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Turbulence in open channels : an experimental study of turbulence structure over boundaries of differing… Nowell, Arthur Ralph Mackinnon 1975

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TURBULENCE IN OPEN CHANNELS: AN EXPERIMENTAL STUDY OF TURBULENCE STRUCTURE OVER BOUNDARIES OE DIFFERING HYDRODYNAtHC ROUGHNESS BY ARTHUR RALPH MacKINNON NOWELL B.A-, UNIVERSITY OF CAMBRIDGE, ENGLAND, 1969. A t h e s i s s u b m i t t e d i n t h e r e q u i r e m e n t D o c t o r o f p a r t i a l f u l f i l m e n t of f o r t h e degree o f P h i l o s o p h y i n t h e Department of Geography We accept 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 g u i r e d s t a n d a r d The U n i v e r s i t y o f B r i t i s h September 1975 Columbia In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shal make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shal not be alowed without my written permission. Department of Geography The University of British Columbia Vancouver 8. Canada Date 16 Sept. 1975 -6 i ABSTRACT R i v e r c h a n n e l s m o d i f y t h e i r b o u n d a r i e s by e n t r a i n m e n t and t r a n s f e r o f s e d i m e n t . The f l o w r e p r e s e n t s a t u r b u l e n t b o u n d a r y l a y e r w h i c h must be e x a m i n e d t o u n d e r s t a n d t h e r e l a t i o n between c h a n n e l m o r p h o l o g y and i t s m o d i f y i n g f l o w . D i s c r e t e r o u g h n e s s e l e m e n t s w i t h v a r i a b l e s p a c i n g s were used, i n f l u m e e x p e r i m e n t s t o s i m p l i f y t h e s t u d y o f t h e r e l a t i o n between f l o w r e s i s t a n c e and b o u n d a r y m o r p h o l o g y . The p a t t e r n s were i n t e n d e d t o model g r a v e l bed c h a n n e l s . The r e s u l t s a r e compared w i t h wind t u n n e l and a t m o s p h e r i c b o u n d a r y l a y e r s t u d i e s In o r d e r t o examine t h e i n f l u e n c e o f t h e f r e e s u r f a c e and t h e a p p l i c a b i l i t y o f t h e e x t e n s i v e r e s u l t s i n a i r t o open c h a n n e l f l o w . S t r e a m w i s e and v e r t i c a l v e l o c i t i e s were m e a s u r e d t h r o u g h o u t t h e d e p t h o f f l o w a t c o n s t a n t R e y n o l d s number o v e r a r o u g h b o u n d a r y . The r o u g h n e s s d ' e n s i t y was m o d e l l e d by p l a s t i c b l o c k s and t h e hot f i l m measurements were s u b s e q u e n t l y d i g i t a l l y a n a l y s e d . D e n s i t y i s d e f i n e d as t h e r a t i o o f t h e s u r f a c e a r e a o f b l o c k s , i n p l a n v i e w , t o t h e t o t a l a r e a o f t h e bad. R e s u l t s on s i x d e n s i t i e s between 1/8 and 1/80, b a s e d on p l a n a r e a s , a r e p r e s e n t e d i n d e t a i l . T h r e e l a y e r s were i d e n t i f i e d i n t h e f l o w : an o u t e r s h e a r l a y e r , a wake l a y e r e x t e n d i n g t o two r o u g h n e s s h e i g h t s a b o v e t h e bed, and a w a l l l a y e r below t h e t o p o f t h e r o u g h n e s s e l e m e n t s . No e x t e n s i v e r a n g e o f l i n e a r c o r r e l a t i o n o f s h e a r v e l o c i t y w i t h t h e l o g a r i t h m o f a r o u g h n e s s d e n s i t y f u n c t i o n i i was f o u n d , p o s s i b l y due t o t h e s u r f a c e between t h e r o u g h n e s s b l o c k s b e i n g h y d r o d y n a r a i c a l l y r o u g h . The g r e a t e s t r e s i s t a n c e t o f l o w was p r e s e n t e d by d e n s i t y 1/12: a t h i g h e r d e n s i t i e s * skimming• f l o w o c c u r r e d i n d i c a t e d by a s h i f t i n t h e 'bed' t o t h e t o p o f t h e r o u g h n e s s e l e m e n t s . The s h a p e o f t h e e n e r g y s p e c t r u m v a r i e d a c r o s s t h e f l o w d e p t h , t h e f r e e s u r f a c e a c t i n g as a b o u n d a r y c o n s t r a i n i n g t h e r a n g e o f i n e r t i a l t r a n s f e r . Most o f t h e t u r b u l e n t e n e r g y was c o n t a i n e d b e l o w 7.25 Hz w h i l e t h e c o s p e c t r u m o f u and v i n d i c a t e d t h a t most o f t h e s t r e s s l a y b e l o s . t h a t f r e q u e n c y . No s i g n i f i c a n t t u r b u l e n t t r a n s f e r s o c c u r r e d b e y o n d 5 Hz w h i l e t h e c o h e r e n c e was b elow. 0.2 beyond 10 Hz. No s i g n i f i c a n t i n e r t i a l s u b r a n g e was d e t e c t e d i n t h e s p e c t r a o r t h e c o s p e c t r a . The moments o f t h e v e l o c i t y d e r i v a t i v e i n d i c a t e d a s t r o n g l y i n t e r m i t t e n t p r o c e s s and e x h i b i t e d a marked c h a n g e a t t h e f r e e s u r f a c e , t h e k u r t o s i s t h e r e b e i n g n e a r l y t w i c e t h a t f o u n d e l s e w h e r e i n t h e f l o w . The p r o b a b i l i s t i c s t r u c t u r e o f t h e v e l o c i t y f l u c t u a t i o n s showed t h a t t h e s e r i e s were n o n - G a u s s i a n i n form and n o t s t r i c t l y s e l f - s i m i l a r . A l l t h e s e r i e s y i e l d e d H u r s t c o e f f i c i e n t s o f a p p r o x i m a t e l y 0.8-An e x p l a n a t i o n o f t h i s r e s u l t i s p o s t u l a t e d i n t e r m s o f t h e s u p e r i m p o s i t i o n o f s e v e r a l i n t e r m i t t e n t p r o c e s s e s . Measurements i n t h e f l o w a p p r o a c h i n g a s i n g l e b l o c k i n d i c a t e d t h a t v o r t i c i t y a m p l i f i c a t i o n o c c u r r e d . When o b s t a c l e w i d t h was i n c r e a s e d t o one t h i r d o f c h a n n e l w i d t h t h e i s o l a t e d b l o c k behaved as a two d i m e n s i o n a l o b s t a c l e i i i ( i . e . l i k a a bar) -R e s u l t s from measurements i n f o u r n a t u r a l g r a v e l channe l s i n d i c a t e d that the f lume reproduced s u c c e s s f u l l y the s p e c t r a l and p r o b a b i l i s t i c s t r u c t u r e of the f l ow. The r e s u l t s are a p p l i e d i n a phenomenolog ica1 e x p l a n a t i o n of the shape of the S h i e l d s ent ra inment f u n c t i o n and i n an e x p l a n a t i o n of the d i f f e r e n c e s i n entra inment and t r a n s p o r t between sand and g r a v e l . The r e s u l t s on the f l ow around an i s o l a t e d b lock a re a p p l i e d to the c o n s t r u c t i o n of a s t a t i s t i c a l f low s t a b i l i t y model f o r g r a v e l en t ra inment . The h i e r a r c h y o f p roces se s {low f requency ' w a v e s ' , b u r s t s , wake shedd ing and d i s s i p a t i o n ) , o p e r a t i n g i n t e r m i t t e n t l y and a c t i n g n o n - l i n e a r l y may be viewed as i n s t a b i l i t i e s which a r e r e s p o n s i b l e f o r the g e n e r a t i o n and d i s s i p a t i o n of t u r b u l e n c e . Such i n s t a b i l i t i e s are d i s c u s s e d i n r e l a t i o n to the h i e r a r c h y of m o r p h o l o g i c a l f e a t u r e s observed i n n a t u r a l c h a n n e l s . i v TABLE OF CONTENTS CHAPTER 1 INTRODUCTION I INTRODUCTION-. . . . 1 R a t i o n a l e and b a s i s - - . - . .----1 O b j e c t i v e s . . . . . -----2 River deformation and dominant frequencies.-- -...-3 Turbulence i n open channel f l o w s . . - - . .....9 I I ASSUMPTIONS . 14 I I I THEORETICAL BASIS AND REVIEW OF PREVIOUS WORK.. . - 1 6 Boundary l a y e r s - . . - - ...... 16 Flow-boundary i n t e r a c t i o n s - . - - - -.-.-17 Turbulence and i s o t r o p y . - . . . . . . . - . 1 8 P r o b a b i l i t y and i n t e r m i t t e n c y . . - . - .-..-21 Turbulence i n r i v e r s . . . . - .....23 IV MEASUREMENT AND ANALYSIS -. - 25 Hot f i l m measurement and re c o r d i n g - . . - ...-25 Flume and f i e l d o p e r a t i o n . . . . . -----28 D i g i t i z a t i o n and e r r o r a n a l y s i s . . . - . .....33 A n a l y s i s . . . - .....34 CHAPTER 2 EXPERIMENTAL RESULTS FOR ROUGHNESS ARRAYS PHYSICAL ANALYSIS I MEAN VELOCITY..... 4 3 Establishment of the f l o w . . . . . ..... 43 Mean v e l o c i t y d i s t r i b u t i o n s , . . . . ...-.48 I I TURBULENCE INTENSITY, DISSIPATION AND SCALES. 65 Turbulence i n t e n s i t y . . - . . ..... 65 D i s s i p a t i o n . . . . . .- ... 73 Microscale and raacroscale----- .....75 Summary of mean flow r e s u l t s and i n t e r p r e t a t i o n of i n t e g r a l measures of turbulence..... ...... 80 I I I ENERGY DISTRIBUTION AND FREQUENCY STRUCTURE ...85 Spectra, s c a l e s and s c a l i n g . . . - - -..-.85 Conclusions f o r s p e c t r a l measures.-... -----107 Cospectra and coherence...-- .-..-108 I s o t r o p y - . . - . -..--115 Conclusions f o r c o s p e c t r a l measures..... .....121 IV CONCLUSIONS --• 122 CHAPTER 3 EXPERIMENTAL RESULTS FOR ROUGHNESS ARRAYS STATISTICAL ANALYSIS I MOMENTS.. .....124 I I STRUCTURE FUNCTIONS. . 133 I I I CORRELATIONS . -146 IV INTERMITTENCY.. 149 V V RESCALED RANGE ANALYSIS ..-.-168 VI CONCLUSIONS ..177 CHAPTER 4 BLOCKS, BARS AND OPEN CHANNELS I BARS... 181 I I SINGLE BAR..... , 186 I I I SINGLE BLOCK .- .....193 IV FIELD RESULTS-.... .....203 CHAPTER 5 BED INSTABILITY AND TURBULENCE INTERACTIONS, IMPLICATIONS AND DISCUSSION I INTRODUCTION.-... ... 214 I I FLOW STRUCTURE INSTABILITY..--. 2 16 Experimental r e s u l t s . . - - - .....216 Measurements over g r a v e l . . . . . .....218 S p a t i a l v a r i a t i o n of i n s t a b i l i t i e s . . . . . .....219 I I I SEDIMENT ENTRAINaENT ..223 Sh i e l d s curve..... ...-.223 P a r t i c l e i n s t a b i l i t y and v i b r a t i o n s . . . . . .....225 Phenomenological d e s c r i p t i o n of S h i e l d s curve-... ..-.228 Dominant p a r t i c l e d i s t r i b u t i o n . . . . . -.-..229 P r o b a b i l i s t i c flow s t r u c t u r e . . . . - ....-232 IV STATISTICAL STABILITY APPROACH TO ENT RAINMENT. -233 B u f f e t i n g of b l u f f bodies-.... ..-..233 Role of p a r t i c l e v i b r a t i o n s . . . . . .....237 Conclusions..... ....-238 V SEDIMENT TRANSPORT.-..- 239 P r o b a b i l i s t i c measures-.--. .....239 S a l t a t i o n . 241 Suspended l o a d . - . . - .....242 Sand and g r a v e l t r a n s p o r t . . . . . .....243 VI HIERARCHIES OF FORM AND PROCESS..-,. 244 I n t r o d u c t i o n . . - . - ....-244 Bedforms i n sand.---. -.-.-245 Bedforms i n g r a v e l . . . - - .....246 Bars i n sand and g r a v e l - - - - - .....247 Hierarchy of i n s t a b i l i t i e s - - . . . .....248 CHAPTER 6 CONCLUSIONS I CONCLUSIONS 253 BIBLIOGRAPHY I -.-..258 APPENDIX I MEAN VALUE BOUNDARY LAYER FORMULATIONS i Power law f o r m u l a t i o n s . . . . . 277 i i Logarithmic laws-.... .....279 i i i Sediment entrainment..... .....282 v i APPENDIX I I PLOW BOUNDARY INTERACTIONS i Roughness a r r a y s - . - - . ....-284 i i I s o l a t e d b l o c k s . . . - . ..-.-287 APPENDIX I I I TURBULENCE' AND ISOTROPY i D e f i n i t i o n - . - . . ..-.-292 i i S p a t i a l s t r u c t u r e . - . . . -----294 i i i Scales and r a t e s of production and d i s s i p a t i o n . . . ---295 i v Rates of energy t r a n s f e r . - . - - --...298 v S p e c t r a l measures i n the atmosphere..... ..-.-301 v i I s o t r o p y - - . . . . - - 3 0 5 APPENDIX IV INTERMITTENCY i I n t e r m i t t e n c y - - - - - ---..309 APPENDIX V TURBULENCE RESEARCH IN OPEN CHANNELS i Open channels...-- --,..313 APPENDIX VI VELOCITY MEASURING TECHNIQUES i Measurement technigues..... .....317 APPENDIX VII HOT FILMS: OPERATION AND LIMITATIONS i Hot f i l m s . - . - - .....322 APPENDIX V I I I TECHNIQUES USED IN CALIBRATION i C a l i b r a t i o n . . . . . -....3 36 APPENDIX IX ANALYSIS i S p e c t r a l a n a l y s i s . - . - - -...-341 i i P r o b a b i l i t y a n a l y s i s - . . . - ,..-.347 v i i L I S T OF TABLES Table I F t e s t f o r cumulative spectra . . . . . 1 0 0 Table I I F t e s t f o r cumulative spectra . . . . . 1 0 0 Table I I I V e l o c i t y increments-excess and d e f i c i t s ...... 140. Table IV Skewness f u n c t i o n and Koliaogorov constant . . . . . 1 4 5 Table V Skewness at small separation and of du/dt .....145 Table VI Moments and value of u from s p e c t r a .....163 Table VII Hurst c o e f f i c i e n t f o r one density . . . . . 1 7 2 v i i i LJJll 0.1 FIGURES Figure 1 Trace of u,v .---.11 Figure 2 Diagram of pump ..-.-2 9 Figure 3 Contour map of mean v e l o c i t i e s -.-.-31 Figure 4 Photograph of po r t a b l e "bridge" p i e r .....32 Figure 5a Block diagram of s p e c t r a l programs .....35 Figure 5b Block diagram of p r o b a b i l i t y programs --...35 Figure 6 P l o t of exceedance s t a t i s t i c s -....38 Figure 7 Rescaled range diagram .....4 0 Figure 8 Shape f a c t o r along the flume .-.--44 Figure 9 Lego roughness blocks i n flume ..... 47 Figure 10a Power law f o r roughness d e n s i t i e s .....49 Figure 10b Logarithmic p l o t of power law ...,.49 Figure 11a n'acd H' p l o t t e d against d e n s i t y .....51 Figure 11b Comparison of n' measured two ways .....52 Figure 12 Skin f r i c t i o n c o e f f i c e n t .....54 Figure 13a V e l o c i t y defect law .....56 Figure 13b Log p l o t of i n n e r and outer laws ..---57 Figure 14 V e l o c i t y p l o t with displacement t h i c k n e s s -.--.60 Figure 15 Shear v e l o c i t y c o r r e l a t i o n s ..-..63 Figure 16a Turbulence i n t e n s i t y fay r e l a t i v e depth -....66 Figure 16b Turbulence i n t e n s i t y c l o s e to bed .....6 8 Figure 17a n'/V a g a i n s t r e l a t i v e depth .--..71 Figure 17b u'/U^£cagainst r e l a t i v e depth .....71 Figure 18a Energy d i s s i p a t i o n -.-..74 Figure 18b Energy d i s s i p a t i o n r a t e -..--76 Figure 19a M i c r o s c a l e by spectrum and analog methods .....77 Figure 19b Mi c r o s c a l e against r e l a t i v e depth ..-.-77 Figure 20 Macroscale against r e l a t i v e depth .....81 Figure 21 Ratio of shear v e l o c i t i e s .....84 Figur e 22a Spectra at seven depths f o r one roughness .....86 Figure 22b Spectra f o r four roughnesses a t one depth ......88 Figure 23 Spectra f o r four depths .....90 Figure 24a V e r t i c a l s p e c t r a on nondimensional axes ..-..91 Figure 24b Streamwise spectra on nondimensional axes ..-..92 Figure 25a Streamwise spectra to show i n f l u e n c e of u* .....92 Figure 25b Streamwise spectra showing s h i f t with depth ....93 Figure 26a Cumulative s p e c t r a at one depth .....96 Figure 26b Cumulative spectra at one depth .....97 Figure 26c Cumulative spectra at one depth -.-..98 Figure 27 M i c r o s c a l e against shear v e l o c i t y ....102 Figure 28a Kolmogorov microscale against r e l a t i v e depth .102 Figure 28b Kolmogorov microscale ....102 Figure 29a U n i v e r s a l s p e c t r a l p l o t s at one depth ....104 Figure 29b U n i v e r s a l s p e c t r a l p l o t s at seven depths ....105 Figure 30a C o s p e c t r a l p l o t s f o r four depths .-..110 Figure 30b C o s p e c t r a l p l o t s f o r three d e n s i t i e s ....110 Figure 31a S p e c t r a l c o r r e l a t i o n f o r four depths .-..113 Figure 31b S p e c t r a l c o r r e l a t i o n for' four d e n s i t i e s ....113 Figure 32 Spectra of u and v overlapped ....117 Figure 33a Angle of s t r a i n ....119 Figure 33b U n i v e r s a l d i s s i p a t i o n r a t e f u n c t i o n ....119 Figure 34a P r o b a b i l i t y d i s t r i b u t i o n and d e n s i t y of u ....125 i x F i gure 34b P r o b a b i l i t y d i s t r i b u t i o n and den s i t y of u 2 ---.126 Figure 34c P r o b a b i l i t y d i s t r i b u t i o n and d e n s i t y of u 3 ....127 Figure 34d P r o b a b i l i t y d i s t r i b u t i o n and d e n s i t y of u* ...-128 Figure 35a Accumulation of variance ....130 Figure 35b Accumulation of skewness ....131 Figure 35c Accumulation of k u r t o s i s ....132 Figur e 36a P r o b a b i l i t y of increments at 0.1 cm --..135 Figure 36b P r o b a b i l i t y of increments at 1.0 cm ...-136 Figure 36c P r o b a b i l i t y of increments at 10-0 cm -.-.137 Figure 36d P r o b a b i l i t y of increments at 100.0 cm ..-,.138 Figure 37a Second order s t r u c t u r e f u n c t i o n --..142 Figure 37b Skewness f u n c t i o n --..142 Figure 37c Fl a t n e s s f u n c t i o n --.-143 Figur e 38a S p a t i a l c o r r e l a t i o n , v e r t i c a l s e p a r a t i o n -...147 Figure 38b S p a t i a l c o r r e l a t i o n , v e r t i c a l s e p a r a t i o n ..--147 Figure 38c S p a t i a l c o r r e l a t i o n , h o r i z o n t a l s e p a r a t i o n ..-.148 Figur e 38d S p a t i a l c o r r e l a t i o n , h o r i z o n t a l s e p a r a t i o n ....148 Figure 39 Coherence against freguency -..-150 Figure 40a Negative peak d i s t r i b u t i o n .---152 Figur e 40b P o s i t i v e peak d i s t r i b u t i o n -..-153 Figure 40c Exceedance s t a t i s t i c s .-..154 Figure 41a Turbulent Reynolds number ....157 Figure 41b Skewness of du/dt against r e l a t i v e depth ....157 Fi g u r e 41c K u r t o s i s of du/dt against r e l a t i v e depth ....157 Figure 42a Skewness against k u r t o s i s of du/dt ....159 Figure 42b K u r t o s i s against t u r b u l e n t Reynolds number ....159 Fi g u r e 42c Skewness against t u r b u l e n t Reynolds number ...-160 Figure 43a Higher order spectra at y =0-1 ....164 Figure 43b Higher order s p e c t r a a t y =1-54 ....165 Figure 43c Higher order s p e c t r a at y - s u r f a c e ....166 Figure 44a Rescaled range p l o t of v e l o c i t y -..-171 Figure 44b Rescaled range p l o t of du/dt -...176 Figure 45 Shear v e l o c i t y c o r r e l a t i o n .....183 Figure 46 P r o b a b i l i t y d e n s i t y f o r bars ......185 Figure 47a S e l f - p r e s e r v i n g p l o t f o r bars ..-.188 Figure 47b S e l f - p r e s e r v i n g p l o t f o r part-bars .-..188 Figure 48a Mean v e l o c i t y f o r va r i o u s bar widths ....190 Figure 48b Turbulence v e l o c i t y f o r va r i o u s bar widths .-..190 Figure 49 D i s s i p a t i o n r a t e s downstream from a bar .....195 Figur e 50a Spectra downstream from a block .....196 Figur e 50b Spectra downstream from a bar .....197 Figure 51a P r o b a b i l i t y d e n s i t i e s : s i n g l e block .-...199 Figure 51b P r o b a b i l i t y d e n s i t i e s : s i n g l e bar .....200 Figur e 52a Spectra at two p o s i t i o n s ahead of a block .....202 Figure 52b Cumulative spectra ahead of a block .....202 Figure 53 Spectra showing e f f e c t of f r e e s u r f a c e .....205 Figure 54 Spectra at f i v e depths f o r Cheekye Creek .-...207 Figur e 55 D i s s i p a t i o n s p e c t r a f o r Cheekye .....208 Figure 56a Cospectra f o r two depths i n Bridge Creek -..--209 Figure 56b S p e c t r a l c o r r e l a t i o n f o r Bridge Creek .....209 Figure 57 P r o b a b i l i t y d e n s i t i e s f o r f i e l d data ..-..211 Figure 58 S h i e l d s curve and k u r t o s i s of du/dt .....224 Figure 59 S t a b i l t y approach f o r entrainment ...--235 Figure 60 E f f e c t i v e roughness against d e n s i t y .....286 Figure 61 S a t u r a t i o n a i r content i n water .....334 xi LIST OF SYMBOLS A Constant A ' Surface area A f F r o n t a l area C Fun c t i o n o f roughness d e n s i t y i n Dvorak's e q u a t i o n D Flow depth {also equal t o 6 ) D • Dynamic h a l f - r a n g e of r e c o r d e r D 90 P a r t i c l e s i z e i n sediment t r a n s p o r t equations H Hurst c o e f f i c i e n t H ' Shape f a c t o r L Turbulent macroscale Ls Averaging l e n g t h i n s p e c t r a l c a l c u l a t i o n s N (y«) Freguency of c r o s s i n g s of value y 1 p Mean pressure Q RMS g u a l i t y of s i g n a l R Adjusted range E U v S p e c t r a l c o r r e l a t i o n c o e f f i c i e n t Re Flow .Reynolds number Re x Turbulent Reynolds number A S Adjusted range SO Slope Sa Surrounding area egual to A'/no Free stream v e l o c i t y u Instantaneous v e l o c i t y u Mean v e l o c i t y o(y) Mean v e l o c i t y at h e i g h t y b» EMS of s i g n a l c C o n c e n t r a t i o n C f Skin f r i c t i o n c o e f f i c i e n t a Zero plane displacement here taken equal to h D i f f u s i o n tensor e' RMS s i g n a l t o n o i s e r a t i o f Non-dimensional freguency g G r a v i t a t i o n a l constant h Height of roughness elements k Wavenumber k « Kolmogorov con s t a n t k' » von Karman cons t a n t kn E g u i v a l e n t sand s i z e ks Kolmogorov m i c r o s c a l e m Number of standard d e v i a t i o n s in j\ P r o d u c t i o n parameter i n t u r b u l e n t energy e q u a t i o n m 2 D i f f u s i o n parameter i n t u r b u l e n t energy eguation n Freguency n' Power law exponent 1/n' no Number of roughness elements i n s u r f a c e area A* P() P r o b a b i l i t y r Length s c a l e ; s e p a r a t i o n d i s t a n c e s Length of r e c o r d ; sample s i z e s i S i l h o u e t t e area u' RL1S v e l o c i t y f l u c t u a t i o n s u 2 Shear v e l o c i t y x i i u, v,w F l u c t u a t i o n i n s t r e a m w i s e , v e r t i c a l and c r o s s - s t r e a m d i r e c t i o n s x,y,z Streamwise, v e r t i c a l and c r o s s - s t r e a m d i r e c t i o n s y' D i s t a n c e from mean of normalized s e r i e s y° S c a l i n g l e n g t h yn Roughness le n g t h y Non-dimensional d i s t a n c e yu %/ v a s P r i n c i p a l a x i s of s t r a i n y 2 Coherence 6 Boundary l a y e r t h i c k n e s s : taken as flow depth 5* Displacement t h i c k n e s s e Rate of energy d i s s i p a t i o n n Kolmogorov m i c r o s c a l e n' S e l f - s i m i l a r parameter f o r wake flow 6 Momentum t h i c k n e s s !$\ T u r b u l e n t macroscale A T a y l o r m i c r o s c a l e X e Downstream spacing of roughness elements Af C o e f f i c i e n t of f r i c t i o n Constant i n equation f o r v a r i a n c e of log-normal d i s t r i b u t i o n v Kinematic v i s c o s i t y p Sediment d e n s i t y p S F l u i d d e n s i t y a) Variance T Time s e p a r a t i o n T" C r i t i c a l shear s t r e s s <f>c Measured one-dimensional energy spectrum ADDITIONAL SYMBOLS USED IN APPENDICES A Area of p a r t i c l e Crj Drag c o e f f i c i e n t E Mean v o l t a g e Eo Mean v o l t a g e at zero flow I Current Nu Nusselt number Pr P r a n d t l number R Resistance S s S p e c i f i c g r a v i t y of p a r t i c l e d s P a r t i c l e diameter f(x) Streamwise v e l o c i t y c o r r e l a t i o n g (x) Transverse v e l o c i t y c o r r e l a t i o n To Wall shear s t r e s s x i i i ACKNOWLEDGEMENTS I should l i k e to thank Mr. A. M o i l l i e t of D E E P Esquimalt, f o r h i s help i n c a l i b r a t i n g the hot f i l m s and Mr. E. Meldrum, Department of Geophysics, U.B.C. f o r a s s i s t a n c e i n d i g i t i z i n g the analog r e c o r d s . Dr. E. Hicken, Simon F r a s e r U n i v e r s i t y , k i n d l y provided access to flume f a c i l i t i e s and Dr. M.C. Quick allowed use of a two channel anemometer and gave f r e e access to the H y d r a u l i c Laboratory F a c i l i t i e s at U.B.C. durin g p r e l i m i n a r y c a l i b r a t i o n . The o r i g i n a l equipment.was purchased from a grant from the P r e s i d e n t ' s Fund to Mr. M.A. Church, while o p e r a t i n g expenses were provided from N.E.C. Grant A7950 to Mr... M.A. Church. Personal funding was provided from U.B.C- and K i l l a a F e l l o w s h i p s . I t i s a plea s u r e to thank Mr. M.A. Church who su p e r v i s e d t h i s d i s s e r t a t i o n . His c o n s c i e n t i o u s i n t e r e s t , s k i l l f u l c r i t i c i s m and i n t e l l e c t u a l impetus have pr o v i d e d the necessary c o n d i t i o n s f o r r e s e a r c h . I should a l s o l i k e to thank my t h e s i s committee who read and c r i t i c i z e d the o r i g i n a l d r a f t of the d i s s e r t a t i o n . I would l i k e t o p a r t i c u l a r l y thank Alan and S h e r r i , E r i c and Diane and Hans and P a t r i c i a ; t h e i r valued f r i e n d s h i p p r o v i d e d the uisge beatha f o r s u r v i v a l . x i v PREFACE Within the past twenty years geomorphology has abandoned i t s t r a d i t i o n a l stance as a g e o g r a p h i c a l l y based d e s c r i p t i v e study of landforms and has given a t t e n t i o n to the more rewarding aspects of the processes t h a t e f f e c t landscape change- I t i s only very r e c e n t l y t h a t the idea of geomorphology as a branch of a p p l i e d physics ( i . e . a p p l y i n g the r e s u l t s and methods of p h y s i c s to the study of landscape) has begun to gain acceptance-Studies of r i v e r form and behaviour occupy a c e n t r a l p a r t of geomorphology s i n c e f l u v i a l processes appear to c o n s t i t u t e the most important t e r r e s t r i a l landscape forming agency- There e x i s t s a reasonable b a s i s f o r s t u d i e s of r i v e r s whose, goal i s to develop a p h y s i c a l l y based understanding of the processes i n v o l v e d (cf. Leopold, Wolman and S i l l e r 1964: Henderson 1966: Scheidegger 1971: Bagnold 1972). The hydrodynamics of f l o w i n open channels would appear to be i n t r a c t a b l e t o a n a l y t i c s o l u t i o n . Open channel flows are h i g h l y t u r b u l e n t shear flows with intense and varying boundary roughness. This research s t u d i e s one aspect of t h i s flow s i t u a t i o n f o c u s s i n g a t t e n t i o n on the s t r u c t u r e of turbulence and i t s r e l a t i o n to the boundary c o n d i t i o n s . The techniques of measurement and a n a l y s i s , and the background l i t e r a t u r e are a l l u n f a m i l i a r i n f l u v i a l geomorphology. Whilst v i t a l to the understanding of the r a t i o n a l e and b a s i s of t h i s research, the lengthy d e t a i l s are placed i n Appendices I to IX, as the m a t e r i a l i s standard w i t h i n the f i e l d s of turbulence research, a p p l i e d meteorology and a p p l i e d aerodynamics- Appendices I t o V deal with the background l i t e r a t u r e and concepts i n more d e t a i l than i s presented In Chapter 1 Section I I I . Each appendix deals with a d i f f e r e n t body of l i t e r a t u r e while the l a s t c o n s i d e r s the work done to date i n open channel f l o w s . Appendices VI t o IX d e a l with the measurements and techniques of a n a l y s i s . They have been described i n great d e t a i l to i n d i c a t e the care that should be taken i n making these measurements f o r they cover much of the • f o l k l o r e ' a s s o c i a t e d with hot f i l m operation-1 CHAPTER _1 I INTRODUCTION Rat i o n a l e and Basis The r a t i o n a l e f o r the present study of the flo w i n open channels i s to gain an understanding of the c o n d i t i o n s under which sediment i s entrai n e d and transported i n r i v e r s - The long range goal r e q u i r e s various stages, beginning with an e v a l u a t i o n of c l a s s i c a l sediment t r a n s p o r t formulae and mean-value h y d r a u l i c t h e o r i e s developed t o understand the behaviour of r i v e r s and ca n a l s . Such s t u d i e s have been undertaken ( c f . Engelund and Hansen 1967: Haddock 1969: Y a l i n 1972) and while such e m p i r i c a l and s e m i - e m p i r i c a l f o r m u l a t i o n s provide reasonable .estimates f o r p r e d i c t i n g average e f f e c t s , the d e t a i l e d behaviour of any s i n g l e channel does not seem t o be p r e d i c t e d by such r e l a t i o n s h i p s . The p o s s i b i l i t y of s u b s t a n t i a l progress i n understanding r i v e r channel form and behaviour l i e s then i n a d e t a i l e d study of the flow s t r u c t u r e and i t s r e l a t i o n to channel morphology. An approach based on pu r e l y s t o c h a s t i c methods i s a f u r t h e r a l t e r n a t i v e (cf. P i t t s b u r g h Symposium 1971: Chiu 1971); but to provide a b a s i s f o r d i s c r i m i n a t i n g between a l t e r n a t i v e models i t i s necessary to r e s o r t to measurements and a p h y s i c a l l y based r a t i o n a l e . 2 It.appears p r o f i t a b l e t o i n v e s t i g a t e the d i s t r i b u t i o n s of v e l o c i t y , energy and s t r e s s i n the flow i n order t o determine the c a p a c i t y of the stream to modify i t s boundaries and the p o i n t s at which t h i s c a p a c i t y i s concentrated. I t i s t h i s step t h a t i s being attempted i n t h i s t h e s i s . I f the r o l e of high frequency, s m a l l s c a l e processes can be evaluated, the r o l e played by intermediate s c a l e s , such as secondary c u r r e n t s , may be evaluated using s i m p l e r measurement techniques. The o v e r a l l s t r u c t u r e of the processes r e s p o n s i b l e f o r channel m o d i f i c a t i o n may be i n c o r p o r a t e d i n t o a model of the f o r c e s o p e r a t i n g i n any f r e e s u r f a c e channel flow. The b a s i s f o r t h i s research grows from an e x t e n s i v e survey of the l i t e r a t u r e from s t u d i e s of a p p l i e d c l a s s i c a l h y d r a u l i c s , wind e f f e c t s on b u i l d i n g s to recent research on the intermitt-ency of t u r b u l e n t f l o w s i n the atmosphere. Two streams of thought are examined. F i r s t l y the s c a l e of f r e q u e n c i e s and the notion of dominant wavelengths i s examined w i t h i n the h i e r a r c h y of f l u v i a l bedforms. Secondly the r o l e of turbulence, i t s mechanics and s t r u c t u r e i s considered. Taken together, these f o c i i provide the b a s i s f o r t h i s stage of the long term research s t r a t e g y of the c o n t i n u i n g i n v e s t i g a t i o n of channel form and process. O b j e c t i v e s The d e t a i l e d o b j e c t i v e s of the present research are f i v e - f o l d : 3 1) to measure the s t r u c t u r e of turbulence i n f r e e s u r f a c e shear flow, using d i g i t a l techniques t o evaluate c e r t a i n s t a t i s t i c a l measures of the v e l o c i t y f l u c t u a t i o n s ; 2) to study the e f f e c t s of v a r i o u s roughness d e n s i t i e s on the mean and f l u c t u a t i n g components of the flow s t r u c t u r e , and t o examine the notion that there e x i s t s a 'dominant* roughness p a t t e r n and thus e l u c i d a t e the flow mechanics f o r roughness d e n s i t i e s t h a t exert the greatest r e s i s t a n c e t o f l o w ; 3) to compare measurements made i n a f r e e s u r f a c e , rough w a l l t u r b u l e n t boundary l a y e r with previous r e s u l t s o b t a i n e d , at s i m i l a r Reynolds numbers, i n zero pressure g r a d i e n t boundary l a y e r s i n a i r ; 4) to apply some of the f i n d i n g s about turbulence to the sediment entrainment problem to provide a p h y s i c a l argument f o r the various forms, of entrainment f u n c t i o n developed by purely dimensional reasoning; 5) to o u t l i n e a research methodology that i n c o r p o r a t e s the notion of a r i v e r as a t u r b u l e n t boundary l a y e r i n t o a s t a b i l i t y a n a l y s i s of the problem of sediment entrainment. Si/Z^r i n f o r m a t i o n and dominant frequencies Open channel flow i s a troublesome phenomenon f o r f l u i d mechanics. I t i s u s u a l l y summarily t r e a t e d a f t e r d e t a i l e d c o n s i d e r a t i o n s of boundary l a y e r s and pipe f l o w . Furthermore, vrhile much d e t a i l i s given about e m p i r i c a l r e s i s t a n c e formulae ( p r i n c i p a l l y Manning's formula) and about flow regimes ( c f . Reynolds 1974), there i s very l i t t l e c o n s i d e r a t i o n given t o the boundary l a y e r parameters and t o the turbulence s t r u c t u r e of t h i s type of f l o w . Even engineering t e x t s devoted s o l e l y to open channel flow (such as Chow 1959: Henderson 1965) give no d e t a i l s about turbulence or i t s r o l e i n open channels, e i t h e r i n d i s s i p a t i n g energy, or i n a f f e c t i n g the shear s t r e s s o p e r a t i n g at the boundary, or i n t r a n s p o r t i n g sediment. Free surface flows are a sub-group of boundary l a y e r f l o w s . Boundary l a y e r s may be defined as flows i n which d i s t a n c e from a s o l i d surface and the p r o p e r t i e s of that surface (such as roughness height and d i s t r i b u t i o n ) determine, and hence may be used to s c a l e , the flow c h a r a c t e r i s t i c s such as v e l o c i t y , turbulence . s t r u c t u r e and shear s t r e s s . Free surface flows i n nature are v i r t u a l l y always t u r b u l e n t ; the .major, t r a n s f e r s of momentum are s i g n i f i e d by the Eeynolds s t r e s s e s . Boundary l a y e r s are cus t o m a r i l y d e l i m i t e d by a f r e e stream v e l o c i t y and a th i c k n e s s which i s defined as tha t d i s t a n c e where the v e l o c i t y reaches 99% of the f r e e stream value. The r e s i s t a n c e to the flow which determines t h i s t h i c k n e s s i s made up of s e v e r a l p a r t s : a) the roughness of the s u r f a c e , which i s f r e q u e n t l y c h a r a c t e r i z e d by an equi v a l e n t sand s i z e roughness(kn); b) form roughness, where flow s e p a r a t i o n may occur and the r e s i s t a n c e i s augmented by shear zones and energy d i s s i p a t i o n i n vortex shedding; c) the p o r o s i t y of the boundary; 5 d) d i s t o r t i o n s of the free s u r f a c e . In open channels t h i s p i c t u r e i s f u r t h e r complicated by the f r e e s u r f a c e , f o r t h i s a l l o w s one f u r t h e r degree of freedom, i n t h a t the flow may be sub- or super- c r i t i c a l . In t h i s t h e s i s we are p r i m a r i l y concerned with the f i r s t two aspects of r e s i s t a n c e , which may be termed p a r t i c l e and form roughness and how t h i s t o t a l boundary roughness i n f l u e n c e s the s t r u c t u r e of the flow. The behaviour of r i v e r channels i s complex viewed at any s c a l e . The l a r g e plan-form meanders which c h a r a c t e r i z e many channels seem to demand study at. the l a r g e s t s c a l e , t h e i r r e g u l a r i t y being suggestive of coherent f o r c e s o p e r a t i n g over l a r g e d i s t a n c e s . Whilst t h e i r i n i t i a t i o n might be examined as a v o r t i c i t . y r e l a t e d phenomenon {Quick 1974) , t h e i r subsequent growth and development r e q u i r e s study of the t h r e e dimensional nature of the bed and banks and i n f o r m a t i o n on the secondary c u r r e n t s that i n t e r a c t with the t u r b u l e n t flow to l o c a l i z e zones of scour and d e p o s i t i o n . The v e r t i c a l r e l i e f of a r i v e r bed, manifested at the largest, s c a l e by the pool and r i f f l e sequence, appears to be a s i m i l a r v o r t i c i t y r e l a t e d phenomenon, but the d i s t a n c e over which such r e g u l a r p e r i o d i c i t i e s occur i n d i c a t e s again t h a t coherent f o r c e s which s c a l e with some m u l t i p l e of channel width are c o n s i s t e n t l y o p e r a t i v e (Leopold, Wolman and M i l l e r 1964). Within any such pool and r i f f l e seguence, given s u i t a b l e sediment c h a r a c t e r i s t i c s , dunes may occur. These f e a t u r e s , which s c a l e with depth, appear to be r e l a t e d to the 6 macro-turbulent s t r u c t u r e of the flow (Velikanov 1936 quoted i n E a u d k i v i 1967: Y a l i n 1972). They propaqate along the stream bed out of phase with the v a r i a t i o n s i n the l e v e l of the f r e e surface. Behind each, when f u l l y developed, i s a s e p a r a t i o n zone where considerable energy d i s s i p a t i o n occurs. Such dunes develop to a maximum height f o r any imposed sl o p e and stream v e l o c i t y and only when the flow i s t u r b u l e n t . T h e i r r e g u l a r i t y of shape and spacing w i l l thus i n f l u e n c e the turbulence s t r u c t u r e of the flow and apparently r e s u l t s i n a s t a b l e morphological f e a t u r e . These f e a t u r e s c o n t r i b u t e t o the o v e r a l l channel r e s i s t a n c e most markedly through form roughness. The r e l a t i o n between macroturbulent s t r u c t u r e and dunes i s not s t r a i g h t f o r w a r d . The d i s c u s s i o n between Kennedy(1971) and Yalin(1971) h i g h l i g h t s the problems of i n t e r p r e t a t i o n of form and process. The c l o s e r e l a t i o n of flow s t r u c t u r e to bed form i s f u l l y d i s c u s s e d i n Chapters 2 though 4 where the measurements are considered and i t s i m p l i c a t i o n s f o r the n o t i o n of dominant roughness and s t a b i l i t y approaches to sediment entrainment and t r a n s p o r t are discussed i n Chapter 5. At the s m a l l e s t s c a l e , independent of channel s i z e , r i p p l e s are observed i n sand bed channels. They appear to be r e l a t e d t o the s t r u c t u r e of the i n n e r boundary l a y e r and have been treated as a d e n s i t y phenomenon (Bagnold 1956: Y a l i n 1972). He claimed that r i p p l e s tend to appear simultaneously over the whole bed, r a t h e r than by downstream propagation, i n d i c a t i n g that they are s t a b i l i t y phenomena which occur when 7 two f l u i d s o f d i f f e r i n g d e n s i t y are sheared past one another. He a l s o claimed that turbulence tends to destroy these f e a t u r e s by causing b u r s t s of momentum t r a n s p o r t i n g f l u i d away from the w a l l and d i s r u p t i o n s t o the simple shear flow r e q u i r e d to maintain such f e a t u r e s . The r o l e of the f r e e surface has hardly been s t u d i e d i n i t s e f f e c t on the flow as a t u r b u l e n t boundary l a y e r , except f o r the pioneering work of Kennedy(1969) i n examining anti-dunes. Aside from the obvious e f f e c t s of s u r f a c e waves the free s u r f a c e a l s o a c t s to r e d i s t r i b u t e the t u r b u l e n t energy from the v e r t i c a l component v i a the pressure v e l o c i t y c o r r e l a t i o n to the downstream and c r o s s stream v e l o c i t y components. The presence of a f r e e s u r f a c e means, f o r a f u l l y developed eguibibrium boundary l a y e r , t h a t there w i l l be a d e f i n i t i o n a l problem with respect to 6;, the t h i c k n e s s of the boundary l a y e r , and U „ the f r e e stream v e l o c i t y . . Gravel bed r i v e r s , l i k e a l l other n a t u r a l channels, e x h i b i t a h i e r a r c h y of forms. The h i e r a r c h y of dunes and r i p p l e s i s mainly studied i n sand bed channels. Gravel bed channels do not deform i n the same manner f o r changes i n flow r a t e s . While g r a v e l dunes are known, the s m a l l e r f e a t u r e s have not been observed. Rather than the boundary form r e s i s t a n c e of r e g u l a r flow-induced p e r i o d i c f e a t u r e s such as dunes and r i p p l e s found i n sand channels, the r e s i s t a n c e to flow i s dominated by i n d i v i d u a l p a r t i c l e roughness. The absence of a uniform viscous sublayer w i l l mean g r a v e l channels are always hydrodynamically rough when sediment 8 movement occurs. The boundary roughness of the g r a v e l cannot be considered i n the c l a s s i c a l manner of Nikuradse as dense packed gr a i n s of uniform s i z e . The range of p a r t i c l e s i z e s and the d i s t r i b u t i o n of these s i z e s on a n a t u r a l r i v e r bed, i n r e l a t i o n t o the flow s t r u c t u r e around them, has not been s t u d i e d . The e a r l y work of Schlichting(1936) and Colebrook and White(1937) i n d i c a t e d t h a t spacing of elements profoundly a f f e c t e d the r e s i s t a n c e . - -Morris(1955) proposed, three types of boundary f l o w i n t e r a c t i o n : skimming, wake i n t e r a c t i o n and i s o l a t e d roughness, which r e s u l t e d from the spacing of the elements and t h e i r i n t e r a c t i o n with the flow. Skimming flow occurs f o r dense packed elements where the c l o s e spacing a f f o r d s mutual p r o t e c t i o n . I s o l a t e d block roughness occurs when the blocks act as i n d i v i d u a l wake shedding b l o c k s i n a rough boundary l a y e r . Wake i n t e r a c t i o n i s the flow s t a t e t h a t r e s u l t s when the wakes from blocks on the boundary i n t e r a c t to i n c r e a s e the r e s i s t a n c e s i g n i f i c a n t l y . Work at the U n i v e r s i t y of Iowa under Rouse ( c f . summary i n Rouse 1965) examined the e f f e c t of spacing on the r e s i s t a n c e at the boundary. Much s i m i l a r work has been c a r r i e d out i n wind tunnels and i n the atmosphere (cf.summary i n Wooding e t a l 1973). No s t u d i e s e x i s t which examine how the spacing a f f e c t s the turbulence s t r u c t u r e of the f l o w . This t h e s i s i s concerned with t h i s aspect, and with how the turbulence s t r u c t u r e may thus be modified and i n t e r a c t with 9 t h e b o u n d a r y t o e n t r a i n c e r t a i n p a r t i c l e s . T u r b u l e n c e i n open c h a n n e l f l o w s T u r b u l e n c e i s a p r o p e r t y o f t h e f l o w and i s u s u a l l y d e s c r i b e d by a ' s y n d r o m e ' o f c h a r a c t e r i s t i c s . The m o s t c o n c i s e p r e s e n t a t i o n o f t h e s e i s g i v e n by B r a d s h a w ( 1 9 7 1 ) T u r b u l e n c e i s a t h r e e d i m e n s i o n a l t i m e - d e p e n d e n t m o t i o n i n w h i c h v o r t e x s t r e t c h i n g c a u s e s v e l o c i t y f l u c t u a t i o n s t o s p r e a d t o a l l w a v e l e n g t h s b e t w e e n a minimum d e t e r m i n e d by v i s c o u s f o r c e s and a maximum d e t e r m i n e d b y t h e b o u n d a r y c o n d i t i o n s o f t h e f l o w . I t i s t h e u s u a l s t a t e o f f l o w a t h i g h R e y n o l d s n u m b e r s . B r a d s h a w (1971, p. 17) A f u l l l i s t o f c h a r a c t e r i s t i c s i s g i v e n i n T e n n e k e s a r i d L u m l e y ( 1 9 7 1 ) and i n L u m l e y and P a n o f s k y (1 964) - F o r t h e p r e s e n t , e m p h a s i s w i l l be l a i d on t h o s e c h a r a c t e r i s t i c s o f t u r b u l e n c e t h a t were m e a s u r e d i n t h i s r e s e a r c h , t h e f i r s t b e i n g t h a t t u r b u l e n c e may be r e g a r d e d a s a s t o c h a s t i c p r o c e s s . C l a s s i c a l work a s s u m e d , a n d e a r l y r e s e a r c h i n d i c a t e d , t h a t t h e p r o c e s s was G a u s s i a n , b u t i t i s b e c o m i n g i n c r e a s i n g l y o b v i o u s t h a t t h e f l u c t u a t i o n s i n t r o d u c e m a r k e d d e p a r t u r e s f r o m a n o r m a l d i s t r i b u t i o n . W h i l s t t h e g e n e r a t i o n o f t u r b u l e n c e i n a b o u n d a r y s h e a r f l o w a p p e a r s t o be d o m i n a t e d by t u r b u l e n t b u r s t s , i n t e r m i t t e n t i n t i m e a n d r a n d o m l y s p a c e d , t h e c o n c e p t o f l o c a l i s o t r o p y a t t h e h i g h e r f r e q u e n c i e s , a d v o c a t e d by K o l m o g o r o v , was b e l i e v e d t o a p p l y . 10 The work o f N o v i k o v and S t e w a r t ( 1 9 6 4 ) and o f G u r v i c h a n d Yaglom(1967) i n d i c a t e d t h a t t h e d i s t r i b u t i o n o f e n e r g y d i s s i p a t i o n was a l s o m a r k e d l y i n t e r m i t t e n t , and h e n c e t h e n o t i o n o f a s i m p l e G a u s s i a n p r o c e s s i s u n s a t i s f a c t o r y ( F i g u r e 1 b ) . A s e c o n d d e f i n i n g c h a r a c t e r i s t i c o f t u r b u l e n c e i s t h a t i t must be s i g n i f i e d i n t e r m s o f a l e n g t h s c a l e . T he m a c r o s c a l e ( L ) o f m o t i o n , r e s p o n s i b l e f o r e x t r a c t i n g e n e r g y f r o m t h e mean f l o w , n o r m a l l y s c a l e s w i t h d i s t a n c e f r o m t h e b o u n d a r y . The e n e r g y i s t h e n c a s c a d e d t o h i g h e r f r e q u e n c i e s where i t i s f i n a l l y d i s s i p a t e d i n t o h e a t . W h i l e t h e m a c r o s c a l e i s w e l l d e f i n e d , t h e r e a r e two m i c r o s c a l e s f r e q u e n t l y u s e d : t h e T a y l o r m i c r o s c a l e ( \) , which c o r r e s p o n d s t o t h a t f r e q u e n c y a t which t h e d i s s i p a t i o n s p e c t r u m p e a k s , a n d t h e K o l m o g o r o v m i c r o s c a l e ( n) » w h i c h i s a b o u t an o r d e r o f m a g n i t u d e s m a l l e r and i s t h e s c a l e a t w h i c h v i s c o s i t y f i r s t e x e r t s a c o n s i d e r a b l e e f f e c t . A t l o w e r f r e q u e n c i e s , ( c o r r e s p o n d i n g t o l a r g e r s c a l e s ) v i s c o s i t y s h o u l d n o t p l a y a s i g n i f i c a n t r o l e , p r o v i d e d t h e R e y n o l d s number i s s u f f i c i e n t l y l a r g e . The r o l e o f t u r b u l e n c e i n f r e e s u r f a c e f l o w s i s p r i m a r i l y as an e n e r g y d i s s i p a t o r . The u s u a l l y r o u g h b o u n d a r y , t h e moderate R e y n o l d s number and t h e s t r o n g l y s h e a r e d f l o w r e s u l t i n t u r b u l e n c e l e v e l s ( t h e r a t i o o f r o o t mean s q u a r e o f t h e f l u c t u a t i o n s (u•) t o mean v e l o c i t y ) c l o s e t o t h e bed i n e x c e s s o f 20% d e c r e a s i n g t o a r o u n d 4% a t t h e f r e e s u r f a c e , d e p e n d i n g on t h e r e l a t i v e r o u g h n e s s . The ELgure l a Trace of u and v components (upper channel re p r e s e n t v lower channel u). T o t a l r e c o r d shown here r e p r e s e n t s 4 seconds. Chart speed 50 mm/sec Figur e lb Trace of u(upper channel) and du/dt. T o t a l r e c o r d shown here represents 4 seconds. 12 e x t r a c t i o n of energy from the mean flow occurs at l a r g e s c a l e s , the macroscale being approximated by flow depth. The Talyor microscale i s of order 5x10~ 3 i but v a r i e s with depth i n c r e a s i n g u n t i l mid-depth and then becoming a constant o r weakly i n c r e a s i n g f u n c t i o n . Wilson(1974) has examined the s c a l e s over which energy was t r a n s f e r r e d and concluded t h a t i n t e r a c t i o n s of s c a l e s i z e s d i f f e r i n g by more than a f a c t o r of ten d i d not s i g n i f i c a n t l y c o n t r i b u t e to the energy t r a n s f e r s . The i n t e r m i t t e n t nature of the energy d i s s i p a t i o n , i n d i c a t e d by k u r t o s i s values of up to 12 f o r the v e l o c i t y d e r i v a t i v e at t u r b u l e n t Reynolds numbers (u»x/ v) around 200, has an e f f e c t at even low . f r e q u e n c i e s , t h a t i s at l a r g e s c a l e s . Recent work f o l l o w i n g Kolmogorov's m o d i f i c a t i o n of h i s o r i g i n a l hypothesis showed that the s p e c t r a l shape depends to some extent on t h i s i n t e r m i t t e n c y . The work of (Torino and Brodkey (1969) , K l i n e (cf. Kim et a l 1971) and of Willmarsh(1974) and h i s co-workers has shown tha t the i n t e r m i t t e n c y phenonomenon has strong r a m i f i c a t i o n s throughout the flow. Results from c o n d i t i o n a l sampling of the record i n d i c a t e t h a t about 70% of the energy of the shear i s associated with bursts from the boundary while the frequency of these b u r s t s i s shown to c o r r e l a t e with some tu r b u l e n t macroscale. Boundary l a y e r flows i n water have been e x t e n s i v e l y used i n v i s u a l i z a t i o n programmes. The work of K l i n e confirmed the presence of s t r e a k s : the importance of the s t r e a k s and 13 i n t e r m i t t e n t b u r s t s of shear s t r e s s generating p a r c e l s of low momentum f l u i d i n the i n i t i a t i o n of motion of f i n e sand p a r t i c l e s was examined by Sutherland (1967). The presence of these s t r e a k s over a rough g r a v e l boundary was confirmed and e x t e n s i v e l y surveyed by Grass (1971) u t i l i z i n g hydrogen bubble techniques to o b t a i n instantaneous values of the v e l o c i t y and shear s t r e s s . Sutherland (1967) and Apperley{1967) q u a l i t a t i v e l y a p p l i e d notions about b u r s t s to the entrainment of sediment. The measurements of Grass (1971) over rough boundaries show that the burst c y c l e may be of great s i g n i f i c a n c e i n e n t r a i n i n g f i n e sediment by causing very l a r g e i n c r e a s e s i n the shear s t r e s s at the boundary. The t r a n s p o r t of sediment depends on an upward t r a n s p o r t of momentum to keep the sediment from being deposited. Few measurements of the v e r t i c a l s t r u c t u r e of turbulence i n water have been made and even fewer spectra of the energy a s s o c i a t e d with the dominant s c a l e s of motion presented. As gr a v e l channels u s u a l l y have a l a r g e range of p a r t i c l e s i z e s , measurements c l o s e to the bed are q u a l i t a t i v e l y comparable with measurements made over a very rough s u r f a c e i n the atmosphere, such as those made over a f o r e s t or i n a model urban area. Data obtained i n t h i s t h e s i s are compared with such measurements, and a l s o with data obtained around i s o l a t e d b l u f f bodies i n a t u r b u l e n t boundary l a y e r -As a range of p a r t i c l e s i z e s might r e s u l t i n a •dominant' p a r t i c l e s p a t i a l d i s t r i b u t i o n on the bed, e q u a l l y 1 4 w e l l each p a r t i c l e may act as a b l u f f body. Measurements i n f r o n t o f , and i n the wake of, b l u f f bodies i n d i c a t e c e r t a i n frequencies may be p r e f e r e n t i a l l y a m p l i f i e d by v o r t i c i t y s t r e t c h i n g and the generation of such dominant eddies may induce some flow i n s t a b i l i t y that could r e s u l t i n s e l e c t i v e sediment entrainment. II -ASSUMPTIONS If. a r i v e r channel i s to be considered as a t u r b u l e n t boundary l a y e r , then any model of the shear flow must e n t a i l s i m p l i f i c a t i o n s of the equations of motion as i n t h e i r most general form the equations have def i e d s o l u t i o n . 