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Wind tunnel simulations of the atmospheric boundary layer De Croos, Kenneth A. 1977

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W I N D T U N N E L S I M U L A T I O N S O F 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 b y K E N N E T H A . ^ d e C R O O S B . S . M . E . , U n i v e r s i t y o f N o t r e D a m e , 1 9 7 4 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e j r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A D e c e m b e r , 1 9 7 6 © ) Kenneth A. de Croos, 1977 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that publication, i n part or i n whole, or the copying of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Kenneth A. de Croos Department of Mechanical Engineering The University of B r i t i s h Columbia, Vancouver, Canada Date 2-1 - M - 7 7  "Big [fikinli, have. tUXlz wklnZM that izad theMt veJLocLty, LAJXIH wbUit!) havz IQA&QA whihJLb, and io to vi&cotxity." L . F . R i c h a r d s o n ABSTRACT The v e l o c i t y p r o f i l e shape and boundary l a y e r t h i c k n e s s o f an e q u i l i b r i u m boundary l a y e r grown o v e r a l o n g f e t c h o f roughness a r e c l o s e l y matched w i t h t h o s e o f a boundary l a y e r a r t i f i c i a l l y t h i c k e n e d u s i n g s p i r e s (by a d j u s t i n g t h e shape and h e i g h t o f t h e s p i r e s ) . Other t u r b u l e n t c h a r a c t e r i s t i c s o f t h e s e two wind t u n n e l s i m u l a t i o n s o f t h e a t m o s p h e r i c wind a r e t h e n compared. A t the same t i m e , more i n f o r m a t i o n on rough w a l l boundary l a y e r s i s o b t a i n e d t o a l l o w f o r a r a t i o n a l c h o i c e o f t h e shape and s p a c i n g o f roughness elements r e q u i r e d t o produce a p a r t i c u l a r s i m u l a t i o n o f t h e f u l l s c a l e boundary l a y e r . A t e c h n i q u e f o r c a l c u l a t i n g t h e shape o f boundary l a y e r s i n e x a c t e q u i l i b r i u m w i t h t h e roughness beneath, u s i n g a d a t a c o r r e l a t i o n f o r t h e w a l l s t r e s s a s s o c i a t e d w i t h v e r y rough b o u n d a r i e s and a semi-e m p i r i c a l c a l c u l a t i o n method, i s examined e x p e r i m e n t a l l y . W a l l shear s t r e s s , measured d i r e c t l y from a d r a g p l a t e , i combined w i t h boundary l a y e r i n t e g r a l p r o p e r t i e s t o show t h a t t h e shear s t r e s s f o r m u l a i s r e a s o n a b l y a c c u r a t e and t h a t t h e boundary l a y e r grown o v e r a l o n g f e t c h o f roughness i s c l o s e t o e q u i l i b r i u m a f t e r p a s s i n g o v e r a streamwise d i s t a n c e e q u a l t o about 350 t i m e s th e rough-ness element h e i g h t . The boundary l a y e r q u i c k l y g e n e r a t e d u s i n g s p i r e s proved t o be a f a i r a p p r o x i m a t i o n t o t h a t grown o v e r a l o n g f e t c h o f r oughness, but d i d not a c c u r a t e l y r e p r e s e n t t h e l o n g i t u d i n a l t u r b u l e n c e i n t e n s i t y o f t h e f u l l s c a l e a t m o s p h e r i c wind o r t h e n a t u r a l l y grown boundary l a y e r . The boundary l a y e r produced here by s p i r e s showed l i t t l e change i n g r o s s c h a r a c t e r i s t i c s a f t e r t r a v e l -l i n g about e i g h t s p i r e h e i g h t s downstream o f t h e s p i r e s . A d i s t a n c e o f s i x o r seven such h e i g h t s has been a d v i s e d by o t h e r w o r k e r s i n t h e p a s t . i v ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation and thanks to Dr. Ian Gartshore. Only his patient supervision and expert guidance made t h i s work possible. ! Thanks also for the invaluable assistance and encouragement rendered by Ashok Malhotra with the i n i t i a l a n a l y t i c a l considerations, and Rick Edey with the experimental work. This research was funded under National Research Council of Canada Grant A-4308. V TABLE OF CONTENTS Page ABSTRACT i l ACKNOWLEDGEMENTS i v LIST OF TABLES v i LIST OF FIGURES v i i NOMENCLATURE i x Chapter 1. INTRODUCTION 1 2. ROUGH WALL BOUNDARY LAYERS 6 3. EXPERIMENTAL ARRANGEMENTS 2 2 4. RESULTS 27 Naturally Grown Boundary Layer 27 A r t i f i c i a l l y Grown Boundary Layer 30 Comparison Between the Two Simulations and F u l l Scale Data 33 5. CONCLUDING DISCUSSION 37 REFERENCES 4 0 APPENDICES 4 4 TABLES 4 6 FIGURES 4 8 v i LIST OF TABLES T a b l e Page I . Dimensions o f M o d i f i e d H a l f - w i d t h S p i r e s . . . 46 I I . D e f e c t Law I n t e g r a l Parameter and von Kantian's C o n s t a n t f o r t h e N a t u r a l l y Grown Boundary L a y e r 47 v i i L I S T OF F I G U R E S F i g u r e P a g e I. S t r u c t u r e o f t h e r o u g h w a l l b o u n d a r y l a y e r 4 8 I I . R o u g h n e s s e l e m e n t s p a c i n g a n d d i m e n s i o n s . . 49 I I I . S p i r e d i m e n s i o n s 50 2 I V . D r a g v s . pU-^; n a t u r a l l y g r o w n b o u n d a r y l a y e r 51 V . S a m p l e l o g - l o g v e l o c i t y p r o f i l e s . . . . . . . 52 V I . W a l l s h e a r s t r e s s e x p e c t e d f o r v a r i o u s r o u g h n e s s g e o m e t r i e s ( D v o r a k ' s c o r r e l a t i o n ) 5 3 V I I . E f f e c t o f r o u g h n e s s g e o m e t r y o n p r o f i l e s h a p e ( D v o r a k ' s c o r r e l a t i o n ) 54 V I I I . V a r i a t i o n o f g r o s s c h a r a c t e r i s t i c s 6 * , a a n d 9 w i t h d i s t a n c e b e h i n d s p i r e s x ' , f o r t h e a r t i f i c i a l l y g r o w n b o u n d a r y l a y e r . . . . 5 5 I X . S h e a r s t r e s s a s m e a s u r e d u s i n g a s l a n t w i r e 5 6 X . P o w e r s p e c t r a l d e n s i t y o f t h e l o n g i t u d i n a l t u r b u l e n c e - H e i g h t 1 5 " a b o v e t h e f l o o r . . . 57 X I . P o w e r s p e c t r a l d e n s i t y o f t h e l o n g i t u d i n a l t u r b u l e n c e - H e i g h t 8 " a b o v e t h e f l o o r . . . 58 v i i i Figure Page XII. Power spectral density of the longitudinal turbulence - Height 3" above the f l o o r . . . . 59 XIII. Longitudinal turbulence i n t e n s i t y 6 0 ix NOMENCLATURE A,A^,B^,C constants Ap f r o n t a l area of one roughness element Ap e f f e c t i v e f l o o r plan area associated with each roughness element C D drag c o e f f i c i e n t of a square bar B Cp drag c o e f f i c i e n t of one roughness element R e slant wire rms voltage E slant wire mean voltage H shape factor = 6*/6 6 — 6 * unusual shape factor = — ^ — K von Kantian's constant Lu in t e g r a l length scale k roughness element height k^ a constant depending on the aspect r a t i o of a hot wire n frequency, cps Re Reynolds number. 2 S u u(n) power spectrum f t /sec Tu in t e g r a l time scale U l o c a l mean ve l o c i t y , ft/sec U^^ free stream v e l o c i t y or gradient wind, ft/sec U T f r i c t i o n v e l o c i t y , = ITQ rms of the turbulence v e l o c i t y i n the x - d i r e c t i o n , f t / s e c streamwise d i s t a n c e ; d i s t a n c e downstream height above ground roughness l e n g t h , z = Z Q at U = 0 H - 1 shape f a c t o r , = —^ —'< power law exponent a power law i s assumed boundary l a y e r t h i c k n e s s displacement t h i c k n e s s bar spacing; d i s t a n c e between square rough ness bars "equivalent" bar spacing a i r d e n s i t y momentum th i c k n e s s angle of yaw, u s u a l l y 45° f o r a s l a n t wire shear s t r e s s surface shear s t r e s s kinematic v i s c o s i t y 1 CHAPTER 1 INTRODUCTION The a t m o s p h e r i c wind has always had a s t r o n g e f f e c t on man and h i s environment. W i t h i n t h e l a s t f i f t e e n y e a r s o r s o , an o v e r a l l c o n c e r n f o r human c o m f o r t and s a f e t y has l e d t o a need f o r more e x t e n s i v e d a t a on wind c h a r a c t e r i s t i c s i n t h e a t m o s p h e r i c boundary l a y e r . Reasons f o r renewed e n g i n e e r i n g i n t e r e s t i n the s t u d y o f wind e f f e c t s i n c l u d e economic l o s s , and l o s s o f l i f e and l i m b caused by wind d e v a s t a t i o n , problems t h a t c o u l d grow a l a r m i n g l y w i t h urban c o n c e n t r a t i o n and p o p u l a t i o n i n c r e a s e s . E n g i n e e r s need t o o b t a i n more d a t a on f o r c e s on g l a s s and c l a d d i n g i n t a l l s t r u c t u r e s , as w e l l as t h e d e f l e c t i o n s o f t h e s t r u c t u r e s t h e m s e l v e s , caused by the a t m o s p h e r i c wind. A n o t h e r c o n c e r n i s human c o m f o r t i n p e d e s t r i a n a r e a s surrounded by t a l l s t r u c t u r e s , and i n t h e upper r e g i o n s o f t h e s e s t r u c t u r e s . A l s o w i n d - i n d u c e d o s c i l l a t i o n s o f b r i d g e s and t r a n s m i s s i o n l i n e s have t o be p r e d i c t e d . The need t o c o n t r o l a i r p o l l u t i o n has caused i n t e r e s t i n t h e d i s p e r s i v e p r o p e r t i e s o f t h e a t m o s p h e r i c boundary l a y e r . Wind e f f e c t s on n u c l e a r power p l a n t s t r u c t u r e s and l a r g e s o l a r c o l l e c t o r s must be c o n s i d e r e d when d e v e l o p i n g a l t e r n a t i v e energy s o u r c e s . 