C K OF

S O L I T A R Y WAVES

by

E h u d N a h e c r

W. M . Keck Laboratory of Hydraulics and Water Resources '

Division of Engineering and Applied Science

C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y

Pasadena, California

K a

r

Postbus 177- OÊLFT

o c

-STABILITY OF BOTTOM ARMORING UNDER THE ATTACK OF SOLITARY WAVES

by Ehud Naheer P r o j e c t Supervisor: F r e d r i c Raichlen Professor o f C i v i l Engineering Supported by N a t i o n a l Science Foundation

Grant Nos'. GK-31802X, ENG71-02367 AOS, and ENG75-15786 AOl

W. M. Keck Laboratory o f Hydraulics and Water Resources D i v i s i o n o f Engineering and Applied Science

C a l i f o r n i a I n s t i t u t e o f Technology Pasadena, C a l i f o r n i a

ACKNOWLEDGMENTS

The w r i t e r wishes t o express h i s g r a t i t u d e t o h i s t h e s i s a d v i s o r , Professor F r e d r i c Raichlen, who suggested t h i s research problem and o f f e r e d h i s advice throughout every phase of the i n v e s t i g a t i o n . The advice and encouragement of Professors V i t o A. Vanoni, Norman H. Brooks, and Peter S. Eagleson are also deeply appreciated.

The w r i t e r also wishes t o thank Dr. Robert C. Y. Koh and Dr. Sasson R. Somekh f o r the long discussions which helped i n developing the procedures used i n the experimental i n v e s t i g a t i o n .

A s p e c i a l thanks i s owed to Mr. E l t o n F. Daly, supervisor of the shop and l a b o r a t o r y , whose assistance d u r i n g the design, c o n s t r u c t i o n , and maintenance of the experimental equipment made i t p o s s i b l e t o solve almost any problem i n the l a b o r a t o r y phase of the i n v e s t i g a t i o n . Thanks are also due to Mr. Joseph J. Fontana, Mr. Robert S h u l t z , Mr. W i l l i a m G. Stone, Mr. Robert L. Greenway, and Mr. Carl A. Green who a s s i s t e d i n the c o n s t r u c t i o n and maintenance o f the experimental equipment; t o Mr. Walter Beckmann and Miss S a l l y Weaks, who helped i n reducing the experimental data; t o Mr. Gregory G a r t r e l l and Mr. P h i l i p J. W. Roberts, who reviewed the o r i g i n a l manuscript, and t o Mrs. Joan L. Mathews, Mrs. S h i r l e y A. Hughes, and Mrs. Linda Rorem who typed i t ; and t o Mr. David Byrum who performed the d r a f t i n g o f the f i g u r e s appearing i n t h i s manuscript.

The w r i t e r also wishes t o thank the C a l i f o r n i a I n s t i t u t e o f Technology f o r f i n a n c i a l assistance and f o r p r o v i d i n g the f a c i l i t i e s f o r t h i s study. The experiments were conducted i n the W. M. Keck Laboratory of Hydraulics and Water Resources. Thanks are also due t o

the Humanities Fund, I n c . , f o r the B o r i s Bakhmeteff Research Fellowship which was awarded t o the w r i t e r i n 1972. The research was supported by N a t i o n a l Science Foundation Grants GK-31802X, ENG71-02367 AOS, and ENG75-15786 AOl.

The deepest g r a t i t u d e i s expressed by the w r i t e r t o h i s w i f e , Daphna, f o r her understanding, p a t i e n c e , and encouragement d u r i n g the p e r i o d o f h i s graduate study.

This r e p o r t was submitted by the w r i t e r on May 5, 1976 as a t h e s i s w i t h the same t i t l e t o the C a l i f o r n i a I n s t i t u t e o f Technology i n p a r t i a l f u l f i l l m e n t o f the requirements f o r the degree o f Doctor o f Philosophy; i t i s reproduced here unchanged.

ABSTRACT

An empirical relationship is presented for the incipient motion of bottom material under solitary waves. Two special cases

of bottom material are considered: particles of arbitrary shape, and isolated sphere resting on top of a bed of tightly packed spheres.

The amount of motion in the bed of particles of arbitrary shape is shown to depend on a dimensionless shear stress, similar to the Shields parameter. The mean resistance coefficient used in estimating this parameter is derived from considerations of energy dissipation, and is obtained from measurements of the attenuation of waves along a channel. A

tbeoretical expression for the mean resistance coefficient is developed for the case of laminar flow from the linearized boundary layer equations and is verified by experiments.

For the case of a single sphere resting on top of a bed of spheres, the analysis is based on the hypothesis that at incipient motion the hydrodynamic moments which tend to remove the sphere are equal to the restoring moment due to gravity which tends to keep it in its place. It is shown that the estimation of the hydrodynamic forces, based on an approach similar to the so-called "Morison's formula", in which the drag, lift, and inertia coefficients are independent of each other, is in-accurate. Alternatively, a single coefficient incorporating both drag, inertia, and lift effects is employed. Approximate values of this co-efficient are described by an empirical relationship which is obtained from the experimental results.

A review of existing theories of the solitary wave is presented and an experimental study is conducted in order to determine which theory should be used in the theoretical analysis of the incipient motion of bottom material.

Experiments were conducted in the laboratory in order to determine the mean resistance coefficient of the bottom under solitary waves, and in order to obtain a relationship defining the incipient motion of bottom material. All the experiments were conducted in a wave tank 40 m long, 110 cm wide with water depths varying from 7 cm to 42 cm. The mean resistance coefficient was obtained from measurements of the attenuation of waves along an 18 m section of the wave tank. Experim~nts were conducted with a smooth bottom and with the bottom roughened with a layer of rock. The incipient motion of particles of arbitrary shape was studied by measuring the amount of motion in a 91 cm x 50 cm section covered with a 15.9 mm thick layer of material. The materials used had different densities and mean diameters. The incipient motion of spheres was observed for spheres of different diameters and densities placed on a bed of tightly packed spheres. The experiments were conducted with various water depths, and with wave height-to-water depth ratios varying from small values up to that for breaking of the wave.

It was found that: (a) The theories of Boussinesq (1872) and McCowan (1891) describe the solitary wave fairly accurately. However, the

differences between these theories are large when used to predict the forces which are exerted on objects on the bottom, and it was not established which theory describes these forces better. (b) The mean resistance coeffici-ent for a rough turbulcoeffici-ent flow under solitary waves can be described as

a f u n c t i o n o f D , h, and H, where D i s the mean diameter of the s s

roughness p a r t i c l e s , h i s the water depth, and H i s the wave h e i g h t , (c) Small e r r o r s i n the d e t e r m i n a t i o n of the dlmensionless shear s t r e s s f o r i n c i p i e n t motion of rocks r e s u l t i n l a r g e e r r o r s i n the e v a l u a t i o n of the diameter of the rock r e q u i r e d f o r i n c i p i e n t motion. However, i t was found t h a t the e m p i r i c a l r e l a t i o n s h i p f o r the i n c i p i e n t motion o f spheres can be used t o determine the s i z e o f rock of a r b i t r a r y shape f o r i n c i p i e n t motion under a given wave, provided the angle of f r i c t i o n o f the rock can be determined a c c u r a t e l y .

(

TABLE OF CONTENTS

Chapter Page 1. INTRODUCTION' 1

1.1 O b j e c t i v e and Scope o f the Present Study 2

2. LITERATURE SURVEY 5 2.1 The I n c i p i e n t Motion o f a Bed o f Rocks 5

2.2 The Resistance C o e f f i c i e n t Under Waves 10

3. THEORETICAL ANALYSIS 18 3.1 The S o l i t a r y Wave 18 3.2 The Hydrodynamic Forces Exerted on Bed

M a t e r i a l Under S o l i t a r y Waves 23 3.2.1 The Damping o f S o l i t a r y Waves 24

3.2.11 C o r r e c t i o n f o r Wall E f f e c t s 36 3.2.2 Shear Stresses i n the Laminar Boundary

Layer 37 3.2.3 The Forces Exerted on a Single Sphere

Resting on a Bed o f Spheres 43 3.3 The I n c i p i e n t Motion o f Bed M a t e r i a l Under

S o l i t a r y Waves 52 3.3.1 The I n c i p i e n t Motion o f P a r t i c l e s o f

A r b i t r a r y Shape 53 3.3.2 The I n c i p i e n t Motion o f a Single Sphere 55

4. EXPERIMENTAL EQUIPMENT AND PROCEDURES 59

4.1 The Wave Tank 59 4.2 The Wave Generator 61 4.3 The Measurement o f Wave Amplitude 63

TABLE OF CONTENTS (Cont'd)

Chapter Page 4.4 Test Sections and Experimental Methods f o r

I n c i p i e n t Motion Experiments 70 4.4.1 The Working Area 70 4.4.2 I n c i p i e n t Motion o f Spheres 72

