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WIND DRAG WITHIN A SIMULATED FOREST CANOf.' FIELD
by
G. H s i and J . H. Nath
P r e p a r e d f o r
U.S. Army M a t e r i e l Command G r a n t No. DA-AMC-28-043-6S-G-20 D i s t r i b u t i o n o f t h i s r e p o r t i s u n l i m i t e d . . • • • | . F l u i d Dynamics and D i f f u s i o n L a b o r a t o r ' T i - r r — r - i n i - ' C o l l e g e o f E n g i n e e r i n g U U b l i i ^ . S l i i J U i ^ C o l o r a d o S t a t e U n i v e r s i t y """"--^ A F o r t C o l l i n s , C o l o r a d o .
\
A u g u s t 1968 CER68-69GH-JHN6 •''XABSTRACT
This report presents an experimental and analysis study of wind
drag within simulated forest canopy fields. The drag coefficients,
aerodynamic roughness and wind velocity profiles are studied for
vari-ous types of forest canopies. Furthermore, a brushy canopy field was
studied which simulated the leafy portion of a forest
witho~tthe tree
trunks present. The wind drag force on a single experimental tree was
also studied. This information is useful to those who are concerned
with diffusion within a canopy in an atmospheric boundary layer.
In the course of this study a shear plate was developed which
reliably measures a drag force from 0.1 gram to 2000 grams. The
func-tion of this plate was successfully validated by testing it in a
tur-bulent flow over a smooth surface.
The drag coefficient of a single model tree, which was the same
as those used in the simulated forest canopy fields, was compared with
prototype conifers. The work found that a plastic model tree has drag
coefficients between those for spruce and Douglas fir trees. The
varia-tion of the tree drag coefficient among trees of the same type is
con-cluded by a statistical study on a number of plastic trees which have
the same amount of foliage but which are different in the arrangement
of the tree branches and the tree leaves.
-M e c h a n i c s P „ , ™ , c « , „ „ , , ^ . « c u „ . . e „ i . , , and i„ p a „ u „ U , . o : B. c = ™ . k , , , „ . , , ^ s u g g e s t i o n s „„ . h , c o n c l u s i o n s Of t h i s r e p o r t . TO o r . „. w. s h c n . „. „. „ „ . „ , ^ , ^ ^^^^^^ ^^^^ » h o . t h e a „ t h o „ r e c e i v e d t e c h n i c s , s u g g e s t i o n s d u , i „ , t h e p „ p . a r a t x o n o f t h i s r e p „ « , a u t h o r s e x p r e s s t h e i r . r . t i t u d e . To Mr. J. P r i s o n , .ho h e l p e d ,„ t h e . r r a n j e . n t s f o r u s i n , t h e Wind t u n n e l s , t o „r. Oo„ C o i i i n s , .ho d i d t h e p r o o f r e a d i n g o f t h i s r e p o r t , a n d t o „iss „. „ a r i , .ho p r e p a r e d t h e d i . , r a „ , t h e a u t h o r s e x p r e s s t h e i r a p p r e c i a t i o n .
T^e a u t h o r s g r a t e f u l l y acknowledge s u p p o r t f o r t h i s work u n d e r t h e U. S. Army R e s e a r c h G r a n t No. DA-AMC-28-043-65-G-20.
V
C h a p t e r Page LIST OF TABLES i x L I S T OF FIGURES x LIST OF SYMBOLS x i v 1 INTRODUCTION 1 1.1 G e n e r a l B a c k g r o u n d 1
1.2 Purpose and Scope 6
1.2.1 S u r f a c e Shear on a Smooth B o u n d a r y . . . 7 1.2 1.2 1.2 1.2 2 D r a g C o e f f i c i e n t o f a S i n g l e E x p e r i -m e n t a l T r e e .- 7 3 D r a g C o e f f i c i e n t o f V a r i o u s Canopies . . 8 4 The V e l o c i t y P r o f i l e s o v e r and w i t h i n t h e S i m u l a t e d F o r e s t Canopy 9 5 The A e r o d y n a m i c Roughness o f C a n o p i e s . . 10 THEORETICAL CONSIDERATIONS
2.1 The R e s i s t a n c e F o r m u l a f o r a Smooth P l a t e and a
U n i f o r m l y Rough P l a t e i n T u r b u l e n t Flow . . . . 12
2.2 The Momentum I n t e g r a l f o r A p p r o x i m a t e T r e e D r a g
C a l c u l a t i o n 22
2.3 S i n g l e Roughness E l e m e n t 24
2.4 T r a n s i t i o n f r o m a Smooth t o a Rough S u r f a c e . , 27 2.5 S i m u l a t i o n o f Model and P r o t o t y p e Canopy F i e l d s
2.6 The A n a l o g y between t h e L o c a l D r a g C o e f f i c i e n t
and t h e D i f f u s i o n C o e f f i c i e n t 32
V I
-EXPERIMENTAL EQUIPNffiNT AND PROCEDURES 3.1 Wind T u n n e l
3.2 I n s t r u m e n t a t i o n
3.2.1 Drag Force Measurement
3.2.2 V e l o c i t y P r o f i l e Measurement 3.2.3 P r e s s u r e G r a d i e n t Measurement 3.3 Smooth Boundary S u r f a c e 3.4 Canopy F i e l d s 3.4.1 S i n g l e T r e e 3.4.2 Orchard-Type C a n o p i e s 3.4.3 F o r e s t - T y p e C a n o p i e s 3.4.4 B r u s h y Canopy
RESULTS OF LABORATORY EXPERIMENTS
4.1 S i n g l e Tree D a t a 4.1.1 T r e e F l e x i b i l i t y 4.1.2 S t a n d a r d D e v i a t i o n o f S i n g l e T r e e D r a g Force * 4.2 V e r i f i c a t i o n o f t h e Shear P l a t e N t e a s u r e i w n t , 4.3 F o r e s t Canopy Data 4.3.1 V e l o c i t y P r o f i l e s o v e r and w i t h i n t h e F o r e s t Canopy F i e l d , a n d t h e A e r o d y -n a m i c Rough-ness 4.3.2 The V a r i a t i o n o f Drag C o e f f i c i e n t w i t h i n t h e F o r e s t Canopy F i e l d 4.4 O r c h a r d Canopy Data v i i