2 Theoretical methods used to predict these flow processes would involve a combination of meteorology, f l u i d mechanics, and s t r u c t u r a l mechanics, and at present require d r a s t i c assumptions and s i m p l i f i c a t i o n s . An increasingly common and very e f f e c t i v e method of obtaining data on wind ef f e c t s on f u l l scale systems i s to test small scale models i n a reasonable simulation of the lower portion of the neutral atmospheric boundary layer. Under strong wind conditions, the intense turbulent mixing in the wind leads to an adiabatic lapse rate, so that a model boundary layer with isothermal conditions i s suitable for these studies. Various methods are used to obtain a s a t i s f a c t o r y model of t h i s neutral atmospheric boundary layer. One technique i s to grow the boundary layer over a long fetch of roughness elements upstream of the test model, as advocated 1 2 by Davenport and Cermak. The dimensions of each roughness element and the roughness spacing geometry can be altered to produce the required v e l o c i t y p r o f i l e shape. This long fetch of roughness elements i s then representative of the f u l l scale roughness (trees, buildings, etc.) that the atmospheric boundary layer.grows over. The disadvantage with t h i s method i s that i t requires a large wind tunnel with a working section about 80 f t long i n order to produce a boundary layer 2 to 4 feet thick. 3 When i t i s n e c e s s a r y t o produce a boundary l a y e r w i t h i n a v e r y much s h o r t e r d i s t a n c e , v a r i o u s t e c h n i q u e s a r e used t o t h i c k e n t h e boundary l a y e r a r t i f i c i a l l y and shape i t t o t h e d e s i r e d v e l o c i t y p r o f i l e . One method i s t o use a s e t o f m e c h a n i c a l d e v i c e s , such as s p i r e s o r wedges l o c a t e d a t t h e e n t r a n c e t o t h e w o r k i n g s e c t i o n o f 3 4 th e t u n n e l . ' These d e v i c e s , u s u a l l y about as t a l l as the d e s i r e d boundary l a y e r t h i c k n e s s , v a r y i n d e s i g n and shape. O t h e r s p r e f e r t o use j e t s w i t h a i r i n j e c t i o n p e r p e n d i c u l a r t o t h e f l o w , ^ o r upstream o r i e n t e d w a l l 6 7 j e t s , o r a c o m b i n a t i o n o f j e t s w i t h g r i d s . g Campbell and Standen, i n an e x h a u s t i v e s t u d y u s i n g s p i r e s , g r i d s and b a r s o f v a r i o u s d i m e n s i o n s and c o m b i n a t i o n s , found t h a t t r i p s c o u l d be d e s i g n e d f o r r e a s o n a b l e s i m u l a t i o n o f most o f the t u r b u l e n t c h a r a c t e r -i s t i c s o f t h e a t m o s p h e r i c boundary l a y e r . However, t h e s e c h a r a c t e r i s t i c s were not a l l s i m u l t a n e o u s l y s i m u l a t e d by one p a r t i c u l a r d e v i c e o r s e t o f d e v i c e s . F o r example, t h e y found t h a t a c e r t a i n s e t o f s p i r e s produced good s i m u l a t i o n o f : t h e f u l l s c a l e v e l o c i t y p r o f i l e and t u r b u -l e n c e i n t e n s i t y , but t h a t t h e s i m u l a t i o n o f t h e a t m o s p h e r i c power spectrum: was poor. . A c o m b i n a t i o n o f s h o r t s p i r e s and a u n i f o r m mesh g r i d produced good power s p e c t r a , b u t o t h e r c h a r a c t e r i s t i c s , were not p r o p e r l y s i m u l a t e d . The g e n e r a l consensus o f o p i n i o n appears t o be t h a t a 4 complete simulation of the atmospheric wind i n a l l respects i s d i f f i c u l t , and a compromise must be achieved whereby i t i s decided beforehand what aspects of the natural wind need be accurately represented, and a t r i p designed for these purposes. over a long fetch of roughened f l o o r , various methods have been used to scale t h i s model boundary layer with the p a r t i c u l a r portion of the atmospheric boundary layer that i s being simulated. In most instances, the geometry of the roughness elements on the f l o o r has been chosen by t r i a l and error. The shape of the v e l o c i t y p r o f i l e which must be produced by these elements i s known i n broad terms, 9 however. Davenport gives an in d i c a t i o n of the values of the exponent a, for a p a r t i c u l a r t e r r a i n , i f a power law of the form If a thick turbulent boundary layer i s grown U a U , 1 • (1) i s assumed. Here U l o c a l v e l o c i t y U 1 free stream v e l o c i t y or gradient wind z height above ground 6 boundary layer thickness. 5 For example, Davenport s u g g e s t s a power law exponent = 0.16 f o r open g r a s s l a n d , a v a l u e o f a=0.28 f o r f o r e s t and suburban a r e a s , and a=0.40 f o r c i t y c e n t r e s . The o b j e c t i v e o f t h i s work i s t o compare t h e c h a r a c t e r i s t i c s o f two boundary l a y e r s , one produced by a l o n g f e t c h o f roughness and t h e o t h e r by a s e t o f s p i r e s . I d e n t i c a l measurements made under s i m i l a r c o n d i t i o n s would a l l o w f o r a b e t t e r comparison t h a n would measurements made by d i f f e r e n t w o r k ers a t s e p a r a t e f a c i l i t i e s . A t t h e same t i m e , more i n f o r m a t i o n on rough w a l l boundary l a y e r s i s o b t a i n e d t o a l l o w f o r a more r a t i o n a l c h o i c e : o f t h e shape and s p a c i n g o f roughness e l e m e n t s . We f i r s t c o n s i d e r here t h e o r y on rough w a l l boundary l a y e r approaches t o e q u i l i b r i u m . The p r e s e n t e x p e r i m e n t a l arrangements a r e then d e s c r i b e d , f o l l o w e d by a d i s c u s s i o n o f r e s u l t s o b t a i n e d f o r b o t h t h e l o n g f e t c h o f roughness and t h e s h o r t e r f e t c h w i t h s p i r e s , known as t h e ' n a t u r a l l y grown' and ' a r t i f i c i a l l y grown' c a s e s r e s p e c t i v e l y . 6 CHAPTER 2 ROUGH WALL BOUNDARY LAYERS P r a n d t l 1 ^ suggested t h a t t h e f l u i d f l o w around o b j e c t s , o r o v e r a f l a t p l a t e , c o u l d be d i v i d e d i n t o two p a r t s : (1) t h e boundary l a y e r , o f t e n d e f i n e d as t h e r e g i o n i n w h i c h t h e v e l o c i t y o f t h e f l u i d i s l e s s t h a n 99 p e r c e n t o f t h e l o c a l maximum v e l o c i t y o f t h e f l o w (the f r e e s t r e a m v e l o c i t y ) , (2) t h e r e g i o n o u t s i d e t h i s boundary l a y e r , where t h e r e i s u s u a l l y no v o r t i c i t y . The boundary l a y e r i s t h e r e f o r e t h e r e g i o n i n whi c h a l m o s t a l l o f t h e v i s c o u s l o s s e s o c c u r . The v e l o c i t y g r a d i e n t i s h i g h , and t h e sh e a r s t r e s s e s a r e i m p o r t a n t . O u t s i d e t h e boundary l a y e r , w i t h l i t t l e v e l o c i t y v a r i a t i o n , t h e r e s u l t i n g shear s t r e s s e s a r e n e g l i g i b l e . When t h e boundary i s a f l a t p l a t e , a l a m i n a r boundary l a y e r i s formed i n i t i a l l y , w i t h t r a n s i t i o n t o a t u r b u l e n t boundary l a y e r o c c u r r i n g i f t h e p l a t e i s l o n g enough o r i f a s m a l l roughness element o r " t r i p " i s p l a c e d on t h e p l a t e . The s o l u t i o n t o t h e m a t h e m a t i c a l e q u a t i o n s d e s c r i b i n g t h e l a m i n a r boundary l a y e r i s an i n f i n i t e s e r i e s , known as t h e B l a s i u s s o l u t i o n . I f t h e v e l o c i t y p r o f i l e s f o r a l a m i n a r boundary l a y e r i n z e r o p r e s s u r e g r a d i e n t a r e p l o t t e d i n a p p r o p r i a t e non-d i m e n s i o n a l form, o n l y one c u r v e , t h e B l a s i u s p r o f i l e , 7 i s obtained, regardless of the Reynolds number of the flow. In turbulent flow, the time-averaged equations cannot provide a solution for the boundary layer v e l o c i t y p r o f i l e , and various semi-empirical p r o f i l e s are used. One such v e l o c i t y d i s t r i b u t i o n has already been mentioned, the power law, derived from Blasius' resistance formula for smooth pipes, but also applicable to boundary layers i n zero pressure gradient. If the non-dimensionalized v e l o c i t y p r o f i l e s for the smooth wall turbulent boundary layer are plotted, they do not collapse, as was the case with the laminar boundary layer, but instead form a family of p r o f i l e s for varying Reynolds number. Clauser} 1 extend-ing t h i s idea to turbulent boundary layers on rough walls, shows that turbulent boundary layers form families of p r o f i l e s dependent on both the Reynolds number and the degree of roughness on the wall. Another semi-empirical p r o f i l e i s the so-called "law of the wall," derivable by dimensional analysis. For smooth walls, the mean v e l o c i t y i s expected to be of the . form JL = f ( _ _ I ) z U which implies that a plot of (—-y-) vs. (y—) for various T flows w i l l y i e l d a single curve. I f , a small distance away from the wall, the 3U v e l o c i t y gradient -K— i s independent of the v i s c o s i t y as a Z z 9 U well, then (— • -5—) i s a constant and the law of the wall U d Z T can be written i n the usual semi-logarithmic form: 1 z U J L = I ( I_) + A U K v ' T where K and A are constants, the former being c a l l e d 12 'von Kantian's constant'. Nikuradse and others have v e r i f i e d that t h i s law holds for both smooth and rough walls, even for moderate pressure gradients. For f u l l y rough walls, U k the v i s c o s i t y i s no longer important, and the law of the wall rel a t i o n s h i p becomes — = f (-) u V T (where k i s a t y p i c a l roughness height) provided that the region d i r e c t l y affected by i n d i v i d u a l roughness elements z z 1 (j- £ 2) and the region far from the wall {-^  >^  -j) are excluded. Again, i f , away from the immediate wall region 9U 3 z we can take the v e l o c i t y gradient — to be determined by 9 U and z o n l y (not k) , t h e n ~- • i s a constant,' and t h e T U T d Z s e m i - l o g a r i t h m i c r e l a t i o n s h i p becomes i f = \ * n <f> + B l ' x I t can be assumed t h a t t h e v i s c o s i t y i s not i m p o r t a n t f o r t h e o u t e r , f u l l y t u r b u l e n t p o r t i o n o f the boundary l a y e r ; i f the v e l o c i t y d e f e c t (U^-U ) depends o n l y on the shear s t r e s s a t t h e w a l l , the boundary l a y e r t h i c k -ness and t h e d i s t a n c e z; t h e n by d i m e n s i o n a l a n a l y s i s we have g<T> . T h i s i s c a l l e d the v e l o c i t y d e f e c t law, w h i c h i s s a t i s f i e d e x p e r i m e n t a l l y f o r z e r o p r e s s u r e g r a d i e n t f o r the o u t e r p o r t i o n o f t h e boundary l a y e r , independent o f w a l l rough-n e s s . C l a u s e r d e s c r i b e s the t u r b u l e n t boundary l a y e r on a smooth w a l l i n terms o f an o u t e r l a y e r and an i n n e r v i s c o u s s u b l a y e r n e x t t o the w a l l . By a n a l o g y , the rough w a l l boundary l a y e r has an i n n e r r e g i o n w h i c h i s dependent on the i n d i v i d u a l roughness e l e m e n t s , t h e i r shape, s p a c i n g and h e i g h t . O u t s i d e t h i s r e g i o n , i t appears t h a t o n l y 10 t h e e f f e c t i v e w a l l s t r e s s a f f e c t s t h e f l o w , t h e m e c h a n i s m c r e a t i n g t h e s h e a r s t r e s s ( w h e t h e r i t b e v i s c o s i t y o r s u r f a c e r o u g h n e s s ) b e i n g i r r e l e v a n t . T h e s t r u c t u r e o f t h e r o u g h w a l l b o u n d a r y l a y e r t h e r e f o r e i s a s f o l l o w s , a n d i s s h o w n i n F i g u r e I: ( a ) a n i n n e r " r o u g h n e s s " s u b l a y e r d i r e c t l y r e l a t e d t o t h e r o u g h n e s s e l e m e n t g e o m e t r y a n d s h a p e , ( b ) a r e g i o n e x t e n d i n g f r o m t h e t o p o f t h i s i n n e r s u b l a y e r u p t o a h e i g h t o f a b o u t o n e - t h i r d t h e b o u n d a r y l a y e r t h i c k n e s s i n w h i c h t h e l a w o f t h e w a l l i s a p p l i c a b l e , ( c ) t h e o u t e r r e g i o n o f t h e b o u n d a r y l a y e r , w h e r e t h e d e f e c t l a w i s v a l i d . R e g i o n s (b ) a n d ( c ) m a y o v e r l a p , a n d t h e o v e r l a p r e g i o n o f t h e d e f e c t l a w w o u l d t h e n b e l o g a r i t h m i c . R e g i o n s ( a ) a n d ( b ) a r e t o g e t h e r s o m e t i m e s c a l l e d t h e c o n s t a n t s t r e s s l a y e r . A l t h o u g h m e a s u r e m e n t s i n s m o o t h w a l l b o u n d a r y l a y e r s s h o w t h a t t h e s h e a r s t r e s s i s 1 3 r o u g h l y c o n s t a n t f o r z/6 _< 0 . 1 ( s e e H i n z e S e c t i o n 7 . 7 ) , t h e s h e a r s t r e s s i s n o t a c t u a l l y c o n s t a n t t h r o u g h o u t r e g i o n s ( a ) a n d ( b ) . I n r o u g h w a l l b o u n d a r y l a y e r s , t h e i n n e r r e g i o n ( a ) m a y b e l a r g e e n o u g h t o o b s c u r e a n y r e g i o n o f c o n s t a n t s h e a r s t r e s s , a s m e a s u r e m e n t s b y A n t o n i a 14 a n d L u x t o n s u g g e s t . 1 5 D v o r a k m a d e a n e x t e n s i v e r e v i e w o f e x i s t i n g d a t a o b t a i n e d u s i n g s q u a r e t w o - d i m e n s i o n a l b a r r o u g h n e s s 16 elements and used t h e form proposed by C l a u s e r t o r e l a t e bar h e i g h t (k) and s p a c i n g (A) t o t h e boundary l a y e r d i s p l a c e m e n t t h i c k n e s s ( 6 * ) and e f f e c t i v e w a l l shear s t r e s s ( T Q ) as f o l l o w s : U l 1 °1 6* I T = I * n ( t T ' TT> + A " C • • • -(2) T T where U < u i ' :: = f r e e s t r e a m v e l o c i t y — a l r e a d y d e f i n e d i n E q u a t i o n (1) p = d e n s i t y o f f l u i d A = c o n s t a n t ~ 4.8 K = c o n s t a n t , u s u a l l y t a k e n t o be 0.41 6* = d i s p l a c e m e n t t h i c k n e s s , a measure o f boundary l a y e r h e i g h t and C i s a c o n s t a n t depending on A and k as f o l l o w s C = 17.35 (.707 £n ^ - 1) f o r £ < 4.68 . . . . ( 3 ) C = -5.95 (.48 In ^ - 1) f o r £ >_ 4.68 . i < 12 From Equation (3) , the value of C i s a maximum when A/k =4.68. Closer bar spacing merely f i l l s the tunnel, r a i s i n g the e f f e c t i v e f l o o r height by producing greater mutual sheltering between the roughness bars. The Equations (2) and (3) are s t r i c t l y v a l i d U k only for values of —^— >_ 70, the " f u l l y rough" condition, which i s s a t i s f i e d i n most atmospheric applications. They allow the f r i c t i o n v e l o c i t y r a t i o UT/U^ to be obtained for any height r a t i o <5*/k and spacing A/k. However, few rough boundaries can be approximated by the square two-dimensional bars used by Dvorak, and 17 Gartshore generalized the res u l t s to three-dimensional roughness elements of various shapes. This was done by defining an e f f e c t i v e spacing between two-dimensional bars which produces surface drag per unit area (or average shear stress) equal to that of the three dimensional roughness pattern. From t h i s work A e k where Ap i s the e f f e c t i v e plan area associated with each element, and A p the fr o n t a l area. The r a t i o ^ e/k i s the equivalent spacing of two-dimensional bars of height k, which w i l l produce wall shear s t r e s s e q u a l t o t h a t produced by t h e g e n e r a l rough-ness o f h e i g h t k. C D and C Q a r e the d r a g c o e f f i c i e n t s B : R o f a b a r and o f one roughness element r e s p e c t i v e l y measured under i d e n t i c a l c o n d i t i o n s . F or square b a r s and cube shaped roughness w i t h s m a l l upstream boundary l a y e r s (compared t o roughness h e i g h t ) , C ~ C - 1 . 2 (see r e f e r e n c e s 18,19) B R and i t i s assumed here t h a t f o r roughness elements whose f r o n t a l shape i s n e a r l y a s q u a r e , T h e r e f o r e , F C l e a r l y , f o r random r o u g h n e s s , C n A p and A would R ' have t o be s t a t i s t i c a l a v e r a g e s . So now ^ e / k from E q u a t i o n (4) above can r e p l a c e A/k i n Dvorak's r e l a t i o n s and a l l o w a g e n e r a l i z a t i o n o f h i s bar roughness c o r r e l a t i o n t o any f u l l y rough geometry. These r e l a t i o n s a r e v a l i d o n l y f o r roughness p r o v i d e d by e s s e n t i a l l y i s o l a t e d 14 roughness elements on c o m p a r a t i v e l y smooth b o u n d a r i e s . Very c l o s e s p a c i n g between roughness elements would have a mutual s h e l t e r i n g e f f e c t s i m i l a r t o t h a t from e x t r e m e l y c l o s e l y spaced b a r roughness, so t h a t t h e r e s t r i c t i o n A e/k > 5 i s p l a c e d on t h e s e r e l a t i o n s . I n o r d e r t o r e l a t e t h e v e l o c i t y p r o f i l e shape a t o t h e f r i c t i o n v e l o c i t y t h r e e e q u a t i o n s a r e needed t o s o l v e f o r t h e unknowns U , 6* and a. Dvorak uses E q u a t i o n (2) above as t h e f i r s t e q u a t i o n , and t h e momentum i n t e g r a l e q u a t i o n 1 ° ^ ' = M (5) D2 dz . . . . -(5) (where 9 i s t h e momentum t h i c k n e s s r e l a t e d t o 6* and a) as t h e second e q u a t i o n . The t h i r d e q u a t i o n w h i c h he used i s a s t a n d a r d e m p i r i c a l c o r r e l a t i o n known as Head's method. E m p i r i c a l l y , Head d e v i s e s f u n c t i o n s and such t h a t e n t r a i n m e n t r a t e i s r e l a t e d t o t h e u n u s u a l shape f a c t o r 5 - 6 * H l a s f o l l o w s : 1 5 [ V 6 - 6 * ) ] = F 1 ( H 1 > . . . . ( 6 ) w h e r e i s r e l a t e d e m p i r i c a l l y t o t h e u s u a l s h a p e f a c t o r H = 6*/9 b y H 1 = G1 (H) . T h e f u n c t i o n s F ^ a n d a r e s p e c i f i e d b y t h e n u m e r i c a l d e s c r i p t i o n g i v e n i n A p p e n d i x I. T h e b a s i s f o r t h e s e c o r r e l a t i o n s i s e x t e n s i v e w o r k o n s m o o t h w a l l t u r b u l e n t b o u n d a r y l a y e r s , a l s o v a l i d 1 5 f o r r o u g h w a l l c a s e s . F o r Ijv = c o n s t a n t ( z e r o p r e s s u r e g r a d i e n t , E q u a t i o n ( 6 ) a b o v e b e c o m e s 4^ (6 - <5*) = F 1 ( H 1 ) . . . . . ( 7 ) 6 — 6 * S i n c e H.