4.4.2.1 The Test Section 72 4.4.2.2 The Measurement o f I n c i p i e n t

Motion 74 4.4.3 The I n c i p i e n t Motion o f P a r t i c l e s o f

A r b i t r a r y Shape 80 4.4.3.1 The Test Section 80

4.4.3.2 The C h a r a c t e r i s t i c s o f t h e

P a r t i c l e s 80 4.4.3.3 The Measurement o f the Motion

o f the P a r t i c l e s 88 4.5 The Measurement o f F l u i d P a r t i c l e V e l o c i t y 95

4.6 Measurements o f Bottom Resistance C o e f f i c i e n t

Under S o l i t a r y Waves 101 5. PRESENTATION AND DISCUSSION OF RESULTS 107

5.1 The S o l i t a r y Wave 107 5.1.1 The Wave P r o f i l e 107

5.1.2 The Wave C e l e r i t y 117 5.1.3 The F l u i d P a r t i c l e V e l o c i t y 121

5.2 The Resistance C o e f f i c i e n t Under S o l i t a r y Waves 125

5.3 The I n c i p i e n t Motion o f Bed M a t e r i a l 144 5.3.1 The I n c i p i e n t Motion o f P a r t i c l e s o f

TABLE OF CONTENTS (Cont'd)

Chapter Page 5.3.2 The I n c i p i e n t Motion o f Spheres 160

6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDIES 203

6.1 Conclusions 203 6.2 Recommendations f o r Future Studies 206

LIST OF REFERENCES 209 LIST OF SYMBOLS 214 APPENDIX I : INERTIA AND LIFT COEFFICIENTS FOR A SPHERE

NEAR THE BOTTOM 220 APPENDIX I I : EXPERIMENTAL DATA 226

LIST OF FIGURES

Figure Page 3.1 D e f i n i t i o n sketch f o r the s o l i t a r y wave 20

3.2 D e f i n i t i o n sketch f o r the f o r c e s exerted on a

sphere under s o l i t a r y waves 45 4.1 D e t a i l s o f 40-meter p r e c i s i o n t i l t i n g fliime

modified f o r wave experiments 60 4.2 O v e r a l l view o f the wave generator 62 4.3 View o f the system c o n t r o l l i n g the motion o f

the wave generator 62 4.4 The displacement o f f l u i d p a r t i c l e s xmder a

s o l i t a r y wave and a t the wave generator p i s t o n 64

4.5 Drawing o f a t y p i c a l wave gage 65 4.6 C i r c u i t diagram f o r wave gages 65 4.7 T y p i c a l c a l i b r a t i o n curve f o r a wave gage 67

4.8 T y p i c a l records o f s t a t i c and djmamlc c a l i b r a t i o n

(wave gage constructed w i t h s t a i n l e s s s t e e l w i r e s ) 69 4.9 O v e r a l l view o f the working area i n the wave tank 71

4.10 View of the bed of spheres 73 4.11 View o f an i s o l a t e d p r e c i s i o n sphere supported on

top of the bed by p r e c i s i o n s t a i n l e s s s t e e l spheres 73 4.12 The p r e c i s i o n spheres whose i n c i p i e n t motion was

i n v e s t i g a t e d 75 4.13 Plan view o f the two p o s l t l o n i n g s o f an i s o l a t e d

sphere on top o f the bed o f spheres 76 4.14 Schematic drawing o f the system used t o detect

i n f i n i t e s i m a l displacements o f a sphere 77 4.15 Schematic drawing o f a t y p i c a l p o s i t i o n i n g of the

l a s e r beam w i t h respect t o the sphere; (a) l a r g e output s i g n a l o f the system; (b) small output s i g n a l of the system ( f o r the same displacement o f the

LIST OF FIGimES (Cont'd)

liSüïË. Page ^•16 View o f the t e s t s e c t i o n d u r i n g an experiment 79

4.17 T y p i c a l record of the motion o f the sphere.

Upper curve d i s p l a y s the wave r e c o r d ; lower curve d i s p l a y s the c a l i b r a t i o n and the motion of the

sphere 79 4.18 View o f the p a r t i c l e s o f a r b i t r a r y shape placed i n

the t e s t s e c t i o n

^•19 Sieve s i z e d i s t r i b u t i o n curves o f the p a r t i c l e s o f a r b i t r a r y shape used i n the experiments o f i n c i p i e n t motion

4.20 S p e c i f i c g r a v i t y d i s t r i b u t i o n curves o f the m a t e r i a l s used i n the experiments; (a) N a t u r a l rock, (b) Coal 4.21 O v e r a l l view o f the system used t o measure the angle

of f r i c t i o n

4.22 View o f the system used t o measure the angle o f f r i c t i o n a t (a) the "packing" angle, and (b) the angle o f " c o l l a p s e "

4.23 Shape-factor d i s t r i b u t i o n curves o f the p a r t i c l e s used i n the experiments o f i n c i p i e n t motion o f p a r t i c l e s of a r b i t r a r y shape

4.24 View of the photographic equipment I n s t a l l e d a t the t e s t s e c t i o n 81 83 84 85 86 89 90 4.25 Overhead photographs o f the bed of p a r t i c l e s o f

a r b i t r a r y shape; (a) negative p i c t u r e of the bed b e f o r e running the wave, (b) p o s i t i v e p i c t u r e o f the bed a f t e r running the wave, (c) alignment o f

p i c t u r e s (a) and (b) 93 4.25d Alignment of Figs. 5.25a and 5.25b (enlarged)

4.26 Photograph of the s e c t i o n o f the bed used t o

estimate the t o t a l number o f p a r t i c l e s seen i n the overhead view o f the e n t i r e t e s t s e c t i o n

'^•27 Schematic drawing of the system used t o measure the f l u i d p a r t i c l e v e l o c i t y

94

96

LIST OF FIGURES (Cont'd)

Figure Page

100 4.28 The scale used t o measure the displacements o f

t r a c e r p a r t i c l e s

4.29 Tracer p a r t i c l e s i n a f l u i d under a s o l i t a r y wave ; (a) a t t=26.04 sec; (b) a t t=26.06 sec; ( c ) alignment

of (a) and (b) 102

4.29d Alignment o f Figs. 4.29a and 4.29b (enlarged t o scale

w i t h F i g . 4.28) 1°^ 4.30 Sieve s i z e d i s t r i b u t i o n curves o f the rock used i n

the experiments o f wave a t t e n u a t i o n 105 5.1 Comparison between a measured s o l i t a r y wave and between

the t h e o r i e s of Botisslnesq, McCowan, and L a i t o n e ;

h=10.0 cm; H/h=0.086; bottom slope=0.0 109 5.2 Comparison between the s o l i t a r y waves measured over t h e

smooth and rough bed sections and between the t h e o r i e s of Bousslnesq, McCowan, and L a i t o n e ; h=30.0 cm; bottom slope^O.O; H/h=0.351 (over the smooth s e c t i o n ) ;

H/h=»0,344 (over the rough s e c t i o n ) 110 5.3 Comparison between the s o l i t a r y waves measured over

the smooth and rough bed sections and between t h e t h e o r i e s o f Bousslnesq, McCowan, and L a i t o n e ; h=10.0 cm; bottom slope=0.0; H/h=0.630 (over the

smooth s e c t i o n ) ; H/h=0.640 (over the rough s e c t i o n ) 111 5.4 Comparison between a measured s o l i t a r y wave and

between the t h e o r i e s of Bousslnesq, McCowan, and

L a i t o n e ; h=26.2 cm; H/h=0.25; bottom slope=l:200 113 5.5 Comparison between a measured s o l i t a r y wave and

between the t h e o r i e s o f Bousslnesq, McCowan, and

L a i t o n e ; h=26.2 cm; H/h=0.60; bottom slope=l:200 114 5.6 Comparison between a measured s o l i t a r y wave and

between the t h e o r i e s of Bousslnesq, McCowan, and

L a i t o n e ; h=26.2 cm; H/h=0.87; bottom slope=l:200 115 5.7 An example of a wave record used t o evaluate the

wave c e l e r i t y 118 5.8 The c e l e r i t y of s o l i t a r y waves 120

LIST OF FIGURES (Cont'd)

liSHIË. Page 5.9 (a) The surface p r o f i l e o f a s o l i t a r y wave;

(b) The f l u i d p a r t i c l e v e l o c i t y near the bottom

under a s o l i t a r y wave 122 5.10 A t t e n u a t i o n o f s o l i t a r y waves along a channel;