C h a p t e r £ M i 4.S B r u s h y Canopy D a t a 62 4.5.1 The V e l o c i t y P r o f i l e s o f t h e B r u s h y Canopy 62 4.5.2 The V a r i a t i o n o f Drag C o e f f i c i e n t i n t h e B r u s h y Canopy, a n d A e r o d y n a t a i c R o u ^ n e s s . 63 4.5.3 The Comparison o f V e l o c i t y P r o f i l e s i n and above t h e F o r e s t Canopy w i t h T h o s e o f
t h e B r u s h y Canopy • 65
5 CONCLUSIONS 66
6 REFERENCES 67
7 APPENDIX 73
T a b l e Page 1.1.1 The a v e r a g e a e r o d y n a m i c r o u g h n e s s e s o f t h e s t a t e o f W i s c o n s i n 3 3.1.1 P e r f o r m a n c e c h a r a c t e r i s t i c s o f t h e Army M e t e o r o l o g i c a l and t h e C o l o r a d o S t a t e U n i v e r s i t y Wind T u n n e l s 74 4.1.1 A r t i f i c i a l t r e e d r a g c o e f f i c i e n t 49 4.5.2.1 The a e r o d y n a m i c r o u g h n e s s e s o f s i m u l a t e d c a n o p i e s 64 4.5.2.2 The a e r o d y n a m i c r o u g h n e s s e s o f low v e g e t a t i v e s u r f a c e s , a f t e r Deacon 65 i x
F i g u r e 2.1.1 Arrangettfönt o f t e s t p l a t e and b a l a n c e , S c h u l t z -Grunow ( 1 9 4 0 ) 2.2.1 C o n t r o l volume a r o u n d a t r e e 2.3.1 Zones o f d i s t u r b e d b o u n d a r y l a y e r , a f t e r P l a t e ( 1 9 6 5 ) 3.1.1 U. S. Army m e t e o r o l o g i c a l w i n d t u n n e l 3.1.2 C o l o r a d o S t a t e U n i v e r s i t y w i n d t u n n e l
3.2.1.1 The s t r a i n gage f o r c e dynamo»eter
3.2.1.2 The c a l i b r a t i o n c u r v e o f t h e s t r a i n gage f o r c e d y ¬ namometer
3.2.1.3 The e l e c t r i c b r i d g e a r r a n g e m e n t f o r t h e s t r a i n gage f o r c e dynamometer . . . .
3.2.1.4 The s t r a i n gage f o r c e dynamometer i n a model o r c h a r d canopy f i e l d 3.2.1.5 I n s t r u m e n t s f o r t h e s t r a i n gage f o r c e dynamometer . . 3.2.1.6 The s c h e m a t i c d i a g r a m o f t h e s h e a r p l a t e 3.2.1.7 The s h e a r p l a t e 3.2.1.8 The c a l i b r a t i o n c u r v e o f t h e s h e a r p l a t e 3.2.2.1 S c h e m a t i c d i a g r a m o f v e r t i c a l v e l o c i t y p r o f i l e n^a-s u r e m e n t 3.3.1 The smooth b o u n d a r y , w i t h t h e s h e a r p l a t e , i n a w i n d t u n n e 1 3.3.2 Two d i m e n s i o n a l i t y o f f l o w c o n d i t i o n o v e r t h e smooth b o u n d a r y 3.4,1,1 C l o s e - u p p h o t o g r a p h o f an a r t i f i c i a l t r e e 3.4.3.1 The f o r e s t - t y p e canopy f i e l d 3.4.3.2 Two d i m e n s i o n a l i t y o f f l o w c o n d i t i o n o v e r t h e f o r e s t -t y p e canopy f i e l d 75 22 76 77 78 79 80 81 82 82 83 84 85 86 87 88 89 90 91
F i g u r e 3.4.4.1 The a r r a n g e m e n t o f b r u s h y canopy on a p i e c e o f p l y w o o d 3.4.4.2 The b r u s h y canopy f i e l d i n a w i n d t u n n e l 3.4.4.3 Two d i m e n s i o n a l i t y o f f l o w c o n d i t i o n o v e r t h e b r u s h y canopy f i e l d 4.1.1 D r a g c o e f f i c i e n t o f a s i n g l e model t r e e 4.1.2 T r e e f l e x i b i l i t y and a c t u a l t r e e d r a g c o e f f i c i e n t s o f s p r u c e and Douglas f i r 4.1.3 D r a g c o e f f i c i e n t o f p r o t o t y p e c o n i f e r t r e e s , a f t e r Raymer ( 1 9 6 2 ) 4.1.4 R e a l c o n i f e r sample t r e e s 4.1.5 The c o m p a r i s o n o f t h e model t r e e w i t h t h e s p r u c e sanqjle t r e e
4.1.1,1 The Douglas f i r i n s t i l l a i r and u n d e r 1750 cm/sec w i n d v e l o c i t y 4.1.2.1 The p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n o f a s i n g l e model t r e e d r a g f o r c e , U : 610 cm/sec s 4.1.2.2 The p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n o f a s i n g l e model t r e e d r a g f o r c e , U : 1220 cm/sec 4.2.1 V e l o c i t y p r o f i l e s a t v a r i o u s s t a t i o n s on a smooth b o u n d a r y , U^ : 1680 cm/sec 4.2.2 Measured l o c a l s k i n f r i c t i o n c o e f f i c i e n t v s . e f f e c -t i v e Reynolds number; s h e a r p l a -t e -t e c h n i q u e , smoo-th b o u n d a r y s u r f a c e , ,
4.2.3 A n a l y t i c s k i n - f r i c t i o n c o e f f i c i e n t v s , e f f e c t i v e R e y n o l d s number, smooth b o u n d a r y s u r f a c e
4.3,1 Method o f v e r i f y i n g t h e d i r e c t measured and a n a l y t i c momentum t h i c k n e s s 9 4.3.1.1 D i m e n s i o n l e s s v e l o c i t y p r o f i l e b e f o r e t h e f o r e s t c a n o p y f i d l d , a t s t a t i o n - l o o cm . 4.3.1.2 The v e l o c i t y p r o f i l e v a r i a t i o n s i n t h e i n i t i a l r e g i o n o f t h e f o r e s t canopy f i e l d U : 915 cm/sec . . , x i