^ i s d e f i n e d a s = — ^ — - , E q u a t i o n ( 7 ) c a n b e w r i t t e n a s ^ (9 Kx) = F1 {E±) . . . . . ( 8 ) Z e r o p r e s s u r e g r a d i e n t s m o o t h w a l l b o u n d a r y l a y e r s c l o s e l y c o n f o r m t o t h e c o n d i t i o n o f e x a c t e q u i l i b r i u m o v e r l i m i t e d s t r e a m w i s e d i s t a n c e s . E q u i l i b r i u m h e r e m e a n s t h a t t h e s t r u c t u r e o f t h e b o u n d a r y l a y e r d o e s n o t c h a n g e s i g n i f i c a n t l y i n t h e s t r e a m w i s e d i r e c t i o n , i . e . , p r o p e r t i e s s u c h a s n o n d i m e n s i o n a l s p e c t r a , t u r b u l e n c e i n t e n s i t i e s , e t c . , r e m a i n t h e s a m e w i t h c h a n g e s i n s t r e a m w i s e d i s t a n c e 2 1 x . E x a c t e q u i l i b r i u m , a l s o k n o w n a s s e l f - p r e s e r v a t i o n , i s o n l y p o s s i b l e f o r z e r o p r e s s u r e g r a d i e n t r o u g h w a l l b o u n d a r y l a y e r s i n r a t h e r e x t r e m e c a s e s : e i t h e r t h e r o u g h -22 n e s s e l e m e n t s a r e h i g h c o m p a r e d t o t h e i r s p a c i n g o r e l s e t h e r o u g h n e s s h e i g h t a n d s p a c i n g c h a n g e w i t h s t r e a m w i s e d i s t a n c e x s u c h t h a t t h e y r e m a i n a c o n s t a n t f r a c t i o n o f 2 3 t h e b o u n d a r y l a y e r t h i c k n e s s . Z e r o p r e s s u r e g r a d i e n t b o u n d a r y l a y e r s d e v e l o p -i n g o v e r a c o n s t a n t r o u g h n e s s g e o m e t r y w i t h A / k ^_ 5 c o n f o r m a p p r o x i m a t e l y t o t h e c o n d i t i o n o f e q u i l i b r i u m a f t e r t r a v e l -l i n g a f a i r l y l o n g d i s t a n c e d o w n s t r e a m o f t h e i r o r i g i n s . E s t i m a t e s o f t h i s d i s t a n c e v a r y f r o m 3 2 0 t i m e s t h e r o u g h n e s s 24 2 5 h e i g h t t o 1 0 0 0 t i m e s t h e r o u g h n e s s h e i g h t . W h a t t h i s m e a n s i s t h a t b o u n d a r y l a y e r c h a r a c t e r i s t i c s s u c h a s H , H-^ a n d a a r e n o t c h a n g i n g s i g n i f i c a n t l y w i t h c h a n g e i n t h e s t r e a m w i s e d i s t a n c e x . I n t h i s c a s e , t h e d i f f e r e n -t i a l E q u a t i o n s ( 5 ) a n d (8 ) b e c o m e a r i t h m e t i c . F o r e q u i l i -b r i u m , E q u a t i o n ( 8 ) b e c o m e s 17 or, de F i ( H i > dx E1 Since H 1 . = G1 (H) , d6 dx function (H) d 6 and i s therefore a constant since H i s a constant, dx H—1 Now we define a new shape factor a = (—j-). This shape factor i s equal to the exponent of the power law (Equation(1)) i f a power law i s assumed. If a power law i s not a good description of the v e l o c i t y p r o f i l e , then a i s simply a non-dimensional description of the boundary layer shape. Now, since (H - 1) a = ~ --=— = function (a) dx and by comparison with Equation (5) the r e s u l t i s : U 4 = function (a) . . . . (9) U l Equation (9) shows that =7- l s a constant for any one equi-librium case ( i . e . , one shape of boundary l a y e r ) , which i s s t r i c t l y possible from Equations (2) and (3) only when Ae/k and 6*/k remain constant, as already mentioned. Using Equation (2), Equation (9) above can be written as 6* a = f n [ X' IT ] ' • • • - d O ) It must again be emphasized here that the shape factor a as used here i s simply defined by where 6* H =_ T • . . . . ( 1 1 ) Also, since x ",!<rfs>' <»» can be used to define a nominal boundary layer thickness 6, equal to the actual boundary layer thickness i f a power law p r o f i l e e x i s t s , Equation (10) can be written as a = fn [ §, ^ ] . (14) 19 E q u a t i o n (14) above i s v a l i d o n l y f o r c o n d i t i o n s a t o r near e q u i l i b r i u m . The r a t i o U,6*/U 6 i s r e l a t e d t o the d e f e c t law, 1 ' x ' 16 as p o i n t e d out by C l a u s e r . By d e f i n i t i o n , 6* = r ( i - TT-) dz . 0 u l (15) I f t h e d i s p l a c e m e n t t h i c k n e s s as g i v e n i n E q u a t i o n (15) above i s combined w i t h t h e d e f e c t law, w h i c h i s U1 - u T = f n (z/6) i t i s e a s i l y shown t h a t : 6* U l r l U, - u 0 °x d ( z / 6 ) f (z/6) d ( z / 6 ) = c o n s t a n t . The p r e s e n t e q u i l i b r i u m c a l c u l a t i o n s from Head's method g i v e a v a l u e o f t h i s parameter (U^6*/U T6) between 3.80 and 3.99, compared t o C l a u s e r ' s e x p e r i m e n t a l v a l u e o f 3.60 f o r boundary l a y e r s d e v e l o p i n g over c o n s t a n t roughness. 20 V a r i o u s methods have been used by o t h e r w o r k e r s t o measure t h e f r i c t i o n v e l o c i t y r a t i o U^/U^, r e l a t e d t o the w a l l shear s t r e s s by U x  U l and t h r e e o f t h e s e methods a r e d e s c r i b e d h e r e : 1. Shear s t r e s s can be o b t a i n e d d i r e c t l y i f t h e d r a g on a s e c t i o n o f t h e f l o o r (a d r a g p l a t e ) i s c a r e f u l l y measured. 2. Shear s t r e s s can be i n f e r r e d i f a l o g law o f the form IT = I *n <f> • • • • <16> i s assumed. T h i s t e c h n i q u e n e c e s s i t a t e s assuming some f i g u r e f o r t h e von Karman c o n s t a n t , u s u a l l y t a k e n t o be 0.41, an assumption w h i c h appears t o be a t b e s t a rough a p p r o x i m a t i o n . 26 Wooding e t a l . p o i n t o u t t h a t t h i s v a l u e o f 0.41 was o b t a i n e d f o r c l a s s i c c y l i n d r i c a l r o u g h -ness shapes. U s i n g a wide range o f d a t a g a t h e r e d u s i n g roughness elements o f many shapes and s i z e s , t h e y deduce t h a t t h e v a l u e o f t h e von Karman c o n s t a n t may l i e between 0.25 and 0.41, depending on the t y p e o f roughness. These v a l u e s w i l l be d i s c u s s e d i n the l i g h t o f new e x p e r i m e n t a l d a t a i n Chapter 4. 3. A n o t h e r method f o r measuring shear s t r e s s t h a t appears t o have p a r t i c u l a r m e r i t i s the s l a n t e d h o t - w i r e anemometer t e c h n i q u e used 27 by P a t e l , and d e s c r i b e d i n Appendix I I . Of c o u r s e , t h e s l a n t w i r e cannot be used v e r y c l o s e t o t h e f l o o r because o f t h e roughness e l e m e n t s , but the c o n s t a n t s t r e s s r e g i o n a l r e a d y d i s c u s s e d i s e x p e c t e d , and s h o u l d s u f f i c e t o g i v e a c l e a r i n d i c a t i o n o f t h e v a l u e o f t h e w a l l shear s t r e s s T Q. A l l t h r e e o f t h e s e measurement t e c h n i q u e s have been used here and t h e r e s u l t s . f r o m a l l o f them w i l l be d i s c u s s e d i n Chapter 4. 22 CHAPTER 3 EXPERIMENTAL ARRANGEMENTS The U.B.C. wind tunnel used to gather experi-mental data i s an open c i r c u i t , blower type tunnel 8 f t wide and i n i t i a l l y 5 f t 2 i n high with a test section 80 f t long. The area contraction r a t i o (ratio of the area of the entrance to the contraction to that of the entrance to the working section) i s 4:1. The test section roof can be adjusted to maintain ambient room pressure throughout the t e s t section, implying zero pressure gradient along the tunnel. Pressure taps located at 8 f t in t e r v a l s along the back wall of the tunnel were connected to a multitube manometer for t h i s purpose. For the "naturally grown" boundary layer, the fl o o r of the working section of the tunnel was covered with uniformly spaced roughness element s t r i p s 1.5 i n high, 0.75 i n wide and 0.041 i n thick, as shown i n Figure II. The s t r i p s were placed 6 in.apart from each other in l i n e s across the tunnel. The l i n e s were also 6 i n . apart from each other, and a staggered pattern was used as shown, so that no roughness element was d i r e c t l y behind another roughness element i n the row ahead of i t . This y i e l d s a v a l u e o f A / A o f 32 f o r E q u a t i o n ( 4 ) , a n d t h e a s s u m p t i o n t h a t t h e d r a g c o e f f i c i e n t s a r e e q u a l g i v e s a v a l u e o f ^ e / k = 32 f r o m E q u a t i o n ( 4 ) . F o r e x p e r i m e n t a l c o n v e n i e n c e , m e a s u r e m e n t s w i t h d i f f e r e n t l e n g t h s o f r o u g h n e s s f e t c h w e r e m a d e a t a f i x e d p o i n t 12 f t u p s t r e a m f r o m t h e o p e n e n d o f t h e w i n d t u n n e l . A d d i n g o r s u b t r a c t i n g r o u g h n e s s s e c t i o n s u p s t r e a m o f t h e f i x e d p o i n t o f m e a s u r e m e n t t h e n y i e l d e d t h e r e q u i r e d r o u g h n e s s f e t c h . 