(a) over a rough bed; (b) over a smooth bed 128 5.11 The mean r e s i s t a n c e c o e f f i c i e n t as a f x m c t l o n o f

the f l o w Rejmolds ninnber 130 5.12 The mean r e s i s t a n c e c o e f f i c i e n t as a f u n c t i o n o f

the r a t i o o f t h e roughness s i z e , D^, t o the displacement o f a f l u i d p a r t i c l e j u s t o u t s i d e

the botindary l a y e r , Z 135 5.13 The mean r e s i s t a n c e c o e f f i c i e n t as a f t m c t l o n o f

the c h a r a c t e r i s t i c dlmensionless a c c e l e r a t i o n 138 5.14 The mean r e s i s t a n c e c o e f f i c i e n t as a f u n c t i o n o f

the dlmensionless l e n g t h s H/h and Dg/h 141 5.15 The packing o f a g r a n u l a r bed by waves 146 5.16 Schematic drawing o f a c r o s s - s e c t i o n a l view o f the

bed o f p a r t i c l e s o f a r b i t r a r y shape, showing the nature o f the bed surface b e f o r e and a f t e r t h e

packing of the bed 149 5.17 The amount o f motion o f p a r t i c l e s o f a r b i t r a r y

shape as a f u n c t i o n o f the dlmensionless shear

s t r e s s 153 5.18 The average amoimt o f motion o f p a r t i c l e s o f a r b i t r a r y

shape as a f u n c t i o n o f t h e dlmensionless shear s t r e s s 155

5.19 The angles (j)^ and (j)^ 163 5.20 A graphic i l l u s t r a t i o n o f the t r i a l and e r r o r

procedure used t o determine the wave t h a t causes

i n c i p i e n t motion 165 5.21 The d i s t r i b u t i o n o f f o r c e s and moments e x e r t e d on a

sphere under s o l i t a r y waves a t i n c i p i e n t motion o f

LIST OF FIGUHES (Cont'd)

Figure Page 5.22 The d i s t r i b u t i o n of f o r c e s and moments exerted on

a sphere under s o l i t a r y waves (using McCowan's

theory) 170 5.23 Comparison between the experimental r e s u l t s of the

i n c i p i e n t motion o f spheres and between the t h e o r e t i c a l p r e d i c t i o n s u s i n g Bousslnesq's and

McCowan's t h e o r i e s 174 5.24 Comparison between the experimental r e s u l t s of the

i n c i p i e n t motion of spheres and the t h e o r e t i c a l p r e d i c t i o n s (using Bousslnesq's theory) assuming t h a t the hydrodynamic f o r c e s a c t e i t h e r a t the top

or a t the center o f the sphere 178 5.25 Experimental r e s u l t s o f the i n c i p i e n t motion o f

spheres 183 5.26 I n c i p i e n t motion o f a sphere. R e l a t i o n s h i p between

the wave h e i g h t , the water depth, the t e s t sphere diameter, the bed sphere diameter, the angle iji, and

the submerged d e n s i t y o f the sphere 185 5.27 I n c i p i e n t motion o f a sphere. R e l a t i o n s h i p between

the wave h e i g h t , the water depth, the angle ^, and

the diameter and submerged d e n s i t y o f the sphere 187 5.28 The resistance c o e f f i c i e n t of a sphere \mder s o l i t a r y

waves 190 5.29 I n c i p i e n t motion of a sphere under breaking waves.

R e l a t i o n s h i p between the water depth, the wave h e i g h t , the angle ^, and the diameter and submerged

density of the sphere 194 5.30 The average amount o f motion o f p a r t i c l e s o f a r b i t r a r y

shape as a f u n c t i o n o f the wave h e i g h t , the water depth, the angle ^, and the mean diameter and submerged

d e n s i t y of the p a r t i c l e s 200 A. 1.1 D e f i n i t i o n sketch f o r the motion o f two spheres i n

LIST OF TABLES

Table Page

3.1 Solutions o f the s o l i t a r y wave due t o Bousslnesq,

McCowan, and Laitone 22 4.1 The measured angles o f packing and collapse f o r

the m a t e r i a l s used i n the i n v e s t i g a t i o n o f

I n c i p i e n t motion 87 5.1 Experimental data o f the average amount o f motion

of p a r t i c l e s o f a r b i t r a r y shape 156 5.2 The values of and y. ( i n Eq. (5.28)) as a f u n c t i o n

o f h/Dg ^ 184 A.2.1 Experimental data of the i n c i p i e n t motion of spheres 226

A.2.1.1 Experimental data o f the i n c i p i e n t motion o f spheres

under breaking waves 231 A.2.2 Experimental data o f the r e s i s t a n c e c o e f f i c i e n t tmder

s o l i t a r y waves 232 A.2.3 C h a r a c t e r i s t i c s o f the m a t e r i a l used i n the

i n v e s t i g a t i o n o f the i n c i p i e n t motion o f p a r t i c l e s

of a r b i t r a r y shape 243 A.2.4 Experimental data o f the i n c i p i e n t motion o f p a r t i c l e s

CHAPTER 1 INTRODUCTION

Offshore s t r u c t u r e s such as sewage o u t f a l l s and thermal discharge pipes which pass from the shore I n t o the ocean are exposed t o ocean waves t h a t shoal on the beach and break I n the s u r f zone. These waves tend t o undermine the pipes by removing the sand and can cause s t r u c -t u r a l f a i l u r e s due -t o d i f f e r e n -t i a l s e -t -t l i n g or by a c -t i n g d i r e c -t l y on the pipes. The pipes are u s u a l l y p r o t e c t e d by p l a c i n g them i n a

trench and armoring t h e i r tops w i t h pavements o f loose rocks. However, i f not designed p r o p e r l y , these rocks can be removed by b i g storm waves. I n order t o determine the s i z e o f the rocks r e q u i r e d f o r adequate p r o -t e c -t i o n a -t a given s i -t e i -t i s impor-tan-t -t o be able -t o p r e d i c -t -the forces and moments exerted on them by the waves. A c c o r d i n g l y , the rocks

should be designed such t h a t they w i l l r e s i s t these hydrodynamic forces and moments.

The design o f the rocks Includes c o n s i d e r a t i o n s o f t h e i r s i z e , weight, shape, grading and placement. I t i s conceivable t h a t l a r g e and heavy rocks are more s t a b l e than small and l i g h t ones. Angular rocks of a r b i t r a r y shape are apparently more s t a b l e than s p h e r i c a l p a r t i c l e s because they tend t o I n t e r l o c k b e t t e r w i t h each o t h e r . A well-graded rock covering a l i m i t e d range o f sizes i s p o s s i b l y b e t t e r than a s i n g l e - s i z e d rock, as the small p a r t i c l e s o f a well-graded rock f i l l i n the holes among the b i g rocks and provide a stronger i n t e r l o c k i n g s t r u c t u r e . F i n a l l y , rocks which are placed I n d i v i d u a l l y , u s u a l l y w i t h the help o f a d i v e r , are more s t a b l e than rocks which are dumped.

parameters r e p r e s e n t i n g the rock a t a c t u a l c o n d i t i o n s ( i . e . , s i z e , weight, shape, grading and placement) and the c h a r a c t e r i s t i c s o f the design wave i s t h e r e f o r e r e q u i r e d f o r a proper design o f bottom armoring.

1.1 OBJECTIVE AND SCOPE OF THE PRESENT STUDY

The o b j e c t i v e of the present study i s t o i n v e s t i g a t e , b o t h t h e o r e t i -c a l l y and e x p e r i m e n t a l l y , the -c o n d i t i o n s r e q u i r e d f o r i n -c i p i e n t motion o f a bed of rocks under s o l i t a r y waves. The i n c i p i e n t motion of a p a r t i c l e i s defined as the event i n which the p a r t i c l e b a r e l y moves, as the hydrodynamic moments f o r c i n g the p a r t i c l e from i t s place are equal t o the r e s t o r i n g moment due t o the weight o f the p a r t i c l e . S o l i t a r y waves were chosen f o r three reasons. F i r s t , the theory of the s o l i t a r y waves i s well-known, so the hydrodynamics o f the f l o w can r e a d i l y be evaluated. Second, long waves s h o a l i n g on a beach have wide troughs and narrow c r e s t s which resemble s o l i t a r y waves. T h i r d , by employing s o l i t a r y waves i n the experimental study the problem of i n t e r a c t i o n between r e f l e c t e d and i n c i d e n t waves i s avoided.

As the motion o f the rocks r e s u l t s from hydrodynamic f o r c e s and moments which are exerted on them by the f l o w , i t i s necessary t o be able to determine these forces and moments. The r e s i s t a n c e c o e f f i c i e n t o f the bottom under s o l i t a r y waves i s t h e r e f o r e I n v e s t i g a t e d , and the

stresses exerted on the bed are determined from t h i s study. The i n v e s t i -g a t i o n of the i n c i p i e n t motion amounts t o the study o f the r e l a t i o n s h i p between the hydrodynamic stresses and the c h a r a c t e r i s t i c s o f the rock

which cause the bed t o be i n a s t a t e o f i n c i p i e n t motion.

The problem o f I n c i p i e n t motion which includes c o n s i d e r a t i o n o f a l l the c h a r a c t e r i s t i c s of the rock, i . e . , s i z e , weight, shape, grada-t i o n , and placemengrada-t, i s q u i grada-t e complex. The f o l l o w i n g s i m p l i f i c a grada-t i o n s have been used i n the present study: a. The considered rock has a narrow s i z e d i s t r i b u t i o n , i . e . , a l l the p a r t i c l e s are f a i r l y u n i f o r m i n diameter; b. A l l the rocks used i n the experimental study are angular, i . e . , have f a i r l y sharp corners, and they a l l have a p p r o x i mately the same shape f a c t o r ; c. A l l the rocks are placed i n the e x p e r i -mental model using the same method o f placement. The i n v e s t i g a t i o n i s thus l i m i t e d t o the problem o f i n c i p i e n t motion o f p a r t i c l e s o f a r b i t r a r y shape c h a r a c t e r i z e d only by weight (or d e n s i t y ) and mean diameter.