Pigur© Page 4.3.1.3 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 above t h e f o r e s t canopy t h i c k b o u n d a r y 1 ayev 109 4.3.1.4 N o n - d i m e n s i o n a l v e l o c i t y p r o f i l e s above t h e c e n t e r r e g i o n o f t h e model f o r e s t canopy f i e l d 110 4.3.1.5 The c o n p a r i s o n o f n o n - d i m e n s i o n a l v e l o c i t y p r o f i l e s w i t h i n t h e c a n o p i e s o f t h e p r o t o t y p e a n d t h e model . m 4.3.1.6 The c o i p a r i s o n o f w i n d v e l o c i t y p r o f i l e s a t t h e e n d r e g i o n o f t h e e x p e r i m e n t a l f o r e s t c a n o p y f i e l d . , , i i 2 4.3.2 TTie c o n p a r i s o n o f momentum t h i c k n e s s 6 b e t w e e n s h e a r p l a t e d a t a a n d 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 . 113 4.3.2.1 D i r e c t measured ( s h e a r p l a t e ) s u r f a c e s t r e s s v s . l o n g i t u d i n a l d i s t a n c e , i n d i m e n s i o n l e s s f o r m , f o r e s t c a n o p y , t h i c k b o m d a r y l a y e r 114 4.4.1 C u B u l a t i v e t r e e d r a g f o r c e f r o m d i r e c t measurement and f r o m c o n t p u t e r c a l c u l a t i o n , o r c h a r d c a n o p y , t h i c k b o u n d a r y l a y e r U S 4.4.2 L o c a l d r a g f o r c e v s . l o n g i t u d i n a l d i s t a n c e i n d i m e n s i o n l e s s f o r m , o r c h a r d c a n o p y , 25.4 cm t r e e -s p a c i n g , t h i c k b o u n d a r y l a y e r 116 4.4.3 S t e a d y decay zone o f o r c h a r d c a n o p y , 25,4 cm t r e e -s p a c i n g , t h i c k b o u n d a r y l a y e r 117 4.5.1.1 D i m e n s i o n l e s s v e l o c i t y p r o f i l e b e f o r e t h e b r u s h y c a n o p y f i e l d , a t s t a t i o n - 12,7 cm, t h i n b o u n d a r y l a y e r 118 4.5.1.2 D i m e n s i o n l e s s v e l o c i t y p r o f i l e b e f o r e t h e b r u s h y c a n o p y , a t s t a t i o n - 12.7 cm, t h i c k b o u n d a r y l a y e r , 118 4.5.1.3 N o n - d i m e n s i o n a l 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 , b r u s h y c a n o p y , t h i n b o u n d a r y l a y e r , . , 119 4.5.1.4 N o n - d i m e n s i o n a l 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 , b r u s h y c a n o p y , t h i c k b o u n d a r y l a y e r 120 4.5.2.1 D i r e c t measured l o c a l d r a g c o e f f i c i e n t v s . x / h , b r u s h y c a n o p y , t h i n b o u n d a r y l a y e r 121 4.5.2.2 D i r e c t measured l o c a l d r a g c o e f f i c i e n t v s , x / h , b r u s h y c a n o p y , t h i c k b o u n d a r y l a y e r 122 x i i
P i g u w
4.S.2.3 A e r o d y n a m i c r o u g h n e s s e s v s . v e g e t a t i o n h e i g h t . .
The c o n p a r i s o n o f v e l o c i t y p r o f i l e i n and above t h e f o r e s t canopy w i t h t h a t o f b r u s h y c a n o p y f i e l d .
"The c o m p a r i s o n o f v e l o c i t y p r o f i l e s o f t h e f o r e s t
canopy and t h e b r u s h y c a n o p y , above t h e i r f u z z y s u r
-c o e f f i -c i e n t o f t h e e x p e r i i M n t a l f o r e s t and b r u s h y canopy i n t h e e s t a b l i s h e d r e g i o n . . . . 4.5.3.1 4.5.3.2 S . l 5.2 D i r e c t w a s u r e n e n t o f s u r f a c e s t r e s s f o r v a r i o u s c a n o p i e s i n ftilly d e v e l o p e d r e g i o n . . . 123 124 125 126 127 X l i i
S y ^ o l D e f i n i t i o n A The l a r g e s t f r o n t a l a r e a o f t h e r o u g h n e s s e l e m n t , o r a t r e e , p e r p e n d i c u l a r t o t h e d i r e c t i o n o f f l o w b The w i d t h o f t h e f e n c e B Roughness f u n c t i o n i n e q u a t i o n 2.1.18 c S p e c i f i c h e a t C o n c e n t r a t i o n o f mass a The f l u c t u a t i o n o f mass c o n c e n t r a t i o n a
S
T r e e d r a g c o e f f i c i e n t c P F r i c t i o n f a c t o r o f t h e p l a t e i n d i s t u r b e d b o u n d a r y l a y e r L o c a l d r a g c o e f f i c i e n t c' L o c a l d r a g c o e f f i c i e n t f r o m a n a l y t i c s o l u t i o n 2 L o c a l d r a g c o e f f i c i e n t f r o m d i r e c t measurement d Z e r o - p o i n t d i s p l a c e m e n t D C o e f f i c i e n t o f d i f f u s i o n Drag f o r c e o f t h e r o u ^ n e s s e l e m e n t , i n e q u a c i o n 2.3,1 T o t a l d r a g f o r c e o f a t r e e The f e n c e d r a g The i n c r e a s e i n d r a g c a u s e d b y t h e f e n c e%
The p l a t e d r a g The p l a t e d r a g u n d e r u n d i s t u r b e d b o u n d a r y l a y e r T o t a l w a l l s k i n f r i c t i o n i n e q u a t i o n 2.2.1 h P h y s i c a l h e i g h t o f a r o u g h n e s s , o r canopy h e i g h t H Shape f a c t o r k The u n i v e r s a l Karman c o n s t a n t , 0.4 x i vLIST 01' SYMIiOLS - C o n t i n u e d S y n b o l U o F i n i t i o n k' Tlic t h e r m a l c o n d u c t i v i t y k g E q u i v a l e n t sand r o u g h n e s s 1 The l e n g t h o f t h e h e a t elenrent i n e q u a t i o n 2.1.22 n C o n s t a n t i n e q u a t i o n 1.1.4 N A c h a r a c t e r i s t i c l e n g t h q Heat f l u x q The s t a g n a t i o n p r e s s u r e a v e r a g e d o v e r t h e h e i d i t o f t h e r o u c h n e s s e l e m e n t ^ T Te^)©rature T' The t e n ^ p e r a t u r e f l u c t u a t i o n u The l o c a l w i n d v e l o c i t y i n x d i r e c t i o n v The l o c a l w i n d v e l o c i t y i n y d i r e c t i o n u ' . v ' The v e l o c i t y f l u c t u a t i o n i n x and y d i r e c t i o n s r e s p e c t i v e l y X The l o n g i t u d i n a l d i s t a n c e i n f l o w d i r e c t i o n y The v e r t i c a l d i s t a n c e z The l a t e r a l d i s t a n c e A m b i e n t w i n d v e l o c i t y z^ A e r o d y n a m i c r o u g h n e s s Mass f l u x a
T