1) I t i s assumed that the instantaneous v e l o c i t y of the stream i s made up of two components " U = U±+ u ± 1=1,2,3 a mean component and a f l u c t u a t i n g one whereby u=0, t h a t i s the mean of the f l u c t u a t i o n s i s zero; 2) The i n t r o d u c t i o n of averaging r e q u i r e s d e f i n i t i o n of the type of averaging used. I t i s necessary f o r l o g i s t i c a l reasons to assume the ergodic hypothesis, t h a t i s that the ensemble average of any random v a r i a b l e can be estimated from an average over space or over time; 3) As a l l measurements are made at one p o i n t through time, the r e p r e s e n t a t i o n of such a random v a r i a b l e as the streamwise v e l o c i t y component r e q u i r e s the assumption of 15 Ta y l o r ' s 'frozen turbulence' hypothesis, notably d_ _ -1 d_ dx U dt The l i m i t s of v a l i d i t y of t h i s assumption have been discussed by Webster{1972) but i t i s necessary to assume T a y l o r ' s hypothesis throughout i f s p a t i a l s c a l e s are to be evaluated i n the streamwise d i r e c t i o n . To avoid t h i s d i f f i c u l t y the wavenumber may be considered to be j u s t the r e c i p r o c a l of l e n g t h ; U) The shear flow i s assumed to be two dimensional and homogeneous i n the h o r i z o n t a l plane; 5) The v e l o c i t y i s i n the streamwise d i r e c t i o n ; and d i r e c t i o n i s i n v a r i a n t with height; t h a t i s 6) a s t a t i o n a r y s t a t e i s assumed, with uniform and steady f l o w ; 7) No l o c a l convergences or divergences e x i s t i n the h o r i z o n t a l plane which i m p l i e s t h a t no secondary c u r r e n t s are present; 8) I t i s assumed that the only body f o r c e i s g r a v i t y and t h a t t r a n s p o r t by molecular a c t i o n i s n e g l i g i b l e by comparison with t u r b u l e n t t r a n s p o r t . While these assumptions are not a l l l i k e l y t o be f u l f i l l e d i n a n a t u r a l channel, most may be held to be v a l i d i n l a b o r a t o r y f l o w s . Reynolds number s i m i l a r i t y i s a l s o i m p l i e d throughout t h i s research. 16 I I I THEORETICAL BASIS AND REVIEW OF PREVIOUS WORK Boundary lay_ers The mean v e l o c i t y p r o f i l e over a rough sur f a c e i s described by a power law ' TT 1/n' U Ks' (1) This equation i s flow Reynolds number dependent. The exponent i s a l s o a f u n c t i o n of the s k i n f r i c t i o n c o e f f i c i e n t c f , so that changes i n surface roughness, i f the flow i s at a constant Reynolds number, should be r e f l e c t e d In v a r i a t i o n s i n the exponent. As t h i s dependence i s not a n a l y t i c a l l y known the more common approach, based upon dimensional arguments, i s to c o n s i d e r rough w a l l boundary l a y e r s as being comprised of two zones. There i s an i n n e r l a y e r , where d i s t a n c e from the w a l l , roughness l e n g t h (yg) and shear v e l o c i t y (u s) are the s i m i l a r i t y parameters; and an outer l a y e r where fr e e stream v e l o c i t y , shear v e l o c i t y and d i s t a n c e from the w a l l are the s c a l i n g parameters. S i m i l a r i t y i s here taken to mean that a length s c a l e and a v e l o c i t y s c a l e are s u f f i c i e n t to determine the s t r u c t u r e of the flow (Stewart and Townsend 1951). The law of the w a l l and the defect law are f u l l y described i n S c h l i c h t i n g (1968) and Reynolds (1 974) . Rough w a l l flows were st u d i e d by Nlkuradse (1933) who modified the law of the w a l l r e p l a c i n g shear v e l o c i t y and 17 v i s c o s i t y by the e q u i v a l e n t sand s i z e of the p a r t i c l e s on t h e bed.. Such an approach can o n l y be a p p l i e d to dense packed uniform s i z e g r a i n s . The e f f e c t o f d i s t r i b u t e d roughness elements i s not amenable to t h i s approach ( S c h l i c h t i n g 1936) and c o n s i d e r a b l e work on t h i s problem has been undertaken because of i t s s i g n i f i c a n c e t o s o i l e r o s i o n and wind break problems (Marshall 1971: P l a t e 1971 f o r summaries). Flow boundary i n t e r a c t i o n s The e a r l y work of S c h l i c h t i n g (1936) and Colebrook and White (1937) i n d i c a t e d t h a t the flow over roughness elements behaved very d i f f e r e n t l y depending on the s p a c i n g o f elements. In open channels Sayre and Albertson{1961) and Koloseus and Davidian (1966) attempted t o summarise the e f f e c t by p l o t t i n g r e s i s t a n c e a g a i n s t roughness c o n c e n t r a t i o n and o b t a i n i n g an e q u i v a l e n t roughness h e i g h t f o r a c e r t a i n range of c o n c e n t r a t i o n s , which c o u l d then be used i n t h e l o g a r i t h m i c law f o r a rough w a l l boundary l a y e r . Perry e t al(1969) i d e n t i f i e d two d i s t i n c t roughness c o n c e n t r a t i o n s : one where the elements were c l o s e l y spaced so t h a t a s t a b l e eddy was contained between the elements. T h i s p a t t e r n would s c a l e with boundary l a y e r t h i c k n e s s . Second they i d e n t i f i e d an h-type roughness where the height of the element (h) i n r e l a t i o n to boundary l a y e r t h i c k n e s s was the important s c a l i n g l e n g t h . Phenomenologically, the best statement was p r o v i d e d by Morris (1955) d e s c r i b e d e a r l i e r who i d e n t i f i e d t hree flow 18 boundary i n t e r a c t i o n types, skimming, i s o l a t e d blocks and wake i n t e r f e r e n c e type roughness. I t would appear l i k e l y t h a t there would be a spacing that presented the g r e a t e s t r e s i s t a n c e to flow i n the wake i n t e r a c t i o n group. Dvorak (1969) s t u d i e d t h i s group and f o r bars found that the most e f f e c t i v e c o n c e n t r a t i o n was at 4.68, based on a r a t i o of f r o n t a l areas t o surrounding bed area. This work i s summarized i n Gartshore (1973). The i s o l a t e d block flow type has been considered from another viewpoint by Counihan et al ( 1 9 7 4 ) , by Bearman (1974) and by Maull and Young(1974). The behaviour of a b l u f f body i n a rough w a l l boundary l a y e r has been l i t t l e s t u d i e d and t h i s work focusses a t t e n t i o n on the wake shedding c h a r a c t e r i s t i c s and the boundary l a y e r s t r u c t u r e downstream of a s i n g l e block. The e f f e c t of a block on the tur b u l e n c e s t r u c t u r e and pressure d i s t r i b u t i o n has been considered by Sadeh and Cerraak{1972) who examined the notion of v o r t i c i t y a m p l i f i c a t i o n - V o r t i c i t y a m p l i f i c a t i o n i s due to the s t r e t c h i n g of vortex f i l a m e n t s i n the d i v e r g i n g s t a g n a t i o n flow around the block. Such a m p l i f i c a t i o n occurs at s e l e c t e d l e n g t h s l a r g e r than the n e u t r a l s c a l e r e s u l t i n g i n a l o c a l c o n c e n t r a t i o n of energy at c e r t a i n low f r e q u e n c i e s , whose s c a l e i s determined by the s i z e of the b l u f f body ( c f . Sadeh et a l 1970 a,b) . Xii£^ iiIsiL£S £LH<1 isotrojvy Turbulence i n a boundary l a y e r may be examined i n two 19 manners. F i r s t l y the d i s t r i b u t i o n s of the f l u c t u a t i n g q u a n t i t i e s across the boundary l a y e r should s c a l e with the same parameters as the mean flow; that i s two region s of fl o w should be i d e n t i f i a b l e . The d i s t r i b u t i o n of roughness elements has the f u r t h e r e f f e c t t h a t , as b l u f f b o d i e s , they w i l l shed v o r t i c e s i n t o the flow. which w i l l r e s u l t i n a d i s t o r t ion.of the b a s i c shear flow and r e s u l t p o s s i b l y i n a region of approximately uniform v e l o c i t y and turbulence c h a r a c t e r i s t i c s . T h i s r e g i o n , the wake zone, w i l l be found at the i n t e r s e c t i o n of the inner and outer l a y e r s . Secondly the turbulence being advected past any point may be regarded as a time s e r i e s and i t s s t r u c t u r e as a continuous random v a r i a b l e examined. This n o t i o n was o r i g i n a l l y p o s t u l a t e d by G.I. Taylor (1935) and i s the basi s f o r the s t a t i s t i c a l theory of turbulence (cf- Lumley . 1970: Panchev 1971: Rosenblatt 1972). The k i n e t i c energy e x t r a c t e d from the mean flow due t o i t s r e t a r d a t i o n caused by the roughness appears as f l u c t u a t i n g energy. The f l u c t u a t i n g v e l o c i t y may be squared and averaged to y i e l d the t u r b u l e n t k i n e t i c energy and the p a r t i t i o n i n g of t h i s energy i n t o i t s frequency components by a F o u r i e r a n a l y s i s y i e l d s the energy spectrum: v i z -(2) J°<j>(k)dk = u 2 (3) 20 where k i s the wavenumber Ls i s a leng t h s c a l e or averaging per i o d and u i s the t u r b u l e n t v e l o c i t y component (See Hinze 1959). The e a r l y n otions of the cascading of energy from low t o high freguency was r e f i n e d by Kolmogorov (1941, 1962) i n h i s three hypotheses. The f i r s t considered that the s m a l l s c a l e components of the turb u l e n t energy were i n e q u i l i b r i u m and independent of the mean fl o w , and here l o c a l i s o t r o p y should be present. This notion may be examined by ta k i n g the r a t i o of the spectra of the v and u components at a frequency where i s o t r o p y should be expected from the c r i t e r i o n of Pond et al ( 1 9 6 3 ) . Kolmogorov's second hypothesis was tha t f o r s u f f i c i e n t l y high Reynolds numbers there should be a range, independent of the 'energy c o n t a i n i n g ' eddies which e x t r a c t the energy from the mean flow, and s t i l l u n a f f e c t e d by v i s c o s i t y , where the turbulence would be defined by E the ra t e at which energy i s being removed. This i s c a l l e d the i n e r t i a l subrange and the spectrum should here have the u n i v e r s a l form • o o = *• e 2 / V 5 / 3 These ideas apply e q u a l l y t o the c r o s s spectrum which i s composed of two p a r t s , the r e a l part or c o i n c i d e n t s p e c t r a l d e n s i t y and the imaginary part or quadrature s p e c t r a l d e n s i t y f u n c t i o n . The area under the co-spectrum of the u and v 21 components re p r e s e n t s the freguency c o n t r i b u t i o n s to the Reynolds s t r e s s . The normalized cospectrum, when no r m a l i z a t i o n i s by the u and v s p e c t r a , or s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t i s a measure of the t r a n s f e r e f f i c i e n c y of energy from one component to the other. P r o b a b i l i t y , and i n t e r m i t t e n c y As t u r b u l e n t motion i s a continuous f u n c t i o n of space and time and i s regarded as a s t o c h a s t i c v a r i a b l e a study of the p r o b a b i l i s t i c s t r u c t u r e of the v e l o c i t y f l u c t u a t i o n s i s a necessary concomitant to a study of the freguency composition. The d i s t r i b u t i o n of f l u c t u a t i o n s was o r i g i n a l l y held to be Gaussian (see Batchelor 1953) but i n c r e a s i n g evidence shows t h a t the s t r u c t u r e departs from Gaussian by e x h i b i t i n g a d i s t i n c t element of s u r p r i s e i n that r a r e events, the t a i l s of the d i s t r i b u t i o n , are more evident than should be expected. The p r o b a b i l i t y d i s t r i b u t i o n s and d e n s i t i e s may be computed, along with the powers of the o r i g i n a l s e r i e s t o examine the ranges t h a t are over-represented and f o r which then t h e r e should be a p h y s i c a l e x p l a n a t i o n . Figure 1a shows a t r a c e of the u and v components of v e l o c i t y i n the flume. Without knowledge of the time s c a l e such a t r a c e could have come from a hypersonic wake or from high a l t i t u d e c l e a r a i r turbulence. This suggests that the s e r i e s may be s e l f - s i m i l a r : that i s s t r e t c h i n g the s e r i e s i n time when accompanied by a m p l i f i c a t i o n i n amplitude i n 22 proportion to the s t r e t c h i n g w i l l maintain the s t a t i s t i c a l s t r u c t u r e . S e l f - s i m i l a r i t y , l i k e s i m i l a r i t y a p p l i e d to. the mean q u a n t i t i e s , gives the s c a l i n g parameter of the s e r i e s . This may be examined by c a l c u l a t i n g the d i s t r i b u t i o n of increments at various l a g s and nor m a l i z i n g them. I f s t r i c t s e l f - s i m i l a r i t y holds the d i s t r i b u t i o n s should be Gaussian, i f the o r i g i n a l s e r i e s i s Gaussian. The r e s c a l e d range may be c a l c u l a t e d , which measures the r a t i o of the range to the standard d e v i a t i o n with i n c r e a s i n g l e n g t h of record. I f the slope of the p l o t i s gr e a t e r than 0.5 t h i s i s a man i f e s t a t i o n of the memory of the s e r i e s or a measure of the i n t e r m i t t e n c y of the processes. As turbulence i s now v i s u a l i z e d as being the su p e r i m p o s i t i o n of many processes which e x h i b i t d i f f e r i n g memories (wake shedding, b u r s t i n g from the w a l l , d i s s i p a t i o n r a t e f l u c t u a t i o n s ) , the r e s u l t a n t s e r i e s may be crudely Gaussian, but i t s c o n s t i t u e n t p a r t s w i l l be non-normal s e r i e s that through the n o n - l i n e a r c o u p l i n g i n the Navier-Stokes equation mean i t may be impossible to ignore the sm a l l s c a l e f l u c t u a t i o n s . The l o g a r i t h m i c normal d i s t r i b u t i o n i s a s s o c i a t e d with the a r r i v a l of r a r e events. The t h i r d Kolmogorov hyopthesis recognised that the energy d i s s i p a t i o n r a t e e i s not constant but v a r i e s with time and space and Novikov and Stewart (1964) and Gurvich and Yaglom (1967) i n d i c a t e d e might be approximated by a l o g a r i t h m i c normal d i s r i b u t i o n of the form a . ^ = A + u l n ( L / r ) (5) where u i s u s u a l l y taken as 0.5 and r as the T a y l o r m i c r o s c a l e . The e f f e c t of t h i s would be to a l t e r the s p e c t r a l shape i n the i n e r t i a l subrange, but more i m p o r t a n t l y the i n t e r r e l a t i o n of the f l u c t u a t i o n s of d i s s i p a t i o n and b u r s t s from the w a l l may be s i g n i f i c a n t through the n o n ^ l i n e a r coupling mechanism. The spectrum i n t u r b u l e n t shear flow may be s t u d i e d using the phenomenological approach of Tchen(1953) and Panchev (1969) . The spectrum equation c o n s i s t s of d i s s i p a t i o n , t r a n s f e r , production and d i f f u s i o n spectrum f u n c t i o n s . A parametric s o l u t i o n using the modified Obukhov approximation ( E l l i s o n 1962) was obtained by Panchev{1969). The production and d i f f u s i o n parameters were shown t o depend on the t u r b u l e n t Reynolds number, which i s the d i mensionless s c a l i n g parameter t h a t i s u s u a l l y considered i n the study of s m a l l s c a l e i n t e r m i t t e n t motion (Wyngaard and Tennekes 1970). Turbulence i n r i v e r s These concepts, b r i e f l y o u t l i n e d , are s i g n i f i c a n t i n r i v e r s . F i r s t , the boundary l a y e r s t r u c t u r e and turbulence must, i n l o c a l e g u i l i b r i u m flow, show s i m i l a r i t y s e a l i n g and thus i t i s important to e l u c i d a t e measures of the flow that respond to the morphological c h a r a c t e r i s t i c s of the boundary. Second, the frequency composition of the t u r b u l e n t k i n e t i c energy and the shear s t r e s s are important i n deterroininq 24 which p a r t i c l e s may be picked up and transp o r t e d i n the f l o w . At present i t appears impossible to obta i n d i r e c t measurements of the f o r c e s operating on the p a r t i c l e s on a very rough boundary and hence a surrogate method of examining the boundary flow c o u p l i n g appears v a l i d . . T h i r d , the p r o b a b i l i s t i c s t r u c t u r e and i n t e r m i t t e n t nature of the flow processes are s i g n i f i c a n t i n determining the s t r e s s e s operating over c e r t a i n s c a l e s , over c e r t a i n p e r i o d s , which w i l l determine which p a r t i c l e s w i l l be eroded. Work c a r r i e d out i n open channels i s sparse. Few measurements of the spectrum e x i s t (McQuivey 1967, 1973: E a i c h l e n 1967: Holley 1971) and these s t u d i e s have j u s t examined the s p e c t r a to see i f a -5/3 slope was present. No measurements e x i s t of the cospectra while only two papers have considered the p r o b a b i l i s t i c s t r u c t u r e of the flow (Nordin et a l 1972: Hansen and Hosbjerg 1974). Although some measures of t u r b u l e n t d i f f u s i o n have been made (McQuivey and Keefer 1972: F i s h e r 1973) the r o l e of turbulence has not been examined at a l l with regard t o sediment entrainment with the exception of the phenomenological d e s c r i p t i o n of Sutherland (1967) , Apperley (1 967) and Grass (1 97 1) . Gyr(1967) and M u l l e r et al(1971) examined the r o l e of vortex motion i n e n t r a i n i n g sediment. The model comprised a s i n g l e vortex impinging on the bed i n s t i g a t i n g p a r t i c l e i n s t a b i l t y . V i s u a l measurements were presented. 25 IV HiaSOSEMENT AND ANALYSIS Hot f i l m measurements and recording The measurement of turbulence i n water to the f r e q u e n c i e s r e q u i r e d i n t h i s research r e q u i r e d the use of a hot f i l m anenometer. Thermo Systems Incorporated (TSI) p a r a b o l i c f i l m s are quartz coated sensors whose s t a b i l t y , rugged design and adeguate freguency response proved s u i t a b l e f o r t h i s study- The sensor, t r e a t e d f o r o p e r a t i o n i n sea water, had a response up to approximately 300 Hz before s e r i o u s drop o f f , while the response of the anemometer (TSI 1058B) was s i g n i f i c a n t l y greater-. The probe was operated with a 45 m lead both i n the l a b o r a t o r y and i n the f i e l d as the anemometer bridge had been modified s p e c i f i c a l l y f o r such a long c a b l e . The system i n t h i s c o n f i g u r a t i o n s e r i o u s l y , attenuated s i g n a l s greater than 3KHz. This was much gr e a t e r than the highest frequency of i n t e r e s t . The s i g n a l was low pass f i l t e r e d p r i o r to any a n a l y s i s . The s e t t i n g of the f i l t e r l e v e l was achieved a f t e r p r e l i m i n a r y s t u d i e s , f i l t e r e d at 1 KHz and d i g i t i z e d at 2-4 KHz, revealed that the. spectrum was s t a r t i n g to r i s e a f t e r 400 Hz. This i n d i c a t e d that noise was by then making a s i g n i f i c a n t c o n t r i b u t i o n to the energy at such f r e q u e n c i e s . The spectrum peaked at 0.1 Hz and a record l e n g t h of over ten times .this wavelength was recorded f o r a n a l y s i s . The- s i g n a l was monitored at a l l times on a Hewlett Packard 1201A v a r i a b l e p e r s i s t e n c e o s c i l l o s c o p e to ensure 26 that the s i g n a l was not contaminated by n o i s e , d i r t or a i r bubbles, a l l of which could s i g n i f i c a n t l y a l t e r the c a l i b r a t i o n and response of the probe. Working i n the l a b o r a t o r y with u n f i l t e r e d water there was i n i t i a l l y a problem with a i r bubbles, but by mixing hot and c o l d water t o room temperature, running the flume f o r t h i r t y minutes p r i o r t o measurement and operating the probe at a 20° C overheat the problem was e l i m i n a t e d . In the f i e l d the s i t e s chosen were i n c l e a n i n t e n s e l y mixed flows and t h e r e were no problems with a i r bubbles or probe contamination. The probe was cleaned between each run, approximately f o u r hours of o p e r a t i o n , by dippi n g i t i n a supersaturated s o l u t i o n of chromic a c i d . At no time was the probe t i p touched by a brush as appears standard p r a c t i c e i n open channel measurements (HcQuivey 1967). The probe was c a l i b r a t e d at the Canadian Defence Research Establishment, Esguimalt B r i t i s h Columbia i n a water t u n n e l . P r i o r to v e l o c i t y c a l i b r a t i o n and the study of the freguency response of each probe, the r e s i s t a n c e temperature curve was obtained so that a 20° C overheat could be s e l e c t e d . The manufacturer's recommended l e v e l (1.10 x r e s i s t a n c e i n ohms) corresponds to approximately a 45° c overheat which w i l l aggravate the a i r bubble problem and w i l l l ead t o a decreased probe l i f e t i m e . I n t e g r a t e d analog measures were obtained on a TSI 1060 RMS meter with a time constant of 100 seconds, while the DC voltage was output to an E s t e r l i n e Angus T171B chart recorder 27 and t h e mean v o l t a g e o b t a i n e d from an a p p r o x i m a t e l y f i v e m i n u t e t r a c e . The s i g n a l c o u l d a l s o be d i f f e r e n t i a t e d a n d t h i s was a l s o o u t p u t t o a RMS meter t o o b t a i n a n a l o g m e a s u r e s o f t h e m i c r o s c a l e and d i s s i p a t i o n r a t e s . The s i g n a l was on o c c a s i o n o u t p u t t o a 2 c h a n n e l B r u s h 222 c h a r t r e c o r d e r s o t h a t a t r a c e o f t h e s i g n a l , w i t h r e s o l u t i o n up t o 30 Hz, c o u l d be r e t a i n e d . F o r two c h a n n e l o p e r a t i o n , w i t h t h e s p l i t f i l m and w i t h two s e p a r a t e p a r a b o l i c p r o b e s , a 2 c h a n n e l T S I 1051 was o p e r a t e d w i t h 5 m c a b l e s . I n t h i s c a s e , and f o r t h e s i n g l e c h a n n e l , t h e anemometer was i n d e p e n d e n t l y g r o u n d e d f r o m a l l t h e o t h e r e q u i p m e n t . F o r s u b s e q u e n t d i g i t a l a n a l y s i s t h e s i g n a l and i t s d e r i v a t i v e were r e c o r d e d on a H e w l e t t P a c k a r d 3960 FM t a p e r e c o r d e r - O p e r a t e d i n FM mode, t h e p a s s b a n d o f t h e r e c o r d e r r u n n i n g a t 3 3/4 i p s was 0-1250 Hz w i t h a s i g n a l t o n o i s e r a t i o o f 48dB. The s i g n a l s were l o w p a s s f i l t e r e d a t 500 Hz and any DC component b a c k e d o f f p r i o r t o a m p l i f y i n g t h e s i g n a l s f o r r e c o r d i n g s o t h a t t h e maximum r e s o l u t i o n o f t h e s i g n a l c o u l d be a c h i e v e d . The t a p e s were new and d e q a u s s e d p r i o r t o r e c o r d i n g , w h i l e t h e r e c o r d i n g head and a l l p a r t s i n c o n t a c t w i t h t h e t a p e were c l e a n e d and d e m a g n e t i z e d a f t e r e ach n i n e t y m i n u t e s o f r e c o r d i n g . To i n d i c a t e t h a t t h e q u a l i t y o f t h e s i g n a l i s s u i t a b l e f o r a l l t h e s u b s e g u e n t a n a l y s i s t h e f o l l o w i n g c a l c u l a t i o n was c a r r i e d o u t . The s i g n a l t o n o i s e r a t i o i s n o m i n a l l y g i v e n a s 48dB. 28 I f we. assume that the k u r t o s i s of the s i g n a l of i n t e r e s t i s 7.0, then Tennekes and Wyngaard (1972) have shown t h a t t h e number of standard d e v i a t i o n s (m) needed to o b t a i n a s t a b l e estimate of the f o u r t h moment would be 6. I f the RMS n o i s e l e v e l i s denoted by e' the RMS s i g n a l t o nois e r a t i o i s g i v e n by Q=b*/e' where b' i s the RHS of the s i g n a l . I f the dynamic h a l f range D* of the r e c o r d e r i s used t o c a p a c i t y ^ D'=mQ=mb'/e' then f o r D»=250=antilog (4.8/2) , m=6 and e»=0.01 the RMS of the s i g n a l should be 0.4 v o l t s which was c l o s e t o the l e v e l used i n a l l cases. Elm® and F i e l d Operation The major p o r t i o n of t h i s r e s e a r c h was c a r r i e d out i n a s m a l l r e c i r c u l a t i n g flume a t Simon F r a s e r u n i v e r s i t y . The flume at S.F.U. i s shown i n e l e v a t i o n view g i v i n g d e t a i l s o f th e pump and t i l t i n g devices (Figure 2). The flume s l o p e c o u l d be a d j u s t e d by two j a c k s so t h a t uniform f l o w c o u l d be e s t a b l i s h e d f o r a l l runs. The i n l e t was covered with s e v e r a l l a y e r s of woven b r a s s mesh and the flow then r e c t i f i e d through a 25.4 cm s t a i n l e s s s t e e l honeycomb b a f f l e of 2.54 cm square openings. At the downstream end of the flume, an a d j u s t a b l e t a i l gate was i n s t a l l e d to minimize the e f f e c t of the s w i r l i n g v o r t e x induced by the pipe o u t l e t . By j u d i c i o u s adjustment of s l o p e and t a i l gate uniform flow was e s t a b l i s h e d f o r any g i v e n roughness on the channel bed. The flume w a l l s were of smooth P l e x i g l a s s with a l l f i t t i n g s r e c e s s e d . The bed of the flume 30 was covered by Lego baseboard over i t s e n t i r e l e n g t h which was glued i n p o s i t i o n a f t e r i n i t i a l measurements had been made over the 'smooth* bed to check f o r u n i f o r m i t y of f l o w . The r e s u l t s of tha t check are presented i n the form of a contour map f o r mean v e l o c i t y (Figure 3). No obvious p a t t e r n e x i s t s and thus the flow i s taken to be s u i t e d f o r t he asssumptions about i t s two dimensional nature o u t l i n e d i n Section I I . The measurement s e c t i o n was taken 4.0 m from the i n l e t and 1.5 m from the t a i l gate. D e t a i l s of the u n i f o r m i t y of the flow and e g u i l i b r i u m c h a r a c t e r i s t i c s are given i n Chapter 2. . The probe t r a v e r s i n g mechanism was of very simple c o n s t r u c t i o n c o n s i s t i n g of an aluminium block supported on two s t a i n l e s s s t e e l r a i l s . The probe holder could s l i d e f r e e l y along the r a i l s and could be secured by two b u t t e r f l y nuts. The v e r t i c a l movement of the probe was operated by a clamp f i x e d to a screw thread with 32 turns per i n c h and thus i t was p o s s i b l e to move the probe i n increments o f approximately 0.04 cm from some r e l a t i v e datum, u s u a l l y taken as the Lego baseboard. When the probe was i n p o s i t i o n i t was f i x e d by a second screw so that the l o c k i n g sleeve would not v i b r a t e . In the f i e l d the hot f i l m was supported on a portable wing shaped s t r u c t u r e that was placed i n the channel as shown i n Figure 4. The support s t r u c t u r e was headed d i r e c t l y i n t o the mean flow and l e v e l l e d with a s m a l l hand l e v e l . Probe l o c a t i o n was measured by tape from the bed of the channel-F i g u r e 3 Contour map o f mean v e l o c i t i e s o v e r smooth metal bed o f the flume a t y/D=0.37.. Contours r e p r e s e n t v a r i a t i o n s above a base o f 25 cm/sec. C r o s s s t r e a m d i s t a n c e i s i n metres: downstream d i s t a n c e i s i n roughness h e i g h t s . Measurement s e c t i o n f o r roughness a r r a y s was at x/h= 400 . 32 Figure 4 Photograph of "bridge" p i e r set up i n Cheekye(a). Probe t i p was l o c a t e d 25cm ahead of p i e r body. The equipment was placed on a foam rubber mattress on the bank and grounded t o copper tubes i n s e r t e d i n the ground surrounded by a s a l t water s o l u t i o n . The power was s u p p l i e d from a truck b a t t e r y and an i n v e r t e r which was independently grounded. D i g i t i z a t i o n and e r r o r a n a l y s i s The analog tapes were played back at the same speed and f i l t e r e d with a Rockland 1121 low pass Butterworth f i l t e r . The f i l t e r was set at 285 Hz and the s i g n a l d i g i t i z e d at 625 Hz g i v i n g a Nyquist frequency of 312.5 Hz which was c l o s e t o , and was u s u a l l y g r eater than, the Kolmogorov m i c r o s c a l e . The s i g n a l was d i g i t i z e d by a 12 b i t plus s i g n analog to d i g i t a l c o nverter i n the Department of Geophysics, Seismology S e c t i o n , U.B.C. and w r i t t e n on 9 t r a c k tape at 800 BPI by a Kennedy formatter with 256 points per block. Two channels were d i g i t i z e d simultaneously to reduce d i g i t i z a t i o n time and o p t i m i z e subsequent data h a n d l i n g . The analog record was 120 seconds but only 112 seconds were d i g i t i z e d a l l o w i n g s u f f i c i e n t time f o r the tape speed t o s t a b i l i z e . Subsequently the f i r s t blocks of d i g i t i z e d data were discarded to ensure no s t r a y voltages or incomplete blocks were used at the beginning of any record. No complete e r r o r a n a l y s i s was c a r r i e d out on the whole system. While i t was being c a l i b r a t e d , the probe was run a t the ambient f l u i d temperature. The f l u i d i n the water t u n n e l was extremely w e l l mixed and thus the probe should not have 34 y i e l d e d any AC s i g n a l as i n t h i s balanced s t a t e the probe i s i n thermal e q u i l i b r i u m . The output then roughly r e p r e s e n t s the noise of the probe anemometer system. The s i g n a l was monitored on an o s c i l l o s c o p e and when pro p e r l y grounded showed a noise l e v e l of l e s s than 10 mV peak to peak. Improper grounding r e s u l t e d i n noise l e v e l s of over 50 mV peak to peak, but such noise was very easy to spot as i t occurred at the l i n e frequency and was e a s i l y d i s t i n g u i s h a b l e from the t u r b u l e n t s i g n a l . A second check on noise was made by d i g i t i z i n g a blank channel from the re c o r d e r . The spectrum of the s i g n a l had wide e r r o r bars and the mean s p e c t r a l estimates over 32 blocks were e s s e n t i a l l y f l a t . A n a l y s i s There were two separate phases t o the a n a l y s i s : f i r s t , a group of programs developed at the I n s t i t u t e of Oceanography U.B.C. performed s p e c t r a l c a l c u l a t i o n s ; the s p e c t r a l estimates were then read by another program developed f o r t h i s reseach to c a l c u l a t e d i s s i p a t i o n r a t e , m i c r o s c a l e , Kolmorgorov microscale and t u r b u l e n t Reynolds number and to p l o t v a r i o u s dimensionless arrangements of the s p e c t r a . Second, a group of programs was obtained and operated t o y i e l d the p r o b a b i l i s t i c measures of the v e l o c i t y f l u c t u a t i o n s . Figure 5a shows the s p e c t r a l programs th a t were used. T h i r t y two blocks of 2048 p o i n t s were analysed y i e l d i n g a s e r i e s of 104.6 seconds. The program FLINOP was run twice on 35 RAW DATA DIGITAL T A P E — FLINOP —Moments of s i g n a l 1 C a l i b r a t i o n c o e f f i c i e n t s FTOR I F o u r i e r c o e f f i c e n t s SCOR | — S p e c t r a l e s t i m a t e s PLOT Non-dimensional and cumulative s p e c t r a l p l o t s . F i g u r e 5a Design of frequency a n a l y s i s programs RAW. DATA DIGITAL TAPE -HURST--EXCEED--Moments Rescaled range a n a l y s i s - P r o b a b i l i t y d i s t r i b u t i o n and d e n s i t i e s f o r s i g n a l and i t s powers up to o r d e r f o u r Second order s t r u c t u r e - f u n c t i o n , skewness and f l a t n e s s f u n c t i o n - D i s t r i b u t i o n and d e n s i t y o f v e l o c i t y increments D i s t r i b u t i o n and d e n s i t y -of n e g a t i v e and p o s i t i v e peaks -Normalized l e v e l c r o s s i n g s F i g u r e 5b Design o f p r o b a b i l i t y a n a l y s i s 36 the data. The f i r s t time the r e c o r d i n g and d i g i t i z a t i o n c a l i b r a t i o n c o e f f i c i e n t s were included and the r e s u l t a n t variance could be compared with the analog measures to examine f o r e g u a l i t y ; then the voltage v e l o c i t y c a l i b r a t i o n was read i n from the prepared c h a r t s drawn up when the probes were c a l i b r a t e d i n the water t u n n e l . The program FTOH read i n the re-blocked data a block a t a time and c a l c u l a t e d , using the Cooley Tukey Fast F o u r i e r a l g o r i t h m , the F o u r i e r c o e f f i c i e n t s which were w r i t t e n out on another tape. F u l l d e t a i l s of the program are provided i n Dobson(1969) while published d e t a i l s and l i s t i n g s are given by Wilson and Boston (1969) and ScKendrick {1972). The F o u r i e r c o e f f i c i e n t s were then read i n and the r e g u i r e d s p e c t r a l f u n c t i o n s c a l c u l a t e d , the bandwidth being e g u a l l y spaced on a l o g a r i t h m i c a x i s so t h a t more and more c o e f f i c i e n t s were grouped f o r the higher freguency e s t i m a t e s . The s p e c t r a l e stimates, bandwidths and frequencies (n) were then read by PLOT to y i e l d the cumulative s p e c t r a , the ' u n i v e r s a l curve' (see Wasmyth 1970) and the spectrum h <J> (n) versus non-dimensional frequency. The second group of programs i s shown i n Figure 5b. One program c a l c u l a t e d the p r o b a b i l i t y d i s t r i b u t i o n s and d e n s i t i e s f o r the time s e r i e s as w e l l as f o r the squares, cubes and f o u r t h powers of the o r i g i n a l s e r i e s . By rearranging the s e r i e s i n ascending order of magnitude the accumulation of the v a r i a n c e , skewness and k u r t o s i s f o r the normalized s e r i e s were c a l c u l a t e d -37 The d i s t r i b u t i o n of increments at various l a g s i s s i g n i f i c a n t i n the examination of the no t i o n of s e l f - s i m i l a r i t y (Van Atta and Park 1972) and the d e n s i t y and d i s t r i b u t i o n of increments were c a l c u l a t e d f o r four l a g s . The s t r u c t u r e f u n c t i o n s are u s u a l l y presented as y i e l d i n g the same in f o r m a t i o n as the spectrum, but the s t r u c t u r e f u n c t i o n s are r e a d i l y c a l c u l a t e d f o r any power as they are j u s t the c e n t r a l moments and thus the second order s t r u c t u r e f u n c t i o n , the skewness and f l a t n e s s f u n c t i o n s were c a l c u l a t e d to y i e l d i n f o r m a t i o n on the nature of the i n t e r m i t t e n c y of the s i g n a l . A f i n a l method of examining the p r o b a b i l i s t i c s t r u c t u r e i s t o look a t exceedance s t a t i s t i c s (Dutton 1969). They have long been used i n a e r o n a u t i c a l engineering i n r e l a t i o n t o wind loadings on b u i l d i n g s . That most commonly used i s shown i n Figure 6. I f the l e v e l y', defined as a d i s t a n c e from the mean of the normalized s e r i e s , and i t s d e r i v a t i v e are considered as independent then the freguency N(y*) of c r o s s i n g s of the value y' with a p o s i t i v e slope i s given by P ( y ' , ! f ) = P ( y ' ) P ( ^ ' ) (6) (7) I f t h i s i s normalized by N(0) which i s the freguency of W W \ o. .0, I j i 1 c •: I D i 2 0 < Y / S I G M A C Y 3 ) * * 2 Figure 6 Test example of exceeclance s t a t i s t i c s . S o l i d l i n e represents e r r o r f u n c t i o n ; u a.nd v v/ere here generated by a pseudo-random number generator. \r represents noise p l u s a 10 Hz s i g n a l . 39 c r o s s i n g s of the mean with p o s i t i v e slope then t h a p r o b a b i l i t y of exceeding y', as a f u n c t i o n of zero c r o s s i n g s i s As we assumed t h a t y' and i t s d e r i v a t i v e were independent then t h i s s i m p l i f i e s to Another approach . to the i n t e r m i t t e n t nature of the t u r b u l e n t time s e r i e s may be taken. Based on the ext e n s i v e s t u d i e s of Mandelbrot and Wallis(1969 a,b,c) the r e s c a l e d range was c a l c u l a t e d f o r the s e r i e s . Long term p e r s i s t e n c e , r e f e r r e d to as the 'Hurst phenomenon' , i s i d e n t i f i e d by the f a c t t h a t the r e s c a l e d range R/S of a time s e r i e s u (t) when 0<t< f o r the period (0,s) I s p r o p o r t i o n a l t o s^ where H i s t y p i c a l l y of order 0.7 (Figure 7) . . For a Gaussian s i g n a l H i s egual to 0.5. The r e s c a l e d range i s given by (8) N(y'-) = p(y' ) N(0) p(.0) (9) R / S = R ( t , s ) / S ( t , s ) ~ s H (10) where | / u ( t ) d t ) fa 0 (11) 40 CD CD CD CO \ CH CD CD 0.789 E(R/5)=0.81S R = 0.992 o .—i CD 10 APPARENT H= 0.78 MOMENTS OF DRTfl 1ST = 0.00 2ND =1.00 3RD = -0.13 4TH =2.3.1 TIME 100 1000 10000 Figure 7 Example of r e s c a l e d range p l o t f o r • u. Time represents cumulative sample s i z e ; sampling r a t e 300 Hz approximately. S o l i d l i n e represents a slope of 0.5. S(t,s) = Var(u(t))= ^ / ( u ( t ) ) 2 d t - ( - / u ( t ) d t ) 2 O2) S » .' . S o I t i s c l e a r that p e r s i s t e n c e may be considered as j u s t the a n t i t h e s i s of i n t e r m i t t e n c y and the use of the r e s c a l e d range permits examination of t h i s phenomenon from another viewpoint. 42 CHAPTER 2 EXPERIMENTAL RESULTS FOR ROUGHNESS ARRAYS PHYSICAL ANALYSIS Measurements were made over seventeen d i f f e r e n t d e n s i t i e s ranging from 1/4 to 1/144- For the sake of b r e v i t y and c l a r i t y i n the diagrams the r e s u l t s from only s i x d e n s i t i e s ranging over the c r i t i c a l r e g i o n of d e n s i t i e s , between 1/8 and 1/80, are presented here. R e s u l t s from the other measurements are i n c l u d e d where p e r t i n e n t . Density I s here defined i n terms of the r a t i o of plan areas of blocks t o the surrounding area. Such a scheme was adopted as the l i m i t i n g values of one and zero have obvious i n t e r p r e t a t i o n s . The measurements were performed at constant f r e e stream v e l o c i t y and uniform depth and as near e q u i l i b r i u m flow as could be e s t a b l i s h e d . The main purposes of these measurements were: 1) to study the mean v e l o c i t y d i s t i b u t i o n over boundaries of v a r i o u s roughnesses; 2) to i n v e s t i g a t e the turbulence i n t e n s i t y v a r i a t i o n , energy d i s s i p a t i o n and l e n g t h s c a l e s of the f l o w ; 3) to explore the t u r b u l e n t energy d i s t r i b u t i o n and turbulence s t r u c t u r e i n r e l a t i o n to r e l a t i v e depth and roughness c o n c e n t r a t i o n . 43 M E A N F L O W S E S D L T S Establishment of the flow A l l measurements were made i n a s u b - c r i t i c a l flow regime with Froude number approximately 0.1. By a d j u s t i n g the slope and the t a i l gate f o r any flow over any su r f a c e roughness uniform depth could be achieved. For a l l experiments the flow Reynolds number was maintained constant at 2.5x10*. Th i s means tha t the r e s u l t s being i n v e s t i g a t e d show the e f f e c t s of the roughness Reynolds number, % h / vwhere h i s roughness height, and hence us should be an important s c a l i n g parameter. Having e s t a b l i s h e d uniform, steady flow the f u r t h e r requirement of e q u i l i b r i u m boundary l a y e r development was d e s i r e d . E q u i l i b r i u m i m p l i e s t h a t the boundary l a y e r has c h a r a c t e r i s t i c s i n pr e c i s e dynamic e q u i l i b r i u m with the l o c a l rouqhness beneath i t , so that p r o p e r t i e s such as t u r b u l e n t i n t e n s i t i e s , s c a l e s , s p e c t r a and so on when nondimensionalized by a s i n g l e l o c a l v e l o c i t y s c a l e and a s i n g l e l o c a l l e n g t h s c a l e do not change i n the streamwise d i r e c t i o n . (Gartshore 1973, p.1) The shape f a c t o r H», being the r a t i o of displacement t h i c k n e s s to momentum t h i c k n e s s , was c a l c u l a t e d at twelve p o i n t s i n the flow over a roughness d e n s i t y of 1/80 (Figure 8). The reason f o r choosing t h i s parameter i s that the gross p r o p e r t i e s of the flow may be obtained by i n t e g r a t i n g the 2 . 8 1.3 0.8 + 4-4. > 4-0 100 2 0 0 3 0 0 4 0 0 5 0 0 Figure 8. V a r i a t i o n of the shape factor(H) down channel: downstream dis t a n c e represents number of roughness heights. Measurements made over a roughness array of density 1/80. 45 d i f f e r e n t i a l t r a n s p o r t equations across the boundary l a y e r . The momentum i n t e g r a l equation i s w r i t t e n as dx dxll. 2 f (13) where H' i s the shape f a c t o r , 0 i s the momentum t h i c k n e s s , S° i s the energy slope and c.'is the w a l l f r i c t i o n c o e f f i c i e n t : t h i s equation i s a l s o c a l l e d the von Karman i n t e g r a l momentum equation. The behaviour of H * along the boundary shows t h a t s t r i c t e g u i l i b r i u m i s not a t t a i n e d , f o r , as Reynolds(1974, p. 396) shows, H' remains a very weak f u n c t i o n of c ^ . The measurement region s t a r t i n g at approximately 4 metres from the entrance, or 400 roughness heights downstream, was s e l e c t e d , as here H' appears e s s e n t i a l l y constant and hence approximate e q u i l i b r i u m may be assumed. This Is important, s i n c e with a constant H• i n a q r a v i t y d r i v e n flow where there should be zero pressure g r a d i e n t , we may make use of the Clauser p l o t t o o b t a i n a measure of the shear v e l o c i t y at the bed. The standard method used to evaluate t h i s term i n flumes i s to use the slope of the flume or water s u r f a c e (SO) , and assuming uniform flow, o b t a i n u * from the r e l a t i o n JgDS 0. However, t h i s technique i s subject to c o n s i d e r a b l e variance as the measurement of slope over the short flume le n g t h must be extremely accurate and, i n any case, the r e s u l t i s an i n t e g r a t e d value over the t o t a l flume l e n g t h and not the e q u i l i b r i u m value i n the measurement s e c t i o n which i s requ i r e d to s c a l e the flow parameters. The, p l o t of H* against p o s i t i o n shows i t to be a w e l l behaved f u n c t i o n and v 46 i t i n d i c a t e s t h a t only 300 roughness heights are r e q u i r e d t o e s t a b l i s h an approximate e q u i l i b r i u m flow. Clauser showed that 6 U ' u» = / I ™ - L L d(|)= Constant • o * (14) which he evaluated as 3.6, where 6* i s the displacement t h i c k n e s s . The r a t i o 6*/6 can be represented as n*/(n' + 1) f o r a power law p r o f i l e and the above equation may be r e w r i t t e n as u.*_'_JL n' «L 3.6 n'+i (15) which then allowed e v a l u a t i o n of u *. Furthermore H' i s r e l a t e d t o the f r i c t i o n c o e f f i c i e n t and w i l l i n d i c a t e the e f f e c t s of d i f f e r i n g boundary roughness. The v a r i a t i o n of H» along the flume was i n v e s t i g a t e d f o r only the lowest roughness d e n s i t y . I t would be reasonable to presume that a s i m i l a r s i t u a t i o n a p p l i e d f o r the other, rougher su r f a c e s over which more in t e n s e mixing would l i k e l y lead to the more ra p i d establishment of e q u i l i b r i u m f l ow. As the homogeneity of the flow at one height i n the cross stream and downstream d i r e c t i o n s had been e s t a b l i s h e d over the smooth bed, i t was considered f u r t h e r checks f o r the g r e a t e r roughness d e n s i t i e s were unnecessary. The roughness arrays c o n s i s t e d of Lego bl o c k s fastened to a Lego base board as shown i n Figure 9. The square blocks are 1.6 cm i n plan l e n g t h with a height of 0.96 cm. They were s e l e c t e d as the top of the block presented a rough a b Figure 9a Lego block roughness In place. 9b Parabolic hot f i l m i n p o s i t i o n , and missing roughness block below the measurement p o s i t i o n . [ 48 su r f a c e s i m i l a r to that on the bed, and they could r a p i d l y be rearranged to d i f f e r i n g d e n s i t i e s . Measurements were made from the bed up t o the f r e e s u r f a c e ; measurements below the top of the blocks were achieved by p o s i t i o n i n g the probe t i p at the point where the next block i n that array should have been. Flow depths were a l l approximately 7.6 cm which meant t h a t the width to depth r a t i o was approximately s i x , c l o s e to the lower l i m i t assumed to ensure two dimensional flow. This depth represented a compromise between a high width-depth r a t i o and the r e l a t i v e roughness r a t i o , which was approximately 8, so t h a t comparison could be made with Counihan et al's{1974) r e s u l t s f o r which D/h=8 J^ eatH V e l o c i t y D i s t r i b u t i o n s In the f u l l y developed flow region the mean v e l o c i t y i n a t u r b u l e n t boundary l a y e r can be represented by a power law. Figur e 10a shows power law p l o t s f o r the s i x roughness a r r a y s . I t was assumed that was the f r e e s u r f a c e v e l o c i t y and thus 6 was s e l e c t e d as the depth of fl o w . This i s l o g i c a l i n t h a t f o r f r e e s u r f a c e flows of such depths as the experiments reported here the boundary l a y e r should extend r i g h t to the sur f a c e . Measurements showed U(y) i n c r e a s i n g a l l the way to the surfa c e . I f the power law p r o f i l e s of Figure 10a are r e p l o t t e d on a double l o g a r i t h m i c s c a l e as i n Figure 10b there i s c l e a r l y a systematic v a r i a t i o n i n the exponent with roughness d e n s i t y . Constant Reynolds number was maintained throughout these experiments and the Figure 10b Logarithmic plot showing v a r i a t i o n i n power law exponents with surface roughness. 1/80 ,2 • v. G . 4 0.B • Figure 10a "Arithmetic plot of power law variation for six roughness den s i t i e s . -1 n 50 v a r i a t i o n s i n the exponent should be a t t r i b u t a b l e d i r e c t l y t o the v a r i a t i o n s i n the w a l l f r i c t i o n c o e f f i c i e n t at the boundary. Using the displacement and momentum t h i c k n e s s and assuming a l o g a r i t h m i c v e l o c i t y p r o f i l e i t may be shown t h a t f o r a boundary l a y e r , the shape f a c t o r takes the form H " 6 " 1-G(c f /2 )V2 (16) and p l o t s of H' and n' against roughness density should show s i m i l a r behaviour. Both r e f l e c t the e f f e c t s of changing s k i n f r i c t i o n f o r varying d e n s i t y . F i g u r e 11a shows the behaviour of H' and n* p l o t t e d , f o r ease of comparison over the range of d e n s i t i e s used, against the logarithm of d e n s i t y . The c l e a r pattern t h a t emerges from these f i g u r e s i s tha t there i s an optimum d e n s i t y , that which presents the g r e a t e s t r e s i s t a n c e to fl o w . This agrees with the n o t i o n s of Morris {1955) and the measurements of Koloseus - and Davidian{1966) and summarized i n Rouse{1965). The shape f a c t o r depends upon the w a l l c o n d i t i o n s . The power law may be reformulated i n terms of n' and H' to y i e l d the r e s u l t H' = 2n' + 1 A p l o t of n' c a l c u l a t e d from the shape f a c t o r and n' evaluated by f i t t i n g a power law to the v e l o c i t y p r o f i l e s should form a 45° l i n e (cf. Figure 11b). The agreement, e s p e c i a l l y given the a r b i t r a r y d e f i n i t i o n s of and 6 , i s most encouraging. 1.7 1.6 1.5 H H • 4 h -1.3 3.0 .005 .05 . 0. density of blocks Figure 11a Plots, of shape factor H', power law exponent 1/n' against the logarithm of density of roughness elements. Also shown is the ef f e c t i v e roughness for cubes on a boundary calculated by Koloseus and Da vidian (.1966). @ EVALUATED OVER TOTAL DEPTH . / X EVALUATED FROM TOP OF ROUGHNESS ELEMENTS Si ® © / X A* y x X / * .2 .3 /n Power law exponent F i g u r e l i b Comparison of power law exponents c a l c u l a t e d by two d i f f e r e n t methods. U l ro 53 The major reason f o r c a l c u l a t i n g H1 and r e l a t i n g t h i s t o n' i s to attempt to get some measure of u*. While i t i s customary p r a c t i c e i n boundary l a y e r work to use u^to express r e s i s t a n c e , i n pipe flow the r e s i s t a n c e i s u s u a l l y presented i n terras of the c o e f f i c i e n t of f r i c t i o n The c o e f f i c i e n t i s defined by the expression §1 = - £f 0.5pU2 dx D (17) The w a l l f r i c t i o n v e l o c i t y I s expressed i n terms of the shear s t r e s s or pressure gradient f o r pipe flow as d l = " D p U * ° 8 ) X f = 8 ( T J * ) 2 ( 1 9 ) As the v e l o c i t y d i s t r i b u t i o n f o r a constant flow Reynolds number i s a f u n c t i o n only of the f r i c t i o n c o e f f i c i e n t , the c o n c l u s i o n i s reached that n' and f u n c t i o n a l l y covary. Nunner(1956) determined n 1 as a f u n c t i o n of A^and Figure 12 shows t h i s f u n c t i o n as he evaluated i t i n the form (20) His data and t h a t of Uikuradse are a l s o i n c l u d e d - The data from a l l experiments show reasonable agreement. The v e l o c i t y defect law i s most commonly used to d e s c r i b e the outer region of the t u r b u l e n t boundary l a y e r . F i r s t formulated by von Karman, i t i s considered as a s p e c i a l case of Reynolds number s i m i l a r i t y which means t h a t the mean n * — v f X OATA OF NIKURADSE (1933) .005 : .01 .02 .05 0.1 0.2 0.5 F i g u r e 12 R e l a t i o n between the exponent from the .power law v e l o c i t y d i s t r i b u t i o n and the f r i c t i o n f a c t o r . ( A f t e r Hinze 1959 p.530) Ul 55 motion does not depend on v i s c o s i t y but only on the f r e e stream v e l o c i t y and the boundary c o n d i t i o n s , c h a r a c t e r i z e d by u *. A p l o t of t h i s parameter with C l a u s e r ' s u n i v e r s a l curve shows qu i t e good accord (Figure 13a). This i s to be expected, f o r i n order t o evaluate. u f t we assumed th a t the area under each curve f o r each p r o f i l e was a constant. A s i m i l a r p l o t on semi-logarithmic axes shows the areas of accord more c l e a r l y (Function G i n Figure 13b) . Clauser (1956 p. 8) showed that i n the ove r l a p p i n g r e g i o n between the outer flow where the s c a l i n g parameters are U m, u and D, and the i n n e r p o r t i o n where u^ , v and y are used the overlap zone must have the form U , ° ~ U = Alog(^) + B (Function G) (21) - = C l o g ( ^ * ) + D ( f u n c t i o n F) (22) These p l o t s are shown i n Figure 13b. The s c a t t e r of po i n t s around the l i n e i n Figure 13a does provide c e r t a i n q u a l i t a t i v e i n s i g h t s i n t o the p o s s i b l e s t r u c t u r e of the boundary l a y e r . The c l o s e s t agreement with the u n i v e r s a l curve i s i n the range 0.12<y/D<0.4 above which there i s a spreading of the data i n a wider band before the r e s u l t s are constra i n e d to f i t at the s u r f a c e . Close to the bed ther e i s a systematic departure from the curve due t o the f a c t t h a t below y/D=0.1 the measurements are below the tops of the roughness elements and the v e l o c i t y d e f e c t i s s m a l l e r than expected. However, the defect law i s . only taken t o DENSITY ° '/B .6. V\2 4> V20 * '/48 e- - '/oo > A > to n 0 > * if •aa &\ 6 n-s-AC —Q" •B-fe.