I t i s conceivable t h a t i f some motion i s expected t o occur under a given wave, the moving rock p a r t i c l e s w i l l be those which emerge above t h e i r neighbors and p r o t r u d e i n t o the f l o w . A s i m i l a r model of a simple geometrical shape can be described by a s i n g l e sphere r e s t i n g on top o f a bed o f s i m i l a r spheres. The i n c i p i e n t motion o f such a model i s also s t u d i e d i n the present i n v e s t i g a t i o n , and the r e s u l t s o f t h i s study are compared t o those obtained w i t h p a r t i c l e s o f a r b i t r a r y shape.

A review o f previous s t u d i e s of the r e s i s t a n c e c o e f f i c i e n t and o f the i n i t i a t i o n o f motion of p a r t i c l e s under waves I s presented i n

Chapter 2. A t h e o r e t i c a l a n a l y s i s i s presented i n Chapter 3 i n which three t h e o r e t i c a l p r e s e n t a t i o n s of the s o l i t a r y wave are compared.

T h e o r e t i c a l c o n s i d e r a t i o n s o f the i n c i p i e n t motion and o f the r e s i s t a n c e c o e f f i c i e n t are also presented i n Chapter 3. The experimental equipment

and procedures are described i n Chapter 4. The r e s u l t s o f the i n v e s t i g a -t i o n are presen-ted and discussed i n Chap-ter 5, and conclusions are s -t a -t e d i n Chapter 6.

CHAPTER 2 LITERATURE SURVEY

2.1 THE INCIPIENT MOTION OF A BED OF ROCKS

There I s a l a r g e number o f s t u d i e s I n the l i t e r a t u r e d e a l i n g w i t h the problem of the I n i t i a t i o n o f motion o f bed m a t e r i a l . However, most of these studies are concerned w i t h the problem as I t occurs I n steady flows I n streams and channels. The c o n d i t i o n s r e q u i r e d f o r i n i t i a t i o n of motion f o r these cases are u s u a l l y described by the s o - c a l l e d

"Shields diagram", or the "Shields curve", which i s named a f t e r Shields (1936) whose i n v e s t i g a t i o n o f the problem was based on s i m i l a r i t y

p r i n c i p l e s . The Shields diagram describes a r e l a t i o n s h i p between a dlmensionless shear s t r e s s , t ^ ^ , and a boundary-particle-Reynolds

number, Re^. The dlmensionless shear s t r e s s , which i s also c a l l e d the "Shields parameter", i s given by

where i s the bottom shear s t r e s s , i s the d e n s i t y o f the f l u i d , g i s the a c c e l e r a t i o n due t o g r a v i t y , and p and D are the mean d e n s i t y

s s

and s i z e (diameter), r e s p e c t i v e l y , o f the p a r t i c l e s . The boundary-particle-Reynolds number i s given by

u^D

= "V"^ . (2.2) where v i s the kinematic v i s c o s i t y o f the f l u i d , and u^ = /r^/p i s c a l l e d

* b w the boundary shear v e l o c i t y .

of motion under waves are q u i t e l i m i t e d . Komar and M i l l e r (1973) used the data obtained e x p e r i m e n t a l l y by Bagnold (1946) and Manohar (1955) t o show t h a t the Shields diagram as i t i s used f o r steady flows cannot be used f o r o s c i l l a t o r y flows. However, Madsen and Grant (1975) used

Bagnold's data t o show t h a t Shields diagram can be a p p l i e d t o o s c i l l a t o r y f l o w s . They noted t h a t the e r r o r i n Komar and M i l l e r ' s r e s u l t s was due to a wrong d e f i n i t i o n of the bottom shear s t r e s s . Komar and M i l l e r (1975) independently recognized the mistake i n t h e i r preceding (1973) study.

Bagnold (1946) and Manohar (1955) simulated the o s c i l l a t o r y f l o w i n t h e i r experiments by o s c i l l a t i n g a granular bed i n s t i l l water. They neglected the i n e r t i a forces a c t i n g on the p a r t i c l e s i n the o s c i l l a t i n g bed, assuming t h a t hydrodynamic drag was dominant. However, i t should be noted t h a t f o r cases where i n e r t i a forces cannot be neglected, the forces a c t i n g on the o s c i l l a t i n g p a r t i c l e s are d i f f e r e n t from those a c t i n g on s t a t i o n a r y p a r t i c l e s i n an o s c i l l a t i n g f l u i d . This i s due t o the d i f f e r e n t masses associated w i t h these f o r c e s .

The dlmensionless shear s t r e s s given by Eq. (2.1) represents the r a t i o between the hydrodynamic f o r c e s a c t i n g on the bed p a r t i c l e s and the g r a v i t a t i o n a l f o r c e t h a t tend t o keep the p a r t i c l e s i n t h e i r a t - r e s t p o s i t i o n s . I n cases of flows i n streams and channels the hydrodynamic forces are considered t o c o n s i s t o f drag and they are assumed t o be p r o p o r t i o n a l t o the shear stresses which are exerted on the bed by the f l o w . L i f t forces a c t i n g i n a d i r e c t i o n perpendicular t o the d i r e c t i o n of the f l o w are u s u a l l y e i t h e r neglected or assumed t o be Included i n the p r o p o r t i o n a l i t y f a c t o r r e l a t i n g the hydrodynamic forces t o Shields

parameter (e.g., see Vanoni (1975) p. 9 2 ) . I n e r t i a forces (due t o t h e a c c e l e r a t i o n o f the f l u i d p a r t i c l e s r e l a t i v e t o the bed p a r t i c l e s ) do not e x i s t i n steady flows over s t a t i o n a r y p a r t i c l e s .

I n cases where i n e r t i a forces cannot be neglected, e.g., under

waves, the most common approach t o the problem c o n s i s t s o f an examination of the f o r c e s a c t i n g on a s i n g l e bed p a r t i c l e . The p a r t i c l e I t s e l f i s u s u a l l y considered t o be a sphere, and the hydrodynamic forces a c t i n g on i t c o n s i s t o f some combination of drag, i n e r t i a , and l i f t e f f e c t s . Grace (1974) presented a few of the formulae which are most commonly used t o evaluate the hydrodynamic forces a c t i n g on a sphere under waves.

The formula which i s most commonly used i n c o a s t a l engineering p r a c t i c e i s t h a t due t o O'Brien and Morison (1952). They assumed t h a t the f o r c e a c t i n g on a sphere r e s t i n g on the bottom i n an unsteady f l o w can be expressed as a l i n e a r combination o f drag and i n e r t i a f o r c e s , i . e . ,

where C^^ and C^^ are the drag and i n e r t i a c o e f f i c i e n t s r e s p e c t i v e l y , assumed t o be constant; A and V are the p r o j e c t e d area and the volume of the sphere r e s p e c t i v e l y ; and u and ^ are the f r e e stream v e l o c i t y and a c c e l e r a t i o n of the f l u i d p a r t i c l e s r e s p e c t i v e l y , estimated a t t h e l e v e l of the sphere i n i t s absence. The d i r e c t i o n o f the v e l o c i t y and a c c e l e r a t i o n near the bottom i s p a r a l l e l t o the bottom plane and so i s the force given by Eq. (2.3). An equation s i m i l a r t o Eq. (2.3) was f i r s t a p p l i e d by Morison e t a l . (1950) t o forces on p i l e s . O'Brien and Morison d i d not consider l i f t forces i n the study. They evaluated C_,

and e x p e r i m e n t a l l y by measuring the wave p r o f i l e and the f o r c e s a c t i n g on t h e sphere simultaneously. The f l u i d p a r t i c l e v e l o c i t y and a c c e l e r a t i o n were estimated from the wave theory (using t h e l i n e a r i z e d equation of m o t i o n ) , and the value o f C^^ was estimated from Eq. (2.3) a t the p o i n t o f zero a c c e l e r a t i o n . S i m i l a r l y , was evaluated a t the p o i n t of zero v e l o c i t y .

Eagleson, Dean and P e r a l t a (1958) I n v e s t i g a t e d t h e f o r c e s a c t i n g on s p h e r i c a l p a r t i c l e s on a s l o p i n g beach a t both i n c i p i e n t motion and established motion c o n d i t i o n s . I n t h e i r t h e o r e t i c a l developments they recognized both drag, i n e r t i a and l i f t e f f e c t s . However, they assumed t h a t l i f t e f f e c t s were n e g l i g i b l e . The major d i f f e r e n c e s between t h e i r a n a l y s i s and t h a t o f O'Brien and Morison are t h a t they considered a higher order wave theory (Stokes waves), and t h a t they also considered the v e l o c i t y d i s t r i b u t i o n i n s i d e the boundary l a y e r f o r t h e cases where the boundary l a y e r t h i c k n e s s , 6, was greater than t h e diameter o f the sphere. O'Brien and Morison a p p l i e d o n l y the f r e e stream v e l o c i t y d i s t r i b u t i o n t o t h e i r c a l c u l a t i o n s .