14 A n t o n i a a n d L u x t o n a l s o f o u n d t h a t t h e d i s -t u r b a n c e i n t r o d u c e d b y t h e f i r s t r o u g h n e s s e l e m e n t h a d a s i g n i f i c a n t e f f e c t o n t h e f l o w d o w n s t r e a m . I n o r d e r t o r e d u c e t h e i m p o r t a n c e o f t h i s f a c t o r , a n d a l s o t o r e d u c e t h e i m p o r t a n c e o f a n y l e n g t h o f s m o o t h f l o o r u p -s t r e a m o f t h e f i r s t r o u g h n e s s e l e m e n t s , a w e d g e a s h i g h a s t h e r o u g h n e s s e l e m e n t s (1 1/2 i n ) w a s p l a c e d a c r o s s t h e t u n n e l . T h i s w e d g e , o f 8 i n c h o r d a n d a s w i d e a s t h e t u n n e l , t h e n c r e a t e d a t w o - d i m e n s i o n a l r a m p a n d t h e n a b a c k w a r d f a c i n g s t e p . T h i s w e d g e i s r e f e r r e d t o i n l a t e r d i s c u s s i o n a s t h e ' t r i p ' f o l l o w i n g c o m m o n u s e o f t h e t e r m . M e a n v e l o c i t y a n d r m s m e a s u r e m e n t s w e r e m a d e u s i n g a l i n e a r i z e d h o t w i r e i n c o n j u n c t i o n w i t h a D I S A t y p e 5 5 D 0 1 a n e m o m e t e r , a D I S A t y p e 5 5 D 1 0 l i n e a r i z e r , a n d a D I S A t y p e 5 5 D 2 5 a u x i l l i a r y u n i t . A K r o h n - H i t e M o d e l 335 low pass f i l t e r * w a s used t o reduce e l e c t r o n i c n o i s e . Drag measurements were made u s i n g a d r a g p l a t e , a f l o a t i n g i s o l a t e d s e c t i o n o f rough f l o o r , about 7' x 3 ' , f a s t e n e d t o an a c c u r a t e wind t u n n e l b a l a n c e . The gap around t h e d r a g p l a t e was c a r e f u l l y made as. s m a l l as p o s s i b l e , b u t n o t so s m a l l as t o a f f e c t t h e r e a d i n g s by c a u s i n g c o n t a c t a t h i g h wind speeds. 2 8 The method o f Cowdrey was f o l l o w e d f o r the s p i r e d e s i g n , w i t h m o d i f i c a t i o n s as made by Campbell and g Standen. The s p i r e s were d e s i g n e d t o produce a boundary l a y e r a t about e i g h t s p i r e h e i g h t s downstream t h a t would match t h e measured measured shape f a c t o r a and measured boundary l a y e r t h i c k n e s s 6 o f t h e " n a t u r a l l y grown" boundary l a y e r w h i c h had been o b s e r v e d 61.6 f t downstream o f t h e t r i p . I t was d e c i d e d t o d e s i g n a s i m p l e a r r a y o f f l a t wooden s p i r e s u s i n g 1/4 i n plywood, t h e advantage w i t h t h i s c o n s t r u c t i o n b e i n g t h a t t h e s p i r e s c o u l d be made s l i g h t l y o v e r s i z e d , and t h e n s m a l l changes t o t h e i r p r o f i l e made u n t i l t h e r e q u i r e d s p e c i f i c a t i o n s were met downstream. Campbell and Standen found t h a t t h e b e s t v e l o c i t y p r o f i l e was g e n e r a t e d by m o d i f i e d h a l f - w i d t h s p i r e s — t h a t i s , w i t h s p i r e w i d t h e q u a l t o h a l f t h e boundary l a y e r t h i c k n e s s . B e f o r e m o d i f i c a t i o n o f t h e s e s p i r e s , Cowdrey 1 s d e s i g n was f o l l o w e d by Campbell and Standen w i t h t h e r e s u l t t h a t the * F i l t e r s e t t i n g e q u a l s 20 kHz. 25 spires were very sharply spiked at the top.; As a r e s u l t , a boundary layer thickness f i v e - s i x t h s the height of the spires was obtained by these workers. Hence, one of the modifications made by Campbell and Standen was to con-struct the modified half-width spires t a l l e r than the required boundary layer thickness. For the purposes of t h i s program, the design of Cowdrey, with the modifications made by Campbell and Standen, was followed. Also a further modification was added here, because the present plywood spires could not be very sharply spiked. The spires were designed to produce the desired shape factor a and boundary layer thickness 6, and were then smoothly rounded o f f at a height equal to about 0.85 6 , where the spire width was very small. The d i s t i n c t i o n between the actual spire height (about 0.85 6 ) and the design spire height (about 1.10 6) must be pointed out, and i s i l l u s t r a t e d i n Figure I I I . C l e a r l y , the boundary layer thickness downstream of the present spires would be larger than the actual spire height, i n contrast to the res u l t s of Campbell and Standen. A short fetch of the same roughness as was used i n the case of the naturally grown boundary layer was placed downstream of the spires. Drag measurements were made as before, using the drag plate. Again, for experi-mental convenience, data was taken at a fixed point, and 26 m e a s u r e m e n t s a t d i f f e r e n t d i s t a n c e s b e h i n d t h e s p i r e s w e r e m a d e b y m o v i n g t h e s p i r e s u p o r d o w n t h e t u n n e l , a n d a d d i n g o r s u b t r a c t i n g r o u g h n e s s s e c t i o n s b e h i n d t h e m . T h u s , i n t h e c a s e s o f b o t h n a t u r a l l y g r o w n a n d a r t i f i c i a l l y g r o w n b o u n d a r y l a y e r s t h e m e a s u r e m e n t s d o n o t r e l a t e p r e c i s e l y t o a s i n g l e b o u n d a r y l a y e r , s i n c e u p -s t r e a m o f e a c h f e t c h o f r o u g h n e s s o f s t r e a m w i s e l e n g t h x t h e r e i s a v a r i a b l e l e n g t h o f s m o o t h f l o o r e q u a l t o t h e t u n n e l l e n g t h m i n u s x . I n b o t h c a s e s , a u t o - c o r r e l a t i o n s o f t h e l o n g i -t u d i n a l t u r b u l e n c e v e l o c i t y a t v a r i o u s h e i g h t s a b o v e t h e f l o o r w e r e o b t a i n e d a t t h e d i s t a n c e s d o w n s t r e a m w h e r e t h e v e l o c i t y p r o f i l e s w e r e m a t c h e d , u s i n g a P A R M o d e l 1 0 1 C o r r e l a t i o n F u n c t i o n C o m p u t e r i n c o n j u n c t i o n w i t h t h e D I S A i n s t r u m e n t s a l r e a d y m e n t i o n e d . F o u r i e r a n a l y s i s o f t h e s e f u n c t i o n s w e r e m a d e u s i n g a P A R M o d e l 1 0 2 F o u r i e r A n a l y s e r . A l l o f t h e s e c o r r e l a t i o n s a n d F o u r i e r a n a l y s e s w e r e f o u n d t o b e r e p e a t a b l e . T h e R e y n o l d s n u m b e r a t w h i c h t h e s e t e s t s w e r e 4 U l k m a d e w a s o f t h e o r d e r o f 3 x 1 0 , w h e r e R e = . 27 CHAPTER 4 RESULTS The Naturally Grown Boundary Layer Drag measurements were made using the drag plate for a wide range of wind v e l o c i t i e s , at various distances downstream of the t r i p . Drag readings were found to be sensitive to tunnel pressure gradient, which was set as accurately as possible to zero by adjusting the tunnel roof. Drag readings were found to be accurately propor-t i o n a l to the square of the free stream v e l o c i t y , as expected, indicating that no Reynolds number e f f e c t s were present. This i s i l l u s t r a t e d i n Figure IV. For c l a r i t y , data points are shown on only one l i n e . A l l other l i n e s were as well described by the experimental points, and a l l values were found to be repeatable. Values of 5* and 9 were obtained by integration from the measured mean v e l o c i t y p r o f i l e s , using the rela t i o n s 6* 0 (1 . (15) and 9 (1 dz . (17) 28 a a n d 6 w e r e t h e n d e d u c e d f r o m 6* a n d 6 a s d e s c r i b e d p r e -v i o u s l y i n E q u a t i o n s ( 1 1 ) a n d ( 1 2 ) . T h e v e l o c i t y d i s t r i b u t i o n s w e r e n o t p a r t i c u l a r l y w e l l d e s c r i b e d b y t h e r e l a t i o n i n E q u a t i o n ( 1 ) , b u t t h e v a l u e s o f : 6 e s t i m a t e d d i r e c t l y f r o m t h e v e l o c i t y p r o f i l e s w e r e c l o s e t o t h e v a l u e s o b t a i n e d b y t h e m o r e p r e c i s e i n t e g r a l m e t h o d . A n e x a m p l e o f t h e p o w e r l a w p r o f i l e p l o t t e d i n t h e f o r m l o g ( U / U ^ ) v s . l o g ( z / 6 ) , s o t h a t a i s d e d u c e d d i r e c t l y f r o m t h e s l o p e o f t h e p l o t , i s s h o w n i n F i g u r e V . M e a s u r e d v a l u e s o f 6* a n d U /U-^ a r e p l o t t e d i n F i g u r e V I , a l o n g w i t h D v o r a k ' s c o r r e l a t i o n o f E q u a t i o n ( 2 ) . T h e m e a s u r e d v a l u e s i n F i g u r e V I a g r e e w i t h t h e t r e n d o f t h e t h e o r y , a n d i n d i c a t e a n e f f e c t i v e v a l u e o f A e / k o f a b o u t 3 0 , w h i c h i s c l o s e t o t h e g e o m e t r i c a l v a l u e o f 32 o b t a i n e d b y E q u a t i o n ( 4 ) u s i n g t h e a s s u m p t i o n t h a t t h e d r a g c o e f f i c i e n t s a r e e q u a l . T h u s , t h e r e i s s o m e v e r i f i -c a t i o n f o r D v o r a k ' s s h e a r s t r e s s c o r r e l a t i o n f o r t w o -d i m e n s i o n a l b a r s a n d i t s e x t e n s i o n t o t h r e e - d i m e n s i o n a l r o u g h n e s s e l e m e n t s . I n F i g u r e V I I , w h e r e t h e s h a p e f a c t o r a i s p l o t t e d a g a i n s t 6 , t h e c a l c u l a t e d c u r v e s a s s u m e e q u i l i b r i u m c o n d i t i o n s , i . e . , t h a t t h e s h a p e f a c t o r i s n o t c h a n g i n g s i g n i f i c a n t l y i n t h e s t r e a m w i s e d i r e c t i o n . A s c a n b e s e e n , t h e m e a s u r e d v a l u e s a p p r o a c h t h e e x p e c t e d e q u i l i b r i u m , a n d a r e w e l l r e p r e s e n t e d b y t h e e q u i l i b r i u m c u r v e o f 29 A Q = 3 0 f o r t h e t w o l a r g e s t v a l u e s o f x / k , i . e . , x / k = 3 9 6 . 8 a n d x / k = 4 9 2 . 