-B- if * t 3*~-a -a-M . . -i 0 ©_ 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 X; = i k l ^ . P X; = constant 1 u* oJ L F i g u r e 13a U n i v e r s a l p l o t o f boundary l a y e r p r o f i l e s . T o t a l depth i s used as the measure of boundary l a y e r t h i c k n e s s as the f r e e stream v e l o c i t y was approximated by the surface v e l o c i t y . Thus D and <f are e q u i v a l e n t throughout t h i s r e s e a r c h . Figure 13 lj Logarithmic p l o t of u n i v e r s a l p r o f i l e s . G represents the defect law G(y/D) and F represents the i n n e r law F(yu%/?),. 58 apply f a r from the w a l l and the departure at t h i s l e v e l i s t o be expected because of the r o l e of roughness and r a p i d l y v a r y i n g shear s t r e s s r e l a t e d to the d i s t r i b u t i o n of roughness elements on the boundary. The l o g a r i t h m i c law was proposed as a s i m i l a r i t y s o l u t i o n f o r boundary l a y e r flow with f r i c t i o n v e l o c i t y and roughness le n g t h as the s i m i l a r i t y parameters. The former I s the square root of the w a l l shear s t r e s s and, as i t s u p p l i e s the energy t o the turbulence, i t i s obv i o u s l y s i g n i f i c a n t . The roughness le n g t h i s r e l a t e d to the height and de n s i t y of the elements on the boundary. For a f u l l y rough f l o w equation 22 above s i m p l i f i e s so t h a t * 0 (23) where yg i s a roughness l e n g t h r e l a t e d to the height of the roughness. Lettau(1969) proposed t h a t i t took the form y n = 0.5 h | T 0 S (24) where h i s the e f f e c t i v e o b s t a c l e height, s i i s the s i l h o u e t t e area and S' i s given by S'=A'/no, where no i s the t o t a l number of roughness elements on a s i t e of t o t a l area A'. The constant r e l a t e s to the average drag c o e f f i c i e n t of the i n d i v i d u a l o b s t a c l e s . Lettau{1969) pointed out that a l i m i t a t i o n to equation 24 occurs when the r a t i o s l / S • approaches u n i t y . From t h i s research the c a l c u l a t i o n of y by equation 24 gives 0.006 f o r a d e n s i t y of 1/80 to 0.06 f o r a de n s i t y of 1/8. For a density of 1/16 the value 0.030 agrees 59 w e l l with that obtained from the l o g a r i t h m i c law which gave a value of 0.027. D e n s i t i e s greater than 1/16 showed a decrease i n y compared with r e s u l t s from the l o g a r i t h m i c law, p o s s i b l y I n d i c a t i n g that the l i m i t of a p p l i c a b i l i t y of Lettau's formula i s not as i t approaches u n i t y but i n f a c t i s an order of magnitude lower. By a d j u s t i n g the v e r t i c a l coordinate o r i g i n a t h i r d s i m i l a r i t y parameter c h a r a c t e r i s t i c of high roughness i s obtained, termed the zero plane displacement (d). By a d j u s t i n g y^ and d the range of l o g a r i t h m i c law f i t can be extended. To avoid the d i f f i c u l t i e s r e l a t e d to the e s t i m a t i o n of zero plane displacement, P l a t e and Q u r a i s h i (1965), working above model t a l l crops i n a i r , assumed th a t d was equal to the height of the roughness elements (h). The r e s u l t s of p l o t t i n g U versus l o g (y-d) are presented i n Figure 14. Two zones of l i n e a r v a r i a t i o n of v e l o c i t y with the logarithm of height are observed, the kink appearing remarkably c o n s i s t e n t l y f o r the d i f f e r i n g d e n s i t i e s at approximately one roughness height above the top of the elements. This c o n c l u s i o n i s not new. Sadeh et al(1971) showed that p r e c i s e l y the same behaviour e x i s t e d over two model, a r t i f i c i a l f o r e s t canopies, one h a l f the d e n s i t y of the other. However i n flows observed over d i s t r i b u t e d roughness i n flumes, such behaviour has been e i t h e r missed or ignored. O'Loughlin and Annambholta (1969), working i n a i r and water, i n t e r p r e t e d t h e i r measurements to show two r e g i o n s , t h a t ' /a '/»«• V 7 ' / 2 ° 74 // " ' Z O N E 2 IV96> CM // / FROM BED tr if d A //,// I VB ill i i / j i / 1 ~ Z O N E I 1 /*)/ ' ' / 8 t 7 10 20 30 v cm/sec 4 0 50 Figure 14 Semi-logarithmic p l o t of mean v e l o c i t y w i t h displacement height(d) set equal to the height of the roughness elements(h). Zone 1 i s the wake l a y e r and Zone 2 represents the outer v e l o c i t y defect zone. 61 below the top of the roughness elements and a second l i n e a r v e l o c i t y region p l o t t e d against In (y/D) above the top of the elements. They po s t u l a t e d the existence of a wake l a y e r and continued: the roughness elements produce an approximately uniform l a y e r of high i n t e n s i t y t u rbulence s i m i l a r to that i n the v i c i n i t y of a s i n g l e obstacle as w e l l as drag on the boundary i t s e l f . The t h i c k n e s s of the l a y e r would be expected t o be comparable t o the ob s t a c l e wake dimension: outside t h i s i n t e g r a t e d wake r e g i o n , the turbulence c h a r a c t e r i s t i c s should be independent of the d e t a i l e d shape of the roughness elements. (O'Loughlin and Annambholta 1969, p.239) In the wake reg i o n the s i z e and spacing of the roughness elements are the s i m i l a r i t y l e n g t h s c a l e s which determine the eddy s i z e s . These eddies v i g o r o u s l y i n t e r a c t t o produce a near constant v e l o c i t y region as evidenced i n Figure 14 f o r the region 0.1<y-h<1.0. Above t h i s , the wake l a y e r should 'blend i n t o ' the flow described by a defect law. O'Loughlin and Annambholta(1969) found t h a t the l o g a r i t h m i c l a y e r was a p p l i c a b l e one roughness height above the top of the roughness elements but they d i d not observe the wake reg i o n found here and by Sadeh. The wake l a y e r may be observed i n other p l o t s of v e l o c i t y but i s u s u a l l y overlooked. Wu{1973) working with a rough boundary with s p h e r i c a l p a r t i c l e s of uniform s i z e 62 observed t h i s constant v e l o c i t y wake l a y e r , and data from the Columbia R i v e r ( S a v i n i and Bodhain 1971) may a l s o be seen t o i n d i c a t e such a zone c l o s e to the bed i n a flow over 12 meters deep. As these r e s u l t s from mean v e l o c i t y measurements i n d i c a t e , u ^ , height and den s i t y of roughness elements are important: s c a l i n g parameters. From the many measurements, made over rough boundaries with d i s t r i b u t e d roughness a r r a y s , Wooding et al(1973) attempted to c o r r e l a t e a l l the data i n an eguation of the form u» = l n ( h h e ) + (25) where ^  represents spacing of the elements. Data of Koloseus and Davidian(1966) f i t t e d t h i s r e l a t i o n with B=0 and A=1/k»'=2.5. Such a l i n e f a l l s c o n s i d e r a b l y below the po s t u l a t e d r e s u l t s of Dvorak (1969) f o r bar roughness — = r~u l n ( — * ) - r-„ l n ( - — s ) + A - C k" v k" v (26) where C depends on the roughness geometry. The r e s u l t a n t f a m i l y of l i n e s i s shown i n Figure 15, evaluated f o r s e v e r a l roughness d e n s i t i e s . The shear s t r e s s c o r r e l a t i o n s of the d e n s i t i e s computed here p a r a l l e l Dvorak's r e s u l t s i n the l i m i t e d range of d e n s i t i e s 1/12 to 1/20 but are s h i f t e d upwards, i n d i c a t i n g a low estimate of u s or a d i f f e r e n c e i n s c a l i n g d e n s i t i e s between bars and i n d i v i d u a l b l o c k s . Dvorak's e v a l u a t i o n was based on t r a n s v e r s e bar, r i b - t y p e roughness which presents a much greater o b s t a c l e t o V'AB ! /60 ^ '/|20 CORRELATION FROM DVORAK (!9£>9)-) —X — — — — X X'/32 >/20 Vie '." ^ - DATA FROM ""^ KOLOSEUS AND DAV1D1AN (19Gfe) 100 1000 h h F i g u r e 15 Wall shear v e l o c i t y c o r r e l a t i o n s ( a f t e r Gartshore 1973)• Xc represents streamwise sp a c i n g . 64 flow r e s u l t i n g i n higher r e s i s t a n c e . The measurements made over bars, discussed i n Chapter 4, showed extremely high turbulence l e v e l s , more than twice those experienced at the same d e n s i t y of three dimensional b l o c k s based on f r o n t a l areas, which would thus be r e f l e c t e d i n a greater u^ . The upward displacement f o r d e n s i t y 1/8 i s e x p l i c a b l e as t h i s d e n s i t y represents a skimming flow s i t u a t i o n i n which c o n s i d e r a b l e p r o t e c t i o n i s afforded by the blocks and hence a lower u j i s t o be expected. The low u x which r e s u l t e d i n the l a r g e divergence of r e s u l t s f o r d e n s i t i e s 1/48, 1/80 and 1/120 may be caused by the r a t i o of the roughness element height t o the height of the s m a l l 'spots' on the Lego baseboard. I t would seem p o s s i b l e that by a den s i t y of 1/4 8 the i n t e r v e n i n g bed i s c o n t r i b u t i n g an equal amount to the r e s i s t a n c e and thus any l i n e on the shear s t r e s s p l o t should f l a t t e n out f o r the lower d e n s i t i e s . I t i s most unfortunate that no d i r e c t measurements of u* could be obtained to check the a p p l i c a b i l i t y of Cla u s e r ' s v e l o c i t y d e f e c t p l o t . C e r t a i n l y the rough e q u a l i t y of slope f o r the l i m i t e d region i n d i c a t e s agreement between the r e s u l t s , but u n t i l d e f i n i t i v e measurements of u x can be made i n a flume over a rough boundary the r e s u l t s must be considered with c a u t i o n . The r e s u l t s of Koloseus and Davidian are very low i n d i c a t i n g an extremely high u s . T h e i r measurements of u s were obtained from the slope of the flume, but as t h e i r measurements were made at high v e l o c i t y the f r e e s u r f a c e was very contorted and i t seems l i k e l y that there was 65 considerable energy d i s s i p a t i o n i n s u r f a c e waves. (The p l a t e s In Koloseus and Davidian show diamond wave p a t t e r n s of considerable s i z e . ) TURBULENCE INTENSITY, DISSIPATION RATES MI> SCALES Turbulence i n t e n s i t y The r e s u l t s of the mean flow survey i n d i c a t e t h a t s e v e r a l regions e x i s t i n the flow: a w a l l region below the tops of the roughness elements, a wake zone above the elements (Figure 14) extending t o approximately two roughness heights from the bed (Zone I) , and above t h i s a shear l a y e r where the defect law a p p l i e s (Zone I I ) . These l a y e r s must be i d e n t i f i a b l e i n the turbulence c h a r a c t e r i s t i c s , f o r i t i s presumably by the process of wake shedding t o produce O'Loughlin's 'approximately uniform l a y e r of high i n t e n s i t y turbulence* that t h i s flow s t r u c t u r e must be maintained. In i t s s i m p l e s t terms boundary l a y e r f l o w has been considered to be composed of two l a y e r s , an inner r e g i o n and an outer f l o w , where the defect law a p p l i e s . Figure 16a i l l u s t r a t e s data on turbulence i n t e n s i t y f o r the whole flow. S i m i l a r i t y of r e s u l t s i s maintained down to approximately y/D=0.25 but then shows a systematic departure c l o s e t o the w a l l , r e g i s t e r i n g widely varying values at y/D<0.15 which i s the zone below the tops of the roughness elements. Thus f o r the region y/D>0.25 u^ and D appear s u i t a b l e s c a l i n g parameters. Figure 16a Comparison between turbulence data f o r d i f f e r e n t d e n s i t i e s . 67 The inner f l o w , r e g i o n , below 0.25, has fewer r e s u l t s c l o s e to the bed due to the problem of l a r g e probe s i z e . The sm a l l e s t y + (yu "'/ v) value was approximately 16 so that the region of decrease of u'/u^ c l o s e to the w a l l could not be measured. The n o t i o n , discussed by Monin and Yaglom (1971) , th a t u 1 + — = Constant as y + <• . u* only a p p l i e s to the inner law where a region of constant s t r e s s e x i s t s . Far from the w a l l y + gets very l a r g e but the l o g a r i t h m i c law no longer a p p l i e s and u„ and D become the s c a l i n g parameters. R e s t r i c t i n g y + to l e s s than 300 i t may be seen (Figure 16b) tha t the r a t i o U'/u tends t o converge t o a constant i n accord with p r e d i c t i o n s . The value 2.3 was obtained as the asymptotic constant from data of Laufer(1950) and Klebanoff (1955)- A s i m i l a r value was found i n the atmospheric boundary l a y e r by Panofsky and Prasad (1965). The spread of r e s u l t s i n c r e a s e s as the w a l l i s approached. The envelope of r e s u l t s widens i n d i c a t i n g t h a t spacing of elements, and not j u s t y* ., i s p o s s i b l y an Important s c a l i n g parameter. Figure 16a may be i n t e r p r e t e d as showing three zones. For y/D<0.15 there i s a broad spread of r e s u l t s where the spacing of the elements Is important. For y/D>0.35 the i n t e n s i t y behaves as a decreasing f u n c t i o n of y/0. Between these two values the i n t e n s i t y appears t o tend towards a constant value. This c o l l a p s e of data shown i n Figure 13 f o r y/D>0.35 i s i n accord with Reynolds number s i m i l a r i t y hypothesis 4.0 3.0 1.0 •a--n-A X * *. 2-3-a o • • V 0 50 100 + 150 200 250 300 F i g u r e 16b Comparison between turbulence i n t e n s i t y values f o r i n n e r r e g i o n of the boundary l a y e r . For symbols see F i g u r e 16a. 69 (Townsend 1956 p .89) concerning the outer f l o w , notably t h a t the motions of the energy c o n t a i n i n g components of the turbulence are determined by the boundary c o n d i t i o n s of the flow alone, and are independent of the f l u i d v i s c o s i t y except so f a r as a change i n the f l u i d v i s c o s i t y may change the boundary c o n d i t i o n s . A recent paper by Perry and Abell(1975) on pipe flow examined the regions of overlap of the inner and outer l a y e r s . The authors s t a t e d that i f a f u n c t i o n f l y / D | d e s c r i b i n g the outer flow i s to extend over a region where the turbulence i s a l s o described by u'/u„ =F)y +| then u'/u must be a u n i v e r s a l constant s i n c e the arguments y + and y/D are a r b i t r a r i l y independent v a r i a b l e s . (Perry and A b e l l 1975, p.268) They p l o t t e d mean v e l o c i t y p r o f i l e s on the same graph as turbulence data and showed that there i s a region of constant turbulence l e v e l corresponding to the zone of overlap of the in n e r and outer flow zones. From the data i n t h i s research there appears a r e g i o n , or at l e a s t an i n f l e x i o n p o i n t , centered a t y/D=0.25 which would correspond to the middle of the wake l a y e r . This i s i n t e r p r e t e d as suggesting t h a t the three l a y e r s i n d e n t i f i e d f o r the mean flow may be t e n t a t i v e l y i d e n t i f i e d i n the turbulence i n t e n s i t y . So f a r we have examined the s c a l i n g laws f o r the turbulence i n t e n s i t y . This has determined the a p p l i c a b i l i t y 70 to streamflow s i t u a t i o n s of r e s u l t s from other boundary l a y e r s t u d i e s and confirmed the use of u , y f o r the wake l a y e r , some measure of d e n s i t y as w e l l as u and y f o r the w a l l r egion and u^and D f o r the outer region as s c a l i n g parameters f o r the v a r i o u s regions of the flow. However, t h i s procedure has removed the e f f e c t s of the d i f f e r i n g boundary roughnesses. I t i s u s e f u l f o r i n t e r p r e t i v e purposes, i n examining the notion of roughness spacing, to p l o t the RMS of the v e l o c i t y f l u c t u a t i o n s , non-dimensionalized by f r e e stream and l o c a l mean v e l o c i t i e s , a g a i n s t r e l a t i v e depth. These are not meant to be s i m i l a r i t y p l o t s but are intended to draw out the e f f e c t s of the roughness a r r a y s : they i n d i c a t e the r o l e t h a t turbulence plays i n determining the o v e r a l l boundary l a y e r s t r u c t u r e . Figure 17a i s a p l o t of u'/0(h) versus r e l a t i v e depth. The data show some s c a t t e r but there seems t o be a c o n s i s t e n t p a t t e r n t h a t may be e x p l a i n e d i n terms of the density of roughness elements. The skimming and i s o l a t e d wake flows as described by Morris might be expected t o behave i n roughly s i m i l a r f a s h ion as regards turbulence g e n e r a t i o n , e s p e c i a l l y i n t h i s i n s t a n c e i n which the s i m i l a r l y rough s u r f a c e s of the tops of the blocks and the bed would be r e s p o n s i b l e f o r most of the r e s i s t a n c e t o f l o w . The data f o r d e n s i t i e s 1/8, 1/48 and 1/80 group very c l o s e together and show c l o s e s t agreement with the smoothed data curve of turbulence i n t e n s i t y measured by Counihan over a . Lego baseboard i n a wind t u n n e l . The most s i g n i f i c a n t departures occur f o r 0.5>y/D>0.15, i n the lower h a l f of the flow. Here S o l i d l i n e r e p r e s e n t s data from C6unihan(i971) 72 the d e n s i t i e s 1/12, 1/16 and 1/20 have turbulence i n t e n s i t i e s double those of the other d e n s i t i e s . Such a s i t u a t i o n i s p l a i n l y a case of 'wake i n t e r a c t i o n flow' whereby the wakes downstream of each element i n t e r a c t shedding v o r t i c e s i n t o the flow, the s t r o n g l y sheared edges of which f l u c t u a t e across a downstream block to give r i s e t o c o n s i d e r a b l e v e l o c i t y g r adients and thus high turbulence p r o d u c t i o n . The d i f f u s i o n of t h i s very i n t e n s e turbulence i n t o the upper p a r t of the flow i s evident f o r the density 1/12 shows a c o n s i s t e n t l y high i n t e n s i t y at a l l depths. This I n t e r p r e t a t i o n i s f u r t h e r s u b s t a n t i a t e d by observing Figure 17b. Density 1/8 c l e a r l y i n d i c a t e s t h a t , when scaled by , the maximum i n t e n s i t y of tur b u l e n c e occurs f a r away from the w a l l . A s i m i l a r s i t u a t i o n w i l l be observed f o r d e n s i t i e s 1/16 and 1/20. I t would be expected t h a t u' would be at a maximum at the wal l as here the v e l o c i t y g r adients are steepest, and here the turbulence should be at a maximum. The data from Counihan (1971) and from the p l a i n Lego board i n the flume show e x c e l l e n t agreement and the den s i t y 1/80 shows a p a r a l l e l behaviour, the displacement p o s s i b l y being due t o the a r b i t r a r y d e f i n i t i o n of &. and U,,,,. Figure 17b shows d e c i s i v e l y t h a t f o r de n s i t y 1/8 the base i s no longer at the Lego board but r a t h e r has s h i f t e d t o above the top of the roughness elements. Such a s i t u a t i o n i s c l e a r l y t o be expected i n skimming flow. The i n t e n s i t i e s f o r 1/12, 1/16 and 1/20 may be i n t e r p r e t e d as re p r e s e n t i n g very a c t i v e wake shedding, as 73 shown i n Figure 17a, with a decrease and eventual c l o s e agreement with the p l a i n Lego board f o r lower and lower d e n s i t i e s . This s i t u a t i o n w i l l become i n c r e a s i n g l y e v i d e n t when the d i s s i p a t i o n p l o t s are considered. D i s s i p a t i o n The d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy i s one of the most s i g n i f i c a n t phenomena i n any study of t u r b u l e n t boundary l a y e r s . The parameter may be used as a s c a l i n g v a r i a b l e i n the s p e c t r a . I t allows e v a l u a t i o n of the m i c r o s c a l e of flow which w i l l determine the lower l i m i t s o f sediment p a r t i c l e s i z e that may be held i n suspension. . More i m p o r t a n t l y , I t g i v e s a d i r e c t measure of where the flow energy I s going, f o r In most n a t u r a l channels the degradation of energy occurs without u s e f u l work being done on the boundary. P l o t s of energy d i s s i p a t i o n are u s u a l l y r a p i d l y decreasing f u n c t i o n s of d i s t a n c e from the w a l l . The c a l c u l a t i o n of e was based on the i s o t r o p i c e v a l u a t i o n as only one v e l o c i t y d e r i v a t i v e was measured. There i s a d i s c u s s i o n l a t e r i n t h i s chapter on the v a l i d i t y of t h i s assumption. As energy d i s s i p a t i o n r a t e has the dimensions L 2 T ~ 3 a dimensionless p l o t of e takes the form e D/u3 which i s shown i n Figure 18a. The f u n t i o n i s w e l l behaved and the s c a l i n g with u $ reduces the e f f e c t of v a r y i n g roughness d e n s i t y throughout the top 15% of the f l o w . The f a i r l y l a r g e s c a t t e r i s due p o s s i b l y to e r r o r s i n computing : as t h i s value i s cubed, only a small change w i l l markedly i n f l u e n c e 74 i.o 0.8 0.6 y/ -D 0.4 0.2 9 o D •a--B-0 2 4 6 8 10 12 14 16 18 20 Figure 18a Estimates of t u r b u l e n t energy d i s s i p a t i o n nondimensionalized by shear v e l o c i t y and depth, p l o t t e d against r e l a t i v e depth. For symbols see Figure 16a. 75 the r a t i o s . Close to the bed i t may be seen t h a t the roughness d e n s i t i e s do have a s i g n i f i c a n t e f f e c t r e s u l t i n g i n a higher l o s s f u r t h e r away from the boundary than would normally be expected f o r a rough boundary l a y e r . T h i s s i t u a t i o n i s h i g h l i g h t e d i n Figure 18b where the p l o t of e p s i l o n i s presented. I t i s c l e a r t h a t the lower boundary has e f f e c t i v e l y s h i f t e d away from the w a l l f o r d e n s i t i e s 1/8 and 1/16 and t h i s e f f e c t i s marginally present up t o 1/20. The height above the bed of the peak of d i s s i p a t i o n shows a s l i g h t i n crease with i n c r e a s i n g d e n s i t y up to 1/20. This peak occurs always w i t h i n the l a y e r of o v e r l a p of the i n n e r and outer flow zones, i n the wake or Zone I as discussed i n the s e c t i o n on mean v e l o c i t i e s . M i c r o s c a l e and macroscale The microscale can be c a l c u l a t e d by two methods. I t may be computed from the energy spectrum as BO 1 _ 4TT_2/n2<j>(n)dn * 2 " U 2 ° (27) or I t may be obtained from analog measures of the d e r i v a t i v e as (28). A p l o t of the microscale computed from the spectrum versus t h a t computed by the d e r i v a t i v e method, Fi g u r e 19a, i n d i c a t e s reasonable agreement e s p e c i a l l y as i t i s necessary 0.0 20.0 80.0 ^> i Q K T r ^ t - i m a t - e s of turbulent energy K a M o n r f t e T o t t e d against relative depth a c : - -I £ 0 I- CJ) 0 ., r a 0 .25 .50 .75 1.0 1.25 1.5 1.75 ^• spec t rum X analog Figure 19a Comparison of Taylor microscale measured by analog devices and from the spectrum. Symbols as In Fig.16a. 15 — — E J - ' &. $ k •" •& -n o___ D 0 £w •e-0 •a-D 0 * y. I -ci—2-th- V p D o C <j>_ X —i on ca X A '~ •a-.02 .04 .06 .03 0.1 .12 .14 .IS Figure 19b Taylor microscale nondimensionalized by depth plotted against r e l a t i v e depth. 78 to i n t e g r a t e the area under the curve n 2 <}> (n) . (This agreement suggests that a l l the s i g n i f i c a n t f r e q u e n c i e s have been included i n the spectrum a n a l y s i s . ) The r e s u l t s of McQuivey(1967) showed that the m i c r o s c a l e approached a maximum at y/d=0.45 and then decreased. The r e s u l t s of N a l l u r i and Novak (1973) d i d not e x h i b i t such obvious behaviour: the microscale i n c r e a s e d away from the w a l l and then became approximately constant. The r e s u l t s of Powe and Townes(1973) f o r pipe flow showed the m i c r o s c a l e t o s l i g h t l y increase or at l e a s t remain constant over the upper 50% of the flow and not t o markedly decrease as McQuivey found.. As the Taylor microscale represents a measure of the peak of the d i s s i p a t i o n spectrum, i t i s expected that, the microscale should show a tendency to decrease throughout the flow as i t i s known that f u r t h e r out i n t o the flow the l o c a l Reynolds number, based on d i s t a n c e from the w a l l and l o c a l mean v e l o c i t y , w i l l be s i g n i f i c a n t l y h i g h er. There should then be a tendency f o r the energy to be cascaded to higher f r e q u e n c i e s , thus r e s u l t i n g i n a decrease i n the m i c r o s c a l e . This c o n c l u s i o n i s tempered by the s i m i l a r i t y parameters u„ and U t h a t w i l l a l s o apply i n the outer l a y e r f l o w . The « CO microscales computed from the d e r i v a t i v e method are presented i n Figure 19b. The spread of the r e s u l t s i n d i c a t e s t h a t the microscale i s i n f l u e n c e d by the amount of energy d e l i v e r e d down the cascade and by the l o c a l Reynolds number. The macroscale of turbulence i s given as the area under the a u t o c o r r e l a t i o n f u n c t i o n and i s t h a t s c a l e which 79 i n t e r a c t s w i t h t h e mean f l o w most e f f i c i e n t l y t o e x t r a c t e n e r g y . I t s measurement f r o m s p e c t r a l i n f o r m a t i o n i s n o t v e r y e a s y . O n l y f o u r d e p t h s i n e a c h p r o f i l e were a n a l y s e d t o o b t a i n t h e a u t o c o r r e l a t i o n f u n c t i o n , w h e r e a s s e v e n p o i n t s were a n a l y s e d f o r t h e s p e c t r a l m e a s u r e m e n t s . The c u s t o m a r y a p p r o a c h t o e v a l u a t e t h e m a c r o s c a l e i s t o assume t h a t t h e s p e c t r u m may be a p p r o x i m a t e d by t h e f o r m g i v e n b y L a u f e r ( 1 9 5 0 ) n o t a b l y , _ J t L / U _ _ ^ - , A , 2 i r L n , a. u 1 + ( ^ ) \T~J (29) t h e n f o r t h e n o r m a l i z e d s p e c t r u m t h e m a c r o s c a l e i s d e f i n e d a s <t> ( n ) = 4L o„ uTy) U (30) The d e r i v a t i o n o f t h e a b o v e e q u a t i o n i s o u t l i n e d i n R i n z e ( 1 9 5 9 , p. 199) and i s p r e d i c a t e d on t h e e x i s t e n s e o f a n i n e r t i a l s u b r a n g e , w h i c h i s t a k e n t o i n d i c a t e t h a t v i s c o s i t y i s no l o n g e r p l a y i n g a s i g n i f i c a n t r o l e . A l l t h e t u r b u l e n t R e y n o l d s numbers d e a l t w i t h i n t h i s r e s e a r c h a r e b e l o w t h e l e v e l a t w h i c h s u c h a r a n g e m i g h t be e x p e c t e d , a n d t h u s s u c h a method i s l i k e l y t o be n o t s t r i c t l y v a l i d . A n o t h e r m e t h o d , g i v e n by K a i m a l (1973) , i s t o u s e t h e w a v e l e n g t h /V c o r r e s p o n d i n g t o t h e peak o f t h e l o g a r i t h m i c power s p e c t r u m . W i t h t h e a s s u m p t i o n o f T a y l o r ' s h y p o t h e s i s , Webb (1 9 5 5 , i n K a i m a l 1973) h a s shown t h a t f o r a n e x p o n e n t i a l l y d e c a y i n g c o r r e l a t i o n f u n c t i o n A = — ; L = A / 2 i r npeak ( 3 1 ) 80 This method was used t o estimate L and the r e s u l t s are given i n Figure 20. The r e s u l t s of McQuivey (1 967) and N a l l u r i and Novak (1973) showed L peaking at approximately 0.6 of the r e l a t i v e depth, while the r e s u l t s of Powe and Townes(1973) showed the macroscale to become constant above t h i s depth. This i s i n agreement with the r e s u l t s here although there i s c o n s i d e r a b l e s c a t t e r i n absolute s c a l e of the macro-eddies which might be a f u n c t i o n of u : that i s the s c a l e i s dependent on the production terra uvdo/dy. Summary of mean flow r e s u l t s and i n t e r p r e t a t i o n of i n t e g r a l Erasures of turbulence Before c o n s i d e r a t i o n of the turbulence s p e c t r a i t should be i n s t r u c t i v e t o summarize the r e s u l t s and i n t e r p r e t the data i n an attempt to e l u c i d a t e the r o l e of roughness. From the measurements of H•, n' and \ (the s k i n f r i c t i o n c o e f f i c i e n t ) i t I s c l e a r that roughness d e n s i t y markedly a f f e c t s these parameters, with a peak i n r e s i s t a n c e appearing at a density of approximately 1/12. This r e s u l t i s i n broad agreement with the data obtained by Koloseus and Davidian(1966). Work c a r r i e d out by Marshall(1971) showed t h a t whether the p a r t i c l e s are randomly or r e g u l a r l y spaced does not seem to have a s i g n i f i c a n t e f f e c t on r e s i s t a n c e when the o v e r a l l d e n s i t y of p a r t i c l e s i s held constant. For a g r a v e l bed channel t h i s i m p l i e s that the g r e a t e s t r e s i s t a n c e to flow would e x i s t f o r a bed on which p a r t i c l e s occupying 1/12 of the area absorb most of the s t r e s s at the boundary 1.0 0.8 0.6 0.4 0 0 0 a 4> X- -fc> A t ) o a > •3- A & 0 A 0rf?£ ({>A x •& In A • 0.2 0.4 0-6 0.8 1.0 1.2 L / D F i g u r e 20 V a r i a t i o n of t u r b u l e n t macro-s c a l e , nondimensionalized by depth, with r e l a t i v e depth. For symbols see F i g u r e 16a 82 and are r e s p o n s i b l e f o r generating most of the tu r b u l e n c e . This i m p l i e s that i n the long term research s t r a t e g y the bed forms w i l l have to be c h a r a c t e r i z e d by a 2-D spectrum a n a l y s i s f o r the spreading of wakes i n the h o r i z o n t a l c r o s s stream d i r e c t i o n may be very important i n c h a r a c t e r i z i n g and determining the turbulence production c l o s e to the bed. From the mean p r o f i l e s three d i s t i n c t zones may be observed: an outer l a y e r i n which the d e f e c t law a p p l i e s and the s c a l i n g parameters are D, u s and D m % a wake l a y e r i n which v e l o c i t y i s a f u n c t i o n of u s , y and some measure of roughness d e n s i t y . I t extends over a narrow zone approximately one roughness height t h i c k above the top of the elements. L a s t , there i s the w a l l l a y e r where spacing, u s on the i n t e r v e n i n g s u r f a c e and the d i s t a n c e from the w a l l a r e a l l important. In the d i s c u s s i o n of turbulence i n t e n s i t y , the s c a l i n g parameters were considered f o r an inner and outer l a y e r : the zone of overlap I s here i d e n t i f i e d as the t h i r d l a y e r or wake zone. a f u r t h e r c o n f i r m a t i o n of these n o t i o n s , and a severe t e s t of the accuracy of the measurements i s provided by c a l c u l a t i n g u s by two methods. The f i r s t method was from the r e l a t i o n u*(shape) = 3 ^ (32) which may a l s o be developed from Rotta's (1962) c r i t e r i a f o r e v a l u a t i n g shear s t r e s s . A second method i n v o l v e s assuming l o c a l e q u i l i b r i u m throughout the flow. In n e u t r a l s t a b i l i t y . 83 the s i t u a t i o n expected i n a highly turbulent water channel, the kinematic stress i s given by 2 _ ,.„„ dU (33) u* = k"y If i t i s assumed that the rate of mechanical production of turbulent energy u s 2du/dy eguals the d i s s i p a t i o n rate then E = u2 dU m 1 y j = u 3 U * dy k" y • ^ y . ( 3 4 ) where k'.' i s von Karman's constant and hence u*(spec) = ( 0 . 4 y e ) 1 / 3 (35) I f the assumption of l o c a l equilibrium i s correct then the r a t i o u „(shape)/u (spec) should be equal to unity across the flow. Figure 21 indicates t h i s i s not the case, but that three c l e a r l y separable regions are v i s i b l e . There i s an outer flow region where the r a t i o drops below one, i n d i c a t i n g non-equilibrium and possibly showing that t h i s region imports the turbulence from the zone below. In the evaluation of e in t h i s study i t has been necessary, as i s indeed customary, to assume isotropy., However, close to the free surface t h i s i s unlikely to be the case and t h i s w i l l lead to an underestimate of e since the gradients in the v e r t i c a l w i l l l i k e l y be quite steep as the surface i s approached, f o r r i g h t at the surface d()/dy=0. Secondly because of the presence of the free surface u 2dD/dy does not go to zero as It would i n a normal boundary layer where the d e r i v a t i v e would become vanishingly small. In t h i s case the boundary . ......y. 6 d •a- 9 43 A3-0Q x D Am ) y A A no Tt. V & net© > Lk~-U/ * * H ; -a r y iTl 0.4 0.6 0.8 1.0 u„(from spectrum) ^ ( f r o m velocity prof i le) F i g u r e 21 Ratio of shear v e l o c i t y c a l c u l a t e d from the v e l o c i t y p r o f i l e and from the spectrum assuming l o c a l e q u i l i b r i u m p l o t t e d a g a i n s t r e l a t i v e depth. For symbols see F i g u r e 16a. 85 l a y e r i n t e r s e c t s the f r e e s u r f a c e and leads t o l a r g e r values f o r the production than would otherwise have been expected. A second zone, which may be termed the inner bed l a y e r , shows s i m i l a r n on-equilibrium c o n d i t i o n s with seemingly s i g n i f i c a n t d i f f u s i o n of t u r b u l e n t k i n e t i c energy. This i s due to the v i o l a t i o n of the assumption that only d e r i v a t i v e s of mean q u a n t i t i e s i n the v e r t i c a l are important. In t h i s zone the wake spreading and i n t e r a c t i o n e f f e c t s may r e s u l t i n a steep l o c a l v e l o c i t y g r adient which w i l l r e s u l t i n con s i d e r a b l e t u r b u l e n t energy production. The t h i r d zone, or wake l a y e r , appears t o be i n approximate e q u i l i b r i u m , with i n some cases s l i g h t excess production which i s then d i f f u s e d to the other zones. The good agreement, between the estimates i n d i c a t e s that t h i s r e gion most c l o s e l y agrees with the s i m i l a r i t y p o s t u l a t e s u s u a l l y advocated f o r boundary l a y e r s , w h i l e the outer zone although showing q u i t e good agreement with the v e l o c i t y d e fect law d i s p l a y s some i n f l u e n c e s of the presence of the fr e e s u r f a c e . ENERGY DISTRIBUTION AND FREQUENCY STRUCTURE Spectra, s c a l e s and scajling; S p e c t r a l measurements were made at seven depths f o r each of the roughness a r r a y s . An example i s shown i n Figure 22a. At the three lowest p o i n t s (y/D equal t o 0.01, 0.02 and 0.13) the spectra e x h i b i t a strong n - 1 r e g i o n which e x i s t s f o r s l i g h t l y over a decade, whereas f o r y/D>0.15 there appears t o be an nS/3 r e g i o n , again f o r approximately one decade. Klebanoff(1955) and Comte-Bellot{1965) found t h a t the -5/3 region became more extensive i n accordance w i t h the r e s u l t s of Laufer(1950) who found t h a t the r e g i o n reached maximum extent at the centre l i n e of the channel. The r e s u l t s from t h i s research i n an open channel show the trend i s a l s o present except f o r the measurement r i g h t at the f r e e s u r f a c e where the -5/3 zone i s p r a c t i c a l l y absent. Such a r e s u l t was common t o a l l the roughness arrays. I t i m p l i e s t h a t the l o n g i t u d i n a l s c a l e of turbulence i n i t i a l l y i n c r e a s e s with d i s t a n c e from the boundary and peaks at j u s t above mid-depth, a f t e r which t h e f r e e s u r f a c e may begin t o c o n s t r a i n the fl o w . This r e s u l t s i n a t r a n s f e r of energy from the v e r t i c a l f l u c t u a t i o n s t o the h o r i z o n t a l components which would be seen as an Increase i n shear. Spectra i n f l u e n c e d i n t h i s way have a tendency to behave more l i k e turbulence s p e c t r a c l o s e t o a w a l l . The presence or otherwise of a -5/3 r e g i o n i s of s i g n i f i c a n c e i n e v a l u a t i n g the concept of l o c a l i s o t r o p y : t h i s point w i l l be f u l l y discussed i n the s e c t i o n on i s o t r o p y . The e f f e c t of u may be examined by l o o k i n g a t spe c t r a f o r f i v e d e n s i t i e s at a constant depth. The widest -5/3 region occurs f o r den s i t y 1/80 while such a zone i s t o t a l l y absent f o r d e n s i t i e s 1/1 and 1/8. (Figure 22b). Spectra i n the atmosphere are u s u a l l y presented i n the form l og n <j> (n) versus frequency or a reduced frequency ny/ L T , iog,.o0(n) I 0 H 10" - 1 0 ,-5 ** x. * X X \ o ° xx X * 0 x ° 0 V Q X \ 0 x \ x \ 1 0 0 0 °0 DENSITY X °o 0 0 >4 0 0 0 * X X X Vs \ X X °0 0 0 0 V43 \ x^ 0 0 0 0 Veo - 0 0 0 0 X X 0 X 0 n 0 0 ( < « 1 0 0 0 X 0 0 0 0 X 0 0 X X X 0 0 0 I0< 10 I0 C log ,0 n I 0 Z 10° IO1 IO 2 I 0 Z 10° 10' F i g u r e 22b Normalized s p e c t r a f o r f i v e d e n s i t i e s at the same height y/D-0.22 f o r streamwise v e l o c i t y f l u c t u a t i o n s . 00 89 where y represents d i s t a n c e from the boundary and U i s the mean v e l o c i t y at that height. There have been many papers r e c e n t l y (e.g. Busch and Panofsky 1968: Kaimal e t a l 1972) reviewing s p e c t r a l measurements and attempting to c o l l a p s e the r e s u l t s from v a r i o u s heights onto a s i n g l e curve. The s p e c t r a are normalized by the variance and p l o t t e d a g a i n s t f/f°, where f i s the reduced frequency and f° i s a frequency r e l a t e d to the c h a r a c t e r i s t i c length s c a l e f o r the parameter. The spectrum i s nearly symmetric with slope of f o n the high freguency s i d e and f + 1 on the low freguency s i d e . Kaimal (1973) showed t h a t fo=0.26f m a xwhere fo i s the peak of the reduced frequency, and that the s p e c t r a take the approximate form n<j.„(n) = 0.164 (f/f°) "ti 1 + 0.l64(f/f°) 5 / 3 (36) where n i s the c y c l i c frequency, n<{>u(n) i s the l o g a r i t h m i c s p e c t r a l d e n s i t y and f 0 i s the reduced freguency at the i n t e r c e p t of the e x t r a p o l a t e d i n e r t i a l subrange slope with the l i n e n ^ (n) /iT 2= 1. The advantage of using f/f° on the a b s c i s s a i s t h a t i t non-dimensionalizes freguency without i n t r o d u c i n g a s p e c i f i c l ength s c a l e a s s o c i a t e d with distance from the boundary. This i s important as the s p e c t r a shown i n F i g u r e s 24a and 25a w i l l show. The r e s u l t s of attempting t o p l o t some of the s p e c t r a l estimates from four depths i n d i c a t e s c o n s i d e r a b l e variance i n the r e s u l t s . None of the data conform to the slope determined by Kaimal, though at y/D=0.57 there i s a Figure 23 Normalized spectra plotted against a modified frequency scale showing approach to universal ttehaviour for four depths. (After Kaimal T972) n 0 v (n) VD o •  0.01 / A A o. a o. x o. 13 22 38 / *0 x n X 3 A % \ A D 0 X . - - —.-n--*< . - 11 X A a 0 U A 0 X (—1—1 0 A U A 0 X ' n A o i . ... .02 .05 O.i 0.2 0.5 1.0 2.0 5.0 ny Figure 24a Normalized spectra for v e r t i c a l velocity fluctuations. S o l i d curve i s from Busch and Panofsky(1968). n 0 u ( n ) 0.1 .01 Vo 0-Oi 0-10 0.3& 0.97 •''7 , s' s • / » / / y \ v - s \ N" X \ * / \ .001 .01 0.1 1.0 10 n y a / g Figure 24b Longitudinal spectra against adjusted nondimension frequency where y was the distance required for spectra to peak at 0.1. n 0 u(n) d u 2 1.0 0.1 .01 DENSITY — Vs ...... i / 2 0 — ' / 8 0 X x X N X X "•. \ \ . \ \ .001 .01 0.1 1.0 10 n y / y Figure 25a Longitudinal spectra for three densities at the same height (y/D=0.36) showing s h i f t i n g peak as a r e s u l t of d i f f e r i n g u s . y represents actual distance from the bed. f 0 u (f ) 10° io" 3 IO"2 10"' 10° IO1 IO 2 f y / u Figure 25b L o n g i t u d i n a l s p e c t r a p l o t t e d against nondimensionalized frequency where y represents a c t u a l distance from the w a l l . tendency f o r the f u n c t i o n t o conform with reasonable closeness t o the d e s i r e d shape (Figure 23a). This confirms-the s u s p i c i o n that the -5/3 range i s l a r g e l y i l l u s o r y . F o l l o w i n g the Monin Obukhov s c a l i n g , the s p e c t r a of the v e r t i c a l f l u c t u a t i o n s were normalized Nby u^and s c a l e d with a c h a r a c t e r i s t i c frequency y/U i n accord with the r e s u l t s of Busch and Panofsky (1968) (Figure 24a). The u s p e c t r a , although having an approximately s i m i l a r ' s h a p e (Figure 24b), do not show simple s c a l i n g with height (Figure 25a). The s p e c t r a separate very c l e a r l y i n e x a c t l y the same manner as noted by Perry and &bell(1975) f o r pipe flow (Figure 25b). The r e s u l t s broadly agree with the work i n atmospheric boundary l a y e r s where the summary by Beriuan (1965) and the measurements of Busch and Panofsky (1968) showed a s l i g h t v a r i a t i o n of the s p e c t r a l peak with i n c r e a s i n g e l e v a t i o n . The degree of s c a t t e r i n the published r e s u l t s i s d i f f i c u l t to estimate as the s p e c t r a l curve a t the peak i s very s m a l l , but the range of r e s u l t s r a r e l y shows more than h a l f a decade s e p a r a t i o n . The r e s u l t s here show w e l l over a decade spread. T h i s d i f f e r e n c e must be a f u n c t i o n of the s i m i l a r i t y arguments used to o b t a i n y/U as the s c a l i n g parameter. I t i s based on the notion that the boundary l a y e r i s a constant s t r e s s l a y e r , which i s c l e a r l y not the case f o r a f r e e surface flow. Secondly i t i s d i f f i c u l t to decide what, e x a c t l y , 'distance from the boundary' would mean. I t i s e a s i e s t , then, f o r comparative purposes t o s e l e c t a s c a l i n g height y°, chosen to match the frequency of the peak of each 95 curve with that of the u n i v e r s a l curve of Busch and Panofsky (1968)- The c o l l a p s e of the data i n t h i s way, as performed by Shaw e t al ( 1 9 7 4 ) , allows c l e a r comparison of the spe c t r a and permits one to see which frequencies are more pronounced at the various depths. In comparison with the v e r t i c a l v e l o c i t y curves (Figure 24b) there i s a much g r e a t e r v a r i a t i o n i n the l o n g i t u d i n a l energy d i s t r i b u t i o n on the low frequency side (Figure 24b) . Having shown that s i m i l a r i t y parameters may be of some value i n c e r t a i n depth zones of the flow Figure 25a shows the e f f e c t of va r i o u s roughness on the s p e c t r a l peak. I t i s comforting t h a t f o r a given height the spect r a should separate so e f f e c t i v e l y , i n d i c a t i n g the dominant r o l e t h a t u must play i n determining the s p e c t r a l s t r u c t u r e of the turbulence. The combined e f f e c t s of r e l a t i v e depth and roughness of the boundary can be examined by studying the cumulative s p e c t r a . Figures 26 a, b and c show the cumulative s p e c t r a normalized by the t o t a l variance while the frequency was standardized by d i v i d i n g by the maximum freguency measured. Figure 26a shows l i t t l e v a r i a t i o n i n energy content with p o s i t i o n i n the boundary, but the most v i s i b l e e f f e c t i s the way the f r e e s u r f a c e separates from a l l the other data. The s t r a i g h t l i n e (see Figure 26a) with i t s a s s o c i a t e d confidence i n t e r v a l s i s the cumulative spectrum . of a white noise s i g n a l . The spectrum of white noise i s approximately f l a t and the cumulative s p e c t r a i s used by J e n k i n s and 96 F i g u r e 2 6 a Cumulative s p e c t r a f o r one roughness d e n s i t y ( 1 / 1 6 ) f o r seven depths. Note that curve that drops below others represents the f r e e s u r f a c e . S o l i d diagonal l i n e i s cumulative spectrum of white noise w i t h a s s o c i a t e d 95% confidence i n t e r v a l s . Figure 2 6 b Cumulative s p e c t r a f o r s i x roughness d e n s i t i e s at•y/D=0.37. 98 0.0 —1— 0.1 I— 0.2 I— 0.3 0.4 — , — ! 0.5 F/FMRX 0.5 — 1 0.7 ~I 0.8 I— 3.9 1.1 F i g u r e 26c Cumulative s p e c t r a f o r s i x roughness d e n s i t i e s at y/D=0...13, e q u i v a l e n t to the top o f the roughness b l o c k s . F o r symbols see F i g u r e 26b 99 Watt (1968) as a t e s t c r i t e r i o n f o r white noise. The confidence i n t e r v a l s are computed i d e n t i c a l l y to the Kolmogorov-Smirnov l i m i t s a p p l i c a b l e to any cumulative d i s t r i b u t i o n f u n c t i o n as i t i s assumed that the new v a r i a b l e represents the sum of independent random v a r i a b l e s with t h e same d i s t r i b u t i o n . The s p e c t r a l estimates are each x 2 v a r i a b l e s , f o r which the cumulative sum of N independent x 2 v a r i a b l e s i s a l s o x 2 d i s t r i b u t e d . The r a t i o of the two cumulative x v a r i a b l e s forms an F d i s t r i b u t i o n so t h a t an F t e s t may be used t o compare the d i f f e r e n c e s between the cumulative s p e c t r a l d i s t r i b u t i o n s . I f the number of degrees of freedom of each v a r i a t e i s taken as the sum of the independent x 2 v a r i a b l e s cumulated t o t h a t p o i n t then the s t a t i s t i c should be reasonably robust. The r e s u l t s of the comparisons between observations f o r two d i f f e r e n t roughness arrays f o r approximately the 0.5 and 0.85 cumulative energy l e v e l s of the normalized energy s p e c t r a are presented i n Table I-The comparison f o r s i x roughness d e n s i t i e s at two depths i s presented i n Table I I . The r e s u l t s add s t a t i s t i c a l c o n f i r m a t i o n to the b e l i e f that the t o t a l energy tends t o be concentrated i n lower freguencies f o r the l e s s dense roughnesses. T h i s patt e r n i s . c o n s i s t e n t at a l l depths. T h i s might be explained by appealing to the r a t i o n a l e of the energy cascade. With greater roughness u i s much l a r g e r , but the Reynolds number u' \/ v remains very c l o s e l y . constant f o r concomitantly the microscale I s reduced i n l e n g t h 100 TABLE Depth 6 7 l 0 T 5 0T9 lTB 2T5 4~76 S u r f a c e Frequency 10 Hz , 1.6? 1.26 1.17 1.02 1.00 1.00 1.20 D e i / 1 2 y 3 0 H z 1 ' 2 0 1 , 1 6 1 * 1 2 1 ' 0 8 1 ' 0 0 1 ' 0 0 1 ' 0 5 Density . 1/48 10 Hz 1.14 1.00 1.00 1.02 1.00 1.02 1.26 30 Hz 1.08 1.02 1.03 .1.01 1.01 1.00.1.06 Table I . F r a t i o s f o r cumulated energy at two f r e q u e n c i e s ( r e p r e s e n t i n g 50 and 85$ of cumulated energy a p p r o x i m a t e l y ) f o r two d i f f e r e n t d e n s i t i e s . Comparisons were made over the depth, o f f low. F t l 6 j l ( > >. 5 =2.98 and thus none of the d i f f e r e n c e s are s i g n i f i c a n t . TABLE I I Density 178 I7l2 l/TE 1720" ITW ±7~W Depth 0.9.. 0.28 0.40 0.38 0.45 0.42 0.48 Frequency 10 Hz 4.0 0.29 0.34 0.37 0.44 0.43 0.53 Table I I Cumulative energy computed from f<j>(f) f o r two depths at one frequency f o r s i x d i f f e r e n t roughness d e n s i t i e s . To convert f i g u r e s to percentage m u l t i D l y by 100. 101 r e s u l t i n g i n a wider spectrum as the peak of the d i s s i p a t i o n spectrum increases i n freguency. In f a c t i f the Taylor microscale i s p l o t t e d a g a i n s t u , there appears a f a i r l y good c o r r e l a t i o n (Figure 27). This p l o t represents the data at y/D=0.22 which i s i n the ov e r l a p zone where s c a l i n g with u i s t o be expected. While a t t e n t i o n has focussed on the r o l e of roughness, on the Taylor microscale and the r o l e of u* i n determining the u n i v e r s a l shape of the spectra on nondimensional freguency axes, the c l a s s i c a l approach to examining the u n i v e r s a l nature of the spectra a t the higher f r e g u e n c i e s i s to u t i l i z e the Kolmogorov s c a l i n g parameters. In Chapter 1 i t was s t a t e d that f o r s u f f i c i e n t l y high Reynolds number 4>(k) = ( e k 5 ) 1 7 4 F(k/ks) where k i s wavenumber, ks i s the Kolmogorov microscale f o r the energy spectrum and k2<J>(k) = k s 2 ( e v 5 ) 1 / i | ( k / k s ) 2 F ( k / k s ) (38) f o r the d i s s i p a t i o n spectrum. P l o t s of ks ( i n f a c t i t s r e c i p r o c a l the Kolmogorov wavelength n ) are shown i n Figures 28 a and b. These p l o t s show pronouncedly the e f f e c t of the u i n Figure 28a while the e f f e c t of the f r e e s u r f a c e i s seen i n Figure 28b. This leads to the c o n c l u s i o n that there are two opposing trends o c c u r r i n g w i t h i n an open channel flow. As one t r a v e r s e s away from the boundary there should be an e v o l u t i o n towards a l e s s s t r o n g l y sheared f l o w and a more X .55 .50 .45 .40 .35 X y X 1.5 2.0 2.5 3.0 1.0 .75 0.5 .25 Figure 27 Taylor microscale plotted against; shear v e l o c i t y at one depth (y/D=0.22) for six roughness densities For. symbols see,Fig. .16a; 0 A L 0 A.' 0 ,n * •a-o na A q 6 •B-i t 1.0 1.5 2.0 1/.D X I ° l Figure 2 8 a ' Kolmogorov microscale divided by depth plotted against r e l a t i v e depth. See Fig, 1 6 a for symbols. 