6 MoTOntum t h i c k n e s s S u r f a c e s h e a r s t r e s s W a l l s h e a r s t r e s s o f a smooth b o u n d a r y W a l l s h e a r s t r e s s o f a r o u g h b o u n d a r y P Mass d e n s i t y o f t h e f l u i d XVS y r i j o l D e f i n i t i o n 6 B o u n d a r y l a y e r t h i c k n e s s 6 D i s p l a c e T O n t t h i c k n e s s \i A b s o l u t e v i s c o s i t y V K i n e m a t i c v i s c o s i t y a T h e r m a l d i f f u s i v i t y
^m' ^ h ' Eddy d i f f u s i v i t y o f momentum, h e a t and mass
Re R e y n o l d s number Re^ R e y n o l d s n u i A e r b e s e d on t h e e f f e c t i v e s t a r t i n g l e n g t h x ROjj R e y n o l d s n u r f j e r b a s e d on t h e t o t a l l e n g t h o f a p l a t e Nu N u s s e l t nunèer, P r P r a n d t l n u r f ) e r , ^ Sh Sherwood n u r i j e r , ? ^ Sc S c h m i d t number, ^ x v i
INTRODUCTION
1.1 G e n e r a l B a c k g r o u n d
The s u r f a c e o f t h e e a r t h i s composed o f s e a and l a n d . L i k e t h e
v i v i d and u n p r e d i c t a b l e v a r i a t i o n o f t h e s e a , t h e a p p e a r a n c e o f l a n d
p r e s e n t s c o m p l e x f e a t u r e s , such as m o u n t a i n s , r i v e r s and p l a i n s . Some
o f t h e l a n d i s shaded w i t h v e g e t a t i o n c o v e r s w h i c h a r e v i t a l l y r e l a t e d
t o human l i f e and a r e t h u s i n t e r e s t i n g t o many s c i e n t i s t s and e n g i n e e r s
i n v a r i o u s f i e l d s . T h i s w o r k i s c o n c e m e d w i t h r e l a t i v e l y h i g h v e g e t a t i v e c o v e r i n g s o f t h e l a n d . A s i n g l e t r e e , o r c h a r d t y p e c a n o p i e s , f o r e s t t y p e c a n o -p i e s a n d b r u s h y c a n o -p i e s a r e t h e o b j e c t s o f m a i n i n t e r e s t . The r e s e a r c h was c o n d u c t e d o n : a. S i n g l e model t r e e and p r o t o t y p e t r e e d r a g c o e f f i c i e n t s , i n w h i c h t r e e f l e x i b i l i t y and s t a n d a r d d e v i a t i o n o f t h e s i n g l e t r e e d r a g f o r c e a r e i n c l u d e d , b. C o n s t r u c t i o n and v e r i f i c a t i o n o f a s h e a r p l a t e w h i c h was u s e d t o measure smooth b o u n d a r y s h e a r a n d t h e l o c a l s u r f a c e d r a g f o r c e s w i t h i n c a n o p i e s . c. The l o c a l d r a g c o e f f i c i e n t o f a s i m u l a t e d o r c h a r d c a n o p y , d. The l o c a l d r a g c o e f f i c i e n t o f a s i m u l a t e d f o r e s t c a n o p y , i n w h i c h v e l o c i t y p r o f i l e s o v e r and w i t h i n t h e canopy w e r e
s t u d i e d . The a e r o d y n a m i c r o u g h n e s s f o r t h e f o r e s t canopy was a l s o
e. The l o c a l d r a g c o e f f i c i e n t o f a s i m u l a t e d b r u s h y c a n o p y , i n w h i c h v e l o c i t y p r o f i l e s o f t h e canopy and t h e a e r o d y n a m i c r o u g h n e s s w e r e s t u d i e d . f . The s i m u l a t i o n o f p r o t o t y p e canopy f i e l d s w i t h m o d e l s i n low v e l o c i t y w i n d t u n n e l s . The w o r k " c a n o p y " w i l l be u s e d f r e q u e n t l y i n t h i s w o r k and i s d e f i n e d as a v e g e t a t i v e f i e l d w h i c h has l a r g e g e o m e t r i c r o u g h n e s s , f l e x -i b l e o r n o n - f l e x -i b l e , a b o u t w h -i c h t h e r e -i s e -i t h e r t w o - d -i m e n s -i o n a l o r t h r e e - d i m e n s i o n a l f l o w c o n d i t i o n s . A s i n g l e t r e e , a row o f t r e e s , a c o m f i e l d , a paddy f i e l d , an o r c h a r d , b r u s h , and a f o r e s t b e l o n g t o t h i s c a t e g o r y . The f o l l o w i n g l i t e r a t u r e r e v i e w i s i n t e n d e d t o p r e s e n t an i n t r o d u c t i o n t o e x i s t i n g k n o w l e d g e o f canopy f i e l d s . Some o f t h e r e f e r e n c e s c i t e d b e l o w a r e r e v i e w e d i n d e t a i 1 i n C h a p t e r 2 w i t h i n s u b -s e c t i o n -s , and -some p r e c i -s e m a t h e m a t i c a l d e f i n i t i o n -s f o r t h e t e r m i n o l o g y w i l l a l s o be i n c l u d e d t h e r e . L e t t a u ( 1 9 6 1 ) s t u d i e d a v e g e t a t i v e s u r f a c e r o u g h n e s s w h i c h d e a l t w i t h r e g i o n a l and s e a s o n a l v a r i a t i o n s . Because o f t h e l a c k o f p e r t i n e n t i n f o r m a t i o n , he s u g g e s t e d t h a t h i s r e s u l t s s h o u l d be c o n -s i d e r e d a-s t e n t a t i v e . L e t t a u ' -s -s t u d y on t h e v e g e t a t i v e c o v e r -s o f t h e s t a t e o f W i s c o n s i n was, n e v e r t h e l e s s , v a l u a b l e . He g r o u p e d v e g e t a t i o n
t y p e i n t o g r a i n s s u c h as c o m , o a t s and h a y , and t r e e s s u c h as o a k ,
ma-p l e and a s ma-p e n . F i e l d c r o ma-p s o t h e r t h a n g r a i n s w e r e s t u d i e d s u c h as i d l e
and n o n - c r o p p e d l a n d , p a s t u r e d w o o d l a n d and b a r e l a n d . He u s e d an
l o g z = - 1.24 + 1.19 l o g h ( 1 . 1 . 1 ) " o
H e r e , t h e a e r o d y n a m i c r o u g h n e s s , , and p l a n t h e i g h t h a r e i n cm.