8 . T h i s t r e n d t o w a r d s e q u i l i b r i u m a t l a r g e v a l u e s o f x / k i s a l s o r e p r e s e n t e d b y t h e c h a n g e i n t h e i n t e g r a l d e f e c t l a w p a r a m e t e r U ^ 6 * / U T 6 , g i v e n i n T a b l e I I . I t i s s e e n t h a t v a l u e s o f t h i s q u a n t i t y d e c r e a s e f r o m a v a l u e o f 4 . 2 6 a t t h e s m a l l e s t v a l u e o f x / k t o a v a l u e o f 3 . 9 6 a t x / k = 4 9 2 . 8 . A s s u m i n g e x a c t e q u i l i b r i u m , t h i s v a l u e i s 3 . 9 9 f o r X / k = 30 a n d a = 0 . 2 6 . e F r o m t h e s e f i g u r e s i t c a n b e c o n c l u d e d t h a t D v o r a k ' s c o r r e l a t i o n o f s h e a r s t r e s s w i t h i n t e g r a l b o u n d a r y l a y e r p r o p e r t i e s a n d r o u g h n e s s g e o m e t r y i s a u s e f u l c o r r e l a t i o n o f r o u g h w a l l b o u n d a r y l a y e r c h a r a c t e r i s t i c s . F r o m F i g u r e V I I a n d t h e d e f e c t l a w p a r a m e t e r U1 6 * / U TS we c o n -c l u d e t h a t t h i s r o u g h w a l l b o u n d a r y l a y e r a p p r o a c h e s e q u i l i b r i u m a t a v a l u e o f x / k * 3 5 0 . T h i s i s i n a g r e e m e n t 24 w i t h t h e v a l u e o f x / k ~ 3 2 0 o b t a i n e d b y A n t o n i a a n d L u x t o n , a n d s u g g e s t s t h a t e q u i l i b r i u m c a n b e a p p r o a c h e d q u i t e c l o s e l y 2 5 f o r x / k l e s s t h a n 1 0 0 0 , t h e v a l u e s u g g e s t e d b y C o u n i h a n , f r o m a n e x t r a p o l a t i o n o f n o n - e q u i l i b r i u m d a t a . I n a d d i t i o n t o t h e d r a g p l a t e m e t h o d , s h e a r s t r e s s o n t h e f l o o r c a n b e i n f e r r e d f r o m t h e l o g l a w , a s p r e v i o u s l y d i s c u s s e d i n C h a p t e r 2 , w i t h s o m e u n c e r t a i n t y , a s t o w h a t v a l u e m u s t b e a s s u m e d f o r t h e v o n K a r m a n c o n s t a n t . 30 If the c l a s s i c a l value of K = 0.41 i s assumed i n the equation jj- = i in (^ ) . . . . (16) T values of U^ /u" were found from the v e l o c i t y p r o f i l e s which were much larger than those obtained d i r e c t l y by the drag plate method. Since t h i s suggested that the value of K was too large, the values of U T/ U2. obtained by the drag plate method for the various distances downstream from the t r i p were substituted i n Equation (16). This, together with plots of the mean v e l o c i t y p r o f i l e , yielded values between 0.33 and 0.38 for von Karman's constant, as shown in Table I I . A r t i f i c i a l l y Grown Boundary Layer The naturally grown boundary layer apparently reaches equilibrium i n these tests for x/k _< 4 00. We now wish to compare t h i s equilibrium boundary layer with one grown a r t i f i c i a l l y , that i s , using spires, and developed i n a shorter length of wind tunnel. For t h i s purpose spires were designed as described i n Chapter 3 to produce a boundary layer at about eight design spire heights (h) downstream whose gross shape (a) and thickness (<5) were 31 the same as the corresponding values measured i n the equilibrium naturally grown case at x/k ~ 4 00. Comparisons of wall stress and turbulent c h a r a c t e r i s t i c s ; i . e . , i n t e n s i t i e s , length scales, spectra etc., could then be made between the two cases. As before, drag measurements were made using the drag plate. Roof adjustments were c a r e f u l l y made to maintain ambient room pressure throughout the length of the tunnel. Once again, drag readings were found to be repeatable, and proportional to the square of the free stream v e l o c i t y , i n d i c a t i n g the absence of Reynolds number e f f e c t s . V e l o c i t y p r o f i l e s were taken at various d i s -tances behind the spires, and as i n the case of the naturally grown boundary layer, values of 6* and 9 were obtained by integration using Equations (15) and (17). Values of a and 6 were then deduced from 6* and 9 using Equations (11) and (12) , as before, and are plotted i n Figure VIII. The measured values i n Figure VIII show that beyond a distance of eight design spire heights, the values of 6*, 9-and a remain roughly constant. With regard to the streamwise gradients i n the flow, t h i s distance appears preferable to the six spire heights commonly used, at least as far as the present spire 32 design i s concerned. It i s not clear from Figure VIII whether equilibrium has been reached at the point where a close match was attempted with the naturally grown equilibrium boundary layer. The measured value of the defect law parameter at t h i s point i s equal to 4.12, compared with the calculated equilibrium value of 3.99, and the value of 3.96 for the natu r a l l y grown boundary layer at x/k s 400. Measured values of 6*, a, 6 and U /U, for the T 1 a r t i f i c i a l l y grown boundary layer are also plotted i n Figures VI and VII for x * 8 design spire heights, the point where a match with the equilibrium n a t u r a l l y grown boundary layer was attempted. In Figure VI, a value of Xfi/k greater than 30, i n fact, closer to 4 0 i s suggested. In Figure VII, where the plotted curves were calculated using the assumption of equilibrium, the value of X/k suggested i s very close to 30. However, t h i s i s not s i g n i f i c a n t , since the spires were designed to produce these very values of a and 6. Equilibrium cannot be assessed by the values i n Figure VII alone since the shear stress r e l a t i o n plotted i n Figure VI must also be v a l i d for the equilibrium;plots of Figure VII to be v a l i d . 33 C o m p a r i s o n B e t w e e n t h e Two S i m u l a t i o n s a n d W i t h F u l l  S c a l e C o n d i t i o n s T h e s h e a r s t r e s s m e a s u r e m e n t s i n t h e a i r u s i n g t h e s l a n t w i r e t e c h n i q u e a r e s h o w n i n F i g u r e I V f o r e a c h c a s e a t t h e s t r e a m w i s e p o s i t i o n s w h e r e a a n d 6 w e r e 2 m a t c h e d . T h e v a l u e s o f x / p l i i n e a c h c a s e a p p r o a c h t h e v a l u e o n t h e f l o o r a s c a l c u l a t e d u s i n g t h e d r a g p l a t e m e t h o d . N e i t h e r o f t h e t w o s i m u l a t i o n s e x h i b i t a r e g i o n o f c o n s t a n t s h e a r s t r e s s e x t e n d i n g u p w a r d s f r o m t h e 1 3 f l o o r . H o w e v e r , a s s h o w n b y H i n z e t h e s h e a r s t r e s s i s c o n s t a n t o n l y i n t h e r e g i o n x/6 < 0 . 1 , a n d n o m e a s u r e m e n t s w e r e t a k e n b e l o w x/6 ~ 0 . 1 h e r e . T h e s l a n t w i r e w a s a l s o u s e d t o m e a s u r e t h e s h e a r s t r e s s a t d i s t a n c e s b e y o n d e i g h t d e s i g n s p i r e h e i g h t s b e h i n d t h e s p i r e s , w h e r e 6*, G a n d a r e m a i n r o u g h l y c o n s t a n t . I t w a s f o u n d t h a t t h e s i n g l e p o i n t s h o w n o n F i g u r e V I i s r e p r e s e n t a t i v e o f t h e a r t i f i c i a l l y g r o w n b o u n d a r y l a y e r m o r e t h a n e i g h t d e s i g n s p i r e h e i g h t s d o w n s t r e a m f r o m t h e s p i r e s . A u t o - c o r r e l a t i o n s o f t h e l o n g i t u d i n a l t u r b u l e n c e v e l o c i t y f r o m t h e t w o s i m u l a t i o n s w e r e t a k e n a t h e i g h t s ( z ) o f 2 k , 5 . 3 k a n d 1 0 k . F o u r i e r a n a l y s i s o f t h e s e f u n c t i o n s w e r e a l s o m a d e . P o w e r s p e c t r a l d e n s i t i e s o f t h e l o n g i t u d i n a l t u r b u l e n c e f r o m t h e t w o s i m u l a t i o n s a t e a c h h e i g h t p l o t t e d i n t h e f o r m 34 n S ( n ) u u TJ2 v s . n • T u a r e s h o w n i n F i g u r e s X t o X I I , a l o n g w i t h t h e w e l l k n o w n v o n K a r m a n s p e c t r u m , o f t h e f o r m w h e r e n S ( n ) u u Tl2 4 n • T u [ 1 + 7 0 . 7 ( n - T u ) 2 ] 5 / 6 w h e r e n S u u < n > T u = f r e q u e n c y i n H z . 2 = p o w e r s p e c t r u m f t / s e c = i n t e g r a l t i m e s c a l e x L u U L u = i n t e g r a l l e n g t h s c a l e , T h e i n t e g r a l t i m e s c a l e w a s f o u n d i n t h e p r e s e n t m e a s u r e -m e n t s b y i n t e g r a t i n g a u t o c o r r e l a t i o n s i n t h e u s u a l w a y . A l s o s h o w n a r e r e p r e s e n t a t i v e r e s u l t s o f f u l l s c a l e m e a s u r e m e n t s t a k e n f r o m r e f e r e n c e s 30 a n d 3 1 . T h e p o w e r s p e c t r u m o f h o r i z o n t a l w i n d s p e e d a t a h e i g h t o f a b o u t 3 3 0 f t s h o w n i n F i g u r e X w a s t a k e n i n 1 9 5 5 - 5 6 a t t h e 37 5 f t m e t e o r o l o g i c a l t o w e r o f t h e B r o o k h a v e n N a t i o n a l L a b o r a t o r y b y p i e c i n g t o g e t h e r v a r i o u s p o r t i o n s o f t h e s p e c t r u m . T h e d a t a a n a l y z e d r a n g e d f r o m 5 - d a y a v e r a g e s p e e d s c o v e r i n g a l m o s t a y e a r t o 2 s e c a v e r a g e s p e e d s 35 covering an hour. The value of the power law exponent a for t h i s set of measurements i s not given i n reference 30, 32 but Templin i n a l a t e r work, deduces that a value of a = 0.28 i s appropriate for t h i s case. The f u l l scale data shown i n Figure XI for Montreal was taken on a tower i n the Botanical Gardens, on the upper l e v e l of Mount Royal, at a height of about 250 f t . In t h i s case, a value of a = 0.28 i s quoted by 31 Davenport. It i s seen that the peak of t h i s Montreal data i s very well defined and coincides c l o s e l y with the peaks of both the t h e o r e t i c a l and experimental curves although the scatter does not enable us to define a slope c l e a r l y . 