Iversen and Balent (1951) and B u g l i a r e l l o (1956) s t u d i e d the

r e s i s t a n c e of an unbounded f l u i d t o the accelerated motion o f disks and spheres ( r e s p e c t i v e l y ) moving i n a u n i d i r e c t i o n a l motion. They suggested t h a t I n e r t i a and viscous e f f e c t s be combined i n t o one c o e f f i c i e n t

C* . This r e s i s t a n c e c o e f f i c i e n t i s then expressed i n t h e form

(2.4)

However, Basset (1888) showed t h a t the f o r c e a c t i n g on a sphere a c c e l e r a -t i n g I n a viscous f l u i d depends also on -the h i s -t o r y of -the f l o w . This means t h a t the forces a c t i n g on two I d e n t i c a l spheres moving I n the same f l u i d a t the same v e l o c i t y and a c c e l e r a t i o n may be d i f f e r e n t f o r

d i f f e r e n t I n i t i a l c o n d i t i o n s o f t h e i r motion. Keulegan and Carpenter (1958) argued t h a t attempts t o c o r r e l a t e t h e r e s i s t a n c e c o e f f i c i e n t , C*, t o the instantaneous Reynolds number, uD/v, and the dlmensionless a c c e l e r a t i o n , D ^ / u ^ , between d i f f e r e n t types o f flows ( i . e . , u n i -d i r e c t i o n a l , o s c i l l a t o r y , e t c . ) were unsuccessful f o r t h i s reason.

I n t h e i r i n v e s t i g a t i o n o f the f o r c e s a c t i n g on c y l i n d e r s and p l a t e s i n an o s c i l l a t i n g f l u i d , Keulegan and Carpenter (1958) assumed average values f o r the drag and i n e r t i a c o e f f i c i e n t s which remain constant throughout the p e r i o d o f o s c i l l a t i o n . They considered the f o r c e t o be given by an equation s i m i l a r t o Eq. ( 2 . 3 ) , but w i t h an a d d i t i o n a l term:

where the f u n c t i o n AR i s used t o account f o r the f a c t t h a t the i n s t a n t a -neous values of C^^ and Cj^ are d i f f e r e n t from t h e i r assumed average values. Keulegan and Carpenter found c o r r e l a t i o n s between the average values of the c o e f f i c i e n t s and a dlmensionless p e r i o d , T^ = Tu /D,

* max ' where T i s the p e r i o d o f o s c i l l a t i o n , u i s the maximum o r b i t a l

max

v e l o c i t y of the f l u i d , and D i s a c h a r a c t e r i s t i c l e n g t h (diameter) o f the o b j e c t . They noted t h a t the dlmensionless p e r i o d , T^, could be replaced by a dlmensionless l e n g t h , A s i m i l a r parameter, C/k ,

where i s the e q u i v a l e n t surface roughness (k^ i s p r o p o r t i o n a l t o the mean diameter, D , of the p a r t i c l e s on a rough s u r f a c e ) , was found t o

s

be s i g n i f i c a n t i n s t u d i e s o f the r e s i s t a n c e c o e f f i c i e n t o f rough surfaces under o s c i l l a t o r y f l o w s . These s t u d i e s are presented i n the f o l l o w i n g s e c t i o n .

2.2 THE RESISTANCE COEFFICIENTS UNDER WAVES

A s i g n i f i c a n t amount o f work has been done i n the past i n order t o estimate the shear stresses exerted by waves on both smooth and rough bottoms. I n most of these s t u d i e s the f l o w was considered t o be o s c i l l a t o r y , and only a few I n v e s t i g a t o r s considered s o l i t a r y waves. A comprehensive review o f s t u d i e s on boundary l a y e r s and f r i c t i o n f a c t o r s under o s c i l l a t o r y flows was given by Jonsson (1966). The present review of such flows w i l l t h e r e f o r e be l i m i t e d and w i l l o n l y demonstrate the various methods used by d i f f e r e n t i n v e s t i g a t o r s .

Of the t h e o r e t i c a l treatments of laminar boundary l a y e r s under o s c i l l a t o r y f l o w s , t h a t due t o L i n (1957) i s o f t e n used i n comparison w i t h experimental s t u d i e s of boundary l a y e r s under waves. L i n considered an o s c i l l a t o r y motion superimposed on a steady stream, where the amplitude of o s c i l l a t i o n and the magnitude o f the stream may v a r y w i t h the x

coordinate. By averaging the equations of motion over the p e r i o d o f o s c i l l a t i o n and assuming h i g h frequency o f o s c i l l a t i o n s , he d e r i v e d a l i n e a r boundary l a y e r equation f o r the o s c i l l a t o r y component o f the f l o w . The a n a l y t i c a l s o l u t i o n o f t h i s equation was then i n t r o d u c e d i n t o the averaged equations o f motion which y i e l d e d an a n a l y t i c a l s o l u t i o n to the mean f l o w i n the boundary l a y e r . For the l i m i t i n g case o f zero

mean stream v e l o c i t y the problem i s reduced t o t h a t o f the o s c i l l a t o r y wave, and the a n a l y t i c a l s o l u t i o n o f the l i n e a r i z e d equations adequately describes the behavior o f the boundary l a y e r .

Turbulent boundary l a y e r s under o s c i l l a t o r y flows over both smooth and rough surfaces were i n v e s t i g a t e d t h e o r e t i c a l l y by K a j i u r a (1968). He subdivided the boundary l a y e r i n t o three r e g i o n s , namely the i n n e r , the overlap, and the outer l a y e r s , and considered d i f f e r e n t forms o f the eddy v i s c o s i t y f o r each o f them. The values of the eddy v i s c o s i t y were assumed t o remain constant throughout the p e r i o d o f o s c i l l a t i o n , and they were obtained from measurements o f steady t u r b u l e n t f l o w s . S u b s t i t u t i n g

the assumed forms o f the eddy v i s c o s i t i e s i n t o the l i n e a r i z e d boundary l a y e r equations ( n e g l e c t i n g convective terms), K a j i u r a obtained the

s o l u t i o n f o r each subdivided r e g i o n i n the boundary l a y e r . The constants of i n t e g r a t i o n i n h i s s o l u t i o n were e l i m i n a t e d by matching the s o l u t i o n s at the boundaries between these regions.

Experimentally, the shear stresses can be evaluated from measure-ments o f shear forces exerted on a p l a t e , or by measuring the v e l o c i t y p r o f i l e s i n the v i c i n i t y o f the boundary and a p p l y i n g some t h e o r e t i c a l considerations which r e l a t e the v e l o c i t y p r o f i l e t o the shear s t r e s s e s . Now, since the shear stresses are the main reason f o r the wave energy d i s s i p a t i o n , and as the wave energy can be expressed i n terms o f the wave h e i g h t , the shear stresses can also be estimated from measurements of the a t t e n u a t i o n o f waves along a channel.

I n a t h e o r e t i c a l study of the a t t e n u a t i o n o f waves, B l e s e l (1949) used the l i n e a r equations o f the laminar boundary l a y e r t o show t h a t the h e i g h t , H, o f an o s c i l l a t o r y progressive wave decays e x p o n e n t i a l l y

along the channel, I . e . ,

H = g-kx/h ^2.7)

where I s the wave h e i g h t a t the coordinate x = 0, k I s the decay c o e f f i c i e n t , and h I s the water depth. The decay c o e f f i c i e n t was shown to be a f u n c t i o n of a form of a Reynolds number, d e f i n e d I n terms o f the wave l e n g t h and the wave speed.

Eagleson (1962) measured the f o r c e s exerted on a p l a t e under o s c i l l a t o r y progressive waves. D e f i n i n g the bottom r e s i s t a n c e co-e f f i c i co-e n t , Cf , t o bco-e givco-en by

b

where i s the bottom shear s t r e s s , i s the d e n s i t y o f the f l u i d , and u i s i n the f r e e stream v e l o c i t y evaluated near the bottom, he obtained a r e l a t i o n s h i p between the decay c o e f f i c i e n t , k, and the

average r e s i s t a n c e c o e f f i c i e n t (averaged over a wave p e r i o d ) . The decay c o e f f i c i e n t s which he obtained were l a r g e r than those p r e d i c t e d by

B l e s e l (1949). A c c o r d i n g l y , the experimental values of the average r e s i s t a n c e c o e f f i c i e n t were l a r g e r than the t h e o r e t i c a l ones.

Iwagaki e t a l . (1965) also measured the forces exerted on a p l a t e . They noted t h a t the discrepancies i n Eagleson's r e s u l t s were probably due t o measurement e r r o r s . They also measured the a t t e n u a t i o n o f waves along a channel. The experimental values which they obtained f o r the decay c o e f f i c i e n t were also l a r g e r than the p r e d i c t e d ones. This discrepancy i s probably due to energy d i s s i p a t i o n a t the f r e e surface i n a d d i t i o n t o the d i s s i p a t i o n near s o l i d boundaries. Van Dorn (1966)

showed t h a t such a d i s s i p a t i o n was p o s s i b l e as a f r e e - s u r f a c e boundary l a y e r could develop due t o contamination.