1.0 0.5 Vii. U i f" -a -a 2 3 4 5 Figure 2&b Kolmogorov microscale showing apread of. results at the free surface. o r u 103 e x t e n s i v e i n e r t i a l subrange as the macroscale w i l l not be c o n s t r a i n e d by the presence of the w a l l . Against t h i s i s the observation t h a t the t u r b u l e n t Reynolds number w i l l be a t best constant or maybe decreasing as the i n t e n s i t y of turbulence i s dropping very r a p i d l y . The e f f e c t s of these opposing trends may be observed i n Figure 29a which shows the s p e c t r a p l o t t e d i n u n i v e r s a l axes of l o g <j,(k) /(.ev )**versus l o g k/ks. The i s o t r o p i c s o l u t i o n i s p l o t t e d and i t can be seen t h a t a l l the spectra f o r a l l the depths depart s i g n i f i c a n t l y from i t , below wavenumbers of l o g k/ks of - 1 . This approximates a frequency of between 24 and 40 Hz depending on the depth i n the flow. The v a r i a t i o n between spect r a at d i f f e r e n t depths can be examined by attempting to f i t the spectrum with a s o l u t i o n that depends on the t u r b u l e n t Reynolds number. As shown by Panchev(1969) there i s no complete s o l u t i o n t o the s p e c t r a l energy equation, but the parametric s o l u t i o n proposed by Kesic(1970) and discussed b r i e f l y i n Appendix I I I a l l o w s examination of the s p e c t r a when production (m^  ) and d i f f u s i o n (m )^ are important. The two parameters nr and m ^ were shown by Kesic(1970) to be dependent on the t u r b u l e n t Reynolds number by examining the s p e c t r a of Tielman {1967) measured i n a boundary l a y e r c l o s e to the w a l l . With i n c r e a s i n g d i s t a n c e from the w a l l Re i n c r e a s e d c o n t i n u o u s l y and the production parameter had the f u n c t i o n a l dependence m l ~ 1 2 /Re x ( 3 9 ) while f o r the same flow 104 - 3 . 2 K/KS F i g u r e 2 9 a U n i v e r s a l spectrum p l o t s f o r s i x roughness d e n s i t i e s at one depth ( y / D = 0 . 2 2 ) . C o n t a i n i n g . c u r v e s c a l c u l a t e d from computed h i g h e s t and lowest t u r b u l e n t Reynolds numbers. For symbols see F i g u r e 2 6 b . 105 CD ^""1 LOG K/KS F i g u r e 29b U n i v e r s a l spectrum p l o t s f o r seven depths f o r one roughness d e n s i t y ( 1 / 2 0 ) . Heavy l i n e r e p r e s e n t s i s o t r o p i c s o l u t i o n , w h ile l i g h t curve r e p r e s e n t s s o l u t i o n u s i n g m o d i f i e d Obukhov approximation based on t u r b u l e n t Reynolds number o f 120. ( S o l u t i o n from K e s i c 1970) For symbols see F i g u r e 26a. 106 m2 ~ 0.5 /Re A (40) These r e s u l t s are based on the assumption that approaching a w a l l , A 2d0/dy i s the same i n any f u l l y developed boundary l a y e r . I t i s assumed that i f Re i s the same f o r two A d i f f e r e n t mean v e l o c i t i e s then ^2dU/dy i s the same. Hence m_and m f o r two d i f f e r e n t boundary l a y e r s are the same i f Re i s the same. X P l o t t e d on Figure 29a are the p o i n t s f o r v a r i o u s patterns at a depth of 1-54 cm. The very good c o l l a p s e of the data confirms the dependence on Re where the upper and lower bounding curves based on computed Re are p l o t t e d and seen to envelop a l l the observations. This would seem to be a very s a t i s f a c t o r y method f o r c o l l a p s i n g the data at any one p a r t i c u l a r height, but i f we consider the r e s u l t s shown i n Figure 29b f o r roughness d e n s i t y 1/20 ( a s i m i l a r p a t t e r n e x i s t s e q u a l l y c l e a r l y f o r a l l other d e n s i t i e s ) the c o l l a p s e i s not n e a r l y so good. The s e p a r a t i o n c e r t a i n l y cannot be explained i n terms of Reynolds number dependence. The lower p o i n t s , r e p r e s e n t i n g the f r e e s u r f a c e , have a t u r b u l e n t Reynolds number of only 95 but they would b e t t e r f i t a curve where the t u r b u l e n t Reynolds number was 25, or m ^  i n s t e a d of being 0.005, was four times l a r g e r . This discrepancy i s to be a n t i c i p a t e d , f o r at the f r e e s u r f a c e i t I s not the gradient dU/dy that i s so important to the turbulence production, but r a t h e r the i n t e r a c t i o n among the t u r b u l e n t v e l o c i t i e s and pressure c o r r e l a t i o n a c t i n g t o make the turbulence i s o t r o p i c . Here, the behaviour i s c o n s t r a i n e d by 107 the c o n t i n u i t y c o n d i t i o n s imposed by the f r e e s u r f a c e . Conclusions f o r s p e c t r a l measures ' • . The c o n c l u s i o n s that may be drawn from t h i s examination of the spectra are that i t i s p o s s i b l e over c e r t a i n ranges of the flow to o b t a i n s c a l i n g parameters that permit the c o l l a p s e of the s p e c t r a to c l o s e conformity. The r o l e of u„ , t h a t i s of the roughness of the boundary, i s s i g n i f i c a n t and the s e p a r a t i o n of the peaks when n<f>(n) i s p l o t t e d a g a i n s t ny/U i s very encouraging. The l a c k of apparent f i t to the Kolmogorov u n i v e r s a l s c a l i n g curve confirms the s u s p i c i o n that the -5/3 s l o p e was t r a n s i t o r y , and t h a t i n e r t i a l subranges are l i k e l y t o be found only i n the l a r g e s t r i v e r s , s i n c e the Reynolds numbers are otherwise too low. The s c a l i n g of the s p e c t r a using Re to estimate the production X and d i f f u s i o n parameters i n the s p e c t r a l energy equation seems to work w e l l only over a l i m i t e d depth range. The e f f e c t of the free s u r f a c e seems to a l t e r the dependence on Re by r e q u i r i n g a much greater d i f f u s i o n term, In accord A with the c o n c l u s i o n s reached on l o c a l e g u i l i b r i u m from the data shown i n Figure 21. Of greater s i g n i f i c a n c e from the p r a c t i c a l p o i n t of view i s the observation that i n • v i r t u a l l y a l l cases 90% of the energy i s below 12 Hz, f r e g u e n t l y 50% being below 3Hz. As most previous measurements of sp e c t r a i n water have not looked below 1 Hz, t h e i r c o n c l u s i o n s as to the d i s t r i b u t i o n of energy must be s p e c u l a t i v e , as t h e i r s p e c t r a d i d not peak. The i m p l i c a t i o n s of t h i s c o n c e n t r a t i o n of 108 energy at low frequencies f o r entrainraent and suspension of sediment are obvious and f u r t h e r d i s c u s s i o n i s reserved u n t i l Chapter 5. Cospectra and coherence The l o g a r i t h m i c c o s p e c t r a , would, f o r high Reynolds number, c o l l a p s e onto a s i n g l e curve when p l o t t e d a g a i n s t f/f°. A -4/3 power law should e x i s t f o r an i n e r t i a l subrange and Hyngaard and Cote (1972) approximated the form of the cospectrum by n Co, (n) _ 0.88(f/f°) nit - fl- p , uv i + i . 5 ( f / f u r ( U 1 ) t o achieve comparison between v e l o c i t y and temperature c o s p e c t r a , In the p l o t of Co (n) there should be a -7/3 uv power law i n the i n e r t i a l subrange. Data from NcBean (1970) confirmed t h i s value while other i n v e s t i g a t o r s have re p o r t e d a value of -8/3 (Panofsky and Mares 1968). In t h i s research the emphasis with the cospectrum i s not on whether i t f i t s any of the asymptotic s o l u t i o n s f o r high Reynolds number, but r a t h e r on examining the frequency c o n t r i b u t i o n s to the o v e r a l l s t r e s s , f o r the area under the cospectrum r e p r e s e n t s the Reynolds s t r e s s . As the s p l i t f i l m probes were not considered to be s u f f i c i e n t l y a c c u r a t e l y c a l i b r a t e d , i t i s not intended to use the c o s p e c t r a l measures i n any but r e l a t i v e terms with a view to comparing energy contents at the v a r i o u s frequencies f o r d i f f e r e n t roughnesses and p o s i t i o n s i n the boundary. 109 I f the sp e c t r a are normalized, p l o t t e d a g a i n s t non-dimensional frequency and compared with the curve from Panofsky and Hares (1968), then f o r any roughness d e n s i t y c e r t a i n p r o p e r t i e s emerge. Four cospectra were computed, two below the tops of the roughness and two i n the wake l a y e r . There i s a systematic i n c r e a s e i n the peak of the curve as the top pf the elements i s approached from above (Figure 30a): ' t h i s holds t r u e f o r the roughness a r r a y s considered. The peak of the cospectrum decreases f o r the measurement c l o s e s t to the bed. A l l the curves peak more than t h a t of Panofsky and Mares (1968). I t i s obvious that t h i s i n c r e a s e must be due to the a d d i t i o n a l turbulent energy r e s u l t i n g from the mechanical mixing as a r e s u l t of wake shedding and inc r e a s e d turbulence generation by the roughness elements. F i g u r e 30a may be compared with s i m i l a r r e s u l t s of c o s p e c t r a l measurements made i n the atmosphere by Shaw et a l ( 1 9 7 4 ) . They a l s o found a l l t h e i r curves to be more peaked but i n t h e i r case found the most peaked curve was above the roughness array (a f o r e s t ) , the next most peaked being above a corn crop while the measurements made w i t h i n the canopy were l e s s peaked but s t i l l above the Panofsky curve. A study of the peaks f o r d e n s i t y 1/8 i n d i c a t e d that the peak was s e n s i b l y constant with h e i g h t , but when an a n a l y s i s was c a r r i e d out f o r a d e n s i t y of 1/5 the same r e s u l t as Shaw was obtained with the gr e a t e s t peak o c c u r r i n g above the top of the elements. This i n d i c a t e s s i g n i f i c a n t s h e l t e r i n g a t d e n s i t i e s greater t h a t 1/8 which i s i n accord with the r e s u l t 110 n 0UV (n) uv 1.0 O.I .01 i ; i i i i i ! ! i ! 1 > > ! I i I D E P T H . - - - o . i 0.9 - - • 1.54 1 1 I ! i 1 : t i 1 1 1 '• i 1 I i 1 * i i ! 1 i - V 2.54 1 i < i \ . I 11 .\-//.. i j i i i i 1 II \ ^ - P A N 0 r 5 X Y A M D M A K E S ( l 9 f e 8 ) \ V-\ .01 0.1 1.0 Figure 30a Normalized cospectra for four depths (y/D=0.01, 0.1, 0.22, 0.37) f o r roughness density 1/20/ .01 O.I 1.0 n y / u Figure 30b Normalized cospectra for three densities and the Lego baseboard at D=0.96cm (y/D=0.1 approx.) 111 from the work on mean values f o r the d i f f e r i n g d e n s i t i e s i n d i c a t e d . The e f f e c t of roughness on the boundary can be examined by p l o t t i n g three d e n s i t i e s at 0.96 cm from the bed, at the top of the roughness elements. Also p l o t t e d i s the curve f o r the base Lego board (Figure 30b). The comparable shape around the s p e c t r a l peak f o r the baseboard and other workers' r e s u l t s lends v a l i d i t y to the c o s p e c t r a l measurements, which are l i k e l y to be those most contaminated by noise and s u b j e c t to e r r o r . One r e s u l t does emerge f o r the p l o t t e d d e n s i t i e s : at t h i s height a l l the curves peak at approximately the same height i n d i c a t i n g t h a t the cospectrum i s not a simple f u n c t i o n of roughness d e n s i t y . The presence of a few p a r t i c l e s on the boundary w i l l cause peaking i n the curves: only with d e n s i t i e s as low as 1/120 was the s t a r t of a decrease towards the value of the p l a i n rough bed observed-The con c l u s i o n that may be drawn from F i g u r e 30b i s tha t even a few l a r g e p a r t i c l e s on a bed seem to cause a c o n c e n t r a t i o n of energy at a p a r t i c u l a r freguency. T h i s may be very s i g n i f i c a n t i f t h a t 'freguency corresponds to a geometric s c a l e of a bed p a r t i c l e , r e s u l t i n g i n s i g n i f i c a n t a l t e r a t i o n i n the mean shear, that the p a r t i c l e might endure over i t s f r o n t a l area. The cospectrum of the uv f l u c t u a t i o n s shows which freguencies c o n t r i b u t e to the Reynolds s t r e s s . The cospectrum, normalized by the square root of the u and v spectra i s termed the s p e c t r a l c o r r e l a t i o n ' c o e f f i c i e n t and 112 examines the t r a n s f e r e f f i c i e n c y between components- P l o t s of the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t i n boundary l a y e r f l o w s are not " very common, the only ones to hand being by McBean(1970) and by Antonia and Luxton{1974). McBean's data f o r the cospectra showed a -4/3 r e g i o n f o r a normalized frequency p l o t and i n t h i s r e g i o n . the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t was l e s s than 0.-2- C o r r s i n (1956 p. 386) , d i s c u s s i n g the shear spectrum, produced p l o t s of the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t and claimed t h a t R (n) ~n~Y3; but uv he then quoted Tchen(1954) as showing t h a t the shear spectrum i n the i n e r t i a ! subrange would be p r o p o r t i o n a l to -7/3. . T h i s l a t t e r r e s u l t i s i n accord with the c o n c l u s i o n s of a l l recent measurements, but r a t h e r than r e s u l t i n g i n a -4/3 slope f o r R U V ( Q ) such an a s s e r t i o n would lead to a -2/3 slope f o r S (n) , f o r i f Co ~-n~ 7/ 3, A ~ n ~ ^ and A —n- 5/ 3 , then / U V U V ^ U v V R ~ n _ 2 A . Such a c o n c l u s i o n gives a much b e t t e r f i t to the uv boundary l a y e r data of Klebanoff and the j e t data of C o r r s i n . Despite the c o n s i d e r a b l e s c a t t e r i t i s seen as a much . b e t t e r f i t t o our data shown i n Figure 31. Some s p e c t r a l c o r r e l a t i o n , data were presented by Weiler(1965), h i s only comment being t h a t they are a l l negative and almost always numerically greater than 0.3 with the l a r g e s t value being -0.73. A l l runs showed th a t l R u v l decreases to q u i t e s m a l l values as frequency i n c r e a s e s . The data presented here seem to show co n s i d e r a b l y l e s s s c a t t e r i n any one run althouqh at any one frequency at d i f f e r e n t depths there i s c o n s i d e r a b l e v a r i a t i o n i n values. The only other data from a boundary D o 0.1 A 0.9 D 1.5 X 2.5 n * , % A ; o A A D o c 1 x X A A -73 X \. X X \ A O D o A A X 10"' 10° 10' 10 n (Hz) Figure 31a Spectral c o r r e l a t i o n c o e f f i c i e n t for four depths for roughness density 1 / 8 0 . , DENSITY o '/12 A VZO D '/&0 CA A a A 8 i N A O i D t ( V 2 / 3 ! * D 0 - A D O O . n 10"' 10° 10* 10 n (Hz) Figure 31b Spectral c o r r e l a t i o n c o e f f i c i e n t for three roughness densities at D=2.54 (y/D=0.37) 1 14 l a y e r where the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t i s presented, by Rntonia and Luxton (1974), behaved i n a very s i m i l a r manner to the form of double l o g a r i t h m i c p l o t s as presented f o r one roughness den s i t y at d i f f e r e n t depths i n F i g u r e 31a. The much f l a t t e r peak of the curve than that of the s p e c t r a or cospectra i n d i c a t e s there i s a broad band of f r e q u e n c i e s , up to around 4 Hz, which c o n t r i b u t e s most e f f e c t i v e l y to the t r a n f e r of energy. This i s to be expected as the u and v s p e c t r a peak at d i f f e r e n t f r e q u e n c i e s (Figure 32). While the highest c o r r e l a t i o n c o e f f i c i e n t i s a t the very lowest f r e q u e n c i e s the broad band of s l i g h t l y lower c o e f f i c i e n t s c o n t r i b u t e s i g n i f i c a n t l y t o the t r a n s f e r . To compare the e f f e c t of roughness on the c o r r e l a t i o n s , F igure 31b shows the s p e c t r a l c o r r e l a t i o n s f o r three roughness a r r a y s at one depth. There seems to be c l o s e agreement f o r a l l the data f o r t h i s depth, which corresponds to the top of the wake l a y e r , suggesting t h a t at t h i s l e v e l the spacing of the elements i s no longer important i n determining the frequencies of momentum t r a n s f e r but only i n determining i t s o v e r a l l magnitude. While the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t provides some i n s i g h t i n t o the s t r u c t u r e of momentum t r a n f e r the coherence (-y. 2) i s more commonly used i n e v a l u a t i n g the i n t e r r e l a t i o n between two s e r i e s . I t behaves very s i m i l a r l y t o the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t : a p l o t of coherence ag a i n s t frequency shows i t to be a s t r o n g l y d e c l i n i n g f u n c t i o n -1 15 S i g n i f i c a n c e t e s t s f o r the coherence have been developed ( c f . Koopmans 1974). The degrees of freedom f o r each coherence estimate change because more F o u r i e r c o e f f i c i e n t s are combined at higher f r e q u e n c i e s ; with 120 degrees of freedom, y2 must be greater than 0.35 t o l e a d to r e j e c t i o n of the n u l l hypothesis ( y 2 - 0 ) . In t h i s research the coherence i s not d i f f e r e n t from zero at frequencies g r e a t e r than 10 Hz. . Following the t a b u l a t i o n of confidence bands i n Koopmans (1974), none of the v a r i a t i o n amongst coherence estimates i s s i g n i f i c a n t at the 90% l e v e l . The seemingly l a r g e v a r i a t i o n s at the low frequencies are not s i g n i f i c a n t because these estimates have r e l a t i v e l y few degrees of freedom. In the present data the d i f f e r e n c e between the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t and the coherence i s s m a l l because the c o n t r i b u t i o n from Quad^ (n) i s always s m a l l . In a l l cases Quad (n) i s an order of magnitude l e s s than uv Co (n). The r a t i o between the quad- and co- s p e c t r a i s uv examined i n the phase r e l a t i o n . The t r a c e of u,v shows an out of phase r e l a t i o n and i n f a c t phase angle remains constant w i t h i n 10° of 180° with v l a g g i n g u, i n d i c a t i n g the d i r e c t i o n of t r a n s f e r of momentum. Isotropy The study of the s p e c t r a and cospectra suggests t h a t more d e t a i l e d c o n s i d e r a t i o n should be given to the nature and presence of i s o t r o p y at the s m a l l e r s c a l e s . The blanket use 116 of the concept based on the presence of a -5/3 slope i s s t i l l common In s t u d i e s of turbulence made i n r i v e r s and only N a l l u r i and Novak (1973) suggest examining other c r i t e r i a t o check f o r i t s presence. The s t r i c t assumption t h a t uv=0 and t h a t u 2=v 2 i s r a r e l y f u l f i l l e d even i n g r i d t u r b u l e n c e , although i f the coherence i s z e r o , l o c a l i s o t r o p y might be i n f e r r e d . In the present case the coherence drops r a p i d l y but even i n the range where a -5/3 s l o p e may be present i n the s p e c t r a and a -7/3 shape may be seen i n the c o s p e c t r a , the coherence i s not zero. There are two t e s t s which may be a p p l i e d . Pond et al(1963) suggested that the lower l i m i t f o r i s o t r o p y was ky>4.5 where k i s radian wavenumber and y i s d i s t a n c e from the w a l l . I t was shown i n Appendix I I I t h a t the r a t i o of the two s p e c t r a <J>v(n) / "J1^") should, i f t h e r e i s an i n e r t i a l subrange, have a value of 4/3. The lower l i m i t f o r i s o t r o p y from the minimum c r i t e r i a a s s e r t e d by Pond would be 9.0 FIz while i f the r a t i o of spectra i s computed at t h a t p o i n t the r e s u l t I s 2.92 i n d i c a t i n g a c o n s i d e r a b l e departure from the p r e d i c t e d value. This i s most l i k e l y due to two phenomena. F i r s t , the turbulence at t h i s l e v e l , as was observed i n Figure 30a, does not conform to the shear-produced turbulence co-spactrum, f o r i t i s more peaked. Second, the high value f o r the v spectrum may be seen i n Figure 32 where the normalized u and v s p e c t r a are p l o t t e d . In attempting to get the cospectra to peak at ny/U^O.1 d i f f e r e n t values of y° had to be used f o r the u and v s p e c t r a : i n the case of the 1/12 density yo adjusted f o r the log,0 0(n) 0 0 U - S P E C T R U M X V - S P E C T R U M • ° ' 0 x ] X 0 j x * x x Y 0 X 0 * v 0 \ O y o x 0 X 0 X 0 X 0 0 * o x 0 > 0 ( X o X -4 -2 0 2 4 log l 0 f r equency Figure 32 Comparison of u and v spectra af t e r normalization at y/D=0.22 118 u component was 2.03 while f o r the v component was 0.83, when the r e a l y was 2.54 cm. This suggests t h a t an i n c r e a s e d amount of turbulence i s being generated i n the v e r t i c a l component, p o s s i b l y by v o r t i c i t y a m p l i f i c a t i o n around the roughness elements, which i s then d i f f u s e d i n t o the wake l a y e r . The d i r e c t i o n of the p r i n c i p a l a x i s of the s t r a i n f o r three arrays based on the d e r i v a t i o n given by C o r r s i n (1956 p. 379) that a = 1 . -1 ,2uv v iuo\ s tan (—p -w) (4<J) u -v i s shown i n Figure 33a. In the e v a l u a t i o n of a , the s magnitude of the var i o u s terms was obtained from the i n t e g r a l s under the r e s p e c t i v e s p e c t r a . Shown a l s o i n the diagram are the average curve f o r a channel and boundary l a y e r based' on Klebanoff»s data and some r e s u l t s computed over and w i t h i n a rough canopy based on data i n Kawatani and Sadeh (1971) and Kawatani and Heroney(1968) . The behaviour of a from the l a t t e r r e s u l t s does not agree with the p r o j e c t i o n s s of Corrsin(1956, p.380) f o r as y/D goes to 1 „ goes to zero _ s f o r c l o s e to the edge of the boundary l a y e r u 2 w i l l always be very much l a r g e r than any other term. T h i s r e s u l t i s confirmed i n the work of H a n j a l i c and Launder (1973) i n an axisymmetric channel. This i s al s o the r e s u l t obtained from Kawatani's data and from the one measurement made i n t h i s r esearch. The second technigue f o r examining the presence of i s o t r o p y and i t s e f f e c t i s to look at the Kolmogorov s c a l i n g 0.6 0.4 0.2 0 DENSITY • X DATA FROM . 0 '/l2 KAWATANl AND & Via SADEH CI97I) • l / 6 0 . . X n X X / IA. x D * o n D i n A 0 ' -40° -20° 0° Figure 33a O r i e n t a t i o n of s t r e s s axes. Line represents boundary l a y e r data from Klebanoff (1950 ) . 1.0 0.8 0.4 0.2 QA DENSITY 0 l/!2 * ' / 2 0 n '/BO V A U 1 a c o oA -of An 0 2 4 6 8 Figure 33b U n i v e r s a l e q u i l i b r i u m parameters S o l i d l i n e represents t o t a l , dashed l i n e the i s o t r o p i c , e valuation of tu r b u l e n t energy d i s s i p a t i o n . Symbols as i n F i g 16a. ( A f t e r Sandborn and Braun 1955). 120 l e n gths and the d i s s i p a t i o n r a t e . Sandborn and Braun(1956) presented Klebanoff's r e s u l t s by e v a l u a t i n g the energy d i s s i p a t i o n based on the i s o t r o p i c r e l a t i o n and a l s o from measuring the a c t u a l v e l o c i t y d e r i v a t i v e s i n two d i r e c t i o n s , assuming a two dimensional boundary l a y e r . I t i s l i k e l y t h a t i n the s i m i l a r s i t u a t i o n here the low Reynolds number means t h a t i s o t r o p y i s not present. Figure 33b shows a p l o t of Klebanoff's r e s u l t s from a zero pressure gradient boundary l a y e r and some of our r e s u l t s . I t may be seen t h a t i n K lebanoff's data the i s o t r o p i c e v a l u a t i o n underestimates the e q u i l i b r i u m r a t i o , meaning that n i s being underestimated or E Is being overestimated. As only du/dt was measured i t was necessary to use the i s o t r o p i c r e l a t i o n . The c l o s e f i t t o the d i r e c t measurement r e s u l t s of Klebanoff, e s p e c i a l l y i n the region 0.2<y/d<0.7, shows that the v e r t i c a l d e r i v a t i v e s are p o s s i b l y much more important and the i s o t r o p i c r e l a t i o n i s n e a r l y f u l f i l l e d i n t h i s boundary l a y e r d e s p i t e the low Reynolds number. The i s o t r o p i c r e l a t i o n may be u s e f u l i n rough boundary l a y e r work, where the a n i s o t r o p y i s counteracted by an increased l e v e l of turbulence i n the v e r t i c a l d i r e c t i o n , i n c r e a s i n g a tendency towards e q u a l i t y of the v e l o c i t y d e r i v a t i v e s . The e x c e p t i o n a l discrepancy at the f r e e s u r f a c e i n d i c a t e s that the a s s e r t i o n made may indeed be true w i t h i n the reqion postulated f o r d /dy/0,except r i g h t a t the f r e e s u r f a c e , and here the i s o t r o p i c r e l a t i o n underestimates the turbulence energy d i s s i p a t i o n . The range of s c a l e s from macroscale to microscale i s 121 from approximately 0.3 to 300 Hz: that i s 1 0 3 - 2 1 0 . A s Wilson(1974) has shown, eddies do not s i g n i f i c a n t l y i n t e r a c t whose s c a l e s are d i f f e r e n t by more than a decade. I f i t i s assumed that t h i s i s the maximum then there would only be three steps i n the cascade process, while i f i t i s assumed that the eddies break i n t o e x a c t l y h a l f s i z e then there are up to ten steps present i n the cascade; the l a t t e r case would, by a simple l i n e a r branching process r e s u l t i n i s o t r o p y a t small s c a l e s while the former would c e r t a i n l y be i n s u f f i c i e n t . The lack of i s o t r o p y suggests t h a t the t u r b u l e n t eddies break up i n t o somewhere between these two extremes. Sii.2iJ2iiLE.y- of. r e s u l t s on cospectra and i s o t r o p y The c r o s s s p e c t r a have been u s e f u l i n two r e s p e c t s . They have shown t h a t the l a r g e s c a l e s , out to' approximately 7 Hz are most important to the Reynolds s t r e s s . While not a l l cumulative cospectra were computed, c a l c u l a t i o n of s e v e r a l i n d i c a t e d that 85% of the s t r e s s lay below 7.25 Hz, which i s a l s o i n d i c a t e d by the h i g h , f l a t peak of the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t which then drops o f f beyond t h i s r e g i o n . The f l u c t u a t i o n s i n u and v are approximately 180° out of phase, as would be the case f o r any vortex s t r e t c h i n g process. The f a c t that they are 180° out of phase means th a t the area under the quadrature spectrum i s very small and hence the coherence and s p e c t r a l c o r r e l a t i o n c o e f f i c e n t qive s i m i l a r r e s u l t s . The a n a l y s i s of the r a t i o of s p e c t r a and 122 the mean angle of s t r e s s showed that i s o t r o p y i s not present, but the e f f e c t of t h i s i s not seen to s i g n i f i c a n t l y a f f e c t the e v a l u a t i o n of the rat e of energy d i s s i p a t i o n - Such an assumption u s u a l l y leads to overestimation of e i n a boundary l a y e r , while In t h i s case the extreme roughness and wake shedding may i n c r e a s e the v e r t i c a l component and i t s d e r i v a t i v e s u f f i c i e n t l y that the i s o t r o p i c e s t i m a t i o n g i v e s a good p r e d i c t i o n of the two dimensional measured r a t e of energy d i s s i p a t i o n . CONCLUSIONS The d e t a i l e d measurements of mean flow i n d i c a t e d a wake zone above the roughness elements, while above t h i s there I s a more conventional shear flow. Such a concept i s important i n determining d i f f u s i o n parameters f o r sediment suspension when the constant s t r e s s l a y e r of the wake may be very e f f i c i e n t i n t r a n s p o r t i n g m a t e r i a l . I n t h i s zone a strong t u r b u l e n t shear, as a r e s u l t of wake shedding, may lead to s i g n i f i c a n t a l t e r a t i o n i n the frequency composition of the v e r t i c a l f l u c t u a t i o n s from that u s u a l l y found i n a boundary l a y e r . The measures of shape f a c t o r and power law exponent i n d i c a t e that there appears to be an optimum spacing of roughness elements on the boundary t o achieve maximum r e s i s t a n c e : at d e n s i t i e s greater t h a t 1/12 the bed i s e s s e n t i a l l y s h i f t e d to the top of the elements, while at l e s s e r d e n s i t i e s the blocks begin to act as independent wake 123 shedding elements and the i n t e r v e n i n g s u r f a c e makes a s i g n i f i c a n t c o n t r i b i t i o n to the o v e r a l l r e s i s t a n c e - Such r e s u l t s were confirmed by examination of the turbulence i n t e n s i t y and rate of energy d i s s i p a t i o n . The r e s u l t s of t h i s chapter have been developed t o achieve the f i r s t three o b j e c t i v e s set out on page 3 of the i n t r o d u c t i o n . The r e s u l t s i n d i c a t e that u s e f u l and d e t a i l e d r e s u l t s on the s t r u c t u r e of the t u r b u l e n t flow f i e l d over rough boundaries can be obtained, and that the r e s u l t s a l l o w c a r e f u l and e n l i g h t e n i n g comparison with measurements made i n other boundary l a y e r s . From the s t a t i s t i c a l measures i t may be seen t h a t i n t e g r a l methods such as s p e c t r a l a n a l y s i s are by themselves not too f r u i t f u l , f o r the spectrum i s a remarkably robust t o o l showing l i t t l e v a r i a t i o n , but cro s s s p e c t r a l a n a l y s i s g i v e s v a l i d i n f o r m a t i o n on the frequencies c o n t r i b u t i n g t o the shear s t r e s s and t h i s w i l l be u s e f u l i n examining the s t a b i l i t y of sediment p a r t i c l e s and t h e i r a b i l i t y to withstand c e r t a i n s t r e s s e s p r i o r to movement. 124 CHAPTER 3 EXPERIMENTAL RESULTS FOR ROUGHNESS ARRAYS STATISTICAL ANALYSIS The a n a l y s i s o f t h e p r o b a b i l i s t i c s t r u c t u r e o f t u r b u l e n t f l u c t u a t i o n s f o r m s one o f t h e two t e c h n i q u e s u s e d t o • t r a n s f o r m t h e i n s c r u t a b l e t i m e or s p a c e h i s t o r i e s o f t u r b u l e n t m o t i o n i n t o c u r v e s t h a t v a r y s m o o t h l y and can t h u s be comprehended' {Dutton and Deaven 1 9 7 1 ) . We have a l r e a d y c o n s i d e r e d s p e c t r a l a n a l y s i s : t h e s e c o n d method i s t o examine t h e a m p l i t u d e p r o b a b i l i t y d e n s i t i e s and t o f o c u s a t t e n t i o n on t h e r e l a t i v e f r e q u e n c y o f o c c u r r e n c e o f v a r i a t i o n s . A l t h o u g h t h e a n a l y s i s was p e r f o r m e d f o r f o u r d e p t h s f o r a l l o f t h e r o u g h n e s s a r r a y s , o n l y sample r e s u l t s a r e p r e s e n t e d f o r t h e s a k e o f b r e v i t y . Moments A l l t h e d a t a were s t a n d a r d i z e d and t h e l i n e a r component was removed. F i g u r e 34a shows t h e d e n s i t y and d i s t r i b u t i o n f u n c t i o n s f o r t h e s t a n d a r d i z e d v e l o c i t y c o m p o n e n t s . F i g u r e s 34b, c and d show t h e s i m i l a r r e s u l t s f o r t h e s g u a r e , c u b e and f o u r t h power o f t h e o r i g i n a l d a t a . I n e a c h g r a p h u and v r e p r e s e n t t h e h o r i z o n t a l and v e r t i c a l f l u c t u a t i o n s w h i l e w y/D=0.01 1/12 y/D=0.01 1/48 y/D=0.37 1/12 y/D=0.37 1/48 J - 4 5 - 2 - 1 0 t IV. V . W J / S I C M A cw. . , , , 1 1 1 -4 -5 -2-1 0 I 2 5 * 5 < U . V . W)/5 I C M A -I—i—r -5 -2 -1 0 cu. v. wj/t I C M A iff,...,, . 1—[— f^cVSfSteT 1 v 5 -t 5 !(•'-'' ' r — I — I — i — i ---^-ae* ! c l tor . J I 1/ - i — r - T T — 7 1 — — 1 — 1 — 1 1 -1 - 1 - 2 - 1 Q I 2 5 * 5 (u. v. wj/s I C U A -5 - r — i — r — i — i — r •*•$ -2 -1 0 I 2 tu. V . » 3 / 5 I C M A 33 20 ' J ^-1 1 1 1 1 1 1 1 1—I ? -•» -5 -2 -1 0 I 2 5 •» 5 < U . V . W l / S I C M A 11 RJ 50 JO 20 J 1 1—r—I—I—i—I—I—I—i—I 1 - 5 - 2 - 1 Q I 2 5 -t J <U. V . V ] / C I C M A Figure 34a Probability densities and d i s t r i b u t i o n s of velocity fluctuations, w represents pseudo-random number generator. Measurements for two depths and two roughness de n s i t i e s . - i 1 1 1 i - i 1 : : r - f 1 ? -j i 1 1 » '• i . T H 1 — ; — i 1 1 0 < 3 1» . 1 4 0 1 5 12 llS 0 4 ! 12 IS 0 . 1 < IJ 1 IS { (U. V. w HI J i M A ) a»i (<U. V. IC *A ).»a . (fu. V. w ) / t 1 C U A ] r i «3 (fu. V. W l / S U M A J . . 8 y/D=0.01 1/12 y/D=0.01 1/48 y/D=0.37 y/D=0.37 •r — I 1 1 1 3 1 8 12 Id ~l '• T" (<u. v. icvA)««a - r 4 s is I(Ut V . V l / C 1 C M A so • ' «« ct 3D 9 • 0 < J 12 I* Figure 34b Probability densities and d i s t r i b u t i o n s of velocity fluctuations w represents pseudo-random number. Measurements for two depths and two roughness dens i t i e s . _^  ( ] ! „ t 64 - 52 0 32 65 ( C U . V . W ] / { I C M A ) • « ! -v—, , q..u. ,u, 64 - 3 2 . 0 52 64 < C U . V . W S / C I C M * ) » • I m -- v — [ V 1 - t | - t f - t ) — , - 6 4 - 3 2 . • 0 32 61 T" - 6 4 - 3 2 0 32 ( ( U . V . • ] / ! I C M A ) » > ; 61 y/D=0.01 1/12 y/D=0.01 1/48 y/D^O.37 1/12 y/D=0.37 1/48 qq • i ; • 5D • JD ' 20 • 5 1 _ — N _ « , , P _ - 6 4 - 3 2 3 32 <CU. V . W3/S tCMA3**5 11 If 53 JO 2D F 1 pp i 1 — i — i — i - 6 4 - 5 2 0 32 64 C C U . V . * 3 / C 1 C M A 3 . M J 5D 2D 1 -61 32 0 52 61 qq 50 ?n 20 F 1 fat**0* - 6 4 I I -32 0 (CU.V.O/EKUA). 6 ' Figure 34c Probability densities and d i s t r i b u t i o n s of velocity fluctuations, w represents output from pseudo-random number generator. Measurements for two depths and two roughness densities. H-1 6 1 1 2 3 1 1 2 ( ( U . V . W J / S ! C M A ] * « 4 25(5 6 4 1 2 8 1 1 2 C tM. V.Hi/t ! C M A 1 » * 4 2 5 6 6 1 1 2 8 1 1 2 t <U. V . W ) / « I C M A ) * « 4 2 9 (S1 1 2 ! 1 1 2 < <U. V , W l / t I C M A ) « « 4 2 5 6 y/D=0.01 1/12 y/D=0.01 1/48 y/D=0.37 1/12 y/D=0.37 1/48 i r 6 < . 1 2 1 1 1 2 liUrV. W l / S J C U A ] x « 4 254 ^ - j 1-6 4 1 2 8 1 1 2 ( OI. V. I C M . - U . a 4 2 5 6 1 1 1 5 3 0 5 0 2 0 9 1 _ _ j — ( j , 6 4 1 2 5 1 1 2 ' 2 5 i i r 6 4 1 2 8 1 1 2 ClU. V . * ] / £ I C U A l n . 4 2 5 6 Figure 34d Probabil i t y densities and distributions of velocity fluctuations, w represents output from pseudo-random number generator. Measurements for two depths and two ruoghness dens i t i e s . i—1 ro oo 129 represents the output from a pseudo-random number generator and the s o l i d l i n e i l l u s t r a t e s the Gaussian d i s t r i b u t i o n f u n c t i o n based upon the e r r o r f u n c t i o n . I t i s c l e a r t h a t there are two regions of departure from Gaussian: t h e r e appears to be an excess of values at the t a i l s f o r y/D=0.3, and there are more zeros than would be p r e d i c t e d , p a r t i c u l a r l y f o r the v component. This c o n c l u s i o n becomes c l e a r e r by examining the higher order p r o b a b i l i t y f u n c t i o n s where the departures from Gaussian form become i n c r e a s i n g l y obvious. For measurements made c l o s e t o the bed, a t y/D o f 0.01, the u f l u c t u a t i o n s accumulate very r a p i d l y so th a t t h e r e seems t o be a paucity of extreme values r i g h t at the bed. While the moments of the v e l o c i t y d e r i v a t i v e show very high k u r t o s i s , the moments of the o r i g i n a l d i s t r i b u t i o n seem to i n d i c a t e a nearly Gaussian process. I f , however, the time s e r i e s i s rearranged so that the observations are ranked by s i z e and then, f o r the a p p r o p r i a t e power of the v a r i a b l e , these values are summed and normalized at each point by the o v e r a l l moment, curves of the accumulation of th a t moment as a f u n c t i o n of len g t h of record may be obtained. I t may be seen from Figure 35a t h a t the accumulated variance seems to show l i t t l e or no departure from the Gaussian curve. However the accumulation of the skewness i n d i c a t e s that both u and v depart from the Gaussian while w the random noise f u n c t i o n shows a very c l o s e f i t ; the d i f f e r i n g d i r e c t i o n s of skewness are a l s o q u i t e apparent 130 < 1/12 y/D=0.01 i I 2 0 4 0 PERCENTAGE dO 3D DP RECORD tn X3 • •3-• a • 1D0 1/48 y/D=0.01 . • I V J . ^ v i i " ' ' ' ' yy. f 1 ~ 2 0 4D PERCENTACE , -j f OF RECORD /—I < t—i y/D=0.37 ~r~r-r.~1. 2 0 4D PERCENTACE i r 2Q 4D PERCENTACE 6U SD OF RECORD 100 Figure 35a Accumulation of variance as a function of length of record f o r two densities at two depths. 131 < C " C 3 1/12 y/D=0.01 20 40 60 30 P E R C E N T A C C QP RECORD 100 to CD C3 I 1/48 y/D=0.01 V W I 1 — r — — i — ! — ) 2D 40 60 SO 1G0 P E R C E N T A G E C F RECORD • L X cr I 1/12 y/D=0.37 V V V V til i 1 1 r~ • 20 40 6U 5D PERCENTACE OP RECORD a NO to a C3 in 1 • to 100 1 1/48 y/D=0.37 i n r ~ n 1 20 TO 60 SO . 100 P E R C E N T A G E 3 C RECORD Figure 35b Accumulation of skewness as a function of length of record for two densities at two depths. 132 C-4 1/12 y/D=0.01 . 1 » y ) ' V ^ ^ ^ » A T T 2D 40 60 3D P E R C C N T A C E OP R E C O R D 1D0 o : sn a to <N C3 1/48 y/D=0.01 iiv  _ „„v a — r i i • r~ 2D 40 .. 6Q 30 P E R C E N T A G E OF R E C O R D i O Q o v . > c a LN Nl - • to cr C 3 1/12 . y/D=0.37 V V C3 a . i . ' i i i i ' i i i - . i l i i . L ^ ' - i i ^ - ' - ^ • Y ?P 1 a 1/48 y/D=0.37 0 20 40 6Q 3D P E R C E N T A C E OF P E C O P O t « i. CJ . -•IDC . • • U i ~ i 1 r ! 2D 40 60 30 P E R C E N T A G E O F RECDP 'O Figure 35c- Accumulation of k u r t o s i s as a f u n c t i o n of l e n g t h of record f o r two d e n s i t i e s at two depths. 133 (Figure 35b). The k u r t o s i s shows how q u i c k l y the f o u r t h moment of v accumulates i n l e s s than 10% of the record (Figure 35c). S t r u c t u r e f u n c t i o n s In the study of random processes the theory of s t a t i o n a r y processes i s w e l l developed. S t a t i o n a r i t y may a P P l y t o the o v e r a l l process, as i t i s u s u a l l y taken to mean, or i t may apply t o the d i s t r i b u t i o n of increments. That i s while the process u (t) i s i t s e l f not s t a t i o n a r y the increments Au(t) = U ( T ) - w(x-t) (43) are s t a t i o n a r y . The s t r u c t u r e f u n c t i o n introduced by Kolmogorov forms the second moment of the d i s t r i b u t i o n of the process A u ( t ) . The s t r u c t u r e f u n c t i o n i s j u s t the c e n t r a l moment; i t may be of any order and i s defined s o l e l y as ( u - u ( r ) ) m where r i s the se p a r a t i o n d i s t a n c e . The s t a t i s t i c a l use of such f u n c t i o n s i s f u l l y discussed i n Panchev(1971) and extensive use i s made of the s t r u c t u r e f u n c t i o n s i n the Russian l i t e r a t u r e where t h e i r c a l c u l a t i o n i s more common than the spectrum. In the i n e r t i a l subrange Kolmogorov showed t h a t Cu - u ( r ) ) m = C m ( e r ) m / 3 (44) where C are u n i v e r s a l constants. In p a r t i c u l a r f o r the m t h i r d moment (u - u ( r ) ) 3 = -4/5 er ( 4 5 ) and as a conseguence the skewness and f l a t n e s s f u n c t i o n s i n 134 the range n <<r«L where n i s the Kolmogorov m i c r o s c a l e len g t h and L i s the macroscale of the energy c o n t a i n i n g eddies should be constant where S - ( u - u ( r ) ) 3 / ((u-u(r)) 2)3/* i s the skewness and F = (u-u{r) ) * / ( < u - u ( r ) ) 2 ) 2 the f l a t n e s s f a c t o r . The only use of the s t r u c t u r e f u n c t i o n In the s t u d i e s of turbulence i n r i v e r s was made by Grinval'd(1971). He. used only the second order s t r u c t u r e f u n c t i o n simply t o examine f o r the presence of a 2/3 range, which he observed, but a t most f o r only h a l f a decade. He concluded t h a t the curves showed a s y s t e m a t i c a l l y i n c r e a s i n g i n e r t i a l subrange as the d i s t a n c e from the bed was i n c r e a s e d . I t i s not the i n t e n t i o n here to study the v a r i a t i o n of the s t r u c t u r e f u n c t i o n over the depth, f o r measurements were r e s t r i c t e d t o the lower 35% of the boundary l a y e r : r a t h e r i t i s t o compare some data from a very rough boundary l a y e r with measurements made i n the atmosphere and to provide a f i r s t look at the higher order s t r u c t u r e f u c t i o n s of skewness and f l a t n e s s , p r i o r t o the c o n s i d e r a t i o n of i n t e r m i t t e n c y . The r e s u l t s of a study of increments of v e l o c i t y f l u c t u a t i o n s are presented i n Figure 36. The r e s u l t s of . the v e l o c i t y increment d i s t r i b u t i o n s i n d i c a t e t h a t at s m a l l l a g s there i s a marked d e v i a t i o n from Gaussian behaviour; more so than at l a r g e r l a g s . Furthermore the increments at s h o r t l a g s show a pronounced excess of both zero and l a r g e v a l u e s . 1——'•vaittosex^ 1 2 5 . 1 5 *c > a <r UJ o o >• Kl IT d _ i CD w < CO d Q cr a. d a d d >- d *-UJ d V to y- d _J < d o c c a. d o d -J -1 — r — ]— p - ^ J a s s c S f I 0 I 2 5 .1 S ^ d < co ca . i * * * ? ^ ^ — i — r — i — r ' ^ - P W 1 - 3 - 2 - 1 0 1 i, S t S y/D=0.01 1/12 y/D=0.01 1/48 y/D=0.37 .1/12 y/D=0.37 1/48 is •SD fC 2D 5 1 1— I I T " 5 - 4 - 3 - 2 -1 T n — i — i — I 1 2 3 1 5 Figure 36a Probabil i t y densities and dis t r i b u t i o n s for velocity increments at a separation of .0.1 cm. w represents output from the preudo-random number generator Measurements made f o r two depths and two roughness, densities... i—• U l -5 -t - J -2 -I 2 3 4 5 y/D=0.01 1/12 I— i—i—r—i—i—i—i—i—r*n -5= — 1 - 5 - 2 - 1 0 1 2 3 * 5 y/D=O.Ol '1/48 f—\—1—r •' 1 1 1 ' i — 1— 1 '1 "i --, -4 -5 -2 -1' 0 1, 2 3 4 ? y/D=0.37 1/12 i i i — 1 — 1 — 1 — 1 — i — 1 — r - ^ ! - 4 - 3 - j - 1 E t 2 3 1 ? y/D=0.37 qq £ q ? -"Z 50 " 5 - 4 -3 -2 -I 0 F i g u r e 36b P r o b a b i l i t y d e n s i t i e s and d i s t r i b u t i o n s f o r v e l o c i t y increments at a s e p a r a t i o n of 1 cm. w re p r e s e n t s output from a pseudo-random number generator. Measurements made f o r two depths and two roughness d e n s i t i e s . t, 00 N -5 -* - J -4 -I D I 3 * 5 y/D=0.01 1/12 y/D=0.01 1/48 y/D=0.37 1/12 y/D=0.37 1/48 i n v, 3D ;Q 20 5 t i — r — t — r -S » -3 -2 - I T — 1 — I — T 1 2 3 * 1 — i — i — r •3-2-1 0 i — r 2 3 1 1 * i T — t — r ~ T — r — T 1 1 1 1 5 - 4 - 3 - 2 - 1 0 1 2 3 * 5 -9 -1 -3 -2 I " I - i a T — i — i — i — I 1 2 3 * 5 Figure 36c Probabil i t y densities and d i s t r i b u t i o n s for velocity increments at a separation of 10 cm. w represents output from a pseudo-random number generator. Measurements made for two depths and two roughness densities. d u\ y/D=0.01 1/12 y/D=0.01 1/48 d US v d y/D=0.37 1/12 - f -1 "5 -2- -1 0 1 2 3 4 ? - 5 - 4 - 5 - 2 - 1 0 1 2 5.4 5 • -4. -S -2 -1 0 1 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 y/D=0.37 1/48 - 5 - 4 - 1 - 2 - 1 0 1 2 3 4 5 Figure 36d Probability densities and d i s t r i b u t i o n s for velocity increments at a separation of 100 cm. w represents output from a pseudo-random number generator. Measurements made for two depths and two roughness densities. OO 1 3 9 This behaviour i s consonant with the no t i o n of a process which i s r e l a t i v e l y s t a b l e at a given l e v e l and then suddenly jumps to a new value: such a view could be expressed i n the notion that the v e l o c i t y g r a d i e n t s tend to become concentrated. Thus the smaller s e p a r a t i o n s should y i e l d p r o b a b i l i t y d i s t r i b u t i o n s that are g u i t e s i m i l a r to those of the time d e r i v a t i v e of the o r i g i n a l s i g n a l . Values of A u /a > 3 . 0 occur c o n s i d e r a b l y more f r e q u e n t l y than expected, i i e s p e c i a l l y f o r the v e r t i c a l component (Table I l i a ) , while w i t h i n the range 1 . 0 < | A U / A | < 3 . 0 there i s a d e f i c i t of i u ± ,. values (Table I l l b ) . Furthermore, the d i s t r i b u t i o n s of the increments e x h i b i t c e r t a i n asymptotic f e a t u r e s . The most probable value f o r A u ^ i s found not at ze r o , but a t a s m a l l negative value. As the s e p a r a t i o n d i s t a n c e i n c r e a s e s , as the diagrams show and the data from the t a b l e s i n d i c a t e s , the d i s t r i b u t i o n s approach Gaussian form. These r e s u l t s have bearing on the notion of s e l f - s i m i l a r i t y . I f the process was s t r i c t l y s e l f - s i m i l a r then the p r o b a b i l i t y d i s t r i b u t i o n s f o r a l l values of sep a r a t i o n should.be i n v a r i a n t when p l o t t e d i n normalized form. T h i s i s c l e a r l y not the case f o r the v e l o c i t y f l u c t u a t i o n s . This i s probably r e l a t e d t o the absence of i n e r t i a l subrange behaviour i n the sp e c t r a but the l a r g e r s e p a r a t i o n s show a strong tendency towards s e l f - s i m i l a r s t r u c t u r e . The separation of 1 0 cm corresponds to 3 . 5 Hz and i s i n the region where the spectra approximate a - 5 / 3 s l o p e . The second order s t r u c t u r e f u n c t i o n was normalized by the Kolmogorov v e l o c i t y s c a l e which had been 140 TABLE I I l a V e l o c i t y component u v Gaussian Lag i n centimeters 0.1 0.35. 0.63 0.16 1.0 0.34 0.43 0.17 10.0 0.24 0.35 0.16 100.0 0.23 0.32 0.17 Table I l i a Percentage o f v a l u e s of the v e l o c i t y increments that are g r e a t e r than 3 . 0 . Gaussian v a l u e computed from pseudo-random number g e n e r a t o r . A l l d i s t r i b u t i o n s has been s t a n d a r d i z e d . TABLE I I l b V e l o c i t y component u v Gaussian Lag i n centimeters 12. 48 10. 50 15. 30 14. 53 13. 32 16. 12 14. 43 13. 87 15. 97 14. 67 13. 96 15. 78 Table I l l b Percentage of v a l u e s of the s t a n d a r d i z e d v e l o c i t y components i n the range l < ^ * - , _ / « r ;<3 • w r e p r e s e n t s a Gaussian s i g n a l from the pseudo-random number generator. 141 p r e v i o u s l y estimated from the spectrum and t h i s was p l o t t e d against the s e p a r a t i o n d i s t a n c e normalized by the Kolmogorov leng t h s c a l e . The'data f o r the three heights y/D 0.01, 0.22 and 0.35 seem to c o l l a p s e onto a s i n g l e curve (Figure 37a) and a b r i e f and probably i l l u s o r y i n e r t i a l subrange i s seen to e x i s t with i t s lower l i m i t near r / n =50. Because of the s h o r t region of 2/3 slope and the p o s s i b i l i t y of e r r o r s i n the computing of v and n i t was not deemed worthwhile t o compute the i n t e r c e p t of the l i n e to o b t a i n an e v a l u a t i o n of the constant C and thus the data are examined only i n terms of s l o p e . The l i n e shown i n F i g u r e 37a i s from data i n Van A t t a and Chen (1970) and the f a c t that the p oints from the flume are c l o s e to t h i s l i n e i s taken as c o n f i r m a t i o n t h a t we may have some f a i t h i n the e v a l u a t i o n of e and n from the spectrum. The r a p i d drop o f f i s c o n s i s t e n t with a r e l a t i v e l y low Reynolds number flow. The skewness f a c t o r , S, r a t h e r than i t s normalized form i s shown i n Figure 37b. I t was shown by Pond et a l (1963) t h a t S = - O . l ( k ' ) " 3 7 2 ( l + 6 ) where k* i s the Kolmogorov constant. The c o n d i t i o n f o r t h i s t o hold i s that the separation d i s t a n c e l i e s i n the i n e r t i a l subrange. I t was thought t h a t i n order to evaluate t h i s f u n c t i o n s e p a r a t i o n s of the order of the peak of the d i s s i p a t i o n spectrum, that i s p r o p o r t i o n a l to the Taylor m i c r o s c a l e , should be used. Only three p o i n t s were computed c l o s e to t h i s s c a l e and the r e s u l t s f o r the two heights are 10' 2 v k 2 10' 10' X s n •^ n OA ' n n o SA n /XL o • i . o A D o A V / D O 0-13 A 0.22 • 0.3k JO 1 10' 10' r/, 10 Figure 3 7 a Second order s t r u c t u r e f u n c t i o n f o r d e n s i t y 1 / 8 0 normalized by Kolmogorov length and v e l o c i t y s c a l e s . S o l i d l i n e gives 2 / 3 range and a l s o represents f i t to data of Van A t t a and Chen ( 1 9 7 0 ) . 0.6 0.4 S 0.2 ( > A o .01 A. .13 n .22 x .3k 0 _x x X D • xc o A O A ° 5 u • ° A A °* x° 5 0 I 2 loglo of separation distance r Figure 37b Skewness f u n c t i o n f o r the streamwise v e l o c i t y f l u c t u a t i o n s f o r a den s i t y of 1 / 8 0 . lh3 log of flatness factor 1.0 0.5 0 ° 0.01 A 0.12 a 0.22 p -0.11 0 U y ° D o % a o 0 . I 2 3 log, of separation distance r Figure 37c. v e l o c i t y f o r Flatne s s f a c t o r f o r streamwise roughness d e n s i t y 1/80. 144 produced i n Table IV. Wilson (1974) reported skewness values of -0.22 to -0.25 while Van fitta and Chen (1970) obtained a value of -0.18. They a l s o presented data from low Reynolds number g r i d turbulence, which i s a s t r o n g l y decreasing f u n c t i o n of s e p a r a t i o n , which e x h i b i t s no t r a c e of a constant S. The values of k* are higher than u s u a l l y found, but t h e l a c k of an extensive subrange means t h a t i t t i e r e l i a n c e should be placed on them. One property of i n t e r e s t that was demonstrated by Wilson (1974) was t h a t as the s e p a r a t i o n goes to zero the skewness f a c t o r tends towards an absolute constant. This constant would be the skewness of the v e l o c i t y d e r i v a t i v e . The data here are somewhat l e s s d e f i n i t i v e : s i x cases f o r three roughnesses and two heights are presented (Table V). The values are i n most cases lower than the a c t u a l measured d e r i v a t i v e value: a s i t u a t i o n t h a t was a l s o found by Wilson perhaps due to the f a c t that the s m a l l e s t s e p a r a t i o n s were 0.1 cm, which i s l a r g e r than the Kolmogorov m i c r o s c a l e , the r e s u l t a n t l o s s of information might be r e s p o n s i b l e f o r the s l i g h t underestimates obtained. The f l a t n e s s f a c t o r was a l s o computed and i s presented f o r two s e t s of data i n Figure 37c. I t i s a smoothly decreasing f u n c t i o n i n accord with the r e s u l t s of Van A t t a and Chen and of Stewart et a l . (1970). The former authors found t h a t the f a c t o r decreases as r-- 1 1 1 over a range of values but i s n e a r l y p r o p o r t i o n a l to r * * 2 2 2 f o r s m a l l sep a r a t i o n s . As s e p a r a t i o n d i s t a n c e s are reduced, i t may be 145 TABLE IV Densi t y 1 / 2 0 D e n s i t y 1/8 Depth Skewness Kolmogorov c o e f f . 2 . 5 1 * - 0 . 1 9 0 . 6 5 1 . 5 4 - 0 . 2 0 0 . 6 3 2 . 5 4 - 0 . 2 3 0 . 5 7 1 . 5 4 - 0 . 2 2 0 . 5 9 Table IV Skewness valu e s from the t h i r d o r d e r s t r u c t u r e f u n c t i o n at a s e p a r a t i o n of order of the T a y l o r m i c r o s c a l e f o r two depths and two roughness d e n s i t i e s . The Kolmogorov c o e f f i c i e n t was c a l c u l a t e d from the r e l a t i o n given by Pond et a l ( 1 9 6 3 ) . TABLE V Densi t y Skew, at 0 . 1 cm D e r i v a t i v e . . s e p a r a t i o n 1/8 - 0 . 