The use o f z w i l l be i l l u s t r a t e d b e l o w i n E q u a t i o n 1.1.2. The a v e r -o age a e r o d y n a m i c r o u g h n e s s e s , o b t a i n e d f r o m t h e v e g e t a t i o n h e i g h t , o f t h e s t a t e o f W i s c o n s i n a r e t a b u l a t e d b e l o w : T a b l e 1.1.1 The a v e r a g e a e r o d y n a m i c r o u g h n e s s e s o f t h e s t a t e o f W i s c o n s i n z
0
z0
n o r t h e a s t 64.96 e a s t 5.24 n o r t h 44.62 s o u t h w e s t 3.19 n o r t h w e s t 38.85 s o u t h 4.29 w e s t 8.12 s o u t h e a s t 6.82 c e n t r a l 12.71 Deacon ( 1 9 5 3 ) c o n d u c t e d r e s e a r c h on v a r i o u s n a t u r a l s u r f a c e s . He f o u n d t h e a e r o d y n a m i c r o u g h n e s s o f n a t u r a l s u r f a c e s b y u s i n g t h e 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 e q u a t i o n : J L . 1 iln ( 1 . 1 . 2 ) w h e r e : w i n d v e l o c i t y as a f u n c t i o n o f y above t h e g r o u n d u ^ = s h e a r v e l o c i t y . T = s u r f a c e s h e a r s t r e s s o p = mass d e n s i t y o f t h e f l u i d k = t h e u n i v e r s a l Karman c o n s t a n t , 0.4= a e r o d y n a m i c r o u g h n e s s .
Deacon's s t u d i e s were m a i n l y c o n d u c t e d on t w o s u r f a c e
c o n d i t i o n s , i . e . , a s u r f a c e w i t h o u t n a t u r a l v e g e t a t i o n and a s u r f a c e
o f low v e g e t a t i o n . The r e s u l t o f h i s s t u d y was w i d e l y r e c o g n i z e d and
u s e d b y many r e s e a r c h e r s i n f l u i d m e c h a n i c s and m e t e o r o l o g y . Deacon
i n d i c a t e d t h a t t h e a e r o d y n a m i c r o u g h n e s s d e c r e a s e s w i t h an i n c r e a s e o f
w i n d v e l o c i t y f o r b o t h cases o f 45 cm and 65 cm h e i g h t mown g r a s s s u r
-f a c e . H i s s t u d y showed t h a t Kung's e m p i r i c a l e q u a t i o n -f o r e s t i m a t e d
a e r o d y n a m i c r o u g h n e s s was good i n t h e s t u d y o f r a t h e r s t i f f v e g e t a t i v e
c o v e r i n g s .
F o r l a r g e c r o p s such as a f o r e s t canopy and a b r u s h y c a n o p y ,
h o w e v e r , t h e d e s c r i p t i o n o f t h e v e l o c i t y d i s t r i b u t i o n b y E q u a t i o n 1.1.2
was n o t s a t i s f a c t o r y b e c a u s e no m e a n i n g f u l l o g a r i t h m i c c u r v e c o u l d be
f o u n d w i t h i n and above t h e canopy. T h u s , Rossby and Montgomery ( 1 9 3 5 )
s u g g e s t e d t h e f o l l o w i n g e q u a t i o n :
1 an ^ . ( 1 . 1 . 3 ) ^ 'o
The z e r o p o i n t d i s p l a c e m e n t , d , i s t h e v e r t i c a l d i s t a n c e w h e r e t h e
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 has u0 , T h u s , d was t h e t h i r d p a
-r a m e t e -r w h i c h was d e t e -r m i n e d e x p e -r i m e n t a l l y and w h i c h h a d t o b e s c a l e d
p r o p e r l y i f t h e d i s t r i b u t i o n o f w i n d above l a r g e c r o p s was t o be s i m u
-l a t e d i n a w i n d t u n n e -l .
S t o l l e r and Lemon ( 1 9 6 3 ) d i d a s t u d y on wheat f i e l d s and
o b t a i n e d w i n d v e l o c i t y d a t a i n d i m e n s i o n l e s s f o r m v e r s u s y/h w i t h i n
t h e w h e a t f i e l d . Tan and L i n g ( 1 9 6 1 ) d i d t h e same k i n d o f s t u d y on
and a r e shown i n F i g . 4.3.1.5.
P l a t e and Q u r a i s h i ( 1 9 6 5 ) d i d a t h o r o u g h s t u d y on c a n o p i e s o f
wood pegs and o f p l a s t i c s t r i p s . The wood peg was 5.08 cm h i g h and
0.475 cm i n d i a m e t e r , a r r a n g e d i n a p a t t e r n o f s q u a r e s , 2.54 cm s p a c i n g
i n b o t h l o n g i t u d i n a l and l a t e r a l d i r e c t i o n s . The f l e x i b l e s t r i p was
0.635 cm w i d e , 0.019 cm t h i c k and 10.2 cm h i g h . S t r i p s were f a s t e n e d
t o wooden s t r i p s . The p l a s t i c s t r i p s w e r e a r r a n g e d t o f a c e t h e d i r e c
-t i o n o f -t h e w i n d w i -t h -t h e i r b r o a d s i d e , w i -t h a s p a c i n g i n -t h e d i r e c -t i o n
n o r m a l t o t h e f l o w o f one e l e m e n t p e r 2.54 cm, and a s p a c i n g i n t h e
d i r e c t i o n o f f l o w o f one row e v e r y 5.08 cm. T h e i r c a n o p i e s were u n i
-f o r m l y and r e g u l a r l y a r r a n g e d on a w i n d t u n n e l -f l o o r o -f 183 cm by 610 cm a r e a and v e l o c i t y d i s t r i b u t i o n s were s t u d i e d u n d e r v a r i o u s a m b i e n t w i n d s p e e d s . They f o u n d t h a t t h e v e l o c i t y d i s t r i b u t i o n i n t h e f l o w above t h e p l a n t c o v e r c o u l d be r e p r e s e n t e d b y 1 ( 1 . 1 . 4 ) w h e r e : n = a c o n s t a n t U = t h e a m b i e n t w i n d v e l o c i t y S - 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 h = p l a n t h e i g h t , o r h e i g h t o f e l e m e n t . The e x p o n e n t n = 3 a g r e e s r e m a r k a b l y w e l l w i t h Moore ( 1 9 5 1 ) and B h a d u r i ( 1 9 6 3 ) f o r r o u g h n e s s e l e m e n t s c o n s i s t i n g o f wooden s t r i p s f a s t e n e d t o t h e w i n d t u n n e l f l o o r a t e q u a l i n t e r v a l s o f 30.48 cm.