The most common procedures a p p l i e d to experimental i n v e s t i g a t i o n s of shear stresses i n t u r b u l e n t o s c i l l a t o r y flows are those which were used by Kalkanis (1957), Jonsson (1963), and Kamphuls (1975). Kalkanis measured the v e l o c i t y p r o f i l e of a f l u i d near a smooth o s c i l l a t i n g p l a t e , where the f l u i d was otherwise a t r e s t . I n h i s experiments, he found

t h a t the amplitude of the f l u i d p a r t i c l e v e l o c i t y i n the t u r b u l e n t boundary l a y e r v a r i e d according t o a power law w i t h a coordinate z which measures the v e r t i c a l d i s t a n c e from the p l a t e . The phase s h i f t between the f l u i d p a r t i c l e v e l o c i t y and the v e l o c i t y of the p l a t e v a r i e d according to a l o g a r i t h m i c law w i t h the coordinate z. These r e s u l t s enabled him to determine the d i s t r i b u t i o n o f the eddy v i s c o s i t y i n the t u r b u l e n t boundary l a y e r . However, he d i d not i n v e s t i g a t e the laminar sub-layer and d i d not provide matching c o n d i t i o n s between the laminar and t u r b u l e n t r e g i o n s , thus, i t appears t h a t h i s study i s incomplete, as f a r as the d e t e r m i n a t i o n o f the boundary shear stresses are concerned.

Jonsson (1963) and Kamphuis (1975) used a closed water t u n n e l i n which the f l u i d o s c i l l a t e d i n a s i n u s o i d a l manner w i t h respect t o time. Such an apparatus could be described as a f l u i d o s c i l l a t i n g i n a "U" shaped tube. Jonsson measured the v e l o c i t y p r o f i l e near the bottom of h i s tank and f i t t e d the data to l o g a r i t h m i c curves, assuming t h a t steady s t a t e t u r b u l e n t boundary l a y e r c o n s i d e r a t i o n s were v a l i d . The constants obtained from the curve f i t t i n g enabled him to estimate the shear

stresses exerted on the bottom.

the average wave r e s i s t a n c e c o e f f i c i e n t i s a f u n c t i o n o f a wave Re3molds number, which Kamphuis (1975) d e f i n e d as Re = a.u /v, where a. i s the

0 max 0 amplitude of a f l u i d p a r t i c l e displacement j u s t o u t s i d e the boundary

l a y e r , u i s the maximum v e l o c i t y of such a f l u i d p a r t i c l e , and v i s the kinematic v i s c o s i t y of the f l u i d . For rough p l a t e s the average r e s i s t a n c e c o e f f i c i e n t i s a f u n c t i o n o f a./k , where k i s the

equiva-0 s s

l e n t surface roughness. Kamphuis (1975) obtained e m p i r i c a l r e l a t i o n s h i p s f o r the r e s i s t a n c e c o e f f i c i e n t s f o r both smooth and rough p l a t e s , and compared h i s r e s u l t s t o those obtained e x p e r i m e n t a l l y by Jonsson (1963), and the t h e o r e t i c a l ones p r e d i c t e d by K a j i u r a (1968). Considering the experimental u n c e r t a i n t i e s due t o measurement e r r o r s , and the t h e o r e t i c a l u n c e r t a i n t i e s due t o the approximations considered by K a j i u r a i n h i s

a n a l y s i s , the r e s u l t s of the t h r e e s t u d i e s appear t o agree reasonably w e l l .

The r e s i s t a n c e of s o l i d boundaries t o the f l o w of s o l i t a r y waves was s t u d i e d t h e o r e t i c a l l y by Keulegan (1948) and Iwasa (1959) f o r the

case of laminar f l o w . So f a r as t u r b u l e n t boundary l a y e r s are concerned, the author has no knowledge of t h e o r e t i c a l s t u d i e s of the cases of f l o w s under s o l i t a r y waves. Experimental s t u d i e s of the r e s i s t a n c e c o e f f i c i e n t under s o l i t a r y waves were conducted by Ippen, K u l l n and Raza (1955), and by Ippen and M i t c h e l l (1957).

Keulegan (1948) considered the l i n e a r i z e d equations o f motion and developed an expression f o r the v e l o c i t y d i s t r i b u t i o n i n the viscous boundary l a y e r f o r the general case of non-uniform d i s t r i b u t i o n of the f r e e stream v e l o c i t y along a s o l i d h o r i z o n t a l boundary. He then

namely

^ z=o

where y I s the dynamic v i s c o s i t y o f the f l u i d , u^^^ I s the h o r i z o n t a l v e l o c i t y component i n the boundary l a y e r , and z i s a v e r t i c a l coordinate w i t h i t s o r i g i n a t the boundary. For the s p e c i a l case of s o l i t a r y waves he assumed t h a t the f r e e stream v e l o c i t y d i s t r i b u t i o n along the wave, u, was given by

u(X) = - ^ 2 ^ , (2.10)

where C i s the wave speed, n i s the f r e e surface e l e v a t i o n above s t i l l water l e v e l , h i s the water depth and X = x-Ct i s a h o r i z o n t a l coordinate moving w i t h the wave, and where x and t are the s t a t i o n a r y h o r i z o n t a l coordinate and time r e s p e c t i v e l y . He also developed a r e l a t i o n s h i p between the shear stresses and the r a t e of wave h e i g h t a t t e n u a t i o n (due

to energy losses) along a channel, and used the experimental data which was observed by Scott-Russell (1844) t o v e r i f y h i s t h e o r e t i c a l develop-ments. The accuracy of h i s r e s u l t s i s , however, d o u b t f u l . The reason i s t h a t the approximate expression o f the v e l o c i t y (Eq. ( 2 . 1 0 ) ) , i s good only f o r small amplitude waves, i . e . , n/h « 1. For waves of l a r g e h e i g h t - t o - d e p t h r a t i o Eq. (2.10) does not describe the v e l o c i t y

a c c u r a t e l y . Furthermore, f o r waves of l a r g e h e i g h t - t o - d e p t h r a t i o the convective terms i n the equations o f motion are not small enough to be neglected compared to the l i n e a r terms, and the f u l l , n o n l i n e a r

equations have to be solved f o r a more accurate d e s c r i p t i o n of the boundary l a y e r .

Iwasa (1959) used a higher approximation (than Eq. (2.10)) f o r the f r e e stream v e l o c i t y . I n order to solve the complete ( n o n l i n e a r )

boundary l a y e r equations he considered the cases where he assumed e i t h e r a l i n e a r or a p a r a b o l i c v e l o c i t y p r o f i l e i n the boundary l a y e r . However, these assumptions were not v e r i f i e d e x p e r i m e n t a l l y .

Ippen, K u l l n , and Raza (1955) used the r e l a t i o n s h i p s which were developed by Keulegan (1948) and measured the a t t e n u a t i o n o f waves over both smooth and rough bottoms. Their r e s u l t s f o r smooth bottoms were I n c o n c l u s i v e , apparently due t o measurement u n c e r t a i n t i e s . For rough bottoms they found t h a t the r e s i s t a n c e c o e f f i c i e n t depended on the absolute value of the roughness s i z e i n a d d i t i o n t o the wave Reynolds number, Re, which they d e f i n e d as

, (2.11)

O

where ? i s the displacement of a f l u i d p a r t i c l e i n the free-stream near the bottom. Ippen and M i t c h e l l (1957) obtained the r e s i s t a n c e c o e f f i c i e n t from d i r e c t measurement o f the forces exerted on a p l a t e . I n t h e i r a n a l y s i s they considered higher approximation f o r the v e l o c i t y than t h a t used by Ippen e t a l . ( 1 9 5 5 ) ( i . e . , Eq. ( 2 . 1 0 ) ) . They also found t h a t the

r e s i s t a n c e c o e f f i c i e n t f o r rough beds depends on the absolute value of the roughness. T h e i r r e s u l t s seem t o be independent of the Reynolds

number, and since the values of the Reynolds numbers i n t h e i r experiments were l a r g e r than those i n the i n v e s t i g a t i o n o f Ippen e t a l . (1955), they assumed t h a t the two d i f f e r e n t s t u d i e s were conducted a t d i f f e r e n t f l o w regimes ( i . e . , t h a t t h e i r experiments were conducted i n the rough t u r b u -l e n t regime, w h i -l e the experiments o f Ippen e t a -l . (1955) were conducted

i n the t r a n s i t i o n t o rough t u r b u l e n t regime).

Results which show dependence on the absolute value of the rough-ness cannot be used i n cases where the roughrough-nesses are d i f f e r e n t from

those t e s t e d . I n order t o o b t a i n a more general r e l a t i o n s h i p f o r the r e s i s t a n c e c o e f f i c i e n t , these r e s u l t s should be examined from other aspects such as dimensional a n a l y s i s . However, n e i t h e r study provided an a n a l y s i s and e x p l a n a t i o n f o r the r e l a t i o n s h i p s which they found.