3 5 2 - 0 . 3 8 Depth 1 / 8 o _ 0 > 2 i | _ 0 > 3 9 2 . 5 4 cm 1 / 1 2 0 -O . 6 5 8 - 0 . 4 8 1/8 - 0 . 2 2 - 0 . 4 3 Depth 1 / 8 0 - 0 . 2 8 - 0 . 3 3 1 . 5 4 cm 1 / 1 2 0 - 0 . 2 7 - 0 . 3 9 Table V Skewness v a l u e s c a l c u l a t e d f o r three d e n s i t i e s at two depth from the s t r u c t u r e f u n c t i o n and d i r e c t l y from the d e r i v a t i v e of the s i g n a l . ns i n f e r r e d from the increase i n f l a t n e s s f u n c t i o n t h a t i n t e r m i t t e n c y becomes i n c r e a s i n g l y important. C o r r e l a t i o n s The cross c o r r e l a t i o n f u n c t i o n between two v e l o c i t y components a t v a r i o u s separations i n the flow was d i r e c t l y determined i n analog form using a c o r r e l a t o r and a RMS meter. The r e s u l t s are presented i n the form of smoothed graphs of the streamwise c o r r e l a t i o n a t zero time l a g with s e p a r a t i o n i n the v e r t i c a l and cross stream d i r e c t i o n s . F i g u r e s 38a and 38b show the c o r r e l a t i o n with v e r t i c a l s e p a r a t i o n ; they never become negative and show a con s i d e r a b l e region of g u i t e high c o r r e l a t i o n f o r the wake l a y e r . Roughnesses 1/12, 1/16 and 1/20 behaved In the same way and the i n i t i a l e x p e c t a t i o n that there would be a sharp break i n c o r r e l a t i o n at the top of the roughness elements was not f u l f i l l e d . T h i s i n d i c a t e s t h a t the flow here i s not a simple shear i n the sense of a uniform l i n e a r shear over the top of the bl o c k s . Rather the wake l a y e r extends down below the tops of the roughness elements, on occasions to h a l f the height of the roughness elements, i n d i c a t i n g a flow s t r u c t u r e dominated by wake shedding and wake flow parameters. The cross stream c o r r e l a t i o n s seemed to s c a l e almost e x a c t l y with d i s t a n c e from the boundary (Figure 39c and 39d), and appeared to be independent of the roughness d e n s i t y on the bed. The f u n c t i o n s go negative and then approach zero from below. 147 R^Q.r .O) 0 \ -\ \ DENSITY J/a 'Aa -V.2 - — Vao V-.\ V. 0 0.2 0.4 0.6 y / D Figure 38a C o r r e l a t i o n s f o r four roughnesses at y/D=0.57 f o r h o r i z o n t a l , s e p a r a t i o n s . S o l i d l i n e represents approximate f i t to c o r r e l a t i o n s at y/D=0.66 by Grant(1958) i n a smooth w a l l boundary l a y e r . R„(0,r,p) 1.0 0.5 0 :/ a - • l : i /#. .• i .* i * t \ -. X "• \ o v : x \ * \ •. x •-D E N S I T Y — - y , 2 ' /BO x / / \ ' ° - . . . o X • > \ q t k 0 . 4 0 .2 0 0 . 2 0 . 4 - i v e + ive towards bed towards s u r f a c e Figure 38b C o r r e l a t i o n s f o r two roughness arrays at y/D=0.22 showing the appearance of the wake l a y e r . 148 Ru (0,0, r) 0.5 0 DENSITY Vs Viz i/eo \ N >•'•• \ \ '- \ \\ N O ® \ \ 0 0.2 0.4 0.6 y / D Figure 38c C o r r e l a t i o n s w i t h s e p a r a t i o n i n the cross-stream d i r e c t i o n at y/D=0.36. S o l i d curves 1 and 2 represent r e s u l t s from Grant(1958) f o r y/D=0.52 and 0.037 r e s p e c t i v e l y . R„ (0,0,r) 1.0 0.5 0 Y/D _ _ _ _ _ 0 - | 0.22 0.36 \x *v \ \ ^ (2) \. v ~ **^ >~<s—— ^ ^ * ^ - - _ _ * " " * — **^*7.'" " ""• "•'' """" *.-— --" 0 0.2 0.4 0-6 Figure 38d C o r r e l a t i o n s In the cross-stream d i r e c t i o n f o r the streamwise v e l o c i t y f l u c t u a t i o n s f o r one roughness array(1/16). S o l i d l i n e s are the same as i n Figure 38c. 149 I t i s i n f o r m a t i v e to examine the c o r r e l a t i o n s between the p o i n t s f o r the various frequencies t h a t comprised the averaged c o r r e l a t i o n c o e f f i c i e n t . Davenport(1961} showed that f o r both v e r t i c a l and l a t e r a l s e p a r a t i o n s the coherence i s a f u n c t i o n of f=ny/0* where y i s the s e p a r a t i o n and U i s the mean v e l o c i t y {across the l a y e r i n the case of v e r t i c a l s e p a r a t i o n s ) . Data from Panofsky's measurements at White Sands (guoted i n P a s g u i l l 1971) y i e l d e d a r e l a t i o n Coherence(n) = exp(-l6ny/U) Panofsky a l s o claimed that i f the coherence i s a simple f u n c t i o n of f , then one may assume the co- and guad-s p e c t r a could a l s o e x h i b i t simple behaviour. The e x p l a n a t i o n of t h i s i n terms of geometric s i m i l a r i t y i s d e t a i l e d i n Appendix I I I . The present r e s u l t s showed t h a t the coherence behaved i n conformity with Panofsky's curve (Figure 39). C a l c u l a t i o n s of the slopes of the approaching eddies d i d not behave i n the simple manner p r e d i c t e d by Panofsky and Singer(1965) who found a uniform slope f o r the approaching eddies. . The measurements made so f a r show t h a t the i n t e g r a l s c a l e s f o r u are approximately twice those of v, and tha t the l a t e r a l s e p a r a t i o n c o r r e l a t i o n i s unaffected by d e n s i t y of roughness, while the v e r t i c a l c o r r e l a t i o n s show a departure from those observed over Lego board roughness i n wind tunnels and from a boundary l a y e r developing over a smooth s u r f a c e . I n t e r m i t t e n c y Throughout t h i s chapter mention has been made of the Coherence 1.0 0.5 DENSITY °^ Vs A , Viz a _ i/ 6 0 o t 1 \ A \ a \ o 3 \ 6 _ I l n o A O A ^ 0 .05 0.1 .15 0.2 U F i g u r e 39 Coherence o f l o n g i t u d i n a l v e l o c i t y f l u c t u a t i o n s p l o t t e d a g a i n s t nondimensional frequency where y r e p r e s e n t s the s e p a r a t i o n d i s t a n c e between the pr o b e s . S o l i d curve r e p r e s e n t s f i t to da t a from Panofsky ( a f t e r P a s q u i l l 197D. 151 seemingly important r o l e that i n t e r m i t t e n c y may play. The l i t e r a t u r e on t h i s one aspect of the flow i s vast and was the sub j e c t of two reviews i n the past two years ( Tennek.es 1973: Mollo-Christensen 1973). I t i s intended to look a t i n t e r m i t t e n c y f i r s t by focussing a t t e n t i o n on the p r o b a b i l i t y s t r u c t u r e of the v e l o c i t y by examining the exceedance s t a t i s t i c s , and then to look at the s t a t i s t i c a l p r o p e r t i e s of the d e r i v a t i v e of the streamwise v e l o c i t y . F i n a l l y , measurements of the spectrum, i t s sguare and f o u r t h powers f o r the v e l o c i t y d e r i v a t i v e are obtained f o r comparison with s i m i l a r measurements made i n a j e t by F r i e h e et al{1971). The n o t i o n s of zero c r o s s i n g s and exceedance of any l e v e l are of obvious p r a c t i c a l relevance i n es t i m a t i n g the p r o b a b i l i t y of exceeding l o a d s , s t r e s s e s or responses of any given magnitude. The c a l c u l a t i o n d e t a i l e d i n Chapter 1 made the assumption, pointed out by at n i p ( i n Pao 1969, p.205), t h a t as the f u n c t i o n N(y')/N(0) i s derived from l e v e l c r o s s i n g s , then the d i s t r i b u t i o n s of N(y') and H(0) are s i m i l a r and possess asymptotic s i m i l a r i t y t o a peak count which i s presented i n Figure 40 f o r negative and p o s i t i v e peaks. Figure 40c shows the normalized c r o s s i n g s ; such a s t a t i s t i c i s important i n r e l a t i o n to the d i s c u s s i o n of the re s c a l e d range a n a l y s i s . The d i s t r i b u t i o n i s ob v i o u s l y non-Gaussian. Dutton(1970) suggests t h a t a simple e x p o n e n t i a l p r o b a b i l i t y d e n s i t y f u n c t i o n would give a d i s t r i b u t i o n that i s greater i n the t a i l s and a l s o g r e a t e r than Gaussian around the o r i g i n and l e s s i n between, a 152 >->-CD < CQ O ce ND LN a cn N l CM 1/12 y/D=0.01 f r 'll£~*~l ! u —I 1 1 IV-i/ - 3 - 2 - 1 D 1 . 2 3 c L O C A L MAX I M A 3 / £ I C M A 4 P T - 4 - 3 - 2 - 1 Q 1 2 3 4 aOCAL MAX I MA 3/£ IC MA 33 r* 7 ,<=> >-CD < CQ CD ce Q_ LN C3 • N l CM a a 1/12 y/D=.0.37 4, vYi? U I - £ - 4 - 3 - 2 - 1 0 1 2 3 4 c L O C A L MAX I MA 3 / 5 I C M A N O a L N N T a CM ft a 1/48 y/D=0.37 vV,, \y\y ~r -4 - 3 - 2 . - 1 D 1 2 3 ( L O C A L MAX I MA 3 / £ I C M A Figure 40a Posi t i v e peak d i s t r i b u t i o n s f o r two roughness densities f or two depths. Note the clear excess of values around the o r i g i n especially for the v component. 153 ca < CQ o cc CL. CO L N N T a C 3 1/12 y/D=0.01 U uv; V V'/ i^^ —ijrajp T-tf]— -4 1/48 . y/D=0.01 u* > vcf/ u W I I I I -5-2-1'D 1 2 5 C L O C A L M I N I M A 3 / S I C M A I - 3 - 2 - 1 0 1 2 3 ( L O C A L M I N I M A J / S I C M A i ~ i 4 5 y/D=0.37 >-o y/D=0.37 - 4 - 5 - 2 - 1 D 1 2 3 ( L O C A L M I N I M A I C M A 4 ?'; " A - 4 - 3 - 2 - 1 0 1 2 3 4 5 ( L O C A L M I N I M A l . ' i TC M A Figure 40b Negative peaks d i s t r i b u t i o n f or two roughness densities at two depths. Note the clear excess of values around the o r i g i n e s p e c i a l l y f or the v component. 154 i.o 0.1 , o 9 . o N(y> N(0) D \ ° A \ D \ o 0.12 A 0.22 a 0.3k * tj— o c / c n \ ° >_ \ c A n • o u 0 5 10 15 20 (y/cr/) 2 Figure 40c. Normalized crossings for stream-wise v e l o c i t y fluctuations for three depths for roughness density 1/80. I 1 5 5 s i t u a t i o n shown i n Figure 36 and i n Table I I I . . An e x p o n e n t i a l p r o b a b i l i t y d i s t i b u t i o n w i l l a l s o r e s u l t i n the behaviour shown i n Figure 40. The formal d e r i v a t i o n f o r t h i s o b s e r v ation has been developed by Dutton{1970) who concludes i f exceedance curves are t r u l y e x p o n e n t i a l and th a t turbulence i s l o c a l l y Gaussian, then the patches are surrounded by a i r i n which there i s no t u r b u l e n t a c t i o n . Thus the exact nature of the patchiness of turbulence and e s p e c i a l l y the d i s t r i b u t i o n of energy found w i t h i n the patches needs to be determined. .An important r e s u l t f o r these a p p l i c a t i o n s would be whether the d i s t r i b u t i o n of variance i n the patches i s a f u n c t i o n of t o t a l v a riance measured by observation of many patches on a long data run. (Dutton 1970, p.88) These notions may seem very e s o t e r i c as f a r as work i n r i v e r s i s concerned but t h i s i s not the case i f one views the patches of turbulence d i s t r i b u t e d i n the flow impinging on a roughness element. I f the patches are s u f f i c i e n t l y f a r apart then the p a r t i c l e may view the patches as a Gaussian f u n c t i o n and hence i t s response, i f i t i s a l i n e a r system, w i l l be Gaussian. Furhermore, depending on the s i z e of the patches and t h e i r advection v e l o c i t y r e l a t i v e t o p a r t i c l e s i z e , the i n t e r m i t t e n t p a t t e r n of the flow w i t h i n the patches or t h e i r Gaussian s t r u c t u r e , then the response may be simply modelled 156 by a Gaussian f o r c i n g f u n c t i o n operating on a l i n e a r system. While there seems to have been r e l a t i v e l y l i t t l e work done on the p r o b a b i l i s t i c s t r u c t u r e of the v e l o c i t y f l u c t u a t i o n s and i t s higher moments, other than by Dutton and h i s co-workers i n comparing turbulence at v a r i o u s l e v e l s i n the atmosphere, there i s a p o s i t i v e deluge of r e s u l t s on the d e r i v a t i v e of the v e l o c i t y f l u c t u a t i o n s . The s m a l l s c a l e s t r u c t u r e of turbulence has been shown to be i n t e r m i t t e n t . The skewness and f l a t n e s s f a c t o r s of du/dt are compared i n t h i s s e c t i o n with r e s u l t s obtained i n other flows a t s i m i l a r or higher t u r b u l e n t Reynolds numbers. The d i s t r i b u t i o n of Re across the l a y e r (Figure 41a) shows that the s m a l l e s t A values of Re occur at the f r e e s u r f a c e as here the i n t e n s i t y i s very low. The skewness remains approximately constant or shows a s l i g h t decrease, i n the i n n e r r e g i o n of the f l o w (Figure 41c) while the f l a t n e s s f a c t o r of the v e l o c i t y d e r i v a t i v e i s very c o n s i s t e n t and shows a systematic i n c r e a s e towards the f r e e s u r f a c e , with the value at the f r e e s u r f a c e being n e a r l y twice that found at any other p o i n t i n the flow (Figure 41b). This would tend to b e l i e H o l l e y ' s (1971) c o n c l u s i o n from an i n t e r p r e t a t i o n of the spectra that the turbulence i s unaffected by the presence of the f r e e s u r f a c e . These graphs may be compared with the r e s u l t s f o r f l o w over a rough w a l l at Re of 4x10 4 obtained by Antonia (1973). As was discussed i n Chapter 1, Wyngaard and Tennekes(1970), on the assumption t h a t e i s lognormally d i s t r i b u t e d with P =0.5, showed that the skewness and DEMSITY 0 J/8 - J/12 n V\b * J/20 X >/48 •a- '/so D 43 ?XOA n .. ~r> u * 1} n • *^ A "• • u Xo • cL & C t 0 A ^ 1 0 _ XJfej.-f*| . - fK N-1 1 Q 100 200 Re Figure 4la Turbulent Reynolds number for six densities.plotted against r e l a t i v e depth. 1.0 -0.8 0.6 0.4 0.2 f\ V 0 DENSITY o l / 3 A l / l 2 n 1/20 X '/so n A [_J v_> n J 6 i.o -0.8 .0.6 0.4 0.2 DENSITY I 0 . l / B A l / l 2 0 J/20 X .J/so A ) Ay 'X AD c §3 A DCA X 0.2 Figure 41 b Flatness d i s t r i b u t i o n for four roughness densities plotted against r e l a t i v e depth. 0.4 -Skewness 0.6 dt Figure 41c Skewness d i s t r i b u t i o n for four roughness densities. 158 k u r t o s i s depended on the turbulence Reynolds number. The d i r e c t r e l a t i o n of S and K i s shown i n F i g u r e 42a. While the general trend seems to hold even down to the s m a l l t u r b u l e n c e Reynolds numbers i n t h i s work there i s c o n s i d e r a b l e s c a t t e r . T h i s s c a t t e r i s q u i t e l a r g e e s p e c i a l l y i n the data f o r the atmosphere from the work of wyngaard and Tennekes{1970). I f the f l a t n e s s i s p l o t t e d to show data from s e v e r a l other sources the simple behaviour seems t o break down. P o i n t s from the present study, from Batchelor and Townsend (1949) and Wygnanski and F i e d l e r (1970) are i n c l u d e d . The f l a t n e s s >/5" f a c t o r i n c r e a s e s as Re (Figure 42b) i m p l y i n g u =0.2 at Reynolds number below 200. The departure of the p o i n t s a t the f r e e surface i n d i c a t e s some other process i s c o n t r i b u t i n g to the parameters c o n t r o l l i n g the d i s t r i b u t i o n of the d e r i v a t i v e f l u c t u a t i o n s , not n e c e s s a r i l y a new process, but a b a r r i e r t h a t a l t e r s the i n f l u e n c e of the turbulence Reynolds number. The vortex s t r e t c h i n g mechanism which tends to steepen v e l o c i t y g r a d i e n t s i s no longer s c a l i n g with X f o r r i g h t at"the surface the v e l o c i t y g r adient must disappear. As the measurements show, the f l a t n e s s seems to depend only weakly on Re . Antonia (1973) found there to be no X dependence i n h i s study of a rough "wall boundary l a y e r , and i n f a c t suggested t h a t f o r 0.2<y/D<0_5 f l a t n e s s was approximately constant. This appears to be the case here a l s o and Re i s a l s o nearly constant as Antonia found i t t o X be i n h i s study over the same depth range. The trend i f •it. p =0.2 would be f o r S~Re * ° , which would g u i t e adeguately 159 0.6 0.5 u o • w ¥ > : ¥ ; : : : : > : - ' ' DENSITY o '/s A l/ji O V»0 ~Z>- FREE SURFACE . POINTS 0.7 1.0 1.3 1.5 1.7 Figure 42a P l o t of the lo g a r i t h m of skewness against the logarithm of k u r t o s i s of du/dt. S o l i d l i n e gives 3/8 slope p r e d i c t e d by Tennekes and Wyngaard(1970). Shaded area represents spread of r e s u l t s from atmospheric measurements. log k I — - F R E E SURFACE •B—-= log Re, F i g u r e 42b' K u r t o s i s o f du/dt as a f u n c t i o n o f t u r b u l e n t Reynolds number. Dashed l i n e has a s l o p e o f 1/2 as p r e d i c t e d by Tennekes and Wyngaard (1970), S o l i d l i n e has a sl o p e of 1/5 . Data from B a t c h e i o r and Townsend(1949) A : from Wyganski and F i e d l e r n : from t h i s r e s e a r c h d e n s i t y 1/20 0 '• Shaded ar e a r e p r e s e n t s data from Wyngaard and Tennekes (1970). . 160 1.0 ' ° g 1 0 S - ( d u / d t ) 0.5 0 2 3 4 l o g l o R e x Figure 42c Skewness of ,u/dt as a f u n c t i o n of t u r b u l e n t Reynolds number. S o l i d l i n e r epresents slope of 3/16 as p r e d i c t e d by Tennekes and Wyrigaard ('19.7.0)..., Shaded area represents spread of data from atmospheric measurements. C i r c l e s represent data f o r d e n s i t y 1/20. 161 d e s c r i b e the data i n the lower Reynolds number reg i o n shown i n Figure 42c. F i n a l l y c o n s i d e r a t i o n must be given i n t h i s s e c t i o n t o e v a l u a t i o n of u. Yaglom(1966) showed t h a t where <j> (k) i s the spectrum of the l o c a l d i s s i p a t i o n r a t e (the s o - c a l l e d k u r t o s i s spectrum). I t was used by Pond and Stewart (1965) , by Van A t t a and Chen (1970) and by Wilson (1974) to o b t a i n measurements of u . Values ranged from 0.4 to 0.85, the l a t t e r at Re =200 i n a curved mixing l a y e r . X (Wyngaard and Tennekes 1970). The most recent r e s u l t s f o r a boundary l a y e r are given by Oeda and Hinze(1975) who found X=0.72 f o r yu ^ v = 3 2 0 and y=0.5 f o r yu^ / v <5 showing t h a t the f i n e s t r u c t u r e of turbulence i s a l s o i n f l u e n c e d by d i s t a n c e from the w a l l as r e f l e c t e d by a' change i n the shape of the energy spectrum as the w a l l i s approached- This i s i n d i c a t e d by a decrease i n the i n e r t i a l subrange (see the r e s u l t s of Tielman 1967). Only three measurements were made of u i n the present research by t a k i n g the d e r i v a t i v e spectrum, i t s sguare and f o u r t h power. As the spectrum of the l o c a l d i s s i p a t i o n can be approximated by the sguare of the v e l o c i t y d e r i v a t i v e (that i s e~(du/dx) 2) Novikov(1966) considered s p e c t r a f o r the higher moments of du/dx and p r e d i c t e d power law subranges given by u-1 ~ k (48) (k) ~ k -1 \$ (n+l)u n = 2 , 3 > 4 . . . (49) 162 which f o r n=2 reduces to Yaglom*s r e s u l t above. The values of the skewness and k u r t o s i s f o r u, du/dt, ( d u / d t ) 2 , and ( d u / d t ) 4 are given i n Table VI. For comparison data from a t u r b u l e n t j e t at Reynolds number of 1.2x10* by Friehe e t al(1971) are i n c l u d e d . Figure 43 shows the spec t r a of du/dt, ( d u / d t ) 2 , and (du/dt)* which were computed f o r the d e n s i t y 1/14. The r e s u l t s d i d not f o l l o w Novikov's p r e d i c t i o n s . While i t may seen t h a t power law . subranges do e x i s t , the slopes are not given by Novikov's y i f u i s constant. I f we take the concensus of r e s u l t s about u based on ( d u / d t ) 2 , then u =0.5 i n which case the spectrum of (du/dt)* should e x h i b i t a +1/2 slope and f o r Jdu/dt| 3 the spectrum would be f l a t . The spectra do show a trend towards a subrange slope approaching zero as the n i n Novikov's eguation incre a s e s . . The r e s u l t s may be summarized and compared with the only other s e t of data f o r which these parameters have been c a l c u l a t e d from the s p e c t r a , notably by F r i e h e et a l . (1971). I f we assume Novikov's eguation t o hold but that u v a r i e s we o b t a i n the r e s u l t s presented i n Table VI. While the frequency range over which these slopes were estimated i s not very e x t e n s i v e the good agreement with the r e s u l t s of F r i e h e suggests that the data i s of a s u f f i c i e n t l y good q u a l i t y and that the simple theory of a c o n s t a n t u may be i n e r r o r . The confidence one may place i n the r e s u l t s i s p r e d i c a t e d on the sample s i z e , over 65000 p o i n t s , the f i l t e r which was set a t 285 Hz (which i s s l i g h l t l y below the Kolmogorov microscale and thus might bia s the r e s u l t s ) , and the technigues of TABLE VI Skew. Kurt. Depth=0.lcm i± Skew. Kurt. Depth=l.54cm £ Skew. Kurt. Depth=Surface i± Skew.. Kurt. Data from F r i e h e ( 1 9 7 D u -0.16 du dt 0.45 4.78 6. 32 182 dt 9.19 h 0.40 du q dt 64.6 5928 0.25 •0.03 0. 30 8.01 2.91 6.09 126 0.50 45.1 3092 0.23 -0.42 0.19 1 3-7 3.04 11.4 355 0.55 71.7 7465 0.33 0.09 0.46 11 70 0.5 0.27 2.85 9.2 215 7650 Table VI Third and fourth moments for the signal, i t s derivative, and the square and fburth power of the derivative at three depths for one density(1/14) for comparison with the results of Friehe et al(1971). M- was calculated from the slope of the spectrum and using Novikov's r e l a t i o n . 1 — 1 1 — — — 1 - 4 . 0 -2 .0 0 . 0 2 .0 . 4.0 LOG10 FREQUENCY DERIVRTIVE D=.l i — ac UJ CJ UJ Q. tO CD I £0 _ J _ 1 1 , 1 -4,U -2.0 O.D 2.0 4.0 LOG10 FREQUENCY DEPTH =.1 SQUARED DERIVRTIVE ct UJ 7 1 5 -CJ UJ - — 1 1 1 -4.0 -2.0 CO 2.0 LOG10 FREQUENCY FOURTH POWER DERIVRTIVE D=.l 2 4 Figure 43a Spectra of du/dt, (du/dt) and (du/dt) i n a r b i t r a r y u n i t s f o r s p e c t r a l d e n s i t y at y/D=0.01 f o r a roughness d e n s i t y of 1/14. i — cc U J C M 7" o LJ cr.m_| L'J 0_ (.") - 4 . 0 T 1— - 2 . 0 0 . 0 2 . 0 LUG10 FREQUENCY D E R I V A T I V E -1 4 . 0 Cd u i (_> UJ ; in S I T i u ui CL. CO I O V I n ! — i — : r— - 4 . 0 '. - 2 . 0 O.D 2 . 0 LOG10 FREQUENCY SQUARED D E R I V A T I V E • 4 . 0 ' t— U J <M ?• U UJ vi >Z<n I Z J . i — 1 u UJ cu to CD I O - 4 . 0 — i - 2 . 0 LOG10 —I 1— O.D 2 . 0 FREQUENCY FOURTH POWER D E R I V A T I V E -1 •4.0 p Figure 43b Spectra of du/dt, (du/dt) and (du/dt) i n arbitrary units for spectral density at y/D=0.22 for a roughness density of 1/14. U l J 1 •f1 rf--I 1 1 r -4.U -2.0 0.0 2.0 LQG10 FREQUENCY DERIVATIVE D=SURFflCE 4.0 U J CM ? • U UJ CO \ <->-,' i eery. i — « u w a . co ON 57-a 7% I ,—. ! — 1 -4.U -2.0 0.0 2.0 4.0 • LOG10 FREQUENCY • SQUARED DERIVATIVE D=SURFACE UJ x U U J V ) .5= u U J a. co 37' o 1ft: 1 ' ^-i r - " — - — n -4.0 -2.0 0.0 2.0 A. LOG 10 FREQUENCY FOURTH POWER'DERIVATIVE D=SU'RFflCE • " p ii . •• -Figure 43c Spectra of du/dt, (du/dt) and"(du/dt) i n arbitrary units for spectral density at y/D=0,97 for a roughness density of 1/14. 167 a n a l y s i s and r e c o r d i n g . With (du / d t ) 4 some c l i p p i n g occurred i n the program as a l l computations were c a r r i e d out i n h a l f word i n t e g e r s , and with a k u r t o s i s of over 7000 some 50 points were c l i p p e d . This should not have markedly a f f e c t e d the spectrum as very l i t t l e power was a s s o c i a t e d with these s p i k e s . The flow Reynolds number was 2.5x10 4 so that the c l o s e comparison with F r i e h e ' s r e s u l t s i s warranted. The i m p l i c a t i o n s are that u , which was used t o o b t a i n the r e l a t i o n s amongst S, K and Re may not be constant. The A c l o s e r e m p i r i c a l f i t with P =0.2 i s consonant with the r e s u l t s obtained from using the f o u r t h power of the d e r i v a t i v e . The i m p l i c a t i o n of these l a s t r e s u l t s i s not f u l l y understood. The dependence of skewness and f l a t n e s s on the t u r b u l e n t Reynolds number may not be so much r e l a t e d to X the space over which the v o r t i c e s are being s t r e t c h e d as to the variance of the f l u c t u a t i o n s that occur i n the t h i n sheets or • s l a b s ' of l o c a l i z e d energy d i s s i p a t i o n t h a t C o r r s i n (1962) conjectured. He assumed s l a b t h i c k n e s s of approximately the Kolmogorov microscale and spacing of approximately the i n t e g r a l s c a l e . I f we r e c a l l t h a t the i n t e g r a l s c a l e spacing corresponds to the zone of n- 1 i n the energy spectrum, which has been shown to be r e l a t e d to the frequency of t u r b u l e n t bursts from the w a l l i n the boundary l a y e r , then the v a r i a n c e of the d i s s i p a t i o n , and i t s i n t e r m i t t e n t nature may be d i r e c t l y a s s o c i a t e d with the variance and i n t e r m i t t e n c y of the t u r b u l e n t b u r s t i n g mechanism now seen to be of paramount 168 importance i n generating most of the shear s t r e s s i n the flow-There i s con s i d e r a b l e d i s c u s s i o n over whether the d i s t r i b u t i o n of d i s s i p a t i o n f l u c t u a t i o n s i s l o g normal ( c f . Mandelbrot 1974). Orzag c l a i m s they cannot be while Mandelbrot shows how a modified l i m i t i n g lognormal process may describe the i n t e r m i t t e n c y . I f we consider turbulence as a superimposition of many processes then . determining the p r o b a b i l i t y f u n c t i o n s of each . of the p o s s i b l y weakly i n t e r a c t i n g i n t e r m i t t e n t processes i s important i f such summary s t a t i s t i c s as the r e s c a l e d range or the spectrum are t o be adequately explained. Rescaled range a n a l y s i s As was suggested above, the departures from Gaussian behaviour may be r e l a t e d to the sporadic s t r u c t u r e or the l a c k of homogeneity of the turbul e n c e . One f i n a l p r o b a b i l i t y measure that may be a p p l i e d to the v e l o c i t y f l u c t u a t i o n s i s the s o - c a l l e d ' r e s c a l e d range' a n a l y s i s . The technique examines the r a t i o of the range to the standard d e v i a t i o n f o r i n c r e a s i n g length of record. For a Gaussian d i s t r i b u t i o n H R/S~s where S i s the standard d e v i a t i o n of the s e r i e s from (0,t) with zero mean, R i s the range and s i s the cumulative sample s i z e . H equals 0.5 f o r a Gaussian d i s t r i b u t i o n . Mandelbrot has examined many geophysical time s e r i e s , from the t h i c k n e s s of sedimentary beds over 60 m i l l i o n years t o f l o o d flows i n the N i l e over 2000 years and a l l e x h i b i t a 169 value of H >0.5. The generating mechanism he used to p r e d i c t such a s e r i e s was a f r a c t i o n a l Brownian noise process with i n f i n i t e memory: tha t i s , while Brownian motion a r i s e s from a white noise process, a f r a c t i o n a l Brownian noise comas from a process where each increment i s a weighted average of a l l past increments i n a white noise process. Such a process i s s e l f - s i m i l a r and i s a l s o s t a t i o n a r y . The use of s e l f - s t a t i o n a r y , s e l f - s i m i l a r concepts i n turbulence was s t u d i e d i n c o n s i d e r a b l e d e t a i l by Dutton and Deaven (1969), while the 'Hurst phenomenon' , t h a t i s the occurrence of H values greater than 0.5, was examined by Nordin et al{1972) and by Hansen and Rosbjerg (1974). S t r i c t s e l f - s i m i l a r i t y w i l l only be present i n the i n e r t i a l subrange: because of the nature of the data we are a n a l y s i n g one cannot expect t o f i n d the s e l f - s i m i l a r i t y parameter of 1/3 which would occur i f the s i g n a l was f i l t e r e d to i n c l u d e only f r e q u e n c i e s i n the i n e r t i a l subrange. While the l a r g e s t s e p a r a t i o n s i n our data may show agreement with s e l f - s i m i l a r p r e d i c t i o n s , does t h i s mean t h a t a f r a c t i o n a l Brownian process can be used to model the v e l o c i t y f l u c t u a t i o n s ? The r e s u l t s of Nordin et al(1972) showed that f o r a l l measurements of v e l o c i t y f l u c t u a t i o n s the value of H was greater than 0.5, ranging up to 0.95 f o r measurements made i n the M i s s i s s i p p i R i v e r . The p l o t s showed R/S against sample s i z e with the l i n e s ^showing no s i g n s of reducing i t s slope to 0.5. This was t r a n s l a t e d as meaning the process had i n f i n i t e memory and i n f a c t they s t a t e 170 the Hurst phenomenon i s observed i n both flumes and r i v e r s . . . . . the i m p l i c a t i o n s are that the i n t e g r a l s c a l e of turbulence does not e x i s t f o r these flows. (Nordin et a l 1972, p.1486) P l o t s of the r e s c a l e d range were produced ( c f . Figu r e 44a) f o r v a r i o u s depths i n the flow and indeed H i s found to l i e between 0.8<H<0.94 with no systematic v a r i a t i o n from bed t o surface ( Table V I I ) . However i t seems unreasonable t o assume that t h e r e f o r e there i s no macroscale. Because H continues to in c r e a s e i t i s u s u a l l y assumed t h a t from the f r a c t i o n Brownian motion model there i s a dependence on a l l previous values and hence there i s p e r s i s t e n c e throughout the data. Such a s t a t e was commented upon by Klemes(1974), who compared the co n c l u s i o n of Mandelbrot and Van Ness (1968): i t i s known t h a t economic time s e r i e s t y p i c a l l y e x h i b i t c y c l e s of a l l orders of magnitude, the slowest c y c l e s being periods of d u r a t i o n comparable t o the sample s i z e , t o t h a t of F e l l e r f o r a c o i n t o s s i n g game of l e n g t h 2n the number of t i e s i n c r e a s e s i n p r o b a b i l i t y o n l y as (2n) 1/ 2; t h a t i s with i n c r e a s i n g d u r a t i o n of the game the frequency of t i e s decreases r a p i d l y and the waves inc r e a s e i n length. Thus the seeming p e r i o d i c i t y of the lowest freguency which would lead t o an undefined macroscale i s j u s t a m a n i f e s t a t i o n of the a r c s i n e law described by F e l l e r ( 1 9 6 6 , p.80). 171 DENSITY # 5 D = 4.0 0.81B E(R/S)=0.70S R = 0.995 APPARENT H= 0.81 MOMENTS OF DATA 1ST =0.00 2ND =1.00 3RD = -0.45 4TH = 3.34 10 TIME 100 1000 10000 Figure 44a Rescaled range of u . Time respresents cumulative sample s i z e ; sampling r a t e 3 0 0 Hz approximately. TABLE V I I D e n s i t y 1/8 1/12 1 / 2 0 Depth 0 . 1 0 . 8 0 0 . 8 1 0 . 7 8 0 . 5 - ; 0 . 8 l 0 . 8 5 0 . 8 0 0 . 9 0 . 8 0 0 . 8 1 0 . 7 8 1 . 5 4 0 . 8 0 0 . 7 9 0 . 8 2 2 . 5 4 0 . 8 3 0 . 8 1 0.84 4 . 0 0 0 . 7 9 0 , 8 2 0 . 8 0 S u r f a c e 0 . 8 0 0 . 7 9 0 . 8 1 T a b l e VII. H v a l u e s f o r the r e s c a l e d range f o r t h r e e d e n s i t i e s at seven depths. A l l exponents gave a f i t with R*in excess o f 0 . 9 9 . 173 The seeming c o n t r a d i c t i o n may be r e s o l v e d by r e a l i z i n g t h a t while f r a c t i o n a l Brownian noise with i n f i n i t e memory can produce H values of 0.75 to 0.95, i n exact agreement with observations over the roughness a r r a y s , a b e t t e r modelling scheme may be developed. F r a c t i o n a l Brownian noise has the property that the s p e c t r a l d e n s i t y takes the form n 1 _ 2 H , where n i s frequency and 0.5<H<1.0. The s p e c t r a f o r the data given i n Table VII may be shown t o be f i t t e d by a curve of the form n" 1 over a c e r t a i n range. However the spectrum of turbulence extends to much higher f r e g u e n c i e s where such a r e l a t i o n does not hold. Another s e l f - s i m i l a r process with H=1/3, y i e l d i n g a s t r u c t u r e f u n c t i o n with a 2/3 s l o p e , may be developed and has been shown t o be approximated f o r the v e l o c i t y with l a g dis t a n c e s i n the p e r i o d corresponding t o the i n e r t i a l subrange by Button and Deaven{1969). But, by using another generating f u n c t i o n , the s o - c a l l e d broken l i n e process, Hurst values of 0.75 to 0.9 may a l s o be generated (Rodriguez-Iturbe et a l 1972). A broken l i n e process r e s u l t s from the l i n e a r i n t e r p o l a t i o n between e q u a l l y spaced independent Gaussian random v a r i a b l e s , and., the random displacement of the s t a r t i n g point (See Mejia et a l 1974). Most s i g n i f i c a n t l y , by adding broken l i n e processes i t i s p o s s i b l e t o reproduce any H value with the decided advantage t h a t there i s a f i n i t e variance f o r each process. Thus i f there were . s e v e r a l i n t e r m i t t e n t processes o p e r a t i n g i n su p e r i m p o s i t i o n , the r e s u l t i n g s e r i e s would e x h i b i t stong p e r s i s t e n c e , and would have a high H value. Kleraes(1974) 1 7 4 indeed discusses such a s i t u a t i o n i n l o o k i n g at streamflow processes made up of wet and dry epochs f o r which H w i l l be 1.0 or 0.5- By mixing the epochs any value of H may be achieved, demonstrating that i n t e r m i t t e n c y i s a process w e l l described by the Hurst phenomenon. While f r a c t i o n a l Brownian motion might be a method of generating the R/S diagram i t i s the mixture of processes t h a t i s more l i k e l y to l e ad t o an adeguate p h y s i c a l e x p l a n a t i o n of the r e s u l t . The presence of i n t e r m i t t e n c y a t the d i s s i p a t i o n f r e q u e n c i e s , wake shedding and of b u r s t i n g from the boundary l a y e r which have been shown t o be r e l a t e d to the spectrum p r e c i s e l y i n the region of n~ l where f r a c t i o n a l Brownian noise i s a p o t e n t i a l model, a l l combine to produce a time s e r i e s whose composite p r o p e r t i e s w i l l not be explained by a simple s e l f - s i m i l a r generating process. A q u a l i t a t i v e d e s c r i p t i o n of the three ranges of the spectrum might be attempted by saying that there i s an n _ 1 region i n which b u r s t i n g from the boundary and wake shedding produce an i n t e r m i t t e n t process.. Secondly t h e r e i s a r e g i o n of steeper s l o p e , the -5/3 r e g i o n , i n which vortex s t r e t c h i n g i s the major process. T h i s may be v i s u a l i z e d as a branching process, or b e t t e r as a random walk i n three dimensions with the c o n d i t i o n that the process e x h i b i t s short memory because s t r e t c h i n g i n one d i r e c t i o n i s followed by s t r e t c h i n g i n the orthogonal d i r e c t i o n s and thus there are two steps between s t r e t c h i n g i n the same d i r e c t i o n . (See recent work by Obukhov and Dolhansky (1975) who have produced a m u l t i - s t o r i e d 175 branching model f o r turbulence which i n c o r p o r a t e s an ' i n s t a b i l i t y t r i p l e t ' that would allow an i n t e r m i t t e n t process to be described f o r the v e l o c i t y f l u c t u a t i o n s . ) The number of steps i n t h i s branching process w i l l be i n f l u e n c e d by a n i s o t r o p i c s t r e s s at low frequency and v i s c o s i t y at the highest frequencies. F i n a l l y there i s a steep slope r e g i o n of energy d i s s i p a t i o n wherein i t i s known t h a t i n t e r m i t t e n c y occurs. In f a c t a re s c a l e d range p l o t of the v e l o c i t y d e r i v a t i v e y i e l d s H=0.37; t h a t i s , a process with a f i n i t e memory but s t r o n g l y i n t e r m i t t e n t . From Figure 44b the memory of the process'may be estimated to y i e l d a break . a t one second, corresponding to bu r s t s of d i s s i p a t i o n a c t i v i t y of the duration of tha t period. The notion of the i n t e r a c t i o n of the v a r i o u s i n t e r m i t t e n t processes would be q u i t e adequate to r e s o l v e the seeming dilemma of H>0.5 implying an i n f i n i t e v a r i a n c e . The r e s u l t s of Nordin et al(1972) were complemented by data c o l l e c t e d by Hansen and given i n Hansen and Bosbjerg(1974). To avoid the unpleasant c o n c l u s i o n of an i n f i n i t e v a r i a n c e , implying an i n f i n i t e i n t e g r a l s c a l e , the l a t t e r authors attempted to show tha t the range w i l l e x h i b i t H=0.5, but only a f t e r a long t r a n s i e n t . I f i n s t e a d of the v e l o c i t y f l u c t u a t i o n s a new process, defined by i n t e g r a t i n g the d e v i a t i o n s from the mean i s used t n ( t ) = / u ( t ) d T (50) o they conclude an asymptotic v a r i a t i o n a ~ t «5 i s reached 176 F i g u r e 44b. Rescaled range of du/dt showing break of slope at 0.3 seconds. Time represents cumulative sample s i z e ; sampling r a t e 300Hz approximately. 177 w i t h i n the period of time l e s s than the l e n g t h of the a v a i l a b l e records. This i m p l i e s a f i n i t e i n t e g r a l time s c a l e . An extremely long t r a n s i e n c e period i s shown to e x i s t before t« 5 v a r i a t i o n of the range and the r e s c a l e d range i s reached. The period of R/S i s found t o be l a r g e r than 1000 times the E u l e r i a n time s c a l e T of the v e l o c i t y . They assumed that the c o r r e l a t i o n f u n c t i o n was e x p o n e n t i a l i n the t a i l and thus i n t e g r a b l e i n the l i m i t , i m p l y i n g a f i n i t e i n t e g r a l s c a l e , thus c o n t r a d i c t i n g the i n i t i a l premise with which they were working. Much sim p l e r i s the notion t h a t an i n t e r m i t t e n t high freguency process operates and appears t o i n t e r a c t with a low frequency b u r s t i n g mechanism which i s a l s o i n t e r m i t t e n t . Furthermore c o n t i n u i t y reguirements and the smoothing a c t i o n of v i s c o s i t y r e s u l t i n vortex s t r e t c h i n g and a tendency towards i s o t r o p y which r e s u l t s i n a process t h a t i s c r u d e l y normal. Such a d i s t r i b u t i o n would be c h a r a c t e r i z e d by a r e s c a l e d range H>0.5 as the variance w i l l much more r a p i d l y converge to a s t a b l e l i m i t while the range w i l l i n c r e a s e because of the p o s s i b l y l o g a r i t h m i c normal d i s t r i b u t i o n of bursts and the d i s t r i b u t i o n of f l u c t u a t i o n s w i t h i n these b u r s t s . Conclusions The r e l a t i o n between t u r b u l e n t Reynolds number the skewness, k u r t o s i s and i s p redicated on the assumption 178 about the u n i v e r s a l nature of p but from the r e s u l t s presented i t i s c l e a r that Re^  plays a c r i t i c a l r o l e as a s c a l i n g parameter. However, the r e l a t i o n i s not f u l l y defined as hoped by Wyngaard and Tennekes{1970). The importance of Re i s emphasized by r e c a l l i n g i t was used as a s i m i l a r i t y parameter i n the parametric s o l u t i o n of the modified Obukhov approximation of the s p e c t r a l energy equation. The t u r b u l e n t bursts which w i l l dominate the skewness and k u r t o s i s seem to be r e l a t e d t o Re f o r the A b u r s t s are a s s o c i a t e d with steep v e l o c i t y g r a d i e n t s and the production parameter takes the form from the i s o t r o p i c r e l a t i o n . Thus the spacing of the vortex tubes at X proposed by Tennekes (1968) may i n d i c a t e a c l o s e connection between i n t e r m i t t e n c y of the production and d i s s i p a t i o n f u n c t i o n s . These notions of a broad c o u p l i n g of i n t e r m i t t e n t f l u c t u a t i o n s are i n agreement with the r e s u l t s of Kim e t al(1971) who found that the small s c a l e f l u c t u a t i o n s which grew r a p i d l y subsequently r e s u l t e d i n major m o d i f i c a t i o n s of the flow at much l a r g e r s c a l e s . These l o c a l i n s t a b i l i t i e s were shown to have a strong s p a t i a l s t r u c t u r e by Gupta et al(1971) i n measurements i n a t u r b u l e n t boundary l a y e r , while Mollo-Christensen{1973) quotes work of Dorman on gust patterns i n the production of turbulence showing how s t r e s s , production of turbulence and waves and s i g n i f i c a n t changes i n where / v.)'* = 1_ X_ 1_ (51) 179 the v e l o c i t y p r o f i l e s a l l occur as j o i n t events which may only occupy 5% of the time at any given l o c a t i o n . He continues a theory that takes i n t o account only weak non l i n e a r i n t e r a c t i o n s of second order may miss the essence of the processes which a c t u a l l y occur i n nature and i n a l l l i k e l i h o o d dominate, namely strong n o n - l i n e a r i n t e r a c t i o n of a wide range of s c a l e s . . . . a wavy boundary a f f e c t s the generation of turbulence and the s t r u c t u r e of the t u r b u l e n t flow f i e l d , and t h a t phase v e l o c i t y of the waves a f f e c t s the generation and response of the t u r b u l e n t f i e l d . I f the -boundary could respond, i t would a f f e c t the dynamics of the s m a l l s c a l e turbulence and produce couplings between s c a l e s . (Mollo-Christensen 1973, p.115) Host s i g n i f i c a n t l y , the r e s u l t s showed t h a t i f the spectrum i s t o be understood i t i s necessary to look at s m a l l s c a l e s as the shape appears to be i n f l u e n c e d by the production and d i f f u s i o n parameters which i n t u r n were found t o depend on the t u r b u l e n t Reynolds number. The study of the p r o b a b i l i t y s t r u c t u r e , of zero c r o s s i n g s and i n t e r m i t t e n c y show that the s m a l l e s t s c a l e s exert a very important i n f l u e n c e on the t o t a l flow s t r u c t u r e . Dutton sounds the o p t i m i s t i c note 180 once the p r o b a b i l i s t i c s t r u c t u r e of the c h a r a c t e r i s t i c k e r n e l s f o r turbulence are known, then the s t a t i s t i c a l p r o p e r t i e s of the responses, whether l i n e a r or n o n - l i n e a r w i l l f o l l o w . (Dutton 1970, p.106) The need then f o r a more d e t a i l e d a n a l y s i s of the f l u c t u a t i o n s i n terms of c o n d i t i o n a l sampling of the s e r i e s when c e r t a i n c o n d i t i o n s or l e v e l s are reached i s obvious. I t i s to be hoped that the work w i l l be undertaken f o r there i s a strong body of theory f o r excursions and i n t e r v a l s between c r o s s i n g s , d e t a i l e d by Cramer and Leadbetter(1967), t h a t could give a u s e f u l b a s i s f o r a model of extreme f o r c e s and periods of these f o r c e s . This would be u s e f u l i n determining entrainment c r i t e r i a f o r p a r t i c l e s of c e r t a i n s i z e s . The r e s c a l e d range confusion i n the l i t e r a t u r e i n d i c a t e s t h a t by d e t a i l e d c o n d i t i o n a l sampling some of the hypotheses put forward could be t e s t e d . The need f o r more measurements of higher order d e r i v a t i v e spectra i s great t o examine Novikov's notions f o r maybe P i s i t s e l f dependent on some parameter of the flow as Mandelbrot suggests or t h a t i t depends on the averaging s i z e used which may i t s e l f be a random v a r i a b l e . But i n a r i v e r flow t o r e s t a t e Mollo-Christensen i f the boundary could respond, i t would a f f e c t the dynamics of the small s c a l e turbulence and produce c o u p l i n g between s c a l e s . 181 CHAPTER 4 BARS, BLOCKS AND OPEN CHANNELS The purpose of t h i s chapter i s to b r i n g together some of the d i s p a r a t e r e s u l t s obtained over d i f f e r e n t roughness types and present some measurements obtained i n n a t u r a l channels. The comparison between bar . type roughness and a r r a y s of i n d i v i d u a l b locks i s presented. The wake behind a s i n g l e bar i s compared with Counihan et al's(1974) theory of the behaviour of a s e l f - p r e s e r v i n g wake. This leads t o a c o n s i d e r a t i o n of two dimensional versus three dimensional o b s t a c l e s i n a w i d t h - l i m i t e d f l ow. The occurrence of v o r t i c i t y a m p l i f i c a t i o n i s considered and i t s p o s s i b l e r e s u l t a n t e f f e c t i n i n t r o d u c i n g a p e r i o d i c i t y i n t o the f l o w ; th a t i s a c o n c e n t r a t i o n of energy at a c e r t a i n frequency r e l a t e d to the s i z e of the elements on the bed i n t e r a c t i n g with the flow by a s p e c i a l type of vortex s t r e t c h i n g . R e sults are presented from measurements made i n open channels with cobble beds t o evaluate the i n f l u e n c e of the f r e e surface and to compare channel s c a l e measurements with l a b o r a t o r y r e s u l t s . Bars Bars i n n a t u r a l channels are v a r i o u s l y d e f i n e d , with many types being d i s c r i m i n a t e d ( A l l e n 1968). Point bars. 182 u s u a l l y a s s o c i a t e d with a tendency to meander, r a r e l y extend over more than one t h i r d of channel width- The bar type roughness used i n the experimental flow was more a k i n t o the wind-break models i n aerodynamics (see Good and j o u b e r t 1968: P l a t e 1971) or the r i f f l e s t h a t extend across the t o t a l width of channels i n g r a v e l streams. To complete the l a b o r a t o r y measurements two spacings of bars* (1/9 and 1/14 based on the height of the bar t o downstream distance) were measured f o r two flow depths, w h i l e a f i f t h measurement was made at a density of 1/17. The r e s u l t s are presented i n Figure 45 and the spread of r e s u l t s i n d i c a t e s t h a t the upper p o i n t s c o r r e l a t e w e l l with Dvorak's r e s u l t s - Here the spacing was l e s s than twice the flow depth. The lower p o i n t s show th a t when the spacing i s approximately three times depth there i s a marked in c r e a s e i n shear v e l o c i t y . Such a spacing b r i n g s these points i n t o c l o s e correspondence with the r e s u l t s of Koloseus and Davidian (1966) i n which f r e e s u r f a c e e f f e c t s are known to have increased r e s i s t a n c e i n the form of s u r f a c e waves. I t would thus seem l i k e l y , and o b s e r v a t i o n during the experiment, i n d i c a t e d , that v a r i a t i o n s i n the f r e e s u r f a c e l e v e l play a s i g n i f i c a n t r o l e i n i n c r e a s i n g r e s i s t a n c e , i m p l y i n g t h a t Froude number should become a s i g n i f i c a n t or dominant parameter. However no simple a n a l y s i s can be c a r r i e d out f o r , as Rouse(1965) pointed out, the bars at such low spacings must be t r e a t e d as i s o l a t e d flow d i s t o r t i n g roughness elements and hence spacing s c a l e s are of l i t t l e value. r h x — •—s h 9 X I T 9 X DATA FROM KOL AND DAVID IAN OSEUS (I9fe6) x h 10 h h 5 0 100 Figure 45 Wall shear velocity correlations for bars i n open channels. Lower points on graph showing correspondence with • data from Koloseus and Davidian(1966) represent conditions when -Ae->2D. 184 The comparison given in.Chapter 2 between r e g u l a r a r r a y s of blocks and bar roughness was achieved by assuming X /h — A/A where A JLs the f r o n t a l area and A i s the e i r s i l h o u e t t e area. Such a procedure assumed that the r a t i o between the drag on a bar and the drag on a three dimensional array with s i m i l a r f r o n t a l areas was u n i t y . However measurements of the turbulence i n t e n s i t y downstream of bars were always found to be greater than f o r apparently corresponding d e n s i t i e s of roughness b l o c k s . The turbulence l e v e l s behind b l o c k s never exceeded 30%, while behind bars a t s i m i l a r d e n s i t i e s (1/14) the i n t e n s i t y rose as high as 55%. (The hot f i l m r e s u l t s must be t r e a t e d with extreme c a u t i o n as the s i m p l i f i e d c a l i b r a t i o n curve i s i n a c c u r a t e at such high l e v e l s . ) I r r e s p e c t i v e l y , the magnitude may be taken to i n d i c a t e . t h a t there i s a much greater turbulence production and concomitantly a much greater r a t e of energy d i s s i p a t i o n than f o r blocks- This i s i n accord with the r e s u l t s of M a r s h a l l (1971) who found bars presented 30% more r e s i s t a n c e t o flow i n comparison with r e g u l a r arrays of b l o c k s at the same d e n s i t y . The maximum value of the d i s s i p a t i o n r a t e i n a v e r t i c a l p r o f i l e occurred above the bed f o r d e n s i t i e s 1/9 and 1/14. The r e s u l t s f o r density 1/17 showed a reattachment point 12 roughness heights downstream from the bar before the s t a g n a t i o n zone i n f r o n t of the next bar caused the maximum value to s h i f t away from the bed. F i g u r e 46 shows the amplitude p r o b a b i l i t y d e n s i t y diagrams of the normalized v e l o c i t y f l u c t u a t i o n s at three 0.10 -. 0.10 -j STANDARDIZED SCORES Figure 46 Probability density diagrams downstream (at x/h= 3-6> 7> lo) within an array of bars spaced at 1/14. 