d i r e c t i o n . The e x p o n e n t n = 3 a l s o a g r e e s w e l l w i t h t h e f o r e s t
canopy s t u d y i n t h i s r e s e a r c h .
P l a t e and Q u r a i s h i a d o p t e d E q u a t i o n 1.1.3 and used t h e
p h y s i c a l h e i g h t o f r o u g h n e s s h i n s t e a d o f t h e z e r o p o i n t d i s p l a c e
-ment d o r i g i n a t e d by Rossby and Montgomery.
u 1 „ ; y-h
- n : = k ^ " ( V ) • ( 1 . 1 . 5 )
. C o n s i d e r i n g t h e above m e n t i o n e d w o r k s , i t i s seen t h a t more
d a t a a r e a v a i l a b l e i n t h e measurement o f v e l o c i t y d i s t r i b u t i o n s t h a n i n
a e r o d y n a m i c r o u g h n e s s and s h e a r v e l o c i t y . F u r t h e r m o r e , none o f them
e l a b o r a t e d upon t h e i n f l u e n c e o f 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 on t h e
s u r f a c e s t r e s s and on t h e a e r o d y n a m i c r o u g h n e s s o f t h e canopy f i e l d .
N e i t h e r d i d t h e y s t u d y t h e i n i t i a l f l o w r e g i o n and t h e end r e g i o n o f
t h e canopy f i e l d s , n o r any d i r e c t measurement o f t h e s u r f a c e s t r e s s
w i t h i n t h e canopy f i e l d s . T h e r e f o r e , i t i s t h e p u r p o s e o f t h i s w o r k t o s t u d y t h e s e t o p i c s e x t e n s i v e l y . The v e l o c i t y p r o f i l e s and a e r o d y n a m i c r o u g h n e s s o f t h e c e n t e r r e g i o n o f t h e c a n o p i e s w i l l be s t u d i e d and com-p a r e d , when com-p o s s i b l e , w i t h t h e d a t a a v a i l a b l e f r o m t h e above m e n t i o n e d r e s e a r c h e r s . 1-2 P u r p o s e and Scope The g e n e r a l p u r p o s e o f t h i s w o r k i s t o p r e s e n t e x p e r i m e n t a l d e t e r m i n a t i o n s o f t h e d r a g c o e f f i c i e n t f o r v a r i o u s t y p e s o f c a n o p i e s by u s i n g a r t i f i c i a l p l a s t i c t r e e s i n a w i n d t u n n e l where t h e s t u d y o f t h e canopy f i e l d can be r e g u l a t e d i n a c a r e f u l f a s h i o n by c o n t r o l l i n g t h e p h y s i c a l h e i g h t o f t h e c a n o p y , t h e d e n s i t y o f t h e c a n o p y , t h e
w i n d V C o e U i e s , „ = . r d i „ , t o tKo p „ . p o „ ,t„d.. „ o „ , p a c i n c a „ , t»o f o U » l „ , t o p i c s s,e s t u d i e d t . o „ u , „ , »„ t H i . „ s e . r c h , d e t a i l s O f w h i c h . 1 1 , be d i s c u s s e d 1„ C h a p t e r s 2 .„d 4. 1-2.1 S u r f a c e Shear on , 5.„„th RounH.,,, 11.0 s u r f a c e s h e a r s t r e s s o n a s . o o t h b o u n d a r y u n d e r i n o o ^ r e s s l b l e , t u r b u l e n t f , o . c o n d i t i o n s .as s t u d i e d I n a . I n d t u n n e l 11.0 P»n.ose ..s t o t e s t t h e u s e o f a d r a , . e . s u r l n . d e v i c e d e v e l o p e d b , N..h «.d d i e d a s h e a r p l a t e , s » o t h b o u n d a r y c o n d i t i o n c r e a t e d a t o s t i n , S i t u a t i o n . h e r e t h e v a l u e o f t h e bounda.-y s h e a r s t r e s s h a s been won e s t a b l i s h e d , ^ u s c o n f i d e n c e . a , ,.i„ed 1„ t h e s u e o f t h e s h e a r P l a t e f o r . e a s u r l n , . . 1 1 s h e a r u n d e r o t h e r s u r f a c e c o n d i t i o n s , ^ e d e s c r i p t i o n o f t h e s h e a r p l a t e „111 b e p r e s e n t e d I n c h a p t e r 3, The S h e a r p l a t e . a s , 1„ oonse,ua.c., used f o r . e . s u r l n g t h e l o c a l s u r f a c e
. h e a r o f t h , . o d e l f o r e s t - t y p e canopy and t h , . o d e , b . ™ h y canopy i„
t h i s w o r k . ^ ^ : g i L C o e m c L e n t o f a S i n . l . I n t h i s w o r k , a s i n g l e a r t i f i c i a l t r e e was s t u d i e d I n a f r e e s t r e a m a n d i n a w e l l submerged b o u n d a r y l a y e r c o n d t i i o n . I t was f e l t t h a t b y d e f i n i n g t h e c h a r a c t e r i s t i c v e l o c i t y as t h e r o o t mean s q u a r e v e l o c i t y o n t h e t r e e f r o n t a l a r e a , t h e t r e e d r a g c o e f f i c i e n t w o u l d be t h e same f o r t h e a r t i f i c i a l t . e e i n a f r e e s t r e a m as f o r t h e a r t i f i c i a l t r e e x n a t h i c k b o u n d a r y l a y e r . D e f i n i t i o n o . t h e t r e e d r a g c o e f f i -c i e n t w i l l be seen i n C h a p t e r 2. By c o m p a r i n g t h e t r e e d r a g c o e f f i c i e n t o f t h e a r t i f i c i a l p l a s t i c t r e e w i t h p r o t o t y p e t r e e s , i t was f e l t tu . a, 11 was t e l t t h a t c o n f i d e n c e w o u l d