CHAPTER 3

THEORETICAL CONSIDERATIONS

I n the course of i n v e s t i g a t i o n o f the i n c i p i e n t motion o f bed m a t e r i a l under s o l i t a r y waves i t i s f i r s t necessary t o study the f l u i d mechanics o f these waves. Second, the hydrodynamic f o r c e s exerted on the bed p a r t i c l e s by the f l o w under s o l i t a r y waves must be determined and, f i n a l l y , the p r o p e r t i e s of the bed p a r t i c l e s have t o be chosen such t h a t they w i l l r e s i s t the hydrodynamic f o r c e s exerted on them. A review o f three e x i s t i n g t h e o r i e s o f the s o l i t a r y wave i s pre-sented i n Section 3.1. T h e o r e t i c a l c o n s i d e r a t i o n f o r the f o r c e s exerted on bed m a t e r i a l under s o l i t a r y waves are presented i n Section 3.2, and the c o n d i t i o n s r e q u i r e d f o r i n c i p i e n t motion o f bed p a r t i c l e s are discussed i n Section 3.3.

3.1 THE SOLITARY WAVE

The existence and the f o r m u l a t i o n s o f the s o l i t a r y wave are very w e l l known, t h e r e f o r e t h e o r e t i c a l developments w i l l not be analyzed here. Three t h e o r e t i c a l s o l u t i o n s of the s o l i t a r y wave equations which are

o f t e n r e f e r r e d t o i n the l i t e r a t u r e are those due t o Bousslnesq (1872) , McCowan (1891), and Laitone (1963). The surface p r o f i l e , the wave c e l e r i t y , and the f l u i d p a r t i c l e v e l o c i t y which are d e r i v e d from these t h e o r i e s are presented here f o r the convenience o f r e f e r e n c e . They w i l l l a t e r be compared t o those o f e x p e r i m e n t a l l y generated waves i n order t o determine which o f the three t h e o r i e s i s most s u i t a b l e f o r use i n con-j u n c t i o n w i t h the experimental study o f s t a b i l i t y o f armored bottoms.

The wave motion i s considered i n a two-dimensional space and i s

I l l u s t r a t e d i n F i g . 3.1. The wave c o n s i s t s of a s i n g l e surface e l e v a t i o n of height H t r a v e l i n g w i t h a speed C over a body o f water of constant depth h. The x coordinate i s located along the bottom of the f l u i d w i t h the z coordinate d i r e c t e d upward. The f l u i d i s unbounded i n the x d i r e c -t i o n . The wave Induces a f l o w f i e l d q ( x , z , -t ) = (u,v) where q i s -the v e l o c i t y vector ( denotes v e c t o r i a l q u a n t i t y ) , u and v are the h o r i z o n t a l and v e r t i c a l v e l o c i t y components r e s p e c t i v e l y and t i s the time. The surface e l e v a t i o n above s t i l l water l e v e l i s denoted by n ( x , t ) . The water away from the wave i s considered t o be a t r e s t .

Considering an Incompressible homogeneous f l u i d and an I r r o t a t i o n a l f l o w , the f l o w f i e l d can be represented by the v e l o c i t y p o t e n t i a l $

(such t h a t q = V$) s a t i s f y i n g Laplace's equation:

= 0 , _(3.1)

w i t h the boundary c o n d i t i o n a t t h e bottom:

I f = 0 ( a t z = 0 ) . _(3.2)

The kinematic c o n d i t i o n a t the f r e e surface i s

9t 8x 8x " 8z 0 ( a t z = h + n ( x , t ) ) , (3.3) and the dynamic c o n d i t i o n , n e g l e c t i n g surface tension, i s

+ j ( V $ ) 2 + gn = 0 ( a t z = h + r i ( x , t ) ) . _(3.4)

where g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n . The pressure a t the f ree

/ / / / / / / / / / / 7 ^

_{^ ^ / / / / / / // // / / / / / / / / / />}

surface i s taken t o be zero f o r convenience. The d i f f i c u l t y of the problem l i e s i n the n o n l l n e a r i t y o f the boundary c o n d i t i o n s a t the f r e e surface, the e l e v a t i o n of which i s unknown a priori and must be d e t e r -mined from the s o l u t i o n of the problem.

Expanding the v e l o c i t y p o t e n t i a l i n a power s e r i e s :

* = S $.z-^ , (3.5) j=o ^

Bousslnesq's (1872) s o l u t i o n to Eq. (3.1) w i t h the boundary c o n d i t i o n s (Eqs. (3.2), ( 3 . 3 ) , and (3.A)) can be considered as the f i r s t term i n the s e r i e s . McCowan (1891) c a r r i e d the s o l u t i o n t o the f i r s t term choosing d i f f e r e n t f u n c t i o n s 0^ t h a t represent the v e l o c i t y p o t e n t i a l . The s o l u t i o n of Laitone (1963) i s s i m i l a r t o t h a t o f Bousslnesq but contains higher order terms. Expressions f o r the surface p r o f i l e , wave c e l e r i t y , and f l u i d p a r t i c l e v e l o c i t y of the s o l i t a r y wave which were derived from these s o l u t i o n s are presented i n Table 3.1 i n terms o f the coordinate system (X,z) where X = x-Ct i s a c o o r d i n a t e system moving w i t h the wave t r a n s f o r m i n g i t t o a s t a t i o n a r y form. McCowan's s o l u t i o n i s shown ( i n Table 3.1) i n dlmensionless terms as presented by Munk (1949).

As can be seen i n Table 3.1, Bousslnesq's p r e s e n t a t i o n of the wave p r o f i l e and the wave speed are the same as the lowest order terms I n Laitone's f o r m u l a t i o n . The h o r i z o n t a l v e l o c i t y as expressed by

Bousslnesq i s d e r i v e d from c o n t i n u i t y c o n s i d e r a t i o n s assuming a uniform v e l o c i t y d i s t r i b u t i o n over the depth. The f i r s t order terms i n the expansion of t h i s expression f o r the v e l o c i t y i n a power s e r i e s of H/h are the same as those appearing i n Laitone's p r e s e n t a t i o n . The

Bousslnesq McCowan L a i t o n e Wave p r o f i l e n =

Hsech

^Vlf

(3 h N s i n M ( l + n / h ) M[cosM{l+n/h) + coshM^] (4

Wave speed C = /gh(l+H/h) l / ^ tanM

r M F l u i d p a r t i c l e v e l o c i t i e s h o r i z o n t a l u = (1 Cn h+n CN^l+cosM^

cosh

^^j

^o-üsM^-

+coshl

^j

( f f ( | i - l ) . e . H ^ ( „ f ) } v e r t i c a l v = (2 CN s i n M | s i n h l ^ ^cosW^ + cosh>^^ Notes 1) u I s averaged over t h e depth a p p l y i n g c o n t i n u i t y c o n s i d e r a t i o n 2) expression f o r the v e r t i c a l v e l o c i t y was not presented by Bousslnesq f o r s o l i -t a r y waves 3) t h e r e l a t i o n s h i p s f o r N and M are ! = 1 - [ l « ( - f ) ] 4)

s i m i l a r i t y and d i f f e r e n c e s between the presentations o f Bousslnesq and McCowan are not immediately seen because o f the complexity o f the expressions i n v o l v i n g the parameters M and N i n McCowan's f o r m u l a t i o n . However, McCowan himself noted t h a t the two s o l u t i o n s are s i m i l a r t o each o t h e r . A comparison between the surface p r o f i l e , the wave c e l e r i t y , and the v e l o c i t y d i s t r i b u t i o n o f an e x p e r i m e n t a l l y generated wave and the three t h e o r e t i c a l f o r m u l a t i o n s shown i n Table 3.1 a r e presented i n Section 5.1. The i n t e r e s t e d reader i s r e f e r r e d t o t h a t s e c t i o n f o r a more d e t a i l e d discussion.

3.2 THE HYDRODYNAMIC FORCES EXERTED ON BED MATERIAL UNDER SOLITARY WAVES

The i n v e s t i g a t i o n o f hydrodynamic forces exerted on s o l i d surfaces u s u a l l y c o n s i s t s o f boundary l a y e r c o n s i d e r a t i o n s , where the c o n d i t i o n s of I n t e r a c t i o n between the surface and the f l o w are taken i n t o account

( i . e . , smooth or rough surface, laminar or t u r b u l e n t f l o w , e t c . ) .

Solutions o f the equations o f motion i n the bottom boundary l a y e r under s o l i t a r y wave may provide a d i r e c t e s t i m a t i o n o f the shear stresses exerted on the bottom. Approximate s o l u t i o n s o f t h i s k i n d are known f o r the case o f smooth laminar f l o w (e.g., Keulegan (1948), Iwasa (1959)). The w r i t e r has no knowledge o f t h e o r e t i c a l s o l u t i o n s t o the cases o f t u r b u l e n t boundary l a y e r s under s o l i t a r y waves. Experimental i n v e s t i g a -t i o n o f -the v e l o c i -t y p r o f i l e i n -the -t u r b u l e n -t boundary l a y e r (see

Section 5.1.3) was unsuccessful. For these cases the shear stresses i n the boundary l a y e r are studied here based on c o n s i d e r a t i o n s o f wave energy d i s s i p a t i o n .