186 p o s i t i o n s downstream from a bar i n an arr a y 1/14 a t 3.6, 7 and 10 roughness heights downstream. The departures from the Gaussian form are very pronounced: at x/h=3.6 the d i s t r i b u t i o n i s sharply l i m i t e d at the upper end, i n d i c a t i n g t h a t the flow i s most f r e q u e n t l y l e s s than the mean with a s i g n i f i c a n t number . of high v e l o c i t y b u r s t s or v o r t i c e s impinging on the bed. At x/h=7.0 the d i s t r i b u t i o n i s . c r u d e l y normal while by x/h=10 the peak has s h i f t e d to a s l i g h t l y p o s i t i v e value, suggesting a p o s s i b l e s l i g h t p u l s i n g i n the flow upstream of the next bar perhaps r e l a t e d t o the wake shedding. In these cases the r e s c a l e d range value was very h i g h , H=0.85, i n d i c a t i n g that the i n t e r m i t t e n t processes discussed i n Chapter 3 are s t r o n g l y evident i n t h i s flow s i t u a t i o n . S i n g l e Bar A cons i d e r a b l e number of measurements was c a r r i e d out on a s i n g l e bar and on bars of various widths to examine when a bar s t a r t s to behave as a three dimensional wake shedding block. The wake behind a bar i n a rough boundary l a y e r has been l i t t l e s t u d i e d . Apart from the work of Counihan et al{1974) most measurements have simulated s h e l t e r b e l t s and fences i n a smooth w a l l boundary l a y e r , thus removing the troublesome aspect of s c a l i n g the r e l a t i v e roughness between the bar and the surrounding bed. The f o l l o w i n g r e s u l t s are presented f o r comparison with Counihan's theory f o r s e l f - p r e s e r v i n g wakes 1 8 7 behind two dimensional o b s t a c l e s . Counihan et al(1974) showed that i f the p r o f i l e s of 0{y) are s e l f - p r e s e r v i n g then the data should p l o t on a s i n g l e curve defined by the parameter -u/D (h) (x/(h-$*)) p l o t t e d against n (y/(h- 6«)) / (Kx/ (h- 6s) r v + 2 where n» i s the power law exponent from the f i t to the i n c i d e n t boundary l a y e r and K i s obtained from K=2k"u 0(h) where k " i s von Karman's s constant, 0(h) i s the v e l o c i t y at the top of the element h, 5*is the displacement t h i c k n e s s , x i s downstream d i s t a n c e and -u i s the d i f f e r e n c e between the v e l o c i t y at the height y with and without the roughness block. K represents the measure of the r a t i o of shear s t r e s s to i n e r t i a l s t r e s s . The r e s u l t s i n Figure 47a show good agreement up to n'-3.0. The l a r g e r values of n' represent greater heights above the element. The drop o f f from t h e . r e s u l t s of Counihan suggests t h a t the f r e e s u r f a c e may play a r o l e i n that the wake does not develop so r a p i d l y v e r t i c a l l y as i n the atmospheric boundary l a y e r case. Such a s i t u a t i o n might be e x p l a i n e d by the higher turbulence l e v e l s i n the outer part of the flow tending to i n h i b i t wake development. The most n o t i c e a b l e s i n g l e e f f e c t of the bar was t h a t the turbulence i n t e n s i t y was up t o 80% greater than f o r the wake behind a block. While the mean v e l o c i t y very r a p i d l y returned to i t s undisturbed value by approximately eighteen roughness heights, the turbulence i n t e n s i t y even twenty four roughness heights downstream (the g r e a t e s t d i s t a n c e measured) had not returned t o i t s o r i g i n a l value. Counihan showed th a t 188 _ v / ( h - y , ) 6 5 4 3 2 I 0 ( K x / ( h - y,)) X / (h-y . ) o 9.5 A 14.0 n 19.0 o ^ J^©^~-A ' A / 0 o \ - ( — ) h) Vh-y,/ •U(h) Figure 47a. Ve l o c i t y defect measurements p l o t t e d as self-preserving p r o f i l e f o r the wake behind a bar. n=0.22 from f i t to incident v e l o c i t y p r o f i l e power . law; K=0.48 h/D=l/7. So l i d l i n e s show region of Counihan et al's(1974) r e s u l t s . 6 5 4 3 2 I .12 .^24 •48 .05 .24-•c ^.3+ .SO -^-^ — (—1 U(h) \ h - y , / Figure 47b V e l o c i t y defect measurements at two heights(y/h=0.1, 1.0) f o r various bar widths. Numbers on graph r e f e r to proportion of channel "barred". 189 the maximum i n c r e a s e i n mean square turbulence v e l o c i t i e s was given by ' l+n' 2 A( U ' ) - n n I, ir 2.+"' v max _ 10.4 K U 2(h) (x/h) • 5 2 where A ( u ' ) 2 represents the d i f f e r e n c e between the squared max values of the RMS v e l o c i t y with and without the bar at any one downstream distance- ( M u * ) 2 /U 2 (h) ) ( x / h ) 2 + n ' should be max a constant value. I t was here evaluated as 2-00 while f o r x/(h-5*) of 9-5, 14 19 and 28 the e m p i r i c a l values were 2.1, 2.5, 2.7 and 2.2. The rough accord of values i n d i c a t e s that the mean square values decrease as approximately (h/x) ^ n'. The major reason f o r computing the s e l f ^ -preserving p r o f i l e s was t o examine i f there was a l i m i t i n g width below which the ob s t a c l e s t a r t e d t o behave as a three dimensional b l u f f body r a t h e r than a two dimensional bar. This was examined i n two ways. F i r s t , the mean v e l o c i t y a t two downstream p o s i t i o n s f o r e i g h t o b s t a c l e widths was measured. The o b s t a c l e widths ranged from 0.05 of channel width to the t o t a l width of the channel. These- data are presented i n Figure 48a where i t may be seen t h a t the mean v e l o c i t y defect reaches i t s maximum value when between 0.25 and 0.5 of the channel width i s 'barred'. This suggests that the block has only to occupy approximately one t h i r d of the channel width before i t begins t o behave as a two dimensional, wake shedding bar. Observations made at the time of measurement . confirmed t h a t when the block occupied 0.34 of the channel width there was a n o t i c e a b l e b u r s t i n g of t u r b u l e n t a c t i v i t y X X o o X o i o o * X V. 0 .25 0 .5 . 7 5 1.0 re lat ive bar width (block width/f lume width) F i g u r e 48b T u r b u l e n c e l e v e l s d i v i d e d by t u r b u l e n c e l e v e l f o r a complete b a r a t two downstream p o s i t i o n s x/h= 9( ) and 14( 1.0 .75 a 0 . 5 . 2 5 "x ° x o X o 1 o y. . 2 5 0 . 5 .75 re lat ive bar width 1.0 F i g u r e 48a Mean v e l o c i t y d i v i d e d by mean v e l o c i t y w i t h o u t any o b s t r u c t i o n f o r v a r i o u b a r w i d t h s a t two downstream p o s i t i o n s as i n F i g u r e 48b. 191 at the surface which r e s u l t e d i n patches of roughened . water t h a t were r a p i d l y d i f f u s e d downstream. A second method of examining t h i s problem was t o p l o t the s e l f - s i m i l a r p r o f i l e s on axes s i m i l a r to those f o r the data of a complete bar, which i s done i n Figu r e 47b. The s i m i l a r i t y to Figure 47a i s good f o r r e l a t i v e bar widths from 1.0 down to 0.24; t h e r e a f t e r the r a p i d d i f f u s i o n of the wake leads to a lower v e l o c i t y d e f i c i t , which e x p l a i n s why the po i n t s congregate at the lower l e f t of the diagram. The turbulence data may be examined i n a s i m i l a r f a s h i o n . In Figure 48b the r a t i o of u' f o r the v a r i o u s bar widths d i v i d e d by u« f o r the complete bar width are presented. They behave i n a fashion s i m i l a r to the mean v e l o c i t y , i n d i c a t i n g that any obs t a c l e under 0.25 of the channel width may be regarded as an i s o l a t e d three dimensional block. Whether the s c a l i n g i s with width of bar to channel width, bar width to flow depth, or bar width t o the macroscale of the flow cannot be determined from these experiments as they were only performed at one flow depth. R e l a t i v e to the macroscale of the f l o w , when the block becomes greater than twice the macroscale i t appears to behave as a two dimensional o b s t a c l e . In t h i s context i t i s i l l u m i n a t i n g to note that the experimental r e s u l t s of P e t t y (presented i n Hunt 1974) showed t h a t f o r a/L (width of body to macroscale) of 1.81 the RMS v e l o c i t y along the s t a g n a t i o n l i n e increased markedly, suggesting t h a t i t i s p o s s i b l y the 192 r e l a t i o n to macroscale that i s the most important: downstream of the block at t h i s width the i n t e n s i t y had r i s e n almost t o i t s peak value. However, when the o b s t a c l e width reached .34% of channel width i t approximated 2.5 times channel depth. From comparison with the experiments over the a r r a y s of b a r s , f r e e surface e f f e c t s may w e l l be s i g n i f i c a n t s i n c e the bar •o w i l l act as a l o c a l nonuniformity causing disturbance of the surface. ' Such I n t e r a c t i o n between depth, macroscale and width I s one of the unresolved problems of f l u v i a l geomorphology: t h i s simple example i n d i c a t e s the i n t e r p l a y of these t h r e e v a r i a b l e s , which i n a n a t u r a l channel could a l l a d j u s t . However, as i n most n a t u r a l channels, width i s the most c o n s e r v a t i v e , and as the experimental observations i n the flume i n d i c a t e d b u r s t i n g at the s u r f a c e , f o r bar p r o p o r t i o n a l widths g r e a t e r than 0.34, i t i s suggested that t u r b u l e n t macroscale i s the l i m i t i n g kinematic parameter. No t h e o r i e s e x i s t t o examine the e f f e c t of blocks when t h e i r dimension equals the macroscale, although asymptotic s o l u t i o n s e x i s t f o r a/L approaching zero and i n f i n i t y . ' When the macroscale ( i t s e l f p a r t l y determined by the depth of flow) i s only h a l f the width of the block, there i s an i n c r e a s e i n turbulence i n t e n s i t y along the s t a g n a t i o n l i n e . T h i s w i l l l i k e l y be coupled with an i n c r e a s e i n the turbulence components downstream as the flow separates, l e a d i n g to a very high i n t e n s i t y and a consequently l a r g e v e l o c i t y d efect i n the wake behind the block. 193 S i n g l e Block The s t r u c t u r e of flow behind and approaching a . s i n g l e roughness block was i n v e s t i g a t e d . There are two reasons f o r studying the flow behaviour around a s i n g l e block. F i r s t i t i s v aluable to know the v a r i a t i o n i n the turbulence s t r u c t u r e downstream, i t s p r o b a b i l i t y s t r u c t u r e and freguency composition r e l a t i v e to t h a t of the i n c i d e n t boundary l a y e r . T h i s would allow an examination of the couple between the flow and the o b s t a c l e , and the p o s s i b l e i n t e r a c t i o n between the flow behind a block as i t approaches another b l u f f body. Second, sediment entrainment mechanics have looked at the f o r c e s and f l u c t u a t i o n s on i n d i v i d u a l p a r t i c l e s (ASCE 1966) and a d e t a i l e d study of the flow around one p a r t i c l e i s thus very p e r t i n e n t to such f o r m u l a t i o n s . The flow around the p a r t i c l e may a l s o undergo v o r t i c i t y a m p l i f i c a t i o n , r e s u l t i n g i n the p r e f e r e n t i a l c o n c e n t r a t i o n of energy at some s c a l e . This might be s i g n i f i c a n t i n determining the p r o b a b i l i t y s t r u c t u r e of f l u c t u a t i o n s r e s u l t i n g i n flow induced p e r i o d i c i t i e s on the bed, or i n p r e f e r e n t i a l l y eroding c e r t a i n p a r t i c l e s i z e s . The study of wakes behind o b s t a c l e s has a l a r g e l i t e r a t u r e , but the study of wakes behind o b s t a c l e s i n a rough w a l l boundary l a y e r i s r a r e . S i m i l a r to the s i t u a t i o n with bar roughness, most experimental measurements have been made i n smooth w a l l boundary l a y e r s , although there i s recent work i n simulated atmospheric boundary l a y e r s on the e f f e c t of neighbouring b u i l d i n g s on the turbulence a f f e c t i n g 194 s t r u c t u r e s (Wind. E f f e c t s on B u i l d i n g s Symp. 1971). A comparison of the wakes behind a bar and behind a s i n g l e block was a l s o c a r r i e d out f o r i t has some relevance f o r the notion of a hierarch y of bedform f e a t u r e s . Bars i n the flume might be l i k e n e d to the r i f f l e s or g r a v e l bars that are u s u a l l y found, whose spacing i s r e g u l a r but at a much lower frequency than the spacing of dominant p a r t i c l e s that are hypothesized t o e x i s t on the g r a v e l bed. The mean v e l o c i t y and turbulence l e v e l s q u i c k l y r e - e s t a b l i s h t h e i r previous values downstream from a s i n g l e block. U n l i k e a bar whose downstream e f f e c t on the mean v e l o c i t y extends to at l e a s t 18 roughness heights, f o r a s i n g l e block the mean v e l o c i t y p r o f i l e has become e s s e n t i a l l y r e - e s t a b l i s h e d by 12 roughness heights while by 24 roughness heights the i n t e n s i t y p r o f i l e i s r e - e s t a b l i s h e d . The d i s s i p a t i o n p r o f i l e , presented as the r a t i o of e (y) at height y d i v i d e d by e at the bed, shown i n Figure 49, i n d i c a t e s t h i s parameter i s the slowest to respond, as one should expect. There are two stages i n the wake: one of production, which occurs i n the mixing region a f t e r the bloc k , and one of decay. The decay p e r s i s t s s i n c e the flow i s not i n l o c a l e q u i l i b r i u m , but energy i s advected i n t o t h i s region from the upstream production zone. The t u r b u l e n t energy d i s t r i b u t i o n i n s p e c t r a l form downstream from the block i s shown i n Figure 50a. I t i s n o t i c e a b l e that there i s a decrease i n the energy at a freguency of 2.25 Hz f o r downstream d i s t a n c e s of x/h of 4,8 3 y / h 2 ' NO ™ = B DLOCK h 7T =12 h 1.0 1.0 1.0 t.o eb e d / e ( h ) F i g u r e 49 R a t i o of d i s s i p a t i o n at v a r i o u s heights d i v i d e d by d i s s i p a t e d r a t e at the bed showing slow change i n d i s s i p a t i o n parameter downstream from a s i n g l e block. NO BLOCK f=24 X X x •> \ X X X 0 0 0 n x 0 X *x X X 0 ° 0 ° 0 0 0 0 o \ X X X X X X X X X 0 0 . X X 0 0 X X X 0 X 0 X X o X 0 X X Cl X • X X 0 X 0 X . X X c * : o n c ' X X 3 X 0 0 X X o 0 xx • 0 0 X X 10° IO 2 10° I O 2 ' 10° IO 2 10° IO 2 10° I O 2 n ( H z ) F i g u r e 50a Energy s p e c t r a o f u at va r i o u s p o s i t i o n s i n the wake of a s i n g l e b l o c k . Measurements made at y/h=1.0. 10° l O " 2 0 I n ) IO" 4 I O " NO BAR -£-=9.(> £ = 14.2 f = 28.4 x * X v X X " X * o 0 o 0 0 X X X X X* * * 0 0 °0 0 0 o o 0 0 'x X X *x y X X * X ) >0 0 0 o 0 1 II ., p\ • *x X X X X I X-X X X X X 0 0 0 0 0 0 0 X X X X X X 0 0 0 0 0 0 X X X X X x X 0 0 X X 0 0 X x X X c c X ft X 0 X —Q- —^ ; X ; o X *x 0 0 X X •10° I0 2 10° 10 2 10° IO 2 10° IO 2 i 0 ° 102 n (Hz) Figure 5 0 b Streamwise velocity spectra at various downstream distances from a bar occupying the t o t a l channel width. Measurements made at y / h = 1 . 0 . Q 198 and 12 while by x/h=24 t h i s s m a ll anomaly has disappeared, and the spectra at t h i s d i s t a n c e and with no block are v i r t u a l l y i d e n t i c a l . (The s l i g h t upturn at the highest f r e q u e n c i e s i s almost c e r t a i n l y due to noise.) The p r o b a b i l i t y d ensity at three downstream d i s t a n c e s and without the block are presented i n Figure 51a. No pattern i s obvious and the departures from Gaussian form do not present any systematic behaviour, although there i s a s l i g h t change i n the t h i r d and f o u r t h moments of the d i s t r i b u t i o n . I t i s obvio.us that the wake a c t s to contain the t a i l s of the d i s t r i b u t i o n , suggesting that as the k u r t o s i s i s l e s s , the i n t e r m i t t e n c y i s reduced: that might be i n t e r p r e t e d as meaning th a t the b u r s t i n g from the w a l l i s r e s t r i c t e d by the increa s e d turbulence l e v e l i n the wake of the block. Before c o n s i d e r i n g the flow approaching a b l o c k , comparison i s made between the s t a t i s t i c a l s t r u c t u r e of flow and the s p e c t r a l d i s t r i b u t i o n of energy of flow i n the wake of s i n g l e blocks and s i n g l e bars. Figure 50b shows the sp e c t r a downstream of a s i n g l e bar. There are no apparent d i f f e r e n c e s among the spect r a except t h a t at the lowest f r e q u e n c i e s , below 1.65 Hz, there i s a greater c o n c e n t r a t i o n of energy than f o r the flow i n the absence of the bar. The percentage of energy below 1.65 Hz was 68, 63 and 61% r e s p e c t i v e l y f o r x/h of 3.8 14.2 and 28.4, while without the bar the value was 60%. The d i f f e r e n c e s among the p r o b a b i l i t y d e n s i t i e s f o r bars and blocks, seen i n Figure 51a and b, show that there i s a 199 - 3 - 2 - 1 0 1 2 3 STIW2S30:X0 SCCES 0.10 - i Figure 51a Probability density diagrams downstream from a single block at x/h= 4, 9, 14 and without any obstacle present. 200 0.50 --. STRNDRRDIZED SCC~.ES F i g u r e 51b P r o b a b i l i t y d e n s i t y diagrams downstream from a s i n g l e bar at x/h= 3.8, 9, 14. 201 d e f i n i t e upper l i m i t with c o n s i d e r a b l e c o n c e n t r a t i o n of events at a value of 1.7 standard d e v i a t i o n s above the mean f o r a bar at x/h of 3.8. This anomaly i s very suggestive of a dominant wake shedding v e l o c i t y . Downstream, t h i s p e r i o d i c i t y decays and the frequency d i s t r i b u t i o n s smooth out. The approach of a t u r b u l e n t boundary l a y e r flow t o a b l u f f body has r e c e n t l y been s t u d i e d by Sadeh and Cerraak (1972) w h i l s t the behaviour i n a uniform t u r b u l e n t stream has been e x t e n s i v e l y s t u d i e d by Bearraan(1974). Re s u l t s from work i n shear flows suggest that on the upstream face of the s t r u c t u r e a h i g h l y t u r b u l e n t three dimensional boundary l a y e r develops. The oncoming turbulence i s a m p l i f i e d and t h i s a m p l i f i c a t i o n depends on a c e r t a i n n e u t r a l wavelength. V o r t i c i t y of a s c a l e smaller than the n e u t r a l s c a l e i s d i s s i p a t e d more r a p i d l y while the l a r g e s c a l e v o r t i c i t y i s a m p l i f i e d by s t r e t c h i n g of the vortex f i l a m e n t s i n the d i v e r g i n g flow as i t approaches the body. The l e v e l of turbulence i n the other components i s augmented around the edges of the o b s t a c l e . The notion of v o r t i c i t y a m p l i f i c a t i o n was examined by i n s e r t i n g a probe through a block with the probe t i p at two p o s i t i o n s i n f r o n t of the block. The r e s u l t s may be seen i n Figure 52a where the sp e c t r a are overlapped. A n e u t r a l frequency appears at 22 Hz which, based on upstream v e l o c i t y at t h i s h e i g h t , would represent a len g t h of 1.2 cm. The block width was 1.6 cm and i t s height was 0.96 cm. Below 202 10° IO" 2 10 - 4 10" X 0.2 CM AHEAD OF BLOCK O INCIDENT T U R B U L E N C E X S < o o 1 Xx O X o xo -x° o x o * o * o X c X X o o Xo *°X 10 - 2 10° I O 2 n ( H z ) F i g u r e 52a C r o s s - o v e r o f s p e c t r a showing c o n c e n t r a t i o n o f energy a t f r e q u e n c i e s below the n e u t r a l f r e q u e n c y a t 20 Hz due to v o r t i c i t y a m p l i f i c a t i o n . 100 % 8 0 6 0 4 0 0 2 0 i A 4 . o i o o A {• ° r u • A ! o n ! a 3 A 0 2 CM O 1.5 CM a INCIDENT AHEAD AHEAD TURBULENCE 10 n ( H z ) 15 2 0 F i g u r e 52b T r a n s f e r o f energy t o l o w e r f r e q u e n c i e s a l o n g t h e f l o w s t a g n a t i o n l i n e . 203 t h i s n e u t r a l frequency there i s a c o n c e n t r a t i o n of energy, while above, the spectrum measured through the block drops more r a p i d l y . The accumulation of t h i s energy, being t r a n s f e r r e d from higher to lower f r e q u e n c i e s i s shown i n Figure 52b where the cumulative s p e c t r a are p l o t t e d a g a i n s t freguency up to the n e u t r a l freguency. I t shows c l e a r l y that as the block i s approached the energy i s being t r a n s f e r r e d to l a r g e r s c a l e s of motion. Furthermore the energy i n the bandwidth of 0.61 Hz centered at 1.65 Hz seems t o be e s p e c i a l l y a m p l i f i e d , i n c r e a s i n g from 9% to 17.3% of the normalized energy spectrum. Such a freguency, corresponding to s l i g h t l y greater than flow depth, could w e l l play a s i g n i f i c a n t r o l e i n r e l a t i n g small s c a l e roughness b l o c k s to l a r g e r s c a l e f e a t u r e s which apparently s c a l e as f u n c t i o n s of a flow macroscale. More importantly the i n c r e a s e i n energy below 22 Hz w i l l markedly i n f l u e n c e the f l u c t u a t i n g f o r c e s any p a r t i c l e on the boundary must withstand, and t h i s c o n c e n t r a t i o n at low freguencies i s p a r t i c u l a r l y s i g n i f i c a n t i n that low frequency f l u c t u a t i o n s are more l i k e l y t o f a c i l i t a t e p a r t i c l e i n s t a b i l i t i e s by inducing the p a r t i c l e t o v i b r a t e on the bed at these low f r e g u e n c i e s . Z i ® M i n s u l t s Measurements of the t u r b u l e n t v e l o c i t y f l u c t u a t i o n s were made at four s e c t i o n s i n small cobble bed channels i n the Squamish area. They were c a r r i e d out i n Cheekye Creek i n Paradise V a l l e y at two p o i n t s , one i n a pool (a) 0.7m deep, 80 204 ni downstream of a r i f f l e . The channel here was 20 m wide and the water surface smooth (Figure 4). A second s i t e (b) 1/2 km downstream was s e l e c t e d where the flow surface was much more d i s t u r b e d , the depth was 0.5 m and the width was 15 m. Mean v e l o c i t i e s at the- two s i t e s were 0.6 and 0.75 ms"1 r e s p e c t i v e l y . Two more s i t e s were s e l e c t e d near one another on the Hamquam River and Bridge Creek; a s m a l l t r i b u t a r y t o i t . At the measurement s i t e on the Mamguam R i v e r the width was over 40 m with depths i n excess of 2 m and hence measurement was r e s t r i c t e d to the zone c l o s e t o one bank where the flow depth was only 0.5 m. Bridge Creek was s e l e c t e d as here the flow surface i s very smooth with no surface disturbances. The bed at a l l four s i t e s was composed of cobble roughness with average diameter of 12 cm. The r o l e of the f r e e surface i n steep mountain streams i s of great importance as many streams are e i t h e r i n tumbling flow or have l a r g e standing waves, some r e l a t e d t o l a r g e l o c a l o b s t a c l e s , t h a t r e s u l t i n l a r g e r a t e s of energy d i s s i p a t i o n . The Mamguam and Cheekye(b) spectra taken at y/D=0.8 i n d i c a t e the r o l e of the f r e e s u r f a c e d i s t u r b a n c e s ( F i g u r e 53). In the former case the spectrum shows no peak whatsoever. The Reynolds number f o r t h i s flow was approximately 10 6. There appears to be an e x t e n s i v e -1 region and even a -5/3 r e g i o n . The spectrum f o r the s u r f a c e of the Cheekye (b) appears very nearly to peak but there i s no suggestion of any extensive subrange. The Reynolds number f o r t h i s flow was 3.7 x10 5. X O C H E E K Y E X M A M Q U A M o X X : o o c o X ° o X ° o o X o V x ° * x X OX X o X 0 o < X X >, ° x IO" 2 10° IO 2 IO 4 n ( H z ) Figure 53 Spectra at the f r e e surfac of two n a t u r a l channels where surface disturbances were present. 206 In order to examine the change i n s p e c t r a l shape with r e l a t i v e p o s i t i o n i n the flow f i v e spectra were c a l c u l a t e d at Cheekye (a). These r e s u l t s are presented i n Figure 54. The s p e c t r a l shapes behaved c o n s i s t e n t l y with the r e s u l t s obtained i n the l a b o r a t o r y , showing only a s m a l l v a r i a t i o n over the depth with the tendency f o r the most ext e n s i v e subrange being noted at y/D=0.7. This measurement s i t e was i n a smooth f r e e s u r f a c e area and i t may be noted that the f r e e surface does not seem to i n f l u e n c e the turbulence s t r u c t u r e even though one hundred depths upstream at a r i f f l e the flow had been i n a s u p e r c r i t i c a l s t a t e with c o n s i d e r a b l e s u r f a c e d i s t o r t i o n . The t u r b u l e n t energy i n t e n s i t i e s v a r i e d from 25% at the bed to 10% at the f r e e surface f o r Cheekye(a), s i m i l a r values being noted at Bridge Creek. The RMS s i g n a l was not s t a b l e f o r the surface measurements at Cheekye(b) or the s u r f a c e of the Mamquam: as f o r the spectra i t may be seen the g r e a t e s t c o n c e n t r a t i o n s of energy are a s s o c i a t e d with f r e q u e n c i e s equal to or g r e a t e r than the time p e r i o d averaging of the meter (Figure 53) . The r a t e s of t u r b u l e n t energy d i s s i p a t i o n were not c a l c u l a t e d , but normalized p l o t s of the d i s s i p a t i o n s p e c t r a are shown i n F i g u r e 55 f o r four depths at Ckeekye (a). The spectra a l l peak at between 30 and 40 Hz. The u and v s p e c t r a and cospectra were measured at Bridge Creek at two depths and Figure 56a shows the cospectra, adjusted to peak at f=0.1 and p l o t t e d with the Panofsky and Mares(1968) curve. 10° y/D=0.1 03 OJ? 07 09 X X X 0 o X *x X X X X x y X X 0 0 0 0 0 0 0 o 0 o o : X X X X X 0 0 0 o 0 X X X X *x X * X X X X X X X 0 0 0 .o-0 0 0 X X X X X X X ° 0 0 0 0 0 0 X X X X X X ; c < x X X X X > 0 0 0 0 X X X X > ; 0 0 0 c X X 10° 10 2 10° 10 2 10° 10° IO 2 10° !0 2 n (Hz) Figure 54 Spectra of u at f i v e r e l a t i v e depth's at Cheekye(a). - 2.4 n20(n) -3.4 V/D = O.i 03 07 Vxx x* X * X X X 0 ° 0 0 ° • 0 ' c X ***x x* X * X X * X opo0 0 ( 0 0 ° ) 0 X X < 0 0 x x x x > 0 X X 0 0 0 0 0 I 0 X D 0 0 0 0 X 0 X 0 X 0 X 0 X 10° IO2 10° IO2 10° IO2 10° 102 n (Hz) F i g u r e 55 D i s s i p a t i o n s p e c t r a f o r 4 r e l a t i v e depths f o r Cheekye(a). n 0 U V /( n) uv 1.0 0.5 0.1 .05 .01 0 . 1 0 . 3 -s- . • * •* —-""^  * V •- N *. S. — N • 1 I I • V . 1,ILL - D A T A FROM PANOFSKY AMD M A R £ S (1965) i l i t .1 in I t i t \ fS V V 1 1 M i * - \ t *. \ i ' .005 .01 .05 0.1 . 0 .5 1.0 f Figure 5 6 a Normalized cospectra f o r two ' depths i n Bridge Creek compared w i t h the curve from Panofsky and Mares ( 1 9 6 8 ) . 10° 1 0 " o • ° * X 0 * xo ' X o > c 3 X 0 ° - ° 0 x 0.1 o 0 . 3 x ° X o iCT' 10° IO1 10 2 n (Hz) Figure 56b S p e c t r a l c o r r e l a t i o n c o e f f i c i e n t f o r two depths i n Bridge Creek. 210 The curves are both more peaked than the curve of Panofsky and Mares, a r e s u l t which agrees with the measurements from the l a b o r a t o r y and from Shaw e t . a l (1974). The f a c t that a t y/D=0.1, w i t h i n the roughness elements, the curve i s more peaked than at y/D=0.3 suggests t h a t the dominant p a r t i c l e s on the bed are spaced at a density lower than 1/8 which was. seen as the break point f o r c o s p e c t r a l measures: at higher d e n s i t i e s the more peaked curve would occur at a point above the roughness array. The s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t was c a l c u l a t e d and i s shown i n Figure 56b. S i m i l a r to the measurements made i n the l a b o r a t o r y , the c o r r e l a t i o n drops below 0.2 at 10 Hz, The quadrature spectrum was a l s o a decade smaller, than the cospectrura and thus made v i r t u a l l y no c o n t r i b u t i o n to the coherence. Ten hertz may then be taken as the highest freguency at which s i g n i f i c a n t energy exchange takes place between the components. The phase i n a l l cases was c l o s e to 180°: at most frequencies v laqged u while f o r the measurements made at the su r f a c e the v component always lagged the u f l u c t u a t i o n s . The p r o b a b i l i t y d e n s i t i e s f o r three p o i n t s are shown i n Figure 57. The most obvious f e a t u r e i s the change i n skewness f o r the point c l o s e to the surfa c e and the excess of values around the mean. Such a p a t t e r n could be w e l l modelled by an exponential d i s t r i b u t i o n as pos t u l a t e d by Dutton (1969) whereby the f l u c t u a t i o n s i n the patches of turbulence are surrounded by areas with an absence of Figure 57 P r o b a b i l i t y d e n s i t y diagrams f o r Cheekye(a) at three r e l a t i v e depths, y/D= 0.1, 0.3 and 0.9 r e s p e c t i v e l y . 212 turbulence, a s i t u a t i o n that would be r e a l i z e d i f the s u r f a c e turbulence i s brought about by waves breaking upstream. The moments of the f l u c t u a t i o n s showed skewness values of -0.13 t o -0-47 and k u r t o s i s values of 2.91 to 3.60 f o r flows with smooth water s u r f a c e s . Such values i n d i c a t e that the summary moments may a l s o be adequately portrayed i n l a b o r a t o r y experiments. Surface waves have the e f f e c t of a l t e r i n g the skewness so that i t e g u a l l e d 0.5 while the k u r t o s i s rose to 4 i n d i c a t i n g a dominant r o l e being played by i n t e r m i t t e n t processes or the i n f l u e n c e of p e r i o d i c waves breaking upstream of the measurement s i t e and the decaying turbulence being most s i g n i f i c a n t i n the f r e e surface measurements. The constancy of the r e s c a l e d range c o e f f i c i e n t f o r the three r e l a t i v e depths was s i m i l a r to that found i n the l a b o r a t o r y , while i t s value, H=0.85 i s a l s o c l o s e to that found i n l a b o r a t o r y measurements, i n d i c a t i n g t h a t t h i s s t a t i s t i c a l c h a r a c t e r i s t i c of the flow i s w e l l modelled by flume experiments. Nordin et al(1972) i n d i c a t e d a systematic i n c r e a s e i n H with the s i z e of the channel, c o v e r i n g a range from a 20 cm flume up to measurements i n the M i s s i s s i p p i ; the l i m i t e d s c a l e here from a 48 cm flume to a channel 25 m wide d i d not show such behaviour. The c o n c l u s i o n s that may be drawn from t h i s study of n a t u r a l channel measurements are that the flow i n the absence of surface waves may be adeguately modelled i n the flume. The s p e c t r a l peak i s p a r t l y i nfluenced by r e l a t i v e p o s i t i o n i n the flow but seems unaffected by the width s c a l e of the 2 13 flow. As s t r a i g h t channel reaches were d e l i b e r a t e l y chosen the r o l e of width should have been h o p e f u l l y e l i m i n a t e d but i n sinuous channels the macroscale might w e l l be i n f l u e n c e d by the c r o s s s e c t i o n a l shape of the channel and, as a r e s u l t , the macroscale might be r e l a t e d to the channel width and depth. 214 CHAPTER 5 TURBULENCE AND BED STABILITY IM1ERACTION,IMPLICATIONS AND DISCUSSION Illtroduc t i o n In h i s study of the motion of coarse p a r t i c l e s along a stream bed, F r a n c i s {1 973) asserted there were three areas of l i t e r a t u r e concerned with sediment motion. F i r s t , he d i s t i n g u i s h e d that corpus of work which seeks a c o r r e l a t i o n between t o t a l s o l i d s flow and the c h a r a c t e r i s t i c s of the f l u i d stream. Second he i d e n t i f i e d t h a t which examines the formation and e f f e c t s of r i p p l e s , dunes and other n o n - u n i f o r m i t i e s at the bed. T h i r d , he d i s c r i m i n a t e d t h a t which aims to determine the t h r e s h o l d c o n d i t i o n of movement. While such a d i v i s i o n may be u s e f u l , the study of the movement of s o l i d p a r t i c l e s by a f l u i d stream has the o v e r a l l o b j e c t i v e of i n t e g r a t i n g models of the separate processes i n these three groups. Such models are o f t e n a mixture of s t o c h a s t i c processes and d e t e r m i n i s t i c q u a n t i t i e s d e l i m i t i n g the boundary of a p p l i c a t i o n . The determination of those parameters i n such processes and t h e i r r e l a t i o n to the flow dynamics i s a primary o b j e c t i v e of t h i s chapter, and indeed of many s t u d i e s of channel processes (cf- Nordin and Richardson 1967: Richardson and Sayre 1967). The mechanics of the processes r e p o n s i b l e f o r sediment 215 entrainraent and t r a n s p o r t must depend on the tu r b u l e n c e s t r u c t u r e . In order to e f f e c t the t r a n s f e r of momentum and sediment, a f l u c t u a t i n g q u a n t i t y , the Reynolds s t r e s s , i s req u i r e d which w i l l r e s u l t i n a p e r s i s t e n t l o c a l c a p a c i t y t o do work and modify the channel boundary. The r e s u l t s obtained i n t h i s t h e s i s are used i n an examination of the problem of sediment entrainraent (Group 3 of F r a n c i s * l i t e r a t u r e ) . The S h i e l d s curve i s p a r t i t i o n e d i n t o t h r e e regions and the measurements made here are discussed i n r e l a t i o n to the l a r g e p a r t i c l e Reynolds number area of the curve. The r o l e of p a r t i c l e spacing, p a r t i c l e i n s t a b i l i t y and v i b r a t i o n are discussed which leads t o a statement of the dynamical d i f f e r e n c e s between sand and g r a v e l channels. The i m p l i c a t i o n s of t h i s are drawn out i n the development of a s t a t i s t i c a l s t a b i l i t y model of entrainraent of coarse p a r t i c l e s which i n c o r p o r a t e s the r o l e of v o r t i c i t y a m p l i f i c a t i o n and flow induced e x c i t a t i o n r e s u l t i n g i n bed i n s t a b i l i t y . The motion of p a r t i c l e s , once e n t r a i n e d , i s determined by the turbulence s t r u c t u r e . The many s t u d i e s of sediment t r a n s p o r t have r a r e l y incorporated measures of tur b u l e n c e , with the notable exceptions of K a l i n s k e (1947) and Ei n s t e i n ( 1 9 5 0 ) . These t h e o r i e s are discussed and the relevance of the measurements made here pointed out, with p a r t i c u l a r r e f e r e n c e to the 'hiding f a c t o r ' , the p r o b a b i l i s t i c s t r u c t u r e and the turbulence i n t e n s i t y measures. Sediment t r a n s p o r t i s f r e q u e n t l y s p l i t i n t o 216 bedload movement and suspended l o a d movement with s a l t a t i o n being a l i n k . The r e s u l t s of t h i s t h e s i s are a p p l i e d t o a c o n s i d e r a t i o n of observations made i n the f i e l d and i n other flume s t u d i e s (Group 1 of F r a n c i s * l i t e r a t u r e . ) The e f f e c t s of no n - u n i f o r m i t i e s of a r i v e r channel bed and banks have r e s u l t e d i n a myriad of morphological s t u d i e s of such f e a t u r e s . The existence of a hierarch y of f e a t u r e s and the d i f f e r e n c e s i n such f e a t u r e s between sand and g r a v e l channels i s dis c u s s e d . The r o l e of dominant roughness spacing, of v o r t i c i t y a m p l i f i c a t i o n and the use of the t u r b u l e n t macroscale as a s i g n i f i c a n t dynamic length s c a l e are a l l considered (Group 2 of F r a n c i s 1 l i t e r a t u r e ) . The primary o b j e c t i v e of t h i s d i s c u r s i v e chapter i s to attempt t o ' i n t e g r a t e these areas of research and s t r e s s the importance of s c a l i n g lengths and turbulence and t h e i r i n t e r a c t i o n with a modifiable boundary. FLOS STRUCTURE AND INSTABILITY Experimenta1 r e s u l t s The experimental r e s u l t s from flow over the roughness elements i n d i c a t e d c e r t a i n s t r u c t u r a l flow f e a t u r e s of importance to sediment entrainraent and t r a n s p o r t . The presence of three zones of fl o w , a w a l l l a y e r , a wake zone and an outer v e l o c i t y defect region i m p l i e s t h a t simple d i f f u s i o n models f o r suspended sediment and d i s p e r s i o n w i l l be i n a c c u r a t e as such models assume that a s i n g l e l o g a r i t h m i c law a p p l i e s throughout the flow. The three zones were a l s o 217 c h a r a c t e r i z e d by d i f f e r e n t r a t i o s of production and d i s s i p a t i o n : the wake l a y e r was seen to export energy towards the f r e e s u r f a c e , which would have considerable import i n t r a n s p o r t i n g suspended sediment. The measurements of turbulence i n t e n s i t y showed that at the 'optimum' spacing of roughness elements, the turbulence i n t e n s i t y at the bed reached 30% while f o r d e n s i t i e s 1/8 and 1/80 the i n t e n s i t y was only 20%. Concomitantly l o c a l d i s s i p a t i o n r a t e s were very high f o r density 1/12, and f o r d e n s i t i e s greater than 1/16 the 'bed' had s h i f t e d t o the t o p of the roughness elements. The s p e c t r a l measurements showed a c o n c e n t r a t i o n of energy at low fre q u e n c i e s , and no s i g n i f i c a n t t r a n s f e r s of momentum, s i g n i f i e d by the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t , above 10 Hz. Other measurements presented i n the l i t e r a t u r e (Raichlen 1967: Bouvard and Pet k o v i c 1973) r a r e l y go down to 1 Hz, although McQuivey's (1967) r e s u l t s give s p e c t r a l d e n s i t i e s down to 0.1 Hz. The measures i n t h i s research reach down to 0.02 Hz by which time the s p e c t r a l energy d e n s i t y i s g e n r a l l y down by two decades i n d i c a t i n g that the s i g n i f i c a n t peak of the energy spectrum has been obtained. An increased c o n c e n t r a t i o n of Reynolds s t r e s s at the peak frequency was observed f o r a l l d e n s i t i e s , even at d e n s i t i e s as low as 1/80. The p r o b a b i l i s t i c a n a l y s i s of the v e l o c i t y f l u c t u a t i o n s showed that u and v were both s i g n i f i c a n t l y non-Gaussian. The s t u d i e s of zero c r o s s i n g s and the d i s t r i b u t i o n s of both 218 negative and p o s i t i v e peaks i n d i c a t e d t h a t i n t e r m i t t e n c y was present, with an excess of values greater than three standard d e v i a t i o n s above the mean and an excess of values c l o s e to the mean. This was i n t e r p r e t e d by Dutton (1969) as suggesting an e x p o n e n t i a l d i s t r i b u t i o n of events. The c o r r e l a t i o n of d i s s i p a t i o n p e r i o d s , portrayed i n the r e s c a l e d range by the break i n slope at approximately one second, suggested the co u p l i n g of macroscale and microscale i n s t a b i l i t i e s . The s t r u c t u r e of the flow approaching a s i n g l e b l u f f body was found t o e x h i b i t v o r t i c i t y a m p l i f i c a t i o n with s i g n i f i c a n t energy t r a n s f e r t.o below the n e u t r a l frequency, which depended on block width. Furthermore, the p r e f e r e n t i a l a m p l i f i c a t i o n of one wavelength led to the r e a l i z a t i o n t h a t such a frequency might be s i g n i f i c a n t i n a l t e r i n g the shear at the boundary. Measurements oyer g r a v e l To t h i s end a separate set of measurements was c a r r i e d out to i n v e s t i g a t e the entrainment of g r a v e l . I t had been noted, by B i s a l and Nielsen(1962) working with s o i l p a r t i c l e s i n a i r , that p r i o r to entrainment they v i b r a t e d : as the windspeed increased the p a r t i c l e v i b r a t i o n increased u n t i l they l e f t the surface as i f e j e c t e d . Work c a r r i e d out i n a wind tunnel by Iyles(1970) a l s o produced t h i s phenomenon and i n d i c a t e d that the frequency of v i b r a t i o n c o r r e l a t e d w e l l with the flow macroscale. For t h i s experiment the flume bed was covered with a l a y e r of angular g r a v e l , sieved so that 219 the sediment was narrowly graded about a mean s i z e of 30 mm, and uniformly packed on the bed. Th i s experiment was performed at much higher v e l o c i t i e s than the flow over the d i s t r i b u t e d roughness elements so that d i r e c t comparison of r e s u l t s was not p o s s i b l e . As the flow v e l o c i t y was increased a few p a r t i c l e s s t a r t e d to v i b r a t e i n t e r m i t t e n t l y at i s o l a t e d places on the bed. As the v e l o c i t y i n c r e a s e d the number of p a r t i c l e s v i b r a t i n g was observed to i n c r e a s e but the freguency of v i b r a t i o n d i d not appear to a l t e r . When the v e l o c i t y was . inc r e a s e d again one or more of the v i b r a t i n g p a r t i c l e s became unstable and s t a r t e d to r o l l downstream u n t i l trapped against another p a r t i c l e . The experiment was s a t i s f a c t o r i l y r epeatable. S p a t i a l v a r i a t i o n of i n s t a b i l i t i e s The sporadic nature of entrainment, i t s i n t e r m i t t e n t time s t r u c t u r e and seemingly random s p a t i a l occurrence, i n d i c a t e s that simple Gaussian models of v e l o c i t y , s t r e s s and pressure f l u c t u a t i o n s are u n l i k e l y t o be of great value i n c o n s t r u c t i n g accurate p h y s i c a l models of sediment entrainment and t r a n s p o r t . The coup l i n g of the i n t e r m i t t e n t s m a l l s c a l e flow f e a t u r e s with the flow macroscale (see Offen and K l i n e 1973) suggests t h a t entrainment i t s e l f might b e t t e r be viewed as an i n s t a b i l i t y mechanism. The wake shedding freguency, v o r t i c i t y a m p l i f i c a t i o n freguency range, the flow macroscale and f l u c t u a t i o n s i n the d i s s i p a t i o n r a t e might be conceived as l e n g t h s c a l e s of p o t e n t i a l i n s t a b i l i t i e s . At a l a r g e r 220 s c a l e , the non-uniformity of the channel boundaries, the presence of secondary c u r r e n t s and the p e r s i s t e n c e of low frequency i n t e r n a l flow f l u c t u a t i o n s generated by channel curvature might a l s o be viewed as p o t e n t i a l i n s t a b i l i t y wavelengths. Hydrodynamic i n s t a b i l i t y i s t r a d i t i o n a l l y a s s o c i a t e d with the t r a n s i t i o n of flow from a laminar to a t u r b u l e n t regime. Recent work (Mollo-Christensen 1970; Kim et a l 1971) suggests that i n s t a b i l i t y i n t e r a c t i o n between v a r i o u s s c a l e s of motion i n the flow i s of paramount importance at a l l values of the Reynolds number. The i n t e r a c t i o n s are by nature i n t e r m i t t e n t : by examination of the Reynolds eguation they may a l s o be seen to be n o n - l i n e a r . The i n s t a b i l i t i e s r e s u l t from the i n t e r a c t i o n of events of d i f f e r e n t wavelength. The t r a d i t i o n a l view i s t h a t a t u r b u l e n t boundary l a y e r i s i n s e n s i t i v e to p e r t u r b a t i o n s because the p r o p e r t i e s of boundary layers, are seen t o be i n s e n s i t i v e to l a r g e v a r i a t i o n s i n Reynolds number. However, the Reynolds number i s a l s o a measure of the r a t i o of macro- to micro-scale: the greater the r a t i o the l a r g e r the number of degrees of freedom, and the greater the number of i n t e r v e n i n g s c a l e s i n the h i e r a r c h y . I f s i m i l a r i t y of l e n g t h s c a l e s f o r i n s t a b i l i t i e s i s assumed, that i s ( X l / X J = A H/X N > )-ct ) then the r a t i o of macro- to micro-scale w i l l be L/ n ^X^/Xi-ct^ so N ~ log Re. For most n a t u r a l channels the Reynolds number w i l l be approximately 10 s suggesting the e x i s t e n c e of f i v e p r i n c i p a l l e v e l s i n the h i e r a r c h y of i n s t a b i l i t y which might 221 s i m i l a r l y be found i n a morphological h i e r a r c h y of f e a t u r e s i n a channel- They could range i n s c a l e from a 'dominant' p a r t i c l e s i z e d i s t r i b u t i o n on the bed t o primary bedforms. The r e s u l t s of t h i s research are germane to the s m a l l e s t s c a l e s of i n s t a b i l i t y , notably i n d i v i d u a l p a r t i c l e s and hence t o sediment entrainraent and d e p o s i t i o n , which c o n s t i t u t e the ba s i c event by which the r i v e r modifies i t s form. The freguency s t r u c t u r e , s p a t i a l s c a l e s of momentum t r a n s f e r and p r o b a b i l i t y s t r u c t u r e have been examined l e a d i n g to the p i c t u r e of a h i e r a r c h y of i n t e r m i t t e n t processes, c r u d e l y Gaussian, with major energy content below 10 Hz and momentum t r a n s f e r below 5 Hz. The r e s u l t s of the r e s c a l e d range a n a l y s i s were i n t e r p r e t e d i n terms of the s u p e r i m p o s i t i o n of s e v e r a l broken l i n e processes which maintained a f i n i t e variance and a v a r i a b l e skewness. T h i s s t a t i s t i c a l s u p e r i m p o s i t i o n of processes with v a r y i n g epoch lengths c o i n c i d e s with K o l l o - C h r i s t e n s e n ' s p h y s i c a l i n t e r p r e t a t i o n of a hi e r a r c h y of i n s t a b i l i t i e s {see a l s o L i g h t h i l l 1969), each with a s i m i l a r i t y of wavelength r a t i o which would maintain a uniform slope f o r the r e s c a l e d range over very long record l e n g t h s . In applying the r e s u l t s of t h i s research to the problem of sediment movement i t i s necesssary to assume that the measurements made i n a c l e a r l i q u i d are s t i l l v a l i d when c o n s i d e r i n g a f l u i d with a r e l a t i v e l y s m a l l c o n c e n t r a t i o n of p a r t i c l e s i n motion i n the f l u i d and on the boundary. The e f f e c t s of d i s c r e t e p a r t i c l e s on t u r b u l e n t flow was s t u d i e d 222 by Hino (1963) and Bouvard and Petkovic (1973). Hino found that there was an increase i n the v e l o c i t y g r adient c l o s e t o the bed and a decrease i n the von Kantian constant. However he quoted work by E l a t a and Ippen as showing that von Karraan's constant decreased. I t was accompanied by an in c r e a s e i n the turbulence I n t e n s i t y . Contrasted with t h i s are the r e s u l t s of Bouvard and Petkovic(1973) who found a decrease i n the t u r b u l e n t i n t e n s i t y by as much as 15%. Saffman {1962) suggested that the r e l a t i v e motion of p a r t i c l e and f l u i d damps turbulence and diminishes the t r a n s p o r t of momentum. I t may a l s o be appreciated t h a t the c o n c e n t r a t i o n gradient of suspended m a t e r i a l should r e s u l t i n a s t a b l e flow s i t u a t i o n with a parameter s i m i l a r to the Richardson number expressing the tendency f o r damping mechanical turbulence production (cf. Monin and Yaglora 1971, p.412). However, the measurements of s p e c t r a l energy content by Bouvard and Petk o v i c (1973) showed no d i f f e r e n c e i n s p e c t r a l shape with and without sediment above 25 Hz, although t h e i r s p e c t r a showed some divergence below t h i s l e v e l . But, the lowest measured frequency was only '3 Hz. The c o n f l i c t i n g evidence (an Increase i n v e l o c i t y g r a d i e n t i s u s u a l l y a s s o c i a t e d with an i n c r e a s e i n turbulence production) of Hino, and Bouvard and P e t k o v i c suggests that the assumption of s i m i l a r i t y of turbulence s t r u c t u r e may not be s i g n i f i c a n t to the o v e r a l l c o n s i d e r a t i o n of f l u c t u a t i o n s at the boundary of n a t u r a l channels. 223 SEDIMENT ENTSAINME_NT S h i e l d s curve S h i e l d s (1936) presented the c l a s s i c a l work on the t r a c t i v e f o r c e necessary to i n i t i a t e the motion of sediment p a r t i c l e s . He presented h i s r e s u l t s i n the form s (53) where p i s sediment d e n s i t y , d i s p a r t i c l e diameter and s T i s the c r i t i c a l shear s t r e s s exerted on the p a r t i c l e t o c i n i t i a t e motion (Figure 58b) . For l a r g e values of u^  d/ v the l e f t hand s i d e of equation (1) assumes a constant value. This i s not s u r p r i s i n g , f o r the t h i c k n e s s of the viscous sublayer i s given by 6'=11.6 v / u s , and f o r p a r t i c l e s with d> s'the viscous sublayer w i l l not play a s i g n i f i c a n t r o l e . The r e s i s t a n c e to flow w i l l then become dominated by the wake shedding from p a r t i c l e s on the boundary. In the domain 3.5<u d/v <11.6 the entrainment f u n c t i o n passes through a minimum, suggesting that here the c r i t i c a l w a l l s t r e s s e s w i l l be a minimum to e n t r a i n c e r t a i n p a r t i c l e s . Such a phenomenon might be r e l a t e d t o the b u r s t i n g of p a r c e l s of f l u i d and sediment away from the boundary f o l l o w i n g the i n r u s h of high momentum f l u i d . These b u r s t s , which are u s u a l l y randomly spaced, might become l o c a l l y f i x e d when p a r t i c l e s i n the s i z e range of f i n e to medium sand act to d i s t o r t the sub l a y e r , thus i n c r e a s i n g the p r o b a b i l i t y of the sediment being e n t r a i n e d . I t has been pointed out by Wallace et a l ( 1 9 7 2 ) , 224 F i g u r e 58 F l a t n e s s f a c t o r s of v e l o c i t y a i t s d e r i v a t i v e and second d e r i v a t i v e p l o t t e d a g a i n s t nondimensional d i s t a n c e . 585 S h i e l d s entrainment f u n c t i o n b p l o t t e d a g a i n s t p a r t i c l e Reynolds number. F i g u r e F i g u r e a b A f t e r Ueda and Hinze 1975 A f t e r Y a l i n 1972. D here r e p r e s e n t s p a r t i c l e diameter. 