b® g a i n e d i n d e t e r a i n i n g t h e t r e e d r a g c o e f f i c i e n t o f a r e a l t r e e u n d e r v a r i o u s f l o w c o n d i t i o n s . The t r e e d r a g c o e f f i c i e n t o f an a r t i ¬ f i c i a l p l a s t i c t r e e was f o u n d t o be b e t w e e n t h a t o f D o u g l a s f i r and s p r u c e . T h i s r e v e a l e d t h a t t h e e x p e r i m e n t a l f o r e s t s t u d i e d i n t h i s w o r k may be c o n s i d e r e d as a s i m u l a t e d D o u g l a s f i r f o r e s t o r a s p r u c e f o r e s t . M o r e o v e r , t h e i n f o r m a t i o n o f a s i n g l e t r e e d r a g c o e f f i c i e n t can be u s e f u l t o t h e s t u d y o f w i n d b r e a k s . The t r e e f l e x i b i l i t y a i d t h e s t a t i s t i c a l v a r i a t i o n o f d r a g f o r c e on an a r t i f i c i a l t r e e were s t u d i e d . T h i s i n f o r m a t i o n can be u s e d t o e s t i m a t e t h e t r e e d r a g c o e f f i c i e n t and t h e s t i f f n e s s o f a s i n g l e t r e e . Drag C o e f f i c i e n t o f V a r i o u s C a n o p i e s The s t a n d a r d i z e d l o c a l s k i n f r i c t i o n d r a g c o e f f i c i e n t s f o r f l o w i n p i p e s and a l o n g p l a t e s a r e u s e d s u c c e s s f u l l y i n e n g i n e e r i n g a p p l i c a ¬ t i o n s . However, t h e d r a g c o e f f i c i e n t s f o r v a r i o u s v e g e t a t i v e c a n o p i e s a r e s c a r c e . T h e r e f o r e , t h i s w o r k i n t e n d s t o do a t h o r o u g h s t u d y on t h i s s u b j e c t r e g a r d i n g e x p e r i m e n t a l t r e e c a n o p i e s i n t h e hope t h a t t h e
cano-p i e s may be g e n e r a l i z e d and a l a b o r a t o r y s t u d y o f c a n o cano-p i e s may s i m u l a t e
p r o t o t y p e c a n o p i e s . I t w i l l be shown how t h i s i n f o r m a t i o n i s u s e f u l f o r d i f f u s i o n s t u d i e s . I n t h i s w o r k , t h e l o c a l d r a g c o e f f i c i e n t was f o u n d f o r v a r i o u s t y p e s o f c a n o p i e s . The e x p e r i m e n t a l r e s u l t s compared f a v o r a b l y w i t h t h e a n a l y t i c a l r e s u l t s . The s i m u l a t e d f o r e s t - t y p e c a n o p i e s s t u d i e d h e r e w e r e composed o f a r t i f i c i a l t r e e s w h i c h were d e s c r i b e d i n p r e v i o u s s u b s e c t i o n 1.2.2. A c l o s e - u p p h o t o . F i g u r e 3 . 4 . 1 . 1 , and t h e d i m e n s i o n s o f an a r t i f i c i a l p l a s t i c t r e e w i l l be s e e n i n C h a p t e r 3.
p l a t e s a r e used by e n g i n e e r s i n e s t i m a t i n g o v e r - a l l d r a g f o r c e o f t h e
p r o b l e m . F o r t h i s r e a s o n , t h e t o t a l d r a g c o e f f i c i e n t o f e x p e r i m e n t a l
f o r e s t - t y p e c a n o p i e s were s t u d i e d . M o r e o v e r , t w o d i f f e r e n t t h i c k n e s s e s
o f b o u n d a r y l a y e r were a p p l i e d t o t h e same e x p e r i m e n t a l b r u s h y canopy
i n o r d e r t o f i n d some r e l a t i o n between t h e t h i c k n e s s o f t h e b o u n d a r y l a y e r and t h e t o t a l d r a g c o e f f i c i e n t o f t h e e x p e r i m e n t a l c a n o p i e s . This i n f o r m a t i o n w i l l p r o v i d e a t l e a s t a q u a l i t a t i v e c l u e when t h e i n -f o r m a t i o n o -f t h e t o t a l d r a g c o e -f -f i c i e n t o -f an e x p e r i m e n t a l canopy i s a p p l i e d t o a p r o t o t y p e canopy w h i c h i s u n d e r a c e r t a i n t h i c k n e s s o f an a t m o s p h e r i c b o u n d a r y l a y e r .
'•^•^ g23Ï£iS£i.ty P r o f i l e s Over and W i t h i n t h e S i m u l a t e d F...
The v e l o c i t y p r o f i l e s o f t h e s i m u l a t e d f o r e s t canopy have an i n t e r e s t i n g phenomenon. The f l o w above t h e t o p o f a f o r e s t canopy
i s e x p e c t e d t o p o s s e s s 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 t o w a r d s t h e cen-t e r p o r cen-t i o n o f cen-t h e canopy f i e l d , and cen-t h e p r o f i l e b e l o w becomes a p p r o x i ¬ m a t e l y u n i f o r m f l o w .