3.2.1 The Damping o f S o l i t a r y Waves

Consider the s o l i t a r y wave i n a channel as presented by Bousslnesq (1872)(see Table 3.1):

Ti(X) = H sech^aX , (3.6)

where n i s the surface e l e v a t i o n above s t i l l water l e v e l , H i s the wave h e i g h t , a=/3H/4h^, w i t h h being the water depth, and X = xCt i s a h o r i -z o n t a l coordinate moving w i t h the wave, i n which x i s a s t a t i o n a r y h o r i z o n t a l c o o r d i n a t e , C i s the wave c e l e r i t y , and t i s the time. The wave c e l e r i t y i s given by:

C = /gh(l+H/h) , (3.7)

where g i s the a c c e l e r a t i o n due t o g r a v i t y . The h o r i z o n t a l component o f the f l u i d p a r t i c l e v e l o c i t y , u, i s expressed as:

1+ — cosh'^aX

n.

Bousslnesq (1872) d i d not present an expression f o r the v e r t i c a l v e l o c i t y component under the s o l i t a r y wave. The t o t a l wave p o t e n t i a l energy per u n i t channel w i d t h , Ep^, can be described by:

00 3/2

E p ^ 4 p , g j ; ^ r i 2 d X = - | ^ p ^ g h 3 ( f ) . (3.9)

where i s the d e n s i t y o f the f l u i d . The t o t a l k i n e t i c energy per u n i t w i d t h , Ej^^, i s given by:

X=+«. z=h+Ti (X)

H/h 2 p „ g h 3 ( W h ) / ^

^ ^ " V 1+H/l

Eqs. (3.9) and (3.10) are s i m i l a r t o the expressions developed by Iwasa (1959). The d i f f e r e n c e s between Iwasa's expressions and Eqs. (3.9) and (3.10) are due t o the d i f f e r e n c e s between h i s p r e s e n t a t i o n o f the s o l i -t a r y wave and -the s o l i -t a r y wave due -t o Bousslnesq (1872) which i s con-sidered here. Iwasa's s o l i t a r y wave i s of higher order approximation and Includes expressions f o r the v e r t i c a l component of the f l u i d p a r t i c l e v e l o c i t y . However, h i s r e s u l t s show t h a t the k i n e t i c energy due t o t h i s v e l o c i t y component i s n e g l i g i b l e compared t o the p o t e n t i a l energy

and the k i n e t i c energy due t o the h o r i z o n t a l v e l o c i t y component.

The t o t a l wave energy per u n i t channel w i d t h , E-^, i s obtained by adding the p o t e n t i a l energy t o the k i n e t i c energy, i . e . , Eqs. (3.4) and

(3.10):

I

1+

J

(3.11) 1+H/h J

The t o t a l wave energy, E, i n a channel o f f i n i t e w i d t h , B, i s g i v e n by:

E = BE^ . (3.12)

As can be seen i n Eqs. (3.11) and (3.12), the t o t a l wave energy o f a given f l u i d (given p^) i n a channel o f constant depth and constant w i d t h i s a f u n c t i o n o f the wave h e i g h t - t o - w a t e r depth r a t i o only ( c o n s i d e r i n g

the g r a v i t a t i o n a l a c c e l e r a t i o n t o be c o n s t a n t ) .

As the wave propagates along a channel, the bottom and the w a l l s e x e r t stresses on the f l u i d . These stresses are the main cause f o r the

dE

d i s s i p a t i o n of wave energy. The r a t e o f energy d i s s i p a t i o n , can be obtained as f o l l o w s :

dE _ 3E d(H/h) , .

d t 3(H/h) d t • ^"^'"^"^^

During the time increment d t the wave t r a v e l s a d i s t a n c e dx = Cdt, thus, Eq. (3.13) becomes:

dE ^ 8E d(H/h)

dt ^ 8(H/h) dx • (^-^^^

S u b s t i t u t i n g Eq. (3.7) f o r C i n Eq. (3.14), and s u b s t i t u t i n g Eq. (3.12) i n t o Eq. (3.11) and d i f f e r e n t i a t i n g i t w i t h respect t o H/h y i e l d s

i

_+JUZiln

J g ( l . H | ) l , / ^ > W h l f h \ hi 2 r-^ l+H/hJ d(HZhI (3.15) d(x/h)

The r a t e of energy d i s s i p a t i o n i s obtained e x p e r i m e n t a l l y from measure-ments o f the a t t e n u a t i o n o f waves along the channel and s u b s t i t u t i o n o f the measured value o f

^(|^^|^j

i n Eq. (3.15).

Consider a shallow wide channel such t h a t the w i d t h i s much greater than the depth, hence the shear forces exerted on the w a l l s are n e g l i g i -ble compared t o those exerted on the bottom. For t h i s case the r a t e o f energy d i s s i p a t i o n i s equal t o the r a t e of work done by the f l u i d on the bottom (considering no energy sources or s i n k s i n the f l o w domain). Assuming t h a t the bottom shear stresses are u n i f o r m l y d i s t r i b u t e d across

the channel, t h i s I s expressed as:

(3.16)

where T, denotes the shear stresses exerted on the bottom. Eq. (3.16) b

describes a simple mechanical law t h a t the r a t e o f energy change o f a body I s equal t o the inner product o f the f o r c e a p p l i e d on the body and i t s v e l o c i t y . The minus sign on the r i g h t - h a n d side o f Eq. (3.16) accounts f o r the f a c t t h a t T, i s considered as the shear s t r e s s exerted

b

on the bottom r a t h e r than t h a t exerted by the bottom on the f l u i d . The bottom shear stress i s defined by means o f a bottom f r i c t i o n c o e f f i c i e n t . Cf , such t h a t

b

^ 2 ^ffePw"' (3.17)

S u b s t i t u t i o n o f Eq. (3.17) i n t o Eq. (3.16) y i e l d s :

^ = -^Bp^ j " C- u3dX . (3.18)

Considering Cf as a mean r e s i s t a n c e c o e f f i c i e n t f o r a wave, i t can be b

taken out o f the i n t e g r a l i n Eq. (3.18). I t i s then evaluated by equating Eq. (3.18) t o Eq. (3.15), i . e . .

7 3 J/2,3/2 g h 1+ Zn

4

H/h -| 1+H/h .-4 H/h 1+H/h d(H/h) d(x/h) (3.19) 1 i ; u3dx

where the bar over Cf denotes a mean value (averaged over the wave) b

I t has to be noted t h a t s u b s t i t u t i o n o f the value o f C^^ as given by Eq. (3.19) i n t o Eq. (3.17) may not n e c e s s a r i l y y i e l d the c o r r e c t d i s t r i b u t i o n of bottom shear stresses under s o l i t a r y wave, since the l o c a l values o f the f r i c t i o n c o e f f i c i e n t may be d i f f e r e n t from the mean r e s i s t a n c e co-e f f i c i co-e n t dco-efinco-ed by Eq. (3.19). I n f a c t , a p p l i c a t i o n o f thco-e mco-ean r e s i s t a n c e c o e f f i c i e n t t o Eq. (3.17) i m p l i e s t h a t the maximum shear s t r e s s occurs under the wave c r e s t , where the v e l o c i t y i s maximum. How-ever, K a j i u r a (1968), i n h i s i n v e s t i g a t i o n o f t u r b u l e n t boundary l a y e r s under o s c i l l a t o r y waves, showed t h a t there i s a phase l a g between the bottom shear s t r e s s p r o f i l e and the free-stream v e l o c i t y p r o f i l e . Keulegan (1948) showed t h a t the maximum bottom shear s t r e s s i n the smooth laminar boundary l a y e r s under s o l i t a r y waves occurs under the wave f r o n t near the c r e s t , but not d i r e c t l y under the c r e s t . Therefore, a p p l i c a t i o n o f the mean r e s i s t a n c e c o e f f i c i e n t may be inaccurate when used t o estimate l o c a l shear stresses and i t can o n l y be a p p l i e d t o problems where wave a t t e n u a t i o n i s concerned. Nevertheless, f o r two channels o f the same w i d t h and depth w i t h waves o f equal h e i g h t s , the stresses i n the channel o f stronger wave a t t e n u a t i o n are l a r g e r than the stresses i n the channel o f weak a t t e n u a t i o n . Therefore, the a p p l i c a t i o n of the mean r e s i s t a n c e c o e f f i c i e n t , although i t may not describe t h e c o r r e c t d i s t r i b u t i o n o f the shear s t r e s s e s , can be used q u a l i t a t i v e l y . When considering a r e p r e s e n t a t i v e shear s t r e s s t o be given by

s u b s t i t u t i o n o f the mean r e s i s t a n c e c o e f f i c i e n t and the maximum v e l o c i t y under the wave i n Eq. (3.17), i t i s expected t h a t the t r u e stresses