225 and from the r e s u l t s on the r e s c a l e d range a n a l y s i s here, that the p e r i o d i c i t y of these sweeps and bursts c o r r e l a t e s with the flow macroscale, s t r o n g l y suggesting a c o u p l i n g between the outer and inner flow and between an i n t e r m i t t e n t phenomenon and a continuous macroscale flow v a r i a b l e - The b u r s t s have been i n v e s t i g a t e d i n r e l a t i o n t o sand movement by Sutherland (1967) and by Grass (1970) but only i n the domain where the entrainraent f u n c t i o n i s a l i n e a r f u n c t i o n of p a r t i c l e Reynolds number, th a t i s where d/ v i s below 3.5. This means t h a t the sand s i z e s were l e s s than 1/3 of the v i s c o u s sublayer t h i c k n e s s , and thus the r e s u l t t h a t x showed l i t t l e v a r i a t i o n over a t w o - f o l d range of sand s i z e c (Grass 1970 p.629) was not s u r p r i s i n g . A l l the values of u d / v w e r e l e s s than three and thus entrainraent would depend on b u r s t i n g freguency and on the t h i c k n e s s of the v i s c o u s sublayer, which would not be i n f l u e n c e d by p a r t i c l e s i z e s t h a t ranged only between 0.15 and 0.3 of the boundary l a y e r t h i c k n e s s . P a r t i c l e i n s t a b i l i t y and v i b r a t i o n The l a r g e r g r a i n s i z e s , g r a v e l and l a r g e r , t o t a l l y d i s r u p t the viscous sublayer. The shear f o r c e necessary to e n t r a i n such p a r t i c l e s w i l l now be a f u n c t i o n of the mean v e l o c i t y and of the frequency and s c a l e of the f l u c t u a t i o n s which could induce p a r t i c l e s to become unstable. For the entrainraent of q r a v e l the t h r e s h o l d of movement i s d i f f i c u l t to d e f i n e . .There are very few observations on such 226 i n i t i a t i o n but from measurements of coarse p a r t i c l e movement i n wind tunnels (Lyles and Woodruff 1971) and from flume s t u d i e s c a r r i e d out i n t h i s r esearch, where an angular g r a v e l was placed on the flume bed, entrainment was seen to be very sporadic. P r i o r to i n i t i a t i o n of motion the p a r t i c l e s would v i b r a t e i n t e r m i t t e n t l y at c e r t a i n spots on the bed. When the v e l o c i t y increased the numbers of p a r t i c l e s v i b r a t i n g i ncreased but the frequency of v i b r a t i o n d i d not appear t o a l t e r . When the v e l o c i t y was again increased a few of the v i b r a t i n g p a r t i c l e s would r o l l or s l i d e from t h e i r p o s i t i o n i n d i c a t i n g that the c r i t i c a l entrainment v e l o c i t y had been' reached. The f e a t u r e s of random impingement of c r i t i c a l s t r e s s e s or c r i t i c a l frequencies on the boundary i s i d e n t i c a l with that noted f o r sand beds where the b u r s t i n g and sweep c y c l e lead t o random l o c a t i o n s of i n i t a l entrainment p r i o r to the t o t a l bed deforming or becoming l i v e . Grass (1971) made observations over 9 mm gravel at u d / v of 85 which would be i n the t r a n s i t i o n a l range of S h i e l d ' s entrainment curve. He concluded that entrainment of f l u i d was extremely v i o l e n t i n the rough boundary case with e j e c t e d f l u i d r i s i n g almost v e r t i c a l l y from between the i n t e r s t i c e s of the roughness elements. The long t w i s t i n g streamwise v o r t i c e s were much l e s s conspicuous (than i n the 2 mm sand case). I t t h e r e f o r e appears p o s s i b l e that d i f f e r e n t dominant modes of i n s t a b i l i t y might p r e v a i l f o r 227 d i f f e r i n g boundary roughness c o n d i t i o n s . (Grass 1971, p.252) The v i b r a t i o n of p a r t i c l e s p r i o r to entrainment would seem to i n d i c a t e a strong i n t e r a c t i o n between flow freguency and a ' p a r t i c l e i n s t a b i l i t y freguency'. For g r a v e l t h i s leads to the proposal of the f o l l o w i n g entrainment mechanism. R e l a t i v e l y l a r g e p a r t i c l e s d i s t r i b u t e d over the bed, which dominate the s t r e s s d i s t r i b u t i o n and might be considered 'dominant' p a r t i c l e s , concentrate energy at low fr e q u e n c i e s by the mechanism of v o r t i c i t y a m p l i f i c a t i o n . In p a r t i c u l a r one freguency i s p r e f e r e n t i a l l y a m p l i f i e d . The v e l o c i t y f l u c t u a t i o n s , should they be sympathetic to the p a r t i c l e ' s n a t u r a l frequency, w i l l induce the p a r t i c l e to v i b r a t e . The maqnitude of t h i s v i b r a t i o n may i t s e l f be s u f f i c i e n t t o r e s u l t i n entrainment. Moreover the v i b r a t i o n of the p a r t i c l e miqht r e s u l t , by a flow-induced feedback, i n an i n c r e a s e i n amplitude of the v i b r a t i o n which w i l l tend t o r e s u l t i n a m p l i f i c a t i o n of th a t frequency i n the wake shedding. Downstream from t h i s p a r t i c l e the wake shedding zone w i l l then c o n t a i n a dominant flow frequency which may r e s u l t i n the s e l e c t i v e erosion of c e r t a i n p a r t i c l e s i z e s . The evidence f o r t h i s hypothesis a r i s e s from the measurements of v o r t i c i t y a m p l i f i c a t i o n through a s i n g l e block and the v i b r a t i o n of g r a v e l p a r t i c l e s observed p r i o r to entrainment. This idea suggests that there w i l l be s i g n i f i c a n t l y d i f f e r e n t entrainment c r i t e r i a f o r sand and g r a v e l bed 228 channels. Entrainment formulae and sediment t r a n s p o r t formulae t h a t attempt to encompass a l l s i z e s of sediment would thus seem to be improperly formulated - i n terms of the processes operating at the boundary that are r e s p o n s i b l e f o r entrainment. Phenomenological d e s c r i p t i o n of S h i e l d s curve The preceding d i s c u s s i o n of entrainment suggests t h a t the S h i e l d s curve may be p a r t i t i o n e d i n t o three r e g i o n s . The dominant processes r e s p o n s i b l e f o r entrainment are d i f f e r e n t i n the three areas and a phenomenological e x p l a n a t i o n of t h i s may be of value i n e x p l a i n i n g the d i f f e r i n g modes of entrainment. In the r e g i o n u' d / v <3 the boundary i s hydrodynamically smooth. P a r t i c l e s of l e s s than 1 mm (medium and f i n e sand) w i l l exert no i n f l u e n c e on the o v e r a l l flow s t r u c t u r e . The r e l a t i o n of the flow macroscale to b u r s t i n g from the w a l l w i l l be determined by a flow i n s t a b i l i t y at the top of the v i s c o u s sublayer. As the burst moves up i n t o the flow the pressure f i e l d w i t h i n i t has been shown (Mollo-Christensen 1970) to s c a l e with the bursts and not with the f l u c t u a t i o n s w i t h i n the burst. Hence c o n s i d e r a b l e entrainment could be f a c i l i t a t e d . In the range 3<u ^ d / v <50 the shear s t r e s s f o r entrainment reaches a minimum. This would appear to be r e l a t e d to the t h i c k n e s s of the viscous sublayer. P a r t i c l e s may o c c a s i o n a l l y d i s r u p t the sublayer even at only 1/3 of i t s 229 average thickness f o r i t i s not a uniform sheet but a c o ntorted wavelike boundary zone. The i n s t a b i l i t i e s may tend to become f i x e d by a few p a r t i c l e s whose s c a l e might d i s t o r t the b u f f e r l a y e r and sublayer s u f f i c i e n t l y to i n c r e a s e and concentrate the flow i n s t a b i l i t i e s . As u sd/v gets l a r g e the viscous sublayer i s no l o n g e r s i g n i f i c a n t . L o c a l flow i n s t a b i l i t i e s r e l a t e d to wake shedding, or v o r t i c i t y a m p l i f i c a t i o n r e s u l t i n g i n a c o n c e n t r a t i o n of v o r t i c i t y at c e r t a i n s c a l e s , w i l l dominate the generation of turbulence. Thus the shear v e l o c i t y r e q u i r e d to e n t r a i n a p a r t i c l e would be p r o p o r t i o n a l t o some power of the p a r t i c l e diameter i n accord with the e a r l i e s t s i x t h power r e l a t i o n s proposed by Brahms, and summarized i n L e l i a v s k y (1959) , between c r i t i c a l v e l o c i t y and p a r t i c l e volume. Dominant p a r t i c l e d i s t r i b u t i o n , v i b r a t i o n and v o r t i c i t y a m p l i f i c a t i o n The concept of flow induced v i b r a t i o n s i s common i n a r c h i t e c t u r a l aerodynamics where wake shedding and the i n t e r a c t i o n between wakes and s t r u c t u r e s i s now i n t e n s i v e l y s t u d i e d (cf. Syrap. Arch. Aerodynamics 1971). I t would a l s o seem to be d i r e c t l y a p p l i c a b l e to open channels with g r a v e l beds. The d i s t r i b u t i o n of p a r t i c l e s i z e s on g r a v e l beds has not been i n v e s t i g a t e d with t h i s approach i n mind, though when attempting to d e s c r i b e p a r t i c l e s i z e f o r coarse m a t e r i a l f o r i n c l u s i o n i n any sediment t r a n s p o r t eguation the 230 s i z e D i s f r e q u e n t l y used to d e s c r i b e the most s i g n i f i c a n t 90 length s c a l e of the boundary. (D i s defined as t h a t s i z e of 90 p a r t i c l e i n a cumulative g r a i n s i z e d i s t r i b u t i o n than which 90% of the p a r t i c l e s are f i n e r ) . Such a l a r g e s i z e i n d i c a t e s that f o r a mixture of p a r t i c l e s a few g r a i n s might be s i g n i f i c a n t i n determining the shear s t r e s s , turbulence s t r u c t u r e and the entrainment c r i t e r i a . This idea was p o s t u l a t e d by White (1940) who proposed tha t the t r a c t i v e f o r c e was d i s t r i b u t e d among the g r a i n s t h a t r e s t e d on top of the bed and from p h y s i c a l reasoning concluded that a packing of approximately 1/8 based on plan areas would d e s c r i b e the d i s t r i b u t i o n of prominent g r a i n s . His experimental observations on exposed g r a i n s gave c l o s e agreement with t h i s value. In t h i s research the d e n s i t y 1/12 f o r angular blocks was shown to be that which presented the g r e a t e s t r e s i s t a n c e t o flow. The grouping of values D and 90 packing of 1/8 f o r smooth pebbles and 1/12 f o r angular b l o c k s suggests t h a t the spacing of elements on the boundary i s h i g h l y s i g n i f i c a n t i n determining the c a p a c i t y of the stream to modify i t s boundary. By removing a few l a r g e r p a r t i c l e s the whole bed would be subjected to a g r e a t e r shear and more entrainment could occur. This packing density would thus seem to represent t h a t s t a t e when the bed i s i n most favourable e q u i l i b r i u m with the imposed f o r c e of the g r a v i t y d r i v e n flow. This leads to the highest r a t e s of t u r b u l e n t energy d i s s i p a t i o n and the raimimura m o d i f i c a t i o n of the channel boundary. 231 Such a c o n c l u s i o n would seem to imply t h a t there should be a c l e a r l y v i s i b l e r e g u l a r spacing, of elements on the bed at a d e n s i t y of approximately 1/12. Such cannot be the case or i t would have been commented upon long ago. R e s u l t s o f M a r s h a l l (1971) show that shear s t r e s s measurements f o r a r r a y s of randomly d i s t r i b u t e d elements and of r e g u l a r l y spaced elements with the same element d e n s i t y are not s i g n i f i c a n t l y d i f f e r e n t . In the extremes elements p a r a l l e l to the flow and at the same density elements normal to the flow show a n e a r l y f o u r f o l d v a r i a t i o n i n shear s t r e s s , but the d i f f e r e n c e between a random and r e g u l a r spacing i s only 8%. Thus i t appears that the spacing of the elements at the c r i t i c a l d e n s i t y may be non-regular and s t i l l exert a s i r a i l a r i n f l u e n c e on the flow s t r u c t u r e . The measurement of energy content at v a r i o u s f r e g u e n c i e s i s s i g n i f i c a n t with respect to the s i z e s of the elements t h a t may be entrained i n the flow. The c o n c e n t r a t i o n of energy a t low frequencies, by v o r t i c i t y a m p l i f i c a t i o n around g r a i n s of a c e r t a i n s i z e , w i l l r e s u l t i n coherent f o r c e s o p e r a t i n g over the t o t a l s u r f a c e of smaller g r a i n s . Such coherent f o r c e s are l i a b l e t o produce i n s t a b i l i t y of the p a r t i c l e s . The c o s p e c t r a l p l o t s of Reynolds s t r e s s showed a c o n c e n t r a t i o n of energy at the peak greater than that expected f o r rough boundary l a y e r s and a s i g n i f i c a n t c o n c e n t r a t i o n of s t r e s s a t the very low frequencies. Both of these c o n d i t i o n s would promote erosion of smaller p a r t i c l e s . The magnitude of the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t out to 5 Hz suggests that 232 c o n s i d e r a b l e momentum t r a n s f e r occurs which could support g r a i n s once they have been e n t r a i n e d . P r o b a b i l i s t i c flow s t r u c t u r e and entrainraent L a s t l y i n c o n s i d e r i n g sediment entrainment, the p r o b a b i l i s t i c s t r u c t u r e r e s u l t s must be considered. The e a r l i e s t study advocating a s t a t i s t i c a l approach to sediment entxainment was by E i n s t e i n (1937). He considered the dependence of the p r o b a b i l i t i e s of movement or d e p o s i t i o n on the flow c h a r a c t e r i s t i c s . He concluded from experimental evidence t h a t p a r t i c l e s were moved along the bed i n a s e r i e s of s t e p s , the l e n g t h and number of them being s t o c h a s t i c v a r i a b l e s . Recent research has shown th a t the d i s t r i b u t i o n of step lengths should be e x p o n e n t i a l (Shen and Todorovic 1971), although experimental measurements have favoured a two parameter Gamma d i s t r i b u t i o n which would approximate an e x p o n e n t i a l d i s t r i b u t i o n w i t h i n the experimental e r r o r of the measurements. The suggestion of an e x p o n e n t i a l d i s t r i b u t i o n , which Todorovic (1970) a l s o a p p l i e d to c o n s i d e r a t i o n of the extremes of a random sum of a set of random v a r i a b l e s , i s important i n that such a d i s t r i b u t i o n was postulated f o r the i n t e r m i t t e n t s t r u c t u r e of turbulence by Dutton to e x p l a i n the form of the exceedance s t a t i s t i c s . F u r t h e r , that such a model can be derived by the superimposition of s e v e r a l random v a r i a b l e s c o r r e l a t e s with the p o s t u l a t e s concerning the s u p e r i m p o s i t i o n of broken l i n e processes as an e x p l a n a t i o n of the slope of the r e s c a l e d range curve. 233 The measurements of the p r o b a b i l i s t i c s t r u c t u r e are important i n that K a l i n s k e (1 947) t r e a t e d the f l u c t u a t i o n s as normally d i s t r i b u t e d , while E i n s t e i n (1942, 1950) re q u i r e d the pressure f l u c t u a t i o n s to be normally d i s t r i b u t e d . As v e l o c i t y and pressure are n o n - l i n e a r l y r e l a t e d both assumptions cannot be c o r r e c t . I t i s seen that u and v components are skewed n e g a t i v e l y and p o s i t i v e l y r e s p e c t i v e l y and t h a t n e i t h e r i s Gaussian from an i n s p e c t i o n of the moments. The r o l e of i n t e r m i t t e n c y would seem to be of paramount importance. The i s o l a t e d v i b r a t i o n s of a few p a r t i c l e s on the bed, the a r r i v a l of t u r b u l e n t sweeps at the bed, and the occurrence of d i s s i p a t i o n periods of a frequency c o r r e l a t i n g with the l a r g e s c a l e flow s t r u c t u r e means that a p r o b a b i l i s t i c approach to sediment entrainment i s l i k e l y t o be extremely complex. h.STATISTICAL STABILITY APPROACH TO ENTRAPMENT IN GRAVEL RIVERS B u f f e t i n g of b l u f f bodies The entrainment of g r a v e l , elements of which act as b l u f f bodies i n a t u r b u l e n t boundary l a y e r , should be approached from a d i f f e r e n t viewpoint than the entrainment of sand. The r o l e of v o r t i c i t y a m p l i f i c a t i o n , of p a r t i c l e v i b r a t i o n s , of coherence of v e l o c i t y f l u c t u a t i o n s over the 234 face of the p a r t i c l e suggest that the average s t r e s s may not be the most s i g n i f i c a n t parameter to p r e d i c t i n i t i a l entrainment. The b u f f e t i n g of b l u f f bodies i n t u r b u l e n t boundary l a y e r s has been s t u d i e d from a s t a t i s t i c a l viewpoint s i n c e Davenport's pioneering s t u d i e s i n the e a r l y s i x t i e s . He suggested that there were three p r i n c i p a l g u a n t i t i e s needed to evaluate the f l u c t u a t i n g f o r c e s on the s t r u c t u r e s : v i z . 1) p r o b a b i l i t y d i s t r i b u t i o n of the three v e l o c i t y components: 2) spectrum of the v e l o c i t y f l u c t u a t i o n s : 3) s p a t i a l c o r r e l a t i o n s of the v e l o c i t y . He proposed that there would be three stages i n a s t a t i s t i c a l approach to the response of a body to f l u c t u a t i n g f o r c e s . Figure 59 shows the approach modified t o i n d i c a t e i t s a p p l i c a t i o n to the problem of sediment entrainment. The measurement of the p r o b a b i l i t y d i s t r i b u t i o n s of two of the v e l o c i t y components has been achieved. The f o r c e f l u c t u a t i o n s w i l l have to i n c l u d e both v e l o c i t y and pressure f l u c t u a t i o n s as both may be s i g n i f i c a n t ( v i z . comments on pressure d i s t r i b u t i o n i n a t u r b u l e n t b u r s t ) . Stage I comprises the spectrum of the v e l o c i t y as i t approaches the p a r t i c l e . I t was shown that the energy tends to become concentrated at lower frequencies as the body i s approached. This spectrum of v e l o c i t y i s then convolved with the f l u i d dynamic admittance spectrum of the boundary to produce the e f f e c t i v e f o r c e spectrum (Stage II) . In the o r i g i n a l s t u d i e s by Davenport t h i s stage represented the aerodynamic 235 STAGE I STAGE Tl STAGE HI VELOCITY SPECTRUM FORCE SPECTRUM RESPONSE SPECTRUM n 0(n) FLUID DYNAMIC ADMITTANCE BED SPECTRUM n Y ( n ) PARTICLE ADMITTANCE n g (n) Figure 5.9 Elements of a s t a t i s t i c a l approach to flow boundary i n t e r a c t i o n . ( A f t e r Davenport 1964 ) . 236 admittance. I t i s u s u a l l y a f l a t f u n c t i o n f o r values of reduced freguency nd/U<0.01, that i s f o r freguencies g r e a t e r than one hundred times the s i z e of the s t r u c t u r e . For eddies much s m a l l e r than the s t r u c t u r e the admittance f u n c t i o n becomes very s m a l l . Between these two extremes there are two opposing trends: reduced v e l o c i t y c o r r e l a t i o n w i l l reduce the admittance while the f l u c t u a t i n g v e l o c i t y w i l l r e s u l t i n aerodynamic c o e f f i c i e n t s l a r g e r than f o r a steady flow s i t u a t i o n . For a r i v e r channel t h i s f u n c t i o n might be taken as the spectrum of the bed roughness, f o r i t w i l l represent the f l u i d dynamic admittance of the o v e r a l l boundary which should i n t e r a c t with the flow t o produce the f o r c e spectrum opera t i n g on any dominant p a r t i c l e . The f i n a l stage corresponds to the c o n v o l u t i o n of the f o r c e spectrum with the mechanical admittance of the p a r t i c l e . Mechanical admittance of the p a r t i c l e must be i n t e r p r e t e d d i f f e r e n t l y f o r a g r a v e l p a r t i c l e l y i n g on a s u r f a c e than f o r an a e r o - e l a s t i c s t r u c t u r e s ubjected t o f l u c t u a t i n g f o r c e s . I t i s suggested that the mechanical admittance represents the v i b r a t i o n frequency of any p a r t i c l e on the bed. I t was observed that p r i o r t o movement, g r a v e l p a r t i c l e s would v i b r a t e i n t e r m i t t e n t l y . T h i s freguency of v i b r a t i o n must be a f u n c t i o n of p a r t i c l e shape and mass. The mechanical admittance might be obtained by p l a c i n g a p a r t i c l e on a low frequency, low amplitude v i b r a t i n g p l a t e and then o b t a i n i n g i t s n a t u r a l i n s t a b i l i t y frequency. The c o n v o l u t i o n of the mechanical admittance and the f o r c e spectrum 237 represents the response spectrum of a p a r t i c l e . Il£°E£ilEce of p a r t i c l e v i b r a t i o n In the context of o b t a i n i n g the mechanical admittance of the p a r t i c l e i t . i s i n t e r e s t i n g to consider the r o l e of v o r t i c i t y a m p l i f i c a t i o n and s e l f c o n t r o l l e d v i b r a t i o n of the p a r t i c l e s . The b a s i s f o r t h i s c o n s i d e r a t i o n r e s t s on the e a r l y work of Naudascher(1963, 1967). As the t u r b u l e n t flow approaches the b l u f f body, because the macroscale i s much gre a t e r than the body dimension, the o v e r a l l turbulence i n t e n s i t y i s reduced but the r e l a t i v e energy content at low fr e g u e n c i e s i s incr e a s e d . The r e s u l t s have shown one wavelength to be p r e f e r e n t i a l l y a m p l i f i e d which, i f i t corresponded to the n a t u r a l frequency of the p a r t i c l e , might w e l l r e s u l t i n a s e l f - c o n t r o l l e d o s c i l l a t i o n . This means t h a t 'the body motion i n f l u e n c e s the frequency of the a l t e r n a t i n g f o r c e s and the a m p l i f i c a t i o n u s u a l l y exceeds t h a t corresponding to the normal resonance c o n d i t i o n s ' (Naudascher 1963). Such a s i t u a t i o n may then l e a d to a flow e x c i t a t i o n at that p a r t i c u l a r freguency. Thus more energy may be concentrated at t h a t one frequency i n the spreadinq wake downstream from the b l u f f body which, i f i t should i n t e r a c t w i t h other p a r t i c l e s downstream, could lead to t h e i r removal. S i m i l a r l y , t h i s flow resonance w i l l mean greater coherence of the v e l o c i t y f l u c t u a t i o n s at s c a l e s l a r g e r than the wake shedding p a r t i c l e , r e s u l t i n g i n the removal of s m a l l e r p a r t i c l e s from the bed. 238 Conclusions The ideas s t a t e d here are r a d i c a l l y d i f f e r e n t from most approaches to sediment t r a n s p o r t f o r they recognise that sand and g r a v e l entrainment cannot be t r e a t e d i n the same manner. Second they recognise the r o l e of low freguency f l u c t u a t i o n s and v o r t i c i t y a m p l i f i c a t i o n r e s u l t i n g i n p a r t i c l e resonance p r i o r to entrainment. I t w i l l r e q u i r e measurement of the spectrum of the bed roughness and of the mechanical admittance of the p a r t i c l e s . T h i r d they i n d i c a t e t h a t the 'dominant roughness spacing' may have s e v e r a l p h y s i c a l conseguences. I t a c t s as the greatest energy d i s s i p a t o r and turbulence producer. I f slope i s imposed on the stream, the best method of energy d i s s i p a t i o n , i n the absence of f r e e s u r f a c e breaking waves, i s shown to be a wake i n t e r a c t i o n spacing of bed p a r t i c l e s . T h i s c o n d i t i o n might then represent the morphological e f f e c t of l o c a l e g u i l i b r i u m being maintained f o r an imposed f o r c e . The spacing of dominant p a r t i c l e s may r e s u l t i n a f l u i d resonant feedback which w i l l l e a d to the s e l e c t i v e t r a n s p o r t of s m a l l e r p a r t i c l e s by t u r b u l e n t entrainment and hence a development and p e r s i s t e n c e of the dominant spacing. The generation of a p r e f e r r e d low freguency may a l s o have s i g n i f i c a n t i n t e r a c t i o n with e i t h e r i n t e r n a l waves or secondary c u r r e n t s , r e s u l t i n g i n the c o u p l i n g of small and l a r g e s c a l e bed f e a t u r e s . 239 SEDIMENT TRANSPORT P r o b a b i l i s t i c measures i n sediment t r a n s p o r t Although the movement of m a t e r i a l has been s t u d i e d i n great d e t a i l under the guise of i n i t i a t i o n of sediment motion, the measurement and p r e d i c t i o n of r a t e s of sediment t r a n s p o r t are at best an e m p i r i c a l a r t and at worst a m y s t i c a l r i t e . There i s a myriad of formulae, some of which apply best to f i n e m a t e r i a l , others to coarse g r a v e l s , depending u s u a l l y on the range f o r which they were flume v a l i d a t e d . Of immediate relevance i s the Meyer Peter and Muller(1948) eguation. They d i v i d e d bed shear i n t o two p a r t s , one that i s necessary to overcome p a r t i c l e r e s i s t a n c e and the other t h a t i s necessary t o overcome the r e s i s t a n c e of bars and channel i r r e g u l a r i t i e s . They concluded that D was 90 the s i z e that was e f f e c t i v e i n e s t a b l i s h i n g the p a r t i c l e r e s i s t a n c e , i n d i c a t i n g a r e c o g n i t i o n of the p o s s i b l e r o l e that 'dominant' p a r t i c l e s might play i n determining r e s i s t a n c e . A p r o b a b i l i s t i c approach advocated by Ka l i n s k e (1947) inc o r p o r a t e d the e f f e c t of v e l o c i t y f l u c t u a t i o n s over the p a r t i c l e s . Based on White *s (1940) a n a l y s i s which inc o r p o r a t e d the notion of 'dominant' p a r t i c l e s on the bed, K a l i n s k e showed th a t x c=12d, which approximates the l i n e a r ''2 range of the S h i e l d s curve. I f x ~ u 2 then u ~ d which was c * * concluded i n the a n a l y s i s of the S h i e l d s curve f o r l a r g e p a r t i c l e s i z e s . The p r o p o r t i o n a l i t y w i l l depend on the 240 turbulence i n t e n s i t y . Re assumed the i n t e n s i t y u '/D was constant at 0 . 25 , while the r e s u l t s of t h i s research showed a great v a r i a t i o n depending on the d e n s i t y of elements on the bed. Furthermore, the i n t e r m i t t e n t generation of shear, of wake shedding and b u r s t i n g from the w a l l suggest t h a t the b a s i c assumption that the v e l o c i t y f l u c t u a t i o n s and the f l u c t u a t i o n s i n t r a c t i v e f o r c e are normally d i s t r i b u t e d i s s e r i o u s l y i n doubt. E i n s t e i n (1950) claimed t h a t i f the r a t e of d e p o s i t i o n balances the r a t e of erosion then the p r o b a b i l i t y of e r o s i o n i s constant and the r a t e of e r s o i o n w i l l depend on the t o t a l d i s t a n c e t r a v e l l e d by each p a r t i c l e . I n t r o d u c i n g t h i s d i s t a n c e , made up of a s e r i e s of d i s c r e t e jumps and r e s t p e r i o d s , i n t o the e q u i l i b r i u m equation of sediment movement he obtained h i s sediment t r a n s p o r t equation. The p r o b a b i l i t y of e r o s i o n he showed to be e x p o n e n t i a l l y d i s t r i b u t e d and occurred whenever l i f t exceeded the p a r t i c l e weight. Rather than using a c r i t i c a l shear v e l o c i t y he used the bed v e l o c i t y by assuming a l o g a r i t h m i c v e l o c i t y d i s t r i b u t i o n . In t h i s l o g a r i t h m i c law he used dfc5 as the length s c a l e . This l e n g t h s c a l e had to be c o r r e c t e d by using a ' h i d i n g ' f a c t o r expressing the s h e l t e r i n g of p a r t i c l e s by l a r g e r elements on the bed. This model works w e l l f o r l a r g e r p a r t i c l e s i z e s . I t i n c o r p o r a t e s a p r o b a b i l i s t i c model fo r entrainment based on the normal d i s t r i b u t i o n of pressure f l u c t u a t i o n s on the boundary, which leads to an e xponential d i s t r i b u t i o n of step 241 lengths and r e s t times. I t u t i l i z e s a l o g a r i t h m i c law f o r v e l o c i t y d i s t r i b u t i o n and recognises the sporadic nature of entrainment over the boundary. The measurements made here i n d i c a t e that the p o s s i b l y e x p o n e n t i a l d i s t r i b u t i o n (or l i m i t e d l o g a r i t h m i c normal) of var i o u s bandlimited parts of the p r o b a b i l i t y d i s t r i b u t i o n s of the v e l o c i t y f l u c t u a t i o n s may w e l l accord with the d i s t r i b u t i o n of step lengths-However, the assumption of a normal d i s t r i b u t i o n of pressures i s u n l i k e l y to be f u l f i l l e d . The non-normal d i s t r i b u t i o n of the v e l o c i t y f l u c t u a t i o n s , e s p e c i a l l y of the v e r t i c a l component, would imply that the pressure w i l l a l s o be non-normally d i s t r i b u t e d . The h i d i n g f a c t o r i n d i c a t e s the importance of p a r t i c l e d i s t r i b u t i o n and the r o l e dominant p a r t i c l e s may play in.determining s u s c e p t i b i l i t y to e r o s i o n . F u rther, the r e s u l t s showing the presence of a wake l a y e r suggest that use of the l o g a r i t h m i c law to ob t a i n the bed v e l o c i t y i s extremely suspect. The departure from n o r m a l i t y f o r v e l o c i t y f l u c t u a t i o n s , the lack of f i t to a s i n g l e l o g a r i t h m i c law are s i g n i f i c a n t i f an attempt i s b e i n g made t o understand the flow boundary i n t e r a c t i o n . The gross assumptions of normality and s i m i l a r s i m p l i f i c a t i o n s n evertheless lead to reasonable bulk estimates of t r a n s p o r t r a t e s , at l e a s t w i t h i n the accuracy of f i e l d measurements. S a l t a t i o n Bedload movement i s not the only type . of sediment t r a n s p o r t . P a r t i c l e s a l s o move i n s a l t a t i o n . The presence 2 42 of a wake l a y e r above d i s t r i b u t e d roughness elements may be most s i g n i f i c a n t i n r e l a t i o n to the s a l t a t i o n of p a r t i c l e s . S a l t a t i o n i s a mode of sediment movement whereby a p a r t i c l e moves i n a s e r i e s of jumps appearing to r e s u l t from momentary contact with the bed. Francis(1973) observed that s a l t a t i o n jumps appeared to be l i m i t e d v e r t i c a l l y t o 2 t o 4 g r a i n diameters and t h a t i f the grain then descended to two diameters above the bed, u p l i f t was r a r e . Such a phenomenon would thus be a f f e c t e d by the uniform wake l a y e r . U p l i f t from the w a l l having occurred, the p a r t i c l e w i l l r e t u r n to the surface or go i n t o suspension i f i t e n t e r s the constant v e l o c i t y region because i t s b a l l i s t i c t r a j e c t o r y (for i t moves l i k e a wheel-barrow being pushed over a pot-hole) would be destroyed by the high l e v e l of turbulence and i t s s p i n would not be maintained as there i s no strong mean shear. Suspended IS^d The turbulence s t r u c t u r e a l s o has a s i g n i f i c a n t e f f e c t on suspended m a t e r i a l . The simple models of sediment suspension assume a l o g a r i t h m i c v e l o c i t y p r o f i l e and a Boussinesg-type assumption, that i s u. c = d. . (—- ) l i j dx. 1 (54) where c represents c o n c e n t r a t i o n and d i s the t u r b u l e n t d i f f u s i o n tensor. However the spreading of small p a r t i c l e s , i f they r e f l e c t the turbulence s t r u c t u r e w i l l not be, as i s always assumed, of Gaussian form. The d i f f u s i o n t e n s o r , 2U3 which i s now thought to depend on the l a r g e s c a l e p r o p e r t i e s of the flow and not j u s t on the l o c a l p r o p e r t i e s of motion, w i l l a l s o not be a simple s c a l a r . I t should be a complete tensor, as the v e l o c i t y p r o f i l e over the d i s t r i b u t e d roughness elements i n t h i s research appeared to be composed of s e v e r a l zones, only one of which appeared to be i n approximate e g u i l i b r i u m . The c o s p e c t r a l measurements i n d i c a t e d t h a t the length s c a l i n g t o non-dimensionalize frequency d i f f e r e d i n the h o r i z o n t a l and v e r t i c a l d i r e c t i o n s , which would suqgest that the tensor cannot be simply reduced t o a s c a l a r d i f f u s i o n c o e f f i c i e n t . Sand and g r a v e l t r a n s p o r t The d i f f e r e n c e s between f i n e m a t e r i a l and coarse g r a v e l s were emphasized i n r e l a t i o n to the i n i t i a t i o n of sediment motion; they are a l s o h i g h l y s i g n i f i c a n t i n r e l a t i o n t o sediment t r a n s p o r t . A g r a v e l bed i s , except at very high f l o w , a non-compliant boundary- "While one of the major f e a t u r e s of sediment motion i n sand channels i s the downstream migration of dunes and r i p p l e s , i n a g r a v e l channel p a r t i c l e motion i s u s u a l l y very i n t e r m i t t e n t . Church (1972) reported the movements of marked cobbles were i r r e g u l a r and that cobbles apparently kept moving over the bed u n t i l caught by a protruding cobble. .Leopold, Emmett and Myrick(1966) reported that there was a tendency f o r cobbles to be l e s s mobile when they were densely d i s t r i b u t e d on the bed (q.v. the r e s u l t s on d i s t r i b u t e d roughness- elements). 244 The s e d i m e n t t r a n s p o r t w i l l l i k e l y be s i g n i f i c a n t l y d i f f e r e n t between sa n d and g r a v e l c h a n n e l s . The r e s u l t s f r o m t h e a r r a y s o f b l o c k s i n t h e f l u m e a r e t h u s r e l e v a n t i n p r o v i d i n g an e x p l a n a t i o n f o r t h e c o n c e n t r a t i o n o f g r a v e l p a r t i c l e s i n r i f f l e s , and f o r p r o v i d i n g a p h e n o m e n o l o g i c a l a c c o u n t o f t h e p a t t e r n o f s a l t a t i o n d e s c r i b e d by F r a n c i s {1 973). HiERftRCHIES OF FORM IN S AND AND GRAVEL CHANNELS I n t r o d u c t i o n The e x i s t e n c e o f an a s s e m b l a g e o f d i f f e r e n t b e d f o r m s a l l i n e q u i l i b r i u m w i t h a s i n g l e o v e r a l l mean f l o w c o n d i t i o n seems t o r e q u i r e t h a t t h e f o r m s c a n be r e l a t e d t o d i f f e r i n g p h y s i c a l l e n g t h s c a l e s i n t h e f l o w . T h i s was d i s c u s s e d by L e l i a v s k y ( 1 9 5 9 ) and by Allen(1968a) , w h i l e B a g n o l d (1956) a l s o r e f e r r e d t o t h e p o s s i b i l i t y t h a t b e d f o r m s o f d i f f e r e n t p h y s i c a l s c a l e s m i g h t c o e x i s t . The m a j o r work i n t h i s a r e a has f o c u s s e d on r i p p l e s , dunes and a n t i d u n e s and on meanders i n s a n d bed c h a n n e l s . F o r sand bed c h a n n e l s t h i s work h a s been summarized i n Y a l i n ( 1 9 7 2 ) w h i l e t h e a s s e m b l a g e o f f o r m s i n g r a v e l c h a n n e l s , n o t s i m p l y b a r s and meanders, has been d i s c u s s e d by C h u r c h (1972) . T h e s e f e a t u r e s were shown t o s c a l e w i t h d i s p l a c e m e n t t h i c k n e s s , d e p t h and w i d t h r e s p e c t i v e l y w h i l e t h e i r i n t e r a c t i o n due to t h e t h r e e d i m e n s i o n a l n a t u r e o f t h e f l o w i n a c u r v e d c h a n n e l r e s u l t s i n a c l o s e c o r r e s p o n d e n c e between meander w a v e l e n g t h s and t h e 245 growth of a l t e r n a t i n g bars. SEIES i l l sand In sand bed channels dunes are t r a d i t i o n a l l y e x plained i n terras of depth, but a more r e a l i s t i c parameter r e l a t e d t o the flow dynamics would be the macroscale of turbulence. A simple argument may be developed i n t h a t , f o l l o w i n g Webb (1 955) , — = A = 2TT L 'Inax where n represents the peak of the energy spectrum (see max Kaimal 1973) and i f L i s approximately equal to the depth then the s p e c t r a l peak w i l l correspond to a wavelength of 2 TT D. Y a l i n (1972) took the macroscale as equal to the channel depth and i f a band-limited process i s assumed so that the a u t o c o r r e l a t i o n has the form K(x) = e ~ a x cos(^) where L i s the macroscale, then the c o r r e l a t i o n wavelength f o r such a process w i l l be 2 ir L or 2 u D i n accord with e m p i r i c a l observations. His r a t i o n a l e f o r proposing t h i s method was based on the notion that the macroscale i s not a s t a b l e s i n g l e frequency, but extends over a c e r t a i n frequency range. He suggested t h e r e f o r e a band-limited process, i g n o r i n g the importance of the t r a n f e r of energy down the cascade. There i s a considerable l i t e r a t u r e on the spectrum 246 a n a l y s i s of dunes (for the most recent summary see J a i n and Kennedy 1974): the v a r i o u s spacings and s p e c t r a l peaks are apparently r e l a t e d to depth, the f o r c i n g f u n c t i o n i n these analyses being seen as surface waves which i n t e r a c t a t c e r t a i n phase l a g s with the bed disturbances. Such s t u d i e s have nowhere attempted to i n v e s t i g a t e the i n t e r a c t i o n and e f f e c t of the dunes on the turbulence as has been done i n wave-air i n t e r a c t i o n s t u d i e s . The important d r i v i n g f o r c e may i n f a c t be dependent upon an i n s t a b i l i t y wavelength i n the flow i n t e r a c t i n g with the l o c a l v e l o c i t y g r a d i e n t s . Bedforms i n g r a v e l In g r a v e l channels h i e r a r c h i e s of bedform a l s o e x i s t . Gravel dunes, or d u n e l i k e f e a t u r e s , r e g u l a r l y spaced, are found along the channel bed (Galay 1967). However, the g r a v e l dunes that have been observed showed the p e c u l i a r i t y t h a t as depth i n c r e a s e d t h e i r spacing d i d not i n c r e a s e (Galay 1967). This i s u n l i k e sand dune behaviour, f o r which i n c r e a s e d depth r e s u l t s i n increased dune spacing: i t suggests that u n l i k e sand dunes the dune-like f e a t u r e s i n g r a v e l cannot be simply r e l a t e d t o flow macroscale. I f a freguency i s p r e f e r e n t i a l l y a m p l i f i e d by v o r t i c i t y s t r e t c h i n g round a b l u f f body (as found i n Chapter 4) , t h i s frequency may represent an important s c a l i n g parameter. I t i s postulated t h a t such v o r t i c i t y a m p l i f i c a t i o n w i l l be unaffected by bulk channel p r o p e r t i e s , such as depth. Such p r e f e r e n t i a l a m p l i f i c a t i o n might e x p l a i n the o b s e r v a t i o n s 247 made i n g r a v e l channels by Galay (1967) that as depth increased the spacing of g r a v e l bars d i d not i n c r e a s e . The spacing should only depend on the frequency a m p l i f i e d i n the flow around the d i s t r i b u t e d roughness elements on the boundary. This t e n t a t i v e c onclusion suggests that there may be i n t e r a c t i o n between the small s c a l e f e a t u r e s on a g r a v e l bed and the l a r g e r f e a t u r e s . This would not appear unreasonable f o r , u n l i k e a sand bed channel f o r which the viscous sublayer e x i s t s and the t u r b u l e n t macroscale i s seen t o c o r r e l a t e w i t h , and may even determine the bursts from the boundary, i n a g r a v e l channel wake shedding and v o r t i c i t y a m p l i f i c a t i o n are l i k e l y t o be the most s i g n i f i c a n t determinants of the freg u e n c i e s of importance w i t h i n the channel. Bars i n sand and g r a v e l channels The r e s u l t s from the work on bars of varying width suggested that once the feature occupied more than 1/3 of the width of the channel the flow s t r u c t u r e markedly a l t e r e d to produce a two dimensional wake downstream. The break p o i n t may depend on the tu r b u l e n t macroscale: when the width became gr e a t e r than t w i c e the macroscale there was a dramatic i n c r e a s e i n turbulence both upstream and downstream which would, i f the boundary was m o d i f i a b l e , tend t o act as a c o n t r o l on the f u r t h e r growth of the bar. Diagonal bars e x i s t i n sand and g r a v e l channels, but they behave i n g u i t e d i f f e r e n t manners. Gravel bars tend t o 248 be permanent f e a t u r e s ; t h e i r s t a b i l i t y may be a t t r i b u t e d t o the c l o s e spacing of the l a r g e p a r t i c l e s which w i l l i n h i b i t f u r t h e r motion. The c e s s a t i o n of motion i s l i k e l y t o be caused by the p a r t i c l e being trapped against other p a r t i c l e s which w i l l r e s u l t i n an imbricated bed that w i l l r e s i s t f u r t h e r e r o s i o n . Sand bars migrate and, r a t h e r than ' a c t i n g as a s t o r e of sediment, they are major c o n t r i b u t o r s to bed t r a n s p o r t . The flow w i l l l i k e l y separate at the downstream edge and the sand, having been en t r a i n e d i n the t u r b u l e n t b u r s t s , i s then deposited downstream r e s u l t i n g i n the migration of the f e a t u r e . Hierarchy of i n s t a b i l i t i e s I n t e r m i t t e n c y seems to play a key r o l e i n t h i s h i e r a r c h y and i n t h i s d i s c u s s i o n considerable emphasis i s placed on the work of Mollo-Christensen(1973) . F i r s t , on a purely h e u r i s t i c l e v e l , the remarkable s i m i l a r i t y of shape of the two f u n c t i o n s shown i n Figure 58 suggests t h a t i n t e r m i t t e n c y , which i s r e l a t e d to the f l a t n e s s f a c t o r , and sediment entrainment c e r t a i n l y covary and may be dynamically r e l a t e d . The minimum In the f l a t n e s s f u n c t i o n corresponds to the peak i n u'/u s , where the highest production and d i s s i p a t i o n occurs. Second, i n t e r m i t t e n c y occurs at v a r i o u s s c a l e s . The wake shedding phenonenon, b u r s t i n g from the w a l l , energy d i s s i p a t i o n and sediment entrainment are a l l s p o r a d i c random v a r i a b l e s . As shown by Kim et al(1971) and by Grass (1971) ,• the energy production process i s dominated by the sweeps of 249 high momentum f l u i d towards the w a l l . Wallace et al(1972) have shown t h a t t u r b u l e n c e d i s s i p a t i o n i s a l s o s t r o n g l y r e l a t e d t o the bu r s t and sweep c y c l e , c h i e f l y because the occurrences of these events corresponds with r e g i o n s of high l o c a l shear r a t e s and consequently high l o c a l d i s s i p a t i o n r a t e s . The o b s e r v a t i o n made here t h a t the p e r s i s t e n c e of the d e r i v a t i v e f l u c t u a t i o n s showed a p e r i o d c o r r e s p o n d i n g t o the macroscale i s taken as c o n f i r m a t i o n of Wallace's o b s e r v a t i o n (Figure 44b). Mollo-Christensen(1973) has shown t h a t i f the energy e g u a t i o n s f o r three widely separated s c a l e s are w r i t t e n and are averaged over the l e n g t h s c a l e s c o r r e s p o n d i n g to t h e d i f f e r e n t v e l o c i t y f i e l d s , then when a bu r s t occurs the d i f f e r e n t s c a l e s of l o c a l i n s t a b i l i t y can i n t e r a c t to enhance one another's growth r a t e s , s m a l l e r s c a l e s o c c u r r i n g p r e f e r e n t i a l l y a t c e r t a i n phases of the l a r g e s c a l e s on which they r i d e t o allow t h e i r Reynolds s t r e s s e s to do work on the l a r g e r s c a l e s , while the l a r g e r s c a l e s , through t h e i r growth, make the s m a l l e r s c a l e i n s t a b i l i t y p e r s i s t . ' Moreover '...the s m a l l s c a l e s seem to a c t i n p a r t as c a t a l y s t s f o r the more e f f i c i e n t e x t r a c t i o n of energy by the next l a r g e r s c a l e s from the s t i l l - l a r g e r s c a l e flow. ( M o l l o - C h r i s t e n s e n 1973, p. 10.1) Such a h i e r a r c h i c a l i n t e r a c t i o n would appear a p p l i c a b l e t o open channel flows. The search f o r s c a l i n g l e n g t h s , using 250 widths f o r meanders, depths f o r dunes and showing the i n t e r r e l a t i o n of width and depth (Leopold, Wolman and M i l l e r 1964; Y a l i n 1972) merely c o n s t i t u t e s s u b s t i t u t i n g a .morphological s c a l e , f o r the s i g n i f i c a n t , dynamical flow s c a l e that w i l l be in f l u e n c e d by the i n t e r a c t i o n and r e l a t i v e values of these morphological s c a l e s - The s t a b i l i t y a n a l y s i s of Engelund and Skovgaard (1973), which showed the a m p l i f i c a t i o n of an i n t e r n a l wavelength l e a d i n g t o meandering depending upon the r a t i o of t h i s wavelength to the r e l a t i v e width-depth r a t i o , i s a case i n p o i n t , while a s i m i l a r a n a l y s i s may be a p p l i e d to bedforms (Fredsoe 1974). However, the assemblage of bedforms with d i f f e r e n t p h y s i c a l s c a l e s t h a t c o e x i s t might most e a s i l y be considered from the viewpoint of the i n t e r a c t i o n of weakly damped waves which might e x i s t i n the shear flow. These waves may t r i g g e r l o c a l i n s t a b i l i t i e s which produce flow r e g u l a r i t i e s that are u s u a l l y s c a l e d with flow macroscale. These may produce bedform p e r i o d i c i t i e s which may then be s u f f i c i e n t to s u s t a i n the i n t e r n a l waves i n the shear flow r e s u l t i n g i n the development of coherent p a t t e r n s of form over c o n s i d e r a b l e d i s t a n c e s . The t u r b u l e n t macroscale may a l s o act as a lower l i m i t f o r these coherent f o r c e s f o r the random vortex s t r e t c h i n g which occurs at s m a l l e r s c a l e s would l i k e l y l e a d to the d e s t r u c t i o n of r e g u l a r l o c a l bed f e a t u r e s . At t h i s s c a l e the i n t e r m i t t e n t processes w i l l a f f e c t the entrainment of sediment by determining the d i s t r i b u t i o n of shear on the p a r t i c l e s . The i n t e r m i t t e n c y of the d i s s i p a t i o n f l u c t u a t i o n s 251 w i l l be s i g n i f i c a n t f o r small p a r t i c l e s . Even i n Cheekye Creek measurements of the Taylor microscale were of order 3.5 mm and thus even coarse sand would be subjected to a s t r o n g l y i n t e r m i t t e n t flow v e l o c i t y . In t h i s research there was t r a n s f e r of energy to the l a r g e r s c a l e s by v o r t i c i t y a m p l i f i c a t i o n and the production of a p r e f e r r e d wavelength might i t s e l f r e s u l t i n i n t e r a c t i o n with a low freguency wavelength i n the flow. S i m i l a r l y the t r i g g e r i n g of b u r s t i n g from the boundary by such waves and the c a t a l y t i c i n f l u e n c e of the s m a l l s c a l e s f o r the more e f f i c i e n t e x t r a c t i o n of energy at the l a r g e s c a l e s suggests th a t the l a r g e s c a l e processes may be i n f l u e n c e d by t h e i r s m aller s c a l e components- In g r a v e l r i v e r s the spacing of g r a v e l bars might depend upon p r e f e r e n t i a l a m p l i f i c a t i o n of a dominant freguency around i s o l a t e d boulders and t h i s frequency may cause i n s t a b i l i t y i n even lower frequency wavelengths. In meandering r i v e r s the presence of i n t e r n a l waves or p e r s i s t e n t secondary c u r r e n t s of low frequency may lead to the propagation of the pools and r i f f l e s . They w i l l occur at higher frequency spacing by impinging on a flow nonuniformity which c o u l d then generate a dominant frequency which might i n t u r n lead to m o d i f i c a t i o n of the bed. The proposed h i e r a r c h y of process r e l a t e s low freguency waves, or secondary c u r r e n t s , to s e v e r a l s c a l e s of dynamic i n s t a b i l i t y . The t r i g g e r i n g of the l o c a l i n s t a b i l i t i e s may be dependent upon the phase v e l o c i t y of the low freguency, l a r g e s c a l e 'waves', but e q u a l l y w e l l i t may depend on the 252 c a t a l y t i c i n f l u e n c e of the sm a l l s c a l e i n s t a b i l i t i e s r e l a t e d to the i n t e r m i t t e n c y at the microscale. This process could be modelled by the superimposition of s e v e r a l d i f f e r e n t d i s t r i b u t i o n s : the r e s u l t a n t r e s c a l e d range would show pe r s i s t e n c e and i n t e r m i t t e n c y . The importance of averaging p e r i o d , or of the s c a l e of i n v e s t i g a t i o n , was emphasized i n the computation of the shape of the higher order d i s s i p a t i o n s p e c t r a . I t was a l s o considered i n the a n a l y s i s of i n t e r m i t t e n c y by Mollo-Christensen(1973) and by Mandelbrot (1974) i n r e l a t i o n to the i n t e r m i t t e n c y of turbulence. I t would seem then t h a t i n t e g r a t e d measures of turbulence, when the averaging period i s chosen to c o i n c i d e with the ' s p e c t r a l gap' may hide the dynamics of the process. S i m i l a r l y , averaging the channel r e s i s t a n c e by c o n s i d e r i n g only a s i n g l e roughness s c a l e (such as Manning's n or the s i z e of roughness p a r t i c l e s ) may hide the i n t e r a c t i o n between the s c a l e s of roughness. I f i t may be taken t h a t the flow shows a hierarchy of processes, some of which are i n t e r m i t t e n t and whose generation depends on s t a b i l i t y c r i t e r i a determined at both l a r g e r and s m a l l e r s c a l e s , then i t would appear reasonable to expect a s i m i l a r c o n d i t i o n to apply to r i v e r channels. The f o o t p r i n t s of the flow processes should s c a l e with d i f f e r i n g dynamic lengths i n the flow and thus the morphological f e a t u r e s must be expected t o be interdependent i n a s i m i l a r manner to the flow p e r i o d i c i t i e s and i n s t a b i l i t i e s . 