9 Davenport gives a t y p i c a l atmospheric boundary layer thickness of 1300 f t for a = 0.28, allowing us to estimate z/6 z 1/4 for Brookhaven, and z/6 ~ 1/5 for Montreal respectively. Both naturally grown and a r t i f i c i a l l y grown boundary layers are close to the t h e o r e t i c a l curve, with both curves exhibiting s i m i l a r slopes to that of the von Karman spectrum over a wide range of reduced frequency n • Tu. The d i s t r i b u t i o n of energy i s therefore roughly correct for both measured cases. The biggest difference found between the two simulations i s shown i n Figure XIII, the plot of l o n g i -36 tudinal turbulence i n t e n s i t y . From a maximum value close to the f l o o r , the rms values for the naturally grown boundary layer decrease sharply with an increase i n distance above the f l o o r z, sim i l a r to the f u l l scale boundary layer. For the a r t i f i c i a l l y grown boundary layer, the values of rms do not change much up to a value of z equal to about 1/2 6 , and thi s i s not representative of f u l l scale conditions. One intere s t i n g point to notice i n connection with the i n t e n s i t i e s measured i n the a r t i f i c i a l l y grown boundary layer, i s that these values increase as x increases for fixed z over a f a i r l y large range of z, roughly z ^> 1/2 6 . This i s possibly due to the transport of turbulent energy by the large scale motions from the wall region, where turbulence energy i s high, and being produced, to the outer region, where i t i s lower, and where production i s also low. The a c t i v i t y of these large scales appears to be too great i n the a r t i f i c i a l l y grown case, insofar as the turbulence l e v e l i n the outer part of t h i s boundary layer i s s i g n i f i c a n t l y higher than the naturally grown case. The f u l l scale atmospheric turbulence i n t e n s i t y was obtained from reference 18, using a value of ZQ ~ 30 cm, where ZQ i s the surface roughness length, as suggested by 9 Davenport for a ~ 0.28. CHAPTER 5 CONCLUDING DISCUSSION Major c o n c l u s i o n s w i t h b r i e f d i s c u s s i o n o f each ar e l i s t e d below: 1. Dvorak's c o r r e l a t i o n , and i t s e x t e n s i o n t o i n c l u d e t h r e e - d i m e n s i o n a l roughness elements appears t o be a u s e f u l method f o r p r e d i c t i n g a roughness geometry w h i c h w i l l produce a d e s i r e d w a l l shear s t r e s s o r shape f a c t o r , once e q u i l i b r i u m has been r e a c h e d . Many problems remain t o be s o l v e d - - f o r example, the a c t u a l boundary l a y e r development i s n o t p r e d i c t e d h e r e . However, t h i s i s a s t e p towards more r a t i o n a l wind t u n n e l s i m u l a t i o n o f t h e a t m o s p h e r i c boundary l a y e r [ f o r i n d i v i d u a l c a s e s , as opposed t o the g e n e r a l ' c i t y c e n t r e ' boundary l a y e r (a = 0.40) o r 'urban f e t c h ' ( a = 0.28) c u r r e n t l y i n common use (see a l s o r e f e r e n c e 1 9 ) ] . 2. The p r e s e n t n a t u r a l l y grown boundary l a y e r i s c l o s e t o e q u i l i b r i u m a f t e r p a s s i n g o v e r u n i f o r m roughness a d i s t a n c e downstream e q u a l t o about 350 t i m e s the roughness h e i g h t . The agreement w i t h the v a l u e o f 38 a b o u t 3 2 0 o b t a i n e d b y A n t o n i a a n d L u x t o n r a i s e s t h e i n t e r e s t i n g p o i n t — D o e s t h e v a l u e o f x / k n e e d e d t o r e a c h e q u i l i b r i u m d e p e n d s i g n i f i c a n t l y o n A / k o r o n 6/k a t e q u i l i b r i u m , o r d o e s i t d e p e n d o n t h e d i m e n s i o n s o f t h e t r i p , o r h a s i t a r e l a t i v e l y u n i v e r s a l v a l u e ? M o r e w o r k n e e d s t o b e d o n e i n t h i s a r e a , a s w e l l a s i n p r e d i c t i n g t h e a c t u a l b o u n d a r y l a y e r t h i c k n e s s w h i c h w i l l b e o b t a i n e d o v e r a n y g i v e n s t r e a m w i s e l e n g t h o f r o u g h n e s s f e t c h . N o n e q u i l i b r i u m c a l c u l a -t i o n s m u s t b e u s e d h e r e , a n d i t r e m a i n s t o b e s e e n w h e t h e r D v o r a k ' s c o r r e l a t i o n a n d H e a d ' s m e t h o d a r e . s u f f i c i e n t l y a c c u r a t e t o p r e d i c t b o u n d a r y l a y e r g r o w t h f a r f r o m e q u i l i b r i u m c o n d i t i o n s . 3 . T h i s b o u n d a r y l a y e r q u i c k l y g e n e r a t e d b y a r t i f i c i a l m e a n s p r o v e d t o b e a f a i r . a p p r o x i m a t i o n t o t h a t g r o w n o v e r a l o n g f e t c h o f r o u g h n e s s . W h e n e v e r p o s s i b l e i t i s a d v i s a b l e t o u s e t h a t p r o d u c e d b y t h e l o n g f e t c h , h o w e v e r , a n d i n a n y c a s e t o a l l o w a s l o n g a f e t c h a s p o s s i b l e d o w n s t r e a m o f t h e s p i r e s f o r t h e b o u n d a r y l a y e r t o r e a c h s t r e a m w i s e s t a b i l i t y . The boundary l a y e r produced here by s p i r e s showed l i t t l e change i n g r o s s c h a r a c t e r i s t i c s 6*, 9 and 8 a f t e r t r a v e l l i n g about e i g h t d e s i g n s p i r e h e i g h t s downstream o f t h e s p i r e s . A d i s t a n c e o f s i x o r seven such h e i g h t s has been a d v i s e d by o t h e r w o r kers i n the p a s t . T h i s s e t o f s p i r e s d i d not a c c u r a t e l y r e p r e s e n t t h e l o n g i t u d i n a l t u r b u l e n c e i n t e n s i t y o f t h e f u l l s c a l e a t m o s p h e r i c wind. The a u t h o r i s i n a g r e e -ment w i t h Campbell and Standen i n t h a t the t h r e e main t u r b u l e n t c h a r a c t e r i s t i c s o f the boundary l a y e r , i . e . , the v e l o c i t y p r o f i l e (shape and t h i c k n e s s ) , the power spectrum, and the t u r b u l e n c e i n t e n s i t y , cannot a l l be a c c e p t a b l y s i m u l a t e d by the same m e c h a n i c a l d e v i c e , i f i n a d d i t i o n a r e a s o n -a b l e degree o f streamwise s t a b i l i t y and moderate s i m p l i c i t y i n t h e d e s i g n and c o n s t r u c t i o n o f t h e s p i r e s i s r e q u i r e d . 4 0 R E F E R E N C E S 1 . D a v e n p o r t , A . G . , a n d I s y u m o v , N . , " T h e A p p l i c a t i o n o f t h e B o u n d a r y L a y e r W i n d T u n n e l t o t h e P r e d i c t i o n o f W i n d L o a d i n g , " P a p e r N o . 7 , P r o c e e d i n g s o f t h e I n t e r - n a t i o n a l R e s e a r c h S e m i n a r , N a t i o n a l R e s e a r c h C o u n c i l , V o l . 1 , O t t a w a ( S e p t e m b e r 1 9 6 7 ) , U n i v e r s i t y o f T o r o n t o P r e s s , ( 1 9 6 8 ) . 2 . C e r m a k , J . E . , " L a b o r a t o r y S i m u l a t i o n o f 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 , " A I A A J o u r n a l , V o l . 9 , ( S e p t e m b e r 1 9 7 1 ) . 3 . C o u n i h a n , J . , " A n I m p r o v e d M e t h o d o f S i m u l a t i n g a n A t m o s p h e r i c B o u n d a r y L a y e r i n a W i n d T u n n e l , " A t m o s p h e r i c E n v i r o n m e n t , V o l . 3 , ( M a r c h 19 6 9 ) . 4 . S t a n d e n , N . M . , " A S p i r e A r r a y f o r G e n e r a t i n g T h i c k T u r b u l e n t S h e a r L a y e r s f o r N a t u r a l W i n d S i m u l a t i o n s i n W i n d T u n n e l s , " R e p o r t L T R - L A - 9 4 , N . A . E . , ( 1 9 7 2 ) . 5 . S c h o n , J . P . , a n d M e r y , P . , " A P r e l i m i n a r y S t u d y o f t h e S i m u l a t i o n o f t h e N e u t r a l A t m o s p h e r i c B o u n d a r y L a y e r U s i n g A i r I n j e c t i o n i n a W i n d T u n n e l , " A t m o s -p h e r i c E n v i r o n m e n t , V o l . 5 , N o . 5 , (May 1 9 7 1 ) . 6 . N a g i b , H ; M . , M o r k o v i n , M . V . , Y u n g , J . T . , a n d T a n -a t i c h a t , J . , " O n M o d e l i n g o f A t m o s p h e r i c S u r f a c e L a y e r s b y t h e C o u n t e r - J e t T e c h n i q u e , " A I A A J o u r n a l , V o l . 1 4 , N o . 2 , ( F e b r u a r y 1 9 7 6 ) . 7 . T e u n i s s e n , H . W . , " S i m u l a t i o n o f t h e P l a n e t a r y B o u n d a r y L a y e r i n t h e M u l t i p l e J e t W i n d T u n n e l , " U T I A S R e p o r t 1 8 2 , ( J u n e 1 9 7 2 ) . 8 . C a m p b e l l , G . S . , a n d S t a n d e n , N . M . , " P r o g r e s s R e p o r t I I o n S i m u l a t i o n o f E a r t h ' s S u r f a c e W i n d s b y A r t i f i -c i a l l y T h i c k e n e d W i n d T u n n e l B o u n d a r y L a y e r s , " R e p o r t L T R - L A - 3 7 , N . A . E . , ( 1 9 6 9 ) . 41 9. Davenport, A.G., "The Relationship of Wind Structure to Wind Loading," Paper No. 2, Symp. 16, Procedures of Conference on Buildings and Structures at NPL, June 1963, HMSO, London (1965). 