253 CHAPTER 6 CONCLUSIONS The experimental work c a r r i e d out f o r t h i s d i s s e r t a t i o n has concentrated on i n v e s t i g a t i n g the i n f l u e n c e of v a r y i n g roughness d e n s i t i e s on the turbulence s t r u c t u r e . The d i s s e r t a t i o n a l s o attempted, by comparison with r e s u l t s from previous research i n a i r , to examine the s i m i l a r i t i e s and d i f f e r e n c e s of r e s u l t s f o r a free surface flow f o r the purpose of avoiding unnecessary d u p l i c a t i o n of measurement. By making measurements i n the l a b o r a t o r y and i n the f i e l d i t was intended to show the s u i t a b i l i t y of l a b o r a t o r y measurements f o r modelling sediment entrainment with p a r t i c u l a r a t t e n t i o n to c o n d i t i o n s i n g r a v e l channels. F i n a l l y t h i s research was performed to c o l l a t e the c o n s i d e r a b l e work on flow induced v i b r a t i o n s , on v o r t i c i t y a m p l i f i c a t i o n and on i n t e r m i t t e n c y and to h e u r i s t i c a l l y apply these notions to open channel flow mechanics with the i n t e n t i o n of examining t h e i r s u i t a b i l i t y f o r e x p l a i n i n g the complex h i e r a r c h i e s of channel f e a t u r e s - This work must n e c e s s a r i l y remain s p e c u l a t i v e and d i s c u r s i v e ; i t s value can only be judged by making f u r t h e r s t r i c t l y c o n t r o l l e d experiments which u n t i l now have not been•considered. I t was shown t h a t a roughness spacing of 1/12 e x h i b i t e d the g r e a t e s t r e s i s t a n c e to flow. At t h i s d e n s i t y turbulence energy production and d i s s i p a t i o n was g r e a t e s t . At higher 254 d e n s i t i e s the 'bed 1 s h i f t s to the top of the elements p r o v i d i n g p r o t e c t i o n f o r the s u r f a c e . At lower d e n s i t i e s the the i n t e r v e n i n g surface makes s i g n i f i c a n t c o n t r i b u t i o n s t o flow r e s i s t a n c e . The r a t i o of roughness element height t o i n t e r v e n i n g roughness surface height was 10:1. No comparable measurements have been made, as a l l other work on roughness arrays has used a smooth i n t e r v e n i n g s u r f a c e . But the adeguate modelling of a i r f l o w over any r u r a l or urban s i t u a t i o n i n the atmosphere or of water flows over elements on a stream bed r e g u i r e s that the i n t e r v e n i n g surface, be rough. Thus the l a r g e range of l i n e a r v a r i a t i o n of shear s t r e s s with d e n s i t y of roughness elements found i n previous work might be i n a p p l i c a b l e i n p r a c t i c a l s i t u a t i o n s . The s t r u c t u r e of the flow over d i s t r i b u t e d roughness elements showed th a t there are three l a y e r s i d e n t i f i a b l e i n the mean flow and turbulence c h a r a c t e r i s t i c s . The most s i g n i f i c a n t f i n d i n g was that there i s a wake l a y e r extending up to two roughness heights above the bed which i s c h a r a c t e r i z e d by a nearly constant v e l o c i t y and where from c a l c u l a t i o n s of the r a t i o of u„ (shape) /u„ (spec) , there appears to be a s l i g h t excess of production which i s tra n s p o r t e d towards the w a l l and up towards the f r e e s u r f a c e . The turbulence s t r u c t u r e showed some v a r i a t i o n from bed to sur f a c e . The f r e e surface a c t s to c o n s t r a i n the flow so that the maximum region of i n e r t i a l t r a n f e r was at 0.5 to 0.7 of the depth. The i n f l u e n c e of the free surface was most s i g n i f i c a n t i n the presence of su r f a c e waves and i n two f i e l d 255 measurements the spectrum c l o s e to the s u r f a c e d i d not peak. This e f f e c t was l i m i t e d to only the top 25% of the flow but even when the flow surface was smooth the p r o b a b i l i s t i c s t r u c t u r e of the v e l o c i t y f l u c t u a t i o n s showed k u r t o s i s values w e l l i n excess of those found elsewhere i n the flow. The lower 50% or more of the flow e x h i b i t s s i m i l a r i t y with measurements made i n the atmosphere and i n wind tunnel s i m u l a t i o n s . Shear s t r e s s c o r r e l a t i o n measurements showed that l o n g i t u d i n a l spacings of bars more than twice the channel depth behaved d i f f e r e n t l y from the c l o s e r spaced elements. The l a t t e r showed good agreement over the l i m i t e d range of measurement with Dvorak's measurements over bars i n a wind t u n n e l . The r o l e of the f r e e surface cannot be so c l e a r l y i d e n t i f i e d i n examining the e f f e c t of bar. width. Measurements behind a complete bar showed f a i r agreement with Counihan et a.l's(1974) s e l f - s i m i l a r wakes behind two dimensional bars and s i m i l a r behaviour was noted f o r bar widths down to 30% of flow width. At t h i s p o i n t , corresponding to approximately twice flow macroscale, the wake became three dimensional with a much more r a p i d decay downstream. The c o s p e c t r a l measures and s p e c t r a l c o r r e l a t i o n measurements showed that frequencies below 5 Hz were mainly s i g n i f i c a n t i n the t r a n f e r of momentum. The cospectrum was more peaked than i s u s u a l l y found with the exception that the r e s u l t s were i n agreement with the measurements made over a 256 f o r e s t by Shaw et al(197'4)» There was s i g n i f i c a n t departure a t the low freguency end of the cospectrum i n d i c a t i n g t h a t low freguencies might play a s i g n i f i c a n t r o l e i n energy exchange. The e f f e c t of varying d e n s i t i e s showed that a t d e n s i t i e s greater than 1/8 the c o s p e c t r a l s t r u c t u r e changed with the peak f o r measurements below the tops of the roughness elements being l e s s than that above the elements, a r e s u l t i n accord with Shaw's measurement. A d e t a i l e d c o n s i d e r a t i o n of i s o t r o p y was presented as most work on turbulence i n r i v e r s has taken the e x i s t e n c e of an i n e r t i a l subrange with a -5/3 slope of even t a n g e n t i a l presence as i n d i c a t i n g l o c a l i s o t r o p y . This was shown not to be the case from the r a t i o s of the s p e c t r a and from the u n i v e r s a l s c a l i n g of the s p e c t r a . The spectra showed reasonable agreement with a s p e c t r a l shape determined by a production and d i f f u s i o n parameter dependent upon the t u r b u l e n t Reynolds number. The work on the p r o b a b i l i s t i c s t r u c t u r e of turbulence i n d i c a t e d that i n t e r m i t t e n c y i s present i n the flow a t s e v e r a l s c a l e s . The conventional e x p l a n a t i o n of the r e s c a l e d range may be more f r u i t f u l l y examined i f i n t e r m i t t e n t generating f u n c t i o n s which might themselves i n t e r a c t are considered, r a t h e r than a process which co n t a i n s i n f i n i t e memory. The comparison of work i n the flume and i n the f i e l d showed th a t the turbulence s t r u c t u r e may be adeguately modelled i n a flume i n that i t s s p e c t r a l c h a r a c t e r i s t i c s and 257 p r o b a b i l i t y s t r u c t u r e were v i r t u a l l y i d e n t i c a l i n a l l cases. The importance of s c a l i n g was f u r t h e r considered i n r e l a t i o n t o sediment entrainment where i t was shown th a t the mechanisms of sand and g r a v e l entrainment might be fundamentally d i f f e r e n t , while the r e l a t i o n between dunes i n sand channels and dune f e a t u r e s i n g r a v e l was discussed i n terms of d i f f e r e n t generating processes r e l a t e d to the h i e r a r c h i e s of channel features that e x i s t . The nature of the flow h i e r a r c h i e s and bedform h i e r a r c h i e s was discussed i n terms of the i n t e r a c t i o n of low freguency 'waves' or secondary c u r r e n t s and d i f f e r e n t s c a l e s of i n t e r m i t t e n t processes. 258 BIBLIOGRAPHY ALLEN J . R . L . 1968 C u r r e n t r i p p l e s : t h e i r r e l a t i o n t o p a t t e r n s o f w a t e r and s e d i m e n t m o t i o n . N o r t h - H o l l a / n d P o l i s h i n g . , JUllsterdam 443pp. ALLEN J . R . L . 1968A The n a t u r e and o r i g i n o f _ b e d - f o r m h i e r a r c h i e s . S e d i m e n t o l q g y V-.8 p. 161-182. ANTONIA R.A. 1973 Some s m a l l s c a l e p r o p e r t i e s o f b o u n d a r y l a y e r t u r b u l e n c e . Phys.. F l u i d s v.__16 £. 1198-1206.. ANTONIA R.A. AND LU.XTO.N R.E. 1974 C h a r a c t e r i s t i c s o f t u r b u l e n c e w i t h i n an i n t e r n a l b o u n d a r y l a y e r . 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Since the t h e o r e t i c a l f o r m u l a t i o n s of the r e s i s t a n c e t o flow f o r boundary l a y e r s p o s t u l a t e d by P r a n d t l and von Karman and the e m p i r i c a l s t u d i e s of Nikuradse(1933) and Keulegan(1938) of rough w a l l t u r b u l e n t boundary l a y e r s , the mechanics of the flow have come under great s c r u t i n y . The impetus of a e r o n a u t i c a l design requirements s t i m u l a t e d e a r l y s t u d i e s , w h i l e the growing awareness of the lower atmosphere as a t u r b u l e n t boundary l a y e r has l e d to a considerable number.of f i e l d experiments and wind t u n n e l s i m u l a t i o n s of flow over rough s u r f a c e s . The n e c e s s i t y t o know f o r engineering design the wind loadings of b u i l d i n g s , bridges and other wind s e n s i t i v e s t r u c t u r e s has r e s u l t e d i n a l a r g e l i t e r a t u r e . I t i s not the i n t e n t i o n to comprehensively review t h i s work, but r a t h e r to s e l e c t apparently u s e f u l n o t i o n s , t h e i r dimensional and a n a l y t i c r a t i o n a l e of development and some of t h e i r e m p i r i c a l v e r i f i c a t i o n s as a b a s i s f o r comparison w i t h open channel flow and as a framework f o r f l u v i a l s t u d i e s should they be found to be a p p l i c a b l e . 277 Power law formula t i o n The mean v e l o c i t y p r o f i l e s w i t h i n a t u r b u l e n t boundary l a y e r over a f l a t surface may be described s a t i s f a c t o r i l y by a power law or a l o g a r i t h m i c law. at t h i s p o i n t i t i s s u f f i c i e n t to d e f i n e 'turbulence' as the s t a t e of flow when exchange of any s c a l a r or vector g u a n t i t y i s dominated by the Reynolds s t r e s s e s which may be considered as a type of 'eddy v i s c o s i t y ' . The power law suggested by P r a n d t l i s based on the assumption that the l o c a l s k i n f r i c t i o n i s p r o p o r t i o n a l t o some power of the Reynolds number based on f r e e stream v e l o c i t y and boundary l a y e r t h i c k n e s s &. That i s U ~ $ A 1 where n' depends on the s k i n f r i c t i o n c o e f f i c i e n t c c f = 6T5P1J2 a 2 The value of 1/n' was o r i g i n a l l y suggested to be 7 but i t was noted by Clauser (1956) that 1/n* ranged from .3 t o 10 depending on the flow Reynolds number . Such a law w i l l give u n r e a l i s t i c p r e d i c t i o n s of v e l o c i t y c l o s e to the w a l l where dimensional arguments show (Reynolds 1974) t h a t U = F ( y , x , p , u ) A 3 As the exponent depends on Reynolds number, as <5 i s an i l l - d e f i n e d q u a n t i t y , t o formulate t h i s dependence the power law may be g e n e r a l i z e d u t i l i z i n g other i n t e g r a l parameters which are w e l l defined, notably the displacement and momentum t h i c k n e s s , where 6*= f d - £ )dy ' e - / £ ( l - £ y dy 278 A4 while t h e i r r a t i o , H'=6 /6 , i s designated as the shape f a c t o r -Thus Equation A1 may be r e w r i t t e n as (Clauser 1956) H'-l U U„ - A 5 i f f o l l o w i n g d e f i n i t i o n A6 and 6 7v2 A7 6 where = ycT72 (1 - G ( | f ) ) 1/2 A8 A9 and H' = (1 - G ( c f ) 1 / 2 ) 1 A 1 0 2 The power law has been c r i t i c i z e d on three major grounds- In atmospheric work the s e l e c t i o n of standard gradient h e i g h t s (see Davenport 1961) such as 10 m, i s seen as s u b j e c t i v e r a t h e r than a n a l y t i c a l l y based- Secondly, as noted above, the exponent of the power law i s not constant but depends on height above the ground-. Such an o b j e c t i o n may be 279 circumvented i f constant heights are used f o r comparative purposes. T h i r d l y the exponent depends on more fundamental parameters. Reynolds (1974) suggested that as c f = C Re -VP A11 then n = 2p - 1 while the exponent should be r e l a t e d to the fundamental parameters such as shear v e l o c i t y u s and roughness l e n g t h which are b a s i c measures of the su r f a c e s t r e s s and roughness. Logarithmic laws The boundary l a y e r may be considered as being made up of two regi o n s , an i n n e r region where the s o - c a l l e d 'law of the w a l l ' a p p l i e s , g e n e r a l i z e d as v A12 where y i s the d i s t a n c e from the w a l l and v i s the kinematic v i s c o s i t y ; and an outer region where the g e n e r a l i z e d • v e l o c i t y d efect law' takes the form When these two regions overlap and are simultaneously v a l i d , the f u n c t i o n a l form i s l o g a r i t h m i c A14 280 For a rough boundary where hu^/v i s greater than 70 , where h i s the roughness height, the roughness height i s the governing f a c t o r of the flow p a t t e r n . The l o g a r i t h m i c law then takes the form U = 1 v The e f f e c t of roughness i s assumed to be confined to a l a y e r where the v e r t i c a l distance from the boundary and the roughness length are comparable when the roughness l e n g t h i s a s c a l e d e s c r i b i n g the i n f l u e n c e of the roughness on the flo w . For greater v e r t i c a l d i s t a n c e s the l o g a r i t h m i c law may be w r i t t e n as u» k" i n y Q A 1 6 whereYg denotes the roughness l e n g t h . In meteorology, t o de s c r i b e the v e l o c i t y d i s t r i b u t i o n over high roughness elements, t h i s eguation i s modified by i n t r o d u c i n g a zero-plane displacement ( c f . Tan and Lin g 1963) to become where d i s the zero plane displacement. This manipulation i s simply an a r t i f i c e to r e - o r i g i n the v e r t i c a l c o - o r d i n a t e so that the maximum range of f i t to a l o g a r i t h m i c law may be achieved. The problem of roughness has been approached almost e x c l u s i v e l y by t h i s method s i n c e Nikuradse's work. The manipulation of the v e r t i c a l co-ordinate to ensure 281 f i t t i n g to the l o g a r i t h m i c law f o r f u l l y rough flow works f o r sand g r a i n roughness; i f the roughness i s of some other form, the methodology adopted s i n c e the analyses of S c h l i c h t i n g (1936) and Keulegan(1938) has been to compute the •eguivalent sand roughness'. P h y s i c a l l y , f o r a known shear v e l o c i t y at the bed, i f a v e l o c i t y at height y i s observed over a w a l l of a r b i t r a r y roughness then th a t same v e l o c i t y w i l l be obtained at that same height i f h i s transformed to i t s e g u i v a l e n t sand s i z e , as given f o r example i n Keulegan (1938 p.722). Monin and Yaglom (1971) reviewed the many papers produced on the c a l c u l a t i o n of displacement height and roughness length but t h e i r only comment on the e f f e c t s of d i f f e r i n g roughness spacings was that the r a t i o h/y 0 w i l l not be s t r i c t l y constant over s e t s of i r r e g u l a r i t i e s of d i f f e r e n t forms. S i m i l a r remarks may be i n f e r r e d to apply t o d as i t i s taken to be of the magnitude y 0 / 2 < d < y 0 The e f f e c t of d i s t r i b u t e d roughness was examined by Schlichting(1936) f o r various shaped elements spaced at varying d e n s i t i e s on a boundary. Ke suggested that the o v e r a l l f o r c e imparted to the rough s u r f a c e could be p a r t i t i o n e d i n t o the force exerted on the roughness elements and the f o r c e exerted on the i n t e r v e n i n g s u r f a c e . The p a r t i t i o n i n g of the boundary l a y e r i n t o two zones recognises t h a t d i f f e r e n t s c a l i n g parameters are important i n d i f f e r e n t zones of flow. Second, the use of the l o g a r i t h m i c laws allows an examination of the e f f e c t of d i f f e r i n g 282 roughuess spacings on separate regions of the flow- This i s important i n open channels where the e f f e c t of the f r e e surface must a l s o be considered. Sediment entrainment f u n c t i o n s The c l a s s i c a l work on sediment entrainment i s tha t of S h i e l d s (1936). Using the l o g a r i t h m i c law to d e s c r i b e the v e l o c i t y d i s t r i b u t i o n f o r the t r a n s i t i o n a l and f u l l y rough w a l l boundary l a y e r , he recast Eg A12 and A14 as £ = 5.75 log £ + p ( 2 L U » ) u * n v A18 In the second term y d where d i s some s e l e c t e d g r a i n s i z e and thus ^ = F ( a _ , — * ) where a , » F(£) 11*^  c. V £ . 1 1 A 19 I f the drag on a p a r t i c l e i s given by F D = C DAp 2 = F ( a , — )p d U A20 where ct i s a g r a i n shape f a c t o r then using the l o g a r i t h m i c law above f o r the v e l o c i t y U V V 2 P ( 0 , < , 2 ' SS*) .21 where u x= Q/ P . - Resistance i s then taken to depend only on the form of the bed and the immersed weight of the p a r t i c l e F R _ a (S s - l ) Y d 3 A 2 2 2 83 where S i s the s p e c i f i c g r a v i t y of the p a r t i c l e and Y i s the s s p e c i f i c weight of the f l u i d . I f the drag and r e s i s t a n c e are equated, then the c r i t i c a l shear s t r e s s x =x i s given by c 0 a 3 ( S s - l ) y d 3 - x c d 2 P(«,« 2, p) A 2 3 _ l c = F( *HiL) Y ( S s - l ) d v A 2 4 S h i e l d s p l o t t e d t h i s r e l a t i o n which y i e l d s the c r i t i c a l s t r e s s f o r any p a r t i c l e s i z e . Colebrook and White(1937) showed that when u«d/v>3.5 the grains behaved l i k e i s o l a t e d o b s t a c l e s and shed wakes. They deduced t h a t prominent g r a i n s l y i n g above the mean surface c a r r i e d most of the drag of the area they occupied and a l a r g e p o r t i o n of t h a t on the area covered by the wake. They estimated t h a t the protected area might extend to 10 to 20 times the area.of the g r a i n . Such an a n a l y s i s ignores the l i f t component completely. L a t e r work by E i n s t e i n and E l Samni(1951) and by C h e p i l i (1959) showed the l i f t component t o be s i g n i f i c a n t because f l u c t u a t i n g v e l o c i t i e s and pressures i n the v e r t i c a l d i r e c t i o n at the w a l l were f r e g u e n t l y up to 70% as la r g e as those i n the streamwise d i r e c t i o n and would thus s i g n i f i c a n t l y i n c r e a s e a tendency towards i n s t a b i l i t y f o r the g r a i n , p r i o r to entrainment. The importance of mixed g r a i n s i z e s was not i n v e s t i g a t e d although Yalin(1972) mentions t h i s as one of the t o p i c s needing i n v e s t i g a t i o n . 284 APPENDIX I I FLOW BOUNDARY INTERACTION Roughness arrays The flow over a rough boundary with spaced roughness elements has received c o n s i d e r a b l e a t t e n t i o n . I t i s necessary t o know the e f f e c t s of such a r r a y s to estimate the boundary l a y e r c h a r a c t e r i s t i c s i n urban and r u r a l areas, and over c e r t a i n crop types and f o r e s t s f o r d i f f u s i o n s t u d i e s . The e f f e c t s of surrounding roughness elements on f o r c e s impinging on a body i n a t u r b u l e n t boundary l a y e r i s of paramount importance to s t r u c t u r a l engineers. Much e m p i r i c a l work has been c a r r i e d out over varying boundary roughnesses s i m u l a t i n g f o r e s t s , wind breaks and also with simple r e g u l a r and random ar r a y s of cubes and c y l i n d e r s . A review of the l a t t e r work was undertaken by Wooding et a l ( 1 9 7 3 ) . They were concerned, as was S c h l i c h t i n g ( 1 9 3 6 ) , with the p a r t i t i o n i n g of drag and concluded that e g u i p a r t i t i o n occurs at q u i t e low conce n t r a t i o n s when the i n t e r a c t i o n between elements i s - s m a l l so that the drag c o e f f i c i e n t of a t y p i c a l roughness element i s nearly constant. They i d e n t i f i e d two l a y e r s from re-examining O'Loughlin's (1965) data: a region below the top of the elements which i s a nea r l y constant s t r e s s l a y e r scale d to the i n t e r v e n i n g surface shear s t r e s s ; and a second region above the elements where the mean v e l o c i t y p r o f i l e i s 285 transformed to a shape r e l a t e d to the t o t a l shear s t r e s s exerted on the roughened surface. Morris (1955) proposed three types of f l o w boundary i n t e r a c t i o n which are c h a r a c t e r i z e d i n Figure 60. The three regions are r e s p e c t i v e l y skimming, wake i n t e r a c t i o n and i s o l a t e d roughness -type flow-boundary exchange. Zone 1, the skimming flow corresponds to the 6-type roughness of Perry et a l ( 1 9 6 9 ) . For t h i s type the f r i c t i o n f a c t o r Reynolds number c h a r a c t e r i s t i c s are i n s e n s i t i v e t o the r e l a t i v e depth s c a l e h/D. This type of roughness i s i d e n t i f i e d as narrow grooves with s t a b l e v o r t i c e s generated by the outer flow e x i s t i n g i n between. K i s t l e r and Tan (1967) made a study of the s t a b i l i t y of these v o r t i c e s while Townes(1965) examined the l i m i t i n g spacing of these bars before the s t a b l e v o r t i c e s became s i g n i f i c a n t l y d i s t u r b e d . Skimming flow can a l s o apply to i n d i v i d u a l blocks when the wake behind each element does not f u l l y develop and the main f l u i d shear occurs above the tops of the elements. Zone I I I corresponds to i s o l a t e d roughness elements and was termed h-D type roughness by Perry et a l ( 1 9 6 9 ) . The drag i s p a r t i t i o n e d between the i s o l a t e d b locks and the i n t e r v e n i n g bed s u r f a c e . Behind each block i s a f u l l y developed wake, but i t s importance i n determining the o v e r a l l shear s t r e s s at the w a l l w i l l depend on the roughness of the i n t e r v e n i n g s u r f a c e . No previous measurements have been made of t h i s c o n d i t i o n , f o r a l l wind tunnel and flume measurements have u n t i l now used a^ smooth w a l l with i s o l a t e d b l o c k s . 5 0: 0.2 0.4 0.5 0.8 1.0 Figure 60 E f f e c t i v e roughness as a function of concentration(Aj. ). kn represents Nikuradse sand size and h symbolizes roughness height. (From Rouse 1965). ro o> 287 Zone I I or h type roughness according t o Perry corresponds to the wake i n t e r a c t i o n flow regime. This type of roughness i s very important as here r e s i s t a n c e to flow approaches a maximum depending on the spacing of the elements. For cubes, the peak r e s i s t a n c e , t r a n s l a t e d i n t o an egu i v a l e n t sand s i z e to roughness height r a t i o , was found t o e x i s t at approximately 15% c o n c e n t r a t i o n based on f r o n t a l area to surrounding area. Other work has aimed at determining the shear s t r e s s a c t i n g on the boundary f o r various d e n s i t i e s of roughness block. Linear w a l l s t r e s s r e l a t i o n s h i p s have been developed by Sayre and Albertson (1961), Dvorak (1969) and Wooding e t al(1973) f o r e g u i l i b r i u m boundary l a y e r s . While e q u i l i b r i u m i s not s t r i c t l y p o s s i b l e f o r h-type roughness as was pointed out by Gartshore (1973} , the boundary l a y e r w i l l be i n approximate e g u i l i b r i u m provided the l e a d i n q edqe of the rouqhness array i s f a r upstream of the re q i o n beinq considered. I s o l a t e g rouqhness elements The study of i s o l a t e d blocks i n a rough t u r b u l e n t boundary l a y e r has r e c e i v e d scant a t t e n t i o n . Despite i t s obvious s i g n i f i c a n c e to m e t e o r o l o g i c a l problems, most wind tunnel experiments have attempted to examine b l u f f bodies i n homogeneous turbulent f l o w s , or at l e a s t over a smooth boundary. Only the recent work by Counihan et al(197'4) has approached the problem i n an a n a l y t i c manner, although 288 s e v e r a l experimental s t u d i e s (see Wind E f f e c t s on B u i l d i n g s Symposia 1967,1969,1971) have been c a r r i e d out on b l u f f bodies with a smooth upstream t u r b u l e n t boundary l a y e r . Counihan's theory leads to the suggestion that the wake of any block can be d i v i d e d i n t o three regions. There i s an e x t e r n a l region i n which the v e l o c i t y f i e l d i s an i n v i s c i d p e r t u r b a t i o n on the i n c i d e n t boundary l a y e r v e l o c i t y which i s assumed to have a power law p r o f i l e . Secondly, there i s a mixing region spreading from the top of the o b s t a c l e where, assuming a l i n e a r eddy v i s c o s i t y i n c r e a s i n g with height can be defined f o r the perturbed f l o w , they show th a t the v e l o c i t y i s of a s e l f - p r e s e r v i n g form: and f i n a l l y there i s a w a l l region where the v e l o c i t y i s shown to be p r o p o r t i o n a l t o l n (y). The maximum increase of the mean square t u r b u l e n t v e l o c i t y i s shown to decay downstream at approximately (x/h) "3/k (where h i s the height of the roughness elements) while the theory suggests the v e l o c i t y d e f i c i t i s a f f e c t e d by the roughness of the surrounding surface i n pro p o r t i o n to l n (h/y) where y i s the roughness l e n g t h . The experimental v e r i f i c a t i o n of t h i s theory l e d to the f o l l o w i n g d e s c r i p t i o n : the v e l o c i t y defect -u and the a d d i t i o n a l turbulence decrease as x/h i n c r e a s e s : the p o s i t i o n of the maximum value of -u occurs at a value of y which i n c r e a s e s as x/h i n c r e a s e s . Also the p o s i t i o n of the maximum i n t e n s i t y of turbulence occurs at values of y>h which suggests t h a t t h i s i n t e n s e turbulence emanates 289 from t h e s h e a r l a y e r t r a i l i n g from t h e t o p o f t h e body. ( C o u n i h a n e t a l 1974, p.549) W h i l e t h i s work was c a r r i e d o u t s p e c i f i c a l l y f o r two d i m e n s i o n a l o b s t a c l e s , n o t a b l y s h e l t e r b e l t s , t h e t h r e e s a l i e n t z o n e s a l s o a p p l y t o t h r e e d i m e n s i o n a l b l o c k s . Work done i n wind t u n n e l s on model f o r e s t s ( K a w a t a n i and Meroney 1968: K a w a t a n i and Sadeh 1971) i n d i c a t e s t h e p r e s e n c e o f l a y e r s e ach s c a l i n g w i t h d i f f e r e n t dynamic p a r a m e t e r s . Below t h e t o p o f t h e r o u g h n e s s e l e m e n t s e x p e r i m e n t a l v e l o c i t y p r o f i l e s have been e m p i r i c a l l y e x p r e s s e d as where U (h) i s t h e v e l o c i t y a t t h e t o p o f t h e e l e m e n t s h, and a i s an e m p i r i c a l l y d e r i v e d c o n s t a n t . D a t a from P l a t e and Q u r a i s h i (1965) a p p e a r t o a g r e e w i t h t h i s f o r m . Such an e m p i r i c a l r e l a t i o n f i t s t h e d a t a w e l l up t o t h e t o p of t h e r o u g h n e s s e l e m e n t s w h i l e C o u n i h a n ' s w a l l l a y e r would be r e s t r i c t e d t o t h e l o w e r h a l f o f t h i s z o n e . However i t i s t o be e x p e c t e d t h a t t h e m i x i n g r e g i o n b e h i n d e a c h b l o c k i n any a r r a y f o r an h t y p e o f r o u g h n e s s would s p r e a d i n t h e v e r t i c a l s o t h a t C o u n i h a n ' s d e f i n i t i o n o f t h e w a l l l a y e r , where t h e v e l o c i t y depends o n l y on d i s t a n c e f r o m t h e b o u n d a r y , would be an a c c u r a t e d e s c r i p t i o n f o r d i s t a n c e s g r e a t e r t h a n a p p r o x i m a t e l y f o u r r o u g h n e s s h e i g h t s back f r o m any s i n g l e b l o c k i n an a r r a y . The m i x i n g l a y e r , o r wake l a y e r , was shown t o e x i s t most m a r k e d l y by Sadeh e t a l (1971). As p o i n t e d o u t e a r l i e r , t h e 290 mean v e l o c i t y w i t h i n the atmospheric boundary l a y e r over very rough s u r f a c e s i s u s u a l l y described by a modified l o g a r i t h m i c law (Eq. A17). I f the displacement t h i c k n e s s i s assumed t o be approximated by the roughness height, f o l l o w i n g P l a t e and Qu r a i s h i (1965), the r e s u l t i n g v e l o c i t y p r o f i l e s show two zones of l i n e a r v a r i a t i o n . Sadeh et al(1971) (see a l s o Figure 14 of t h i s t h e s i s ) suggested that Zone I i n d i c a t e s the extent of the canopy top wake whereas Zone I I i s a f f e c t e d by the t h i c k n e s s of the boundary l a y e r and s c a l e s with u ( and U. * r c o The s e l f - p r e s e r v i n g v e l o c i t y p l o t s of Counihan et al(1974) cannot be d i r e c t l y a p p l i e d i n t h i s context but s u f f i c e t o note that the wake zone, of Sadeh, and mixing zone, of Counihan, have the same s t r e s s balance between i n e r t i a l and shear s t r e s s e s and both are s e l f - p r e s e r v i n g . That means 'the motion at d i f f e r e n t s e c t i o n s d i f f e r s only i n v e l o c i t y and length s c a l e s and are dynamically s i m i l a r i n those aspects of motion c o n t r o l l i n g mean v e l o c i t y and Reynolds s t r e s s ' (Townsend 1956). Thus i n t h i s region the only l e n g t h s c a l e i s the height of the obstacle (and f o r the arrays of elements some measure of t h e i r spacing). Such a view of the behaviour of the inner l a y e r of a rough boundary i s al s o advocated by Wu(1973) who claimed that the height of the .blocks and the f r i c t i o n v e l o c i t y are the le n g t h and v e l o c i t y s c a l e s , i n a s e l f - s i m i l a r p l o t . This region i s c h a r a c t e r i z e d by a very steep v e l o c i t y p r o f i l e , that i s , there i s l i t t l e change i n mean v e l o c i t y i n comparison with the outer l a y e r , p o s s i b l y due to the shedding of eddies from the i s o l a t e d b l o c k s 291 v i g o r o u s l y i n t e r a c t i n g to produce a near constant v e l o c i t y r e g i o n . This review has concentrated on the i n t e r p r e t i v e c o n c l u s i o n s of previous work rather than i t s t h e o r e t i c a l d e r i v a t i o n i n order to develop an understanding of the p o s s i b l e mechanisms by which flows and boundaries i n t e r a c t through the mechanism of turbulence. Besides the work of Sadeh et al{1971) few have s t u d i e d the turbulence s t r u c t u r e and how i t i s a f f e c t e d by d i f f e r i n g roughness a r r a y s . I t i s by the operation of turbulence generation, wake shedding and v o r t i c i t y a m p l i f i c a t i o n that the boundary roughness elements exert an e f f e c t on the flow which r e s u l t s i n the region s described above. 292 APPENDIX I I I TDHBULENCg AND ISOTROPY D e f i n i t i o n For t h i s research turbulence i s most e a s i l y defined i n terms of c e r t a i n p r o p e r t i e s . A f u l l l i s t of the p r o p e r t i e s i s given by Lumley and Panofsky (1964). P l o t s of v e l o c i t y f l u c t u a t i o n s i n d i c a t e that they may be considered as a s t o c h a s t i c process, which i s weakly s t a t i o n a r y with a w e l l defined mean- The process operates i n t h r e e dimensions and p l o t s of the u and v components show no fundamental d i f f e r e n c e s , i f the amplitudes are egual. Because i t i s a random process operating i n three dimensions i t i s a l s o d i f f u s i v e and thus can t r a n s p o r t any s c a l a r . This d i f f u s i v e property i s one of the most important i n t h a t i t i s r e s p o n s i b l e f o r suspension and d i s p e r s i o n of sediment throughout the -flow. Turbulence i s a d i s s i p a t i v e process; unless energy i s c o n t i n u o u s l y e x t r a c t e d from the mean flow the turbulence w i l l decay. E a r l y phenomenoloqical views of turbulence were based on the notion of an eddy v i s c o s i t y . The Navier Stokes equations contained a new term, when expanded f o r (U + u) and subsequently averaged, which could be thought of as e x t r a c t i n g energy from the mean flow and l e a d i n g to the more r a p i d d i f f u s i o n of the r e s u l t i n g momentum d e f i c i t throughouht the flow. Turbulence must also be defined i n terms of a length and 293 a time s c a l e . P l o t s of f l u c t u a t i o n s from stream f l o w , from high v e l o c i t y wind tunnels and the atmospheric boundary l a y e r a l l look remarkably s i m i l a r even though v i s c o s i t y , v e l o c i t y and the magnitude of flow are t o t a l l y d i f f e r e n t . The turbulence can be defined i n terms of the Reynolds number, which i s simply the r a t i o of a macroscale to a m i c r o s c a l e : the macroscale i s l i m i t e d by the s i z e of the experiment to pipe diameter or a f u n c t i o n of boundary l a y e r t h i c k n e s s ; while the microscale i s r e l a t e d to the d i s s i p a t i o n of energy by the a c t i o n of v i s c o s i t y on the l o c a l v e l o c i t y g r a d i e n t . Furthermore, turbulence i s now being examined as a s e l f - s i m i l a r process. By s t r e t c h i n g the time s e r i e s (u (t)) by any amount then by a s u i t a b l e a m p l i f i c a t i o n u (t)P of the amplitude the r e s u l t i n g s e r i e s w i l be s e l f - s i m i l a r with the s e l f - s i m i l a r parameter p. The two s e r i e s w i l l then have the same normalized p r o b a b i l i t y d e n s i t y . The f l u c t u a t i o n s around the mean are regarded as a r e a l i z a t i o n of a s t o c h a s t i c process. The most e x t e n s i v e l y developed theory of s t o c h a s t i c processes deals with the Gaussian p r o b a b i l i t y density which may be thought of as the sum of a l a r g e number of independent events or the outcome of a l a r g e number of independent processes. Turbulence i s made up of many op e r a t i v e processes: vortex s t r e t c h i n g i n three d i r e c t i o n s , v i s c o s i t y operating on v e l o c i t y g r a d i e n t s and eddies of various s c a l e s i n t e r a c t i n g . However the processes are not t o t a l l y independent due t o the n o n - l i n e a r nature of the Navier Stokes equation. The s t a t i s t i c a l parameters that 294 may be developed to describe t h i s s t o c h a s t i c process are l e g i o n but break clown i n t o four major groups according to Bradshaw(1971 p. 23) 1) the s p a t i a l d i s t r i b u t i o n of the t u r b u l e n t k i n e t i c energy; 2) the s c a l e s and r a t e s at which such energy i s being produced, tran s p o r t e d or destroyed at any p o i n t ; 3) the c o n t r i b u t i o n of d i f f e r e n t s i z e s of eddy t o the t u r b u l e n t k i n e t i c energy and Reynolds s t r e s s e s ; 4) the c o n t r i b u t i o n of d i f f e r e n t s i z e s of eddy to the r a t e s mentioned i n 2) and the rat e at which energy i s t r a n s f e r r e d from one range of eddy s c a l e s t o another. S p a t i a l s t r u c t u r e The s p a t i a l d i s t r i b u t i o n of t u r b u l e n t k i n e t i c energy through a boundary l a y e r has been w e l l documented s i n c e the e a r l y work of Klebanoff {1955). I f a two dimensional boundary l a y e r with zero pressure gradient i s assumed, the i n t e n s i t y of turbulence when non-dimensionalized by shear v e l o c i t y w i l l reach an approximately constant value. The magnitude of the i n t e n s i t y w i l l be Reynolds number dependent and a l s o w i l l vary with the roughness of the boundary. Such v a r i a t i o n s are removed by nondimensionalizing p o s i t i o n i n the boundary l a y e r bY u s / v • As Monin and Yaglom(1971) p o i n t out, f a r away from the w a l l the f l u c t u a t i o n s should be independent of v i s c o s i t y , and the c e n t r a l moments ( those independent of the mean v e l o c i t y ) w i l l tend to a constant value as y + , where • y + =yu / v , approaches i n f i n i t y . I f the v e l o c i t y p r o f i l e 295 may be represented by a logarithmic law, then the shear ve l o c i t y may be fu n c t i o n a l l y defined as uv = uf F ( y + ) A26 F'is'equal to unity and the other functional r e l a t i o n s f o r the turbulent k i n e t i c energy terms % = u * F i ( y + ) ° v = u * F 2 ( y + ) A 2 7 become constants p (•) = A P 2(-) = B A28 where A and B are 2.3 and 1.7 (See Wonin and Yaglora 1971, p. 278-280). The s p a t i a l d i s t r i b u t i o n of flu c t u a t i o n s may also be examined by looking at the s p a t i a l c o r r e l a t i o n s . While an analy t i c theory of the behaviour of s p a t i a l c o r r e l a t i o n s was developed for homogeneous turbulence many years ago (Karman and Howarth 1938) their behaviour i n a boundary layer i s s t i l l the subject of speculation. I f measurements are made at two points s u f f i c i e n t l y close together, the c o r r e l a t i o n c o e f f i c i e n t s , w i l l approach unity and as the measurement uv points are moved further apart, i f the time s e r i e s represents a random process, the co r r e l a t i o n should f a l l to -zero _ u u V A 29 q where v i s the velocity at the fixed point p and u i s the P q ft vel turbulent v e l o c i t y at a distance r from p. Scales and r a t e s of production and d i s s i p a t i o n I f t h i s c o r r e l a t i o n c o e f f i c i e n t i s p l o t t e d a g a i n s t 296 s e p a r a t i o n , a length s c a l e may defined by L = fni:L(r) dr A.30 This i s the i n t e g r a l s c a l e of turbulence. Because of the problem of wake i n t e r f e r e n c e i n the streamwise d i r e c t i o n R^(r) i s u s u a l l y found by applying Taylor's hypothesis to the v e l o c i t y at a f i x e d p o i n t . Townsend (1956) argued t h a t motion i n the outer part of the boundary l a y e r where the l o g a r i t h m i c laws apply i s c o n t r o l l e d by a group of eddies very s i m i l a r t o those -known to e x i s t i n wake flows. He presented i n t e g r a t e d transverse c o r r e l a t i o n s Oft J R , , (y:0,r,0) dy / fn dy o x ± o A31 / R T , (y:0,0,r) dy / / u 2 dy A 3 2  0 o as a f u n c t i o n of non-dimensional s e p a r a t i o n . Such curves show that c o r r e l a t i o n s with v e r t i c a l s e p a r a t i o n s are always p o s i t i v e , while those with s e p a r a t i o n i n the h o r i z o n t a l plane d i p below the zero l i n e as r e g u i r e d by c o n t i n u i t y . From the c o r r e l a t i o n curves, another s c a l e may be obtained c a l l e d the Taylor microscale. Hinze(1959,p.36) showed that i t was p o s s i b l e to express the shape of the c o r r e l a t i o n f u n c t i o n f o r a zero separation i n terms of the v e l o c i t y d e r i v a t i v e s at that point by means of a Taylor s e r i e s expansion which r e s u l t e d i n a d e f i n i t i o n of the microscale as 297 y i e l d i n g 1/2 M 3 u/3 x ) 2 / A34 The microscale may be regarded as a measure of the mean square r a t e of change of u and i s u s u a l l y thought of as the average dimension of the smallest eddies. This i s not the case, but r a t h e r X i s the s i z e of the eddies which at the same i n t e n s i t y produce the same d i s s i p a t i o n as the turbulence considered. I t would correspond t o the peak of the d i s s i p a t i o n spectrum discussed below. The r a t e at which energy i s being produced, t r a n s p o r t e d or d i s s i p a t e d at any point i n the flow i s the second group of s t a t i s t i c a l measures to be considered. The d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy occurs because of the i n a b i l i t y of the flow to maintain steep v e l o c i t y g r a d i e n t s . As the random v o r t i c i t y l i n e s become s t r e t c h e d , the v e l o c i t y g r a d i e n t s become steeper u n t i l the a c t i o n of v i s c o s i t y i s s u f f i c i e n t t o reduce the g r a d i e n t . From the energy equation ^ 3 t 3 x j 3 x i 3 x j 3 X i 3 x j 3 x i 3 x i which i s obtained by m u l t i p l y i n g the Reynolds equations by u (a f u l l e x p l a n a t i o n of each terra i s given i n Hinze 1959, p.65), i f we assume i s o t r o p y and consider only the l a s t term which represents the d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy, then • . ' 1 = .— 2 = — 3 and — 1 = — 3 = — 2 5 9 x^> 3 3 x^ ^ X-j^  3 x ^  The expansion of the l a s t term i n the energy equation which 298 represents the d i s s i p a t i o n of tur b u l e n t k i n e t i c energy becomes 2 - 1 c fi u V E - 15v (—) The r e l a t i o n to the microscale i s obvious by simple manipulation 2 e =. 15v -\* The d i s t r i b u t i o n of the microscale across a boundary l a y e r with a rough w a l l was found by Antonia (1973) t o be an i n c r e a s i n g f u n c t i o n with d i s t a n c e while the r e s u l t s of Laufer(1950) i n d i c a t e d only a very weakly i n c r e a s i n g f u n c t i o n i n h i s c o n s i d e r a t i o n of flow i n a two dimensional channel. The r a t e at which energy i s being produced by the working of the Reynolds s t r e s s e s on the mean v e l o c i t y g r adient and the r a t e at which i t i s being transported can best be considered a f t e r d e t a i l e d study of the spectrum of turbulence. Pistes of energy t r a n s f e r : Spectra and Cross-spectra Turbulent motion may be represented ' by the su p e r i m p o s i t i o n of p e r i o d i c eddies, or i n wavenumber space, as waves of d i f f e r e n t length and time s c a l e s with d i f f e r i n g o r i e n t a t i o n s i n space. The t h i r d and f o u r t h groups of s t a t i s t i c a l parameters mentioned, the processes of s t r e s s d i f f u s i o n and of energy transm i s s i o n between various s c a l e s of motion can best be described i n terms of i n t e r a c t i o n s between these eddies. 299 The one dimensional streamwise energy spectrum i s defined so tha t •(k) = i ( j u ( x ) e l k x d x ) ( / u ( x ) e ~ i k X dx) J J S o ft A35 _ u 2 = J%(k) dk 0J A36 where k i s wavenumber and u 2 i s the mean sguare of the v e l o c i t y f l u c t u a t i o n s or the t u r b u l e n t k i n e t i c energy and Ls i s a length s c a l e . For computational purposes i t i s assumed t h a t the v e l o c i t y s i g n a l s can be represented over the data i n t e r v a l by a d i s c r e t e F o u r i e r s e r i e s , t h a t i s u ( x ) Z u k . - j £ sinCkx + e k) a 3 7 where c i s the amplitude of the k Four i e r c o e f f i c i e n t and k Q i s the phase of the k c o n s t i t u e n t , k T h i s d e s c r i b e s the d i s t r i b u t i o n of energy i n e i t h e r wavenumber or freguency space, s i n c e k = 2*n / U and by assuming T a y l o r ' s hypothesis t h i s may be r e l a t e d to a len g t h L = U / n The d e s c r i p t i o n of the spectrum was given c o n s i d e r a b l e t h e o r e t i c a l currency by Kolmogorov, who p o s t u l a t e d t h a t at la r g e Reynolds numbers the motion i s l o c a l l y i s o t r o p i c and independent of the an i s o t r o p y of the l a r g e s c a l e s ,and f u r t h e r , f o r s u f f i c i e n t l y high Reynolds numbers, there i s a sub-range w i t h i n the energy spectrum where the i n e r t i a l 300 t r a n s f e r of energy i s the dominant process- Within t h i s zone the turbulence i s s t a t i s t i c a l l y independent of the energy c o n t a i n i n g eddies and of strong d i s s i p a t i o n . This i n t r o d u c e s the important notion of the energy cascade with l a r g e s c a l e s of motion being c o n s t a n t l y degraded i n t o s maller and s m a l l e r s c a l e s . The 'Kolomogorov hypotheses* have formed the b a s i s of con s i d e r a b l e research s i n c e they were p o s t u l a t e d i n 1941. The f i r s t hypothesis s t a t e s that f o r high Reynolds number, the s t a t i s t i c a l p r o p e r t i e s are uniquely determined by the mean rat e of energy d i s s i p a t i o n per u n i t mass and by kinematic v i s c o s i t y . By dimensional arguments the forms y i e l d e d f o r the energy and d i s s i p a t i o n s p e c t r a are *(k) = U v 5 ) 1 / 2 , F ( k / k s ) A 3 8 and <j> k 2 (k) = k s 2 ( E v 5 ) 1 / i , ( k / k s ) 2 F ( k / k s ) R 3 g r e s p e c t i v e l y where 1/ks i s the Kolmogorov microscale and ks- { e / v 3) V*. For very high Reynolds number F|k/ks| should be a u n i v e r s a l f u n c t i o n . The energy d i s s i p a t i o n spectrum de s c r i b e s the d i s t r i b u t i o n according t o wavenumber of the r a t e of decay of t u r b u l e n t k i n e t i c energy i n t o heat. The second Kolmogorov hypothesis s t a t e s t h a t there w i l l be a range of eddies which are i s o t r o p i c and i n l o c a l steady s t a t e but f o r which v i s c o s i t y i s not important. Here the p r o p e r t i e s of the t u r b u l e n t flow are determined by e alone and by dimensional a n a l y s i s t h i s r e s u l t s i n • Oc) = k ' e 2 / V 5 / 3 A40 which i d e n t i f i e s the i n e r t i a l subrange. Spectral, measurements i n the atmospheric boundary, l a y e r The reasons f o r o u t l i n i n g the st a n d a r d m e t e o r o l o g i c a l methods f o r n o n - d l m e n s l o n a l i z i n g the spectrum i s t h a t i t i s necessary i n comparing r e s u l t s from open channel f l o w s t o c a s t the r e s u l t s i n u n i v e r s a l l y s c a l e d axes i n order t o remove the e f f e c t s of d i s t a n c e from the boundary and the l o c a l mean v e l o c i t y . T h i s procedure should a l l o w d i s t i n c t i o n s to be made between a f r e e s u r f a c e boundary l a y e r and a f u l l y developed e g u i l i b r i u m atmospheric boundary l a y e r f l o w . The i n v e s t i g a t i o n s of s p e c t r a l c h a r a c t e r i s t i c s i n wind t u n n e l s and i n the atmosphere have attempted to o b t a i n s i m i l a r i t y c r i t e r i a so t h a t the s p e c t r a can be made t o c o l l a p s e to u n i v e r s a l curves i r r e s p e c t i v e of h e i g h t o r v e l o c i t y at the measurement p o i n t . By e x t e n s i o n of the Konin-Obukhov s c a l i n g theorem, the p r o p e r l y s c a l e d l o g a r i t h m i c s p e c t r a n ${n) should only be a f u n c t i o n of the reduced freguency f=ny/0 and a s t a b i l i t y parameter, the l a t t e r o f which would not apply, except under c o n d i t i o n s o f high sediment t r a n s p o r t , f o r n a t u r a l channels where th e r e should be no d e n s i t y s t r a t i f i c a t i o n and the flow can be regarded as being i n a ' n e u t r a l ' c o n d i t i o n . Busch{1973), 302 using r e s u l t s from McBean(1970) and Kaimal et a l ( 1 9 7 2 ) , suggested that most near n e u t r a l v e r t i c a l v e l o c i t y s p e c t r a are w e l l described by ( l t l . 5 ( W / ) O T * .„„ where f=yn/U which i s the reduced freguency f o r which n A(n) m has i t s maximum. Values of f vary from 0.2 to 0.6 and those m f o r a from 1 to 5/3. The l o n g i t u d i n a l spectra behave s i m i l a r l y to the v e r t i c a l s p e c t r a though t h e i r shape i s d i f f e r e n t . , The spectra have t h e i r maxima at f=0.05 whi l e Busch and m Panofsky (1968) found f to range from 0.025 to 0.06. They m a l s o found t h a t the u spectra do not obey the Monin-Obukhov hypothesis, f o r a d d i t i o n a l height and t e r r a i n e f f e c t s can be seen when the spe c t r a are p l o t t e d according to s i m i l a r i t y c o r r e l a t e s . But i f the v e r t i c a l s p e c t r a f o l l o w s i m i l a r i t y laws and i f an i n e r t i a l subrange e x i s t s then the h o r i z o n t a l v e l o c i t y spectra must comply with the same s c a l i n g i n t h i s r e g i o n . The c r o s s spectrum between two v e l o c i t y components should a l s o be capable of u n i v e r s a l s c a l i n g . The cross-spectrum i s made up of two p a r t s : the c o i n c i d e n t spectrum and the guadrature spectrum. The i n t e g r a l of the co-spectrum i s p r o p o r t i o n a l to the s t r e s s and represents the d i s t r i b u t i o n of Reynolds s t r e s s according to freguency. I t i s d i f f i c u l t to measure because the c o r r e l a t i o n between v a r i a b l e s i s sometimes very s m a l l and instrument noise 303 becomes s i g n i f i c a n t . The g u a d - s p e c t r u m i s r a r e l y p r e s e n t e d a s i t u s u a l l y makes o n l y a s m a l l c o n t r i b u t i o n t o t h e c o h e r e n c e a n d t h e s m a l l a b s o l u t e m a g n i t u d e i s t h u s s u b j e c t t o c o n s i d e r a b l e e r r o r . The c o s p e c t r a s h o u l d show a -7/3 p o w e r l a w i n t h e i n e r t i a l s u b r a n g e . The d i m e n s i o n a l a r g u m e n t s f o r t h i s a r e p r e s e n t e d by Wyngaard a n d C o t e (1972) a n d e x p e r i m e n t a l v e r i f i c a t i o n was p r o v i d e d by McBean(1970) i n t h e a t m o s p h e r i c b o u n d a r y l a y e r . F o r n e u t r a l s t a b i l i t y , t h a t i s s t r e a m s w i t h no t e m p e r a t u r e o r d e n s i t y s t r a t i f i c a t i o n , t h e c o s p e c t r u m s h o u l d show a f u n c t i o n a l d e p e n d e n c e o f t h e f o r m n C o u v ( n > ^ n ~ V 3 A42 2 C o h e r e n c e i s d e f i n e d a s y2 = Co ( n ) 2 + Q u u y ( f t ) 2 uv <f>u(n) 4> v ( n ) A j , 3 a n d i s r e g a r d e d as a k i n d o f s p e c t r a l c o r r e l a t i o n . As t h e q u a d - s p e c t r u m i s much s m a l l e r t h a n t h e c o - s p e c t r u m , f o r t h e i n e r t i a l s u b r a n g e t h e c o h e r e n c e s h o u l d be s o l e l y a f u n c t i o n o f f r e q u e n c y 2 -4/3 'uv d r o p p i n g r a p i d l y t o z e r o i n t h e i n e r t i a l s u b r a n g e a s s t r i c t l y i s o t r o p y i s d e f i n e d a s uTu" = 0 i ? f j W h i l e t h e s p e c t r a l b e h a v i o u r o f t h e s t r e s s may be p r e d i c t e d on t h e b a s i s o f K o l m o g o r o v t y p e n o t i o n s , any p i c t u r e o f t h e 304 v e r t i c a l s t r u c t u r e must inc l u d e c r o s s s p e c t r a l measurements of the v e l o c i t i e s with v e r t i c a l and l a t e r a l s e p a r a t i o n . The ba s i s f o r t h i s i s Davenport's r e s u l t (1961) that coherence i s a u n i v e r s a l f u n c t i o n of a reduced freguency f'= An/0 where Arepresents probe s e p a r a t i o n . Panofsky and Singer(1965) suggested that i f coherence i s a simple f u n c t i o n of f ' then the g u a n t i t i e s Qu u v/ (f > u <f > v and Conv/^v^y w i l l i n d i v i d u a l l y be simple f u n c t i o n s of f . They s t a t e that t h i s i m p l i e s a kind of geometric s i m i l a r i t y and conclude f o r a l l height d i f f e r e n c e s , one f i n d s g e o m e t r i c a l l y s i m i l a r eddies with the same slopes i n t o the wind and the same r a t i o of h o r i z o n t a l to v e r t i c a l dimensions. (Panofsky and Singer 1965, p.278) One f i n a l parameter c a l c u l a t e d from the cross s p e c t r a l s t u d i e s i s the s p e c t r a l c o r r e l a t i o n c o e f f i c i e n t defined as D / \ Co (n) Ru v( n ) = uv  U u ( n H v ( n ) ) 1 / 2 A 4 4 I f there i s a l o t of t u r b u l e n t motion and l i t t l e t r a n s f e r R w i l l be c l o s e to zero while i f there i s a high r a t e of uv t r a n s f e r R w i l l approach u n i t y ; t h i s term i s thus sometimes c a l l e d the t r a n s f e r e f f i c i e n c y . Under Kolmogorov type assumptions one should expect i t to behave as R < n ) ~ n - 2 / 3 U V 305 I s o t r o p y The c o n s i d e r a t i o n of s p e c t r a l shape so f a r has been pr e d i c a t e d on the notion of l o c a l i s o t r o p y , which i s u s u a l l y taken to e x i s t , i n a c c u r a t e l y , when a spectrum shows a -5/3 sl o p e . The importance of t h i s assumption i s emphasized because i s o t r o p y i s assumed i n the-standard e v a l u a t i o n of the k i n e t i c ener