10. Prandtl, L., "Uber Flussigkeitsgewegung bei sehr kleiner Reibung," Proc. 3rd Intern. Math. Congr., Heidelberg, (1904), (English Translation NACA TM 452, 1928). 11. Clauser, F.H., "The Turbulent Boundary Layer," Advances  i n Applied Mechanics, Vol. 4, (1956). 12. Nikuradse, J . , "Stromungesgesetze i n rouhen Rohren," VEI - Forschungsheft, No. 361, (1933). 13. Hinze, J.O., Turbulence, McGraw-Hill Inc., 2nd ed. (1975). 14. Antonia, R.A., and Luxton, R.E., "The Response of a Turbulent.Boundary Layer to a Step Change i n Surface Roughness," Journal of F l u i d Mechanics, Vol. 48, (1971), p. 721. 15. Dvorak, F.A., "Calculation of Turbulent Boundary Layers on Rough Surfaces i n Pressure Gradient," AIAA Journal, Vol. 7, (1969), p. 1752. 16. Clauser, F.H., "Turbulent Boundary Layers i n Adverse Pressure Gradients," Journal of the Aero. Sciences, Vol. 21, (1954), p. 91. 17. Gartshore, I.S., "A Relationship Between Roughness Geometry and Velocity P r o f i l e Shape for Turbulent Boundary Layers," Report LTR-LA-140, N.A.E., (October 1973) . 18. Engineering Sciences Data Unit, Item 70015, " F l u i d Forces and Moments on F l a t Plates," (October 1972). Item 71016, " F l u i d Forces, Pressures and Moments on Rectangular Blocks," (July 1963). 42 19. G a r t s h o r e , I . S . , and de Cro o s , K.A., "Roughness Element Geometry R e q u i r e d f o r Wind Tunnel S i m u l a t i o n s o f the Atmos p h e r i c Wind." To be p u b l i s h e d by A.S.M.E. ( p r e -p r i n t 76-WA/FE-18). 20. M e l l o r , G.L., and G i b s o n , D.M., " E q u i l i b r i u m T u r b u l e n t Boundary L a y e r s , " " J o u r n a l o f F l u i d M e c h a n i c s , V o l . 2 4 , ( 1 9 6 6 ) , p. 255. 21. Townsend, A.A., The S t r u c t u r e o f T u r b u l e n t Shear. Flow, Cambridge U n i v e r s i t y P r e s s , 2nd ed., (1956). 22. P e r r y , A.E., S c h o f i e l d , W.H., and J o u b e r t , P.N., "Rough W a l l T u r b u l e n t Boundary L a y e r s , " J o u r n a l o f F l u i d  M e c h a n i c s , V o l . 37, ( 1 9 6 9 ) , p.383-23. R o t t a , J.C., " P r o g r e s s i n Aero S c i e n c e s , " V o l . 2 (ed. F e r r i , Kuchemann and S t e r n e ) , Pergammon P r e s s , N.Y., (1962) 24. A n t o n i a , R.A., and L u x t o n , R.E., "The Response o f a T u r b u l e n t Boundary L a y e r t o an U p s t a n d i n g Step Change i n S u r f a c e Roughness," ASME, Paper No. 7 0 - F E - l , (May 1970). D e l i v e r e d a t F l u i d s E n g i n e e r i n g , Heat T r a n s f e r and L u b r i c a t i o n C o n f e r e n c e , D e t r o i t , M i c h i g a n . 25. Counihan, J . , "Wind Tunnel D e t e r m i n a t i o n o f the Roughness Length as a F u n c t i o n o f t h e F e t c h and the Roughness:Density o f T h r e e - D i m e n s i o n a l Roughness E l e m e n t s , " Atmospheric Environment, V o l . 5, (1 9 7 1 ) , p. 637. 2 6 . Wooding, R.A., B r a d l e y , E.F., and M a r s h a l l , J.K., "Drag Due t o R e g u l a r A r r a y s o f Roughness Elements o f V a r y i n g Geometry," Boundary-Layer M e t e o r o l o g y , V o l . 5, (1 9 7 3 ) , P. 285. 27. P a t e l , R.P., "Reynolds S t r e s s e s i n F u l l y Developed T u r b u l e n t : Flow i n a C i r c u l a r P i p e , " M c G i l l U n i v . R e p o r t , No. 68-7, (1968). 4 3 2 8 . C o w d r e y , C . F . , " A S i m p l e M e t h o d f o r t h e D e s i g n o f W i n d T u n n e l V e l o c i t y P r o f i l e G r i d s , " N a t . P h y s i c a l L a b , A e r o N o t e 1 0 5 5 , (May 1 9 6 7 ) . 2 9 . C h a m p a g n e , F . H . , " T u r b u l e n c e M e a s u r e m e n t s w i t h I n -c l i n e d H o t W i r e s , " B . S . R . L . F l i g h t S c i e n c e s L a b R e p o r t , N o . 1 0 3 , ( 1 9 6 5 ) . 3 0 . V a n d e r H o v e n , I., " P o w e r S p e c t r u m o f H o r i z o n t a l W i n d S p e e d i n t h e F r e q u e n c y R a n g e f r o m 0 . 0 0 0 7 t o 9 0 0 C y c l e s p e r H o u r , " J o u r n a l o f M e t e r o l o g y , V o l . 1 4 , ( 1 9 5 7 ) , p p . 1 6 0 - 1 6 4 . 3 1 . D a v e n p o r t , A . G . , " T h e D e p e n d e n c e o f W i n d L o a d s o n M e t e r o l o g i c a l P a r a m e t e r s , " P a p e r N o . 2 , P r o c e e d i n g s  o f t h e I n t e r n a t i o n a l S e m i n a r o n W i n d E f f e c t s o n  B u i l d i n g s a n d S t r u c t u r e s , N a t i o n a l R e s e a r c h C o u n c i l , O t t a w a , ( S e p t e m b e r 1 9 6 7 ) , p p . 1 1 - 1 5 . 3 2 . T e m p l i n , R . J . , " I n t e r i m P r o g r e s s N o t e o n S i m u l a t i o n o f E a r t h ' s S u r f a c e W i n d s b y A r t i f i c i a l l y T h i c k e n e d W i n d T u n n e l B o u n d a r y L a y e r s , " R e p o r t L T R - L A - 2 2 , N . A . E . , ( F e b r u a r y 1 9 6 9 ) . APPENDIX 1 Head's empirical functions can be numerically 15 defined as follows: F(H 1) = exp (-3.512 - 0.617 In (H 1 - 3) :) where H 1 = (6 - 6*) / 6 G(H) = 3.3 + exp (.4667 - 2.722 £n-(H-.6798) for H £ 1.6 G(H) = 3.3 + exp (.4383 - 3.064 £n(H-. 67.98) for H > 1.6 where H = 6*/6 . i 45 A P P E N D I X 2 T h e s l a n t e d h o t w i r e a n e m o m e t e r t e c h n i q u e f o r m e a s u r i n g s h e a r s t r e s s u s e d b y P a t e l f o l l o w s t h e w o r k o f 2 9 C h a m p a g n e a n d o t h e r s . B a s i c a l l y , t h e t e c h n i q u e u s e s - 2 t w o r e a d i n g s o f t h e h o t w i r e r m s v o l t a g e ( e ) . T h e s l a n t w i r e i s s e t a t a n a n g l e +ty i n t o t h e m e a n w i n d d i r e c t i o n - 2 a n d t h e f i r s t s e t o f r m s r e a d i n g s ( e ^ ) , t a k e n . Now t h e s l a n t w i r e i s t u r n e d 18 0 ° i n t h e p l a n e d e f i n e d b y t h e w i r e a n d m e a n w i n d v e l o c i t y , t o a n a n g l e -i|>, a n d t h e s e c o n d s e t o f r e a d i n g s t a k e n . w h e r e e , E a r e t h e r m s a n d m e a n v o l t a g e r e a d i n g s t a k e n b y t h e s l a n t w i r e a t e a c h o r i e n t a t i o n r e s p e c t i v e l y . U s u a l l y , = 4 5 ° a n d k 1 = 0 . 2 , s o t h a t t h e e q u a t i o n a b o v e r e d u c e s t o . T h e s h e a r s t r e s s i s g i v e n a s T 4 (1 - k 2 ) C o t 2 T A B L E I Dimensions o f M o d i f i e d H a l f - W i d t h S p i r e s H e i g h t ( i n ) S p i r e Width ( i n ) 0.0 16.00 0.5 8.80 2.5 6. 30 5.0 4. 95 7.5 4. 05 10.0 3. 35 15. 0 2.25 20.0 1. 35 24. 0 0. 00 TABLE II Variation of the Defect Law Integral Parameter and von Karman's Constant with Distance Downstream for the Naturally Grown Boundary Layer X k U 6 T K 140.8 4.26 0. 33 268.8 4. 24 0. 35 332.8 4.13 0.38 396.8 3. 99 0. 38 492.8 3.96 0. 34 F i g u r e I S t r u c t u r e o f t h e rough w a l l boundary l a y e r F- tguAe I I Rougknte* oZmznt spacing and dantn&loni, actual spires design of spires (a) design spire height (32 in) (b) boundary layer thickness (29 in) (c) actual spire height (24 in) Figure III Spire dimensions 51 DRAG (lbs) 0.3200 0.2800 0.2400 0.2000 0.1600 x/k= 492.8 " = 396.8 " = 332.8 " = 268.8 : " = 140.8 0.1200 0.800 0.400 10 12 PU Figure IV Drag vs. pU^; naturally grown boundary layer 52 z / 6 x/k = 492.8 F i g u r e V Sample l o g - l o g v e l o c i t y p r o f i l e s Figure VI Wall shear stress correlation) expected for various roughness geometries (Dvorak's F i g u r e V I I E f f e c t o f r o u g h n e s s geometry on p r o f i l e shape ( D v o r a k ' s c o r r e l a t i o n ) 5.0 10 20 30 (x- f t ) 3.0 L 10 I (x' f t ) 1 F i g u r e V I I I V a r i a t i o n o f g r o s s c h a r a c t e r i s t i c s 6*, a and 9 w i t h d i s t a n c e b e h i n d s p i r e s x', f o r t h e a r t i f i c i a l l y grown b o u n d a r y l a y e r 56 n a t u r a l l y grown boundary l a y e r - A - a r t i f i c i a l l y grown boundary l a y e r P U , v a l u e on f l o o r f o r n a t u r a l l y grown boundary l a y e r v a l u e on f l o o r f o r a r t i f i c i a l l y grown boundary l a y e r 004 h-002 U-(z i n ) F i g u r e IX .Shear s t r e s s as measured u s i n g a s l a n t w i r e 1 . ART IF IC IALLY GROWN BOUNDARY LAYER, x/h = 8 BROOKHAVEN, z/& = 1/3 • » 1 10 100 nTu Ln —i Figure X Power spectral density of the longitudinal turbulence - Height 15" above the floor \ . V \ X N . > v \ \ — \ > . VON KARMAN - NATURALLY GROWN BOUNDARY LAYER, x/k = 500, z/6 - 1/4 ARTIFICIALLY GROWN BOUNDARY LAYER, x/h = 8 777 MONTREAL, z/S = 1/5 I i 1 10 100 g u r e X I P o w e r s p e c t r a l d e n s i t y o f t h e l o n g i t u d i n a l t u r b u l e n c e - H e i g h t 8" a b o v e t h e f l o o r 1.1 0.1 nS U I I (n) o . o i ^ ^^^^^^^^^^^ N >X _ w VON KARMAN NATURALLY GROWN BOUNDARY LAYER, x/k = 500 ARTIFICIALLY GROWN BOUNDARY LAYER, x/h = 8 J L 1 n T u 1 0 F i g u r e X I I Power s p e c t r a l d e n s i t y o f t h e l o n g i t u d i n a l t u r b u l e n c e - H e i g h t 3" above t h e f l o o r 60 a r t i f i c i a l l y grown b o u n d a r y l a y e r a t v a r i o u s d i s t a n c e s b e h i n d t h e s p i r e s x 1 ( f t ) n a t u r a l l y grown e q u i l i b r i u m b o u n d a r y l a y e r F i g u r e X I I I L o n g i t u d i n a l t u r b u l e n c e i n t e n s i t y 

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