The s t u d y o f t h e w i n d v e l o c i t y p r o f i l e s o v e r and w i t h i n t h e
s i m u l a t e d f o r e s t c a n o p i e s has two p u r p o s e s : one i s r a t h e r f o r r e
-s e a r c h i n t e r e -s t . T h e r e a r e no f o r e -s t w i n d p r o f i l e d a t a w h i c h c o v e r
f r o m t h e b e g i n n i n g t o t h e end o f a f o r e s t c a n o p y , and o n l y b y s i m u l a t i n g
a f o r e s t canopy i n a w i n d t u n n e l was such a s t u d y p o s s i b l e and
meaning-f u l . The o t h e r i s meaning-f o r p r a c t i c a l a p p l i c a t i o n s . By k n o w i n g t h e w i n d
c h a r a c t e r , t h a t i s t h e l o c a l w i n d v e l o c i t y , o v e r a n d w i t h i n r h e f o r e s t
e f f i c i e n t l y when c o n s i d e r i n g t h e c h a r a c t e r i s t i c s o f t h e w i n d . The f o r e s t f i r e , snow p a c k , desease c o n t r o l , e t c . , a r e p r o b l e m s w h i c h a r e c l o s e l y r e l a t e d t o t h e w i n d c h a r a c t e r o f a f o r e s t . I n t h i s w o r k , t h e w i n d v e l o c i t y p r o f i l e s o v e r t h e c e n t e r r e g i o n o f a s i m u l a t e d f o r e s t canopy were f i t t e d t o t h e 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 E q u a t i o n 1.1.3 and t o t h e power v e l o c i t y d i s t r i -b u t i o n E q u a t i o n 1.1.4 f o r t h e p u r p o s e o f c o m p a r i n g w i t h o t h e r s ' work. The w i n d v e l o c i t y p r o f i l e s w i t h i n t h e c e n t e r p o r t i o n o f t h e s i m u l a t e d f o r e s t canopy were u n i f o r m up t o 0.6 t r e e h e i g h t f r o m t h e f l o w , t h a t i s , no v e l o c i t y g r a d i e n t e x i s t e d i n t h e v e r t i c a l d i r e c t i o n . These d a t a are p r e s e n t e d i n C h a p t e r 4, t o g e t h e r w i t h t h e w i n d p r o f i l e s a t t h e i n i
-t i a l and -t h e end r e g i o n s o f -t h e s i m u l a -t e d f o r e s -t canopy.
1-2.5 The A e r o d y n M i i c Roughness o f C a n o p i e s Th© a e r o d y n a m i c r o u g h n e s s , , i s one o f t h e p a r a m e t e r s i n E q u a t i o n 1.1.5 w h i c h was u s e d t o i n t e r p r e t t h e v e l o c i t y p r o f i l e s a t t h e c e n t e r r e g i o n o f t h e s i m u l a t e d f o r e s t and b r u s h y c a n o p i e s . I t was f e l t t h a t t h e m a g n i t u d e o f t h e a e r o d y n a m i c r o u g h n e s s i s n o t o n l y p r o -p o r t i o n a l t o t h e -p h y s i c a l h e i g h t o f t h e s t u d i e d c a n o -p i o s b u t a l s o a f u n c t i o n o f t h e d r a g f o r c e f o r f l o w o v e r a t r a n s i t i o n r e g i o n o r o v e r a c o m p l e t e l y r o u g h r e g i o n o f a canopy. The t e r m i n o l o g y o f " t r a n s i t i o n r e g i o n " and " c o m p l e t e l y r o u g h r e g i o n " were a d o p t e d f r o m S c h l i c h t i n j ( 1 9 6 8 ) . The p u r p o s e o f s t u d y i n g t h e a e r o d y n a m i c r o u g h n e s s i n t h i s w o r k was t o f i n d o u t w h e t h e r E q u a t i o n 1.1.1 w o r k e d a l s o f o r t h e s i m u l a t e d f o r e s t and b r u s h y c a n o p i e s . T h u s , t h e l o c a l s h e a r s t r e s s o f c a n o p i e s c o u l d be a p p r o x i m a t e d t h r o u g h E q u a t i o n s 1.1.1 and 1.1.5 by j u s t m e a s u r i n g
t h e l o c a l v e l o c i t y u . T h i s k n o w l e d g e i s p a r t i c u l a r l y i m p o r t a n t
t o t h e s t u d y o f t h e p r o t o t y p e f o r e s t c a n o p i e s , b e c a u s e t h e n t h e l o c a l
s h e a r s t r e s s , o r t h e l o c a l d r a g c o e f f i c i e n t , o f t h e p r o t o t y p e f o r e s t
canopy c a n be c a l c u l a t e d f r o m E q u a t i o n s 1.1.1 and 1.1.5.
The r e s u l t s o f t h e a e r o d y n a m i c r o u g h n e s s o f t h e s i m u l a t e d
f o r e s t and b r u s h y c a n o p i e s were compared w i t h t h o s e o f Deacon, L e t t a u , and Kung.
C h a p t e r 2
Tl IHÜRli r I CAL CONS I DüUAT IONS
2.1 The R e s i s t a n c e F o r m u l a f o r a Smooth P l a t e and a U n i f o r m Rough P l a t e i n T u r b u l e n t Flow The t u r b u l e n t b o u n d a r y l a y e r on a f l a t p l a t e a t z e r o i n c i d e n c e w i t h and w i t h o u t z e r o p r e s s u r e g r a d i e n t i s o f g r e a t p r a c t i c a l i m p o r t a n c e . However, l i k e t h e case w i t h l a m i n a r f l o w , t h e s k i n f r i c t i o n c o e f f i c i e n t s i n a m i l d p r e s s u r e g r a d i e n t a r e n o t m a t e r i a l l y d i f f e r e n t f r o m t h o s e f o r z e r o p r e s s u r e g r a d i e n t , p r o v i d e d t h e r e i s no s e p a r a t i o n . The o n l y me-t h o d s a v a i l a b l e a me-t me-t h e p r e s e n me-t me-t i m e f o r me-t h e m a me-t h e m a me-t i c a l me-t r e a me-t m e n me-t o f t u r b u l e n t b o u n d a r y l a y e r s a r e a p p r o x i m a t e m e t h o d s .
A method f o r t h e smooth p l a t e i s b a s e d on t h e momentum I n t e g r a l
e q u a t i o n f o r a t w o - d i m e n s i o n a l i n c o m p r e s s i b l e b o u n d a r y l a y e r w h i c h i s p r e s e n t e d i n S c h l i c h t l n g ( 1 9 6 8 ) a s : where t h e momentum t h i c k n e s s 9 i s ^ = f " ^ ( 1 - Ü - ) ''y ^'-'-^^ Jo a a
ft
and t h e d i s p l a c e m e n t t h i c k n e s s 6 i s ^ OO 6* - / 1 - g - 1 dy . ( 2 . 1 . 3 ) •Jo a ' The a m b i e n t w i n d v e l o c i t y U i s a f u n c t i o n o f x because t h e E q u a t i o n 2.1.1 i s d e r i v e d f o r t h e f l o w w i t h p r e s s u r e g r a d i e n t , andv a r i e s a c c o r d i n g t o t h e m a g n i t u d e o f t h e p r e s s u r e g r a d i e n t . By i n t r o d u c i n g t h e shape f a c t o r H w h i c h i s t h e r a t i o o 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 and t h e momentum t h i c k n e s s ,
ft
" " — • ( 2 . 1 . 4 ) The E q u a t i o n 2.1.1 can be e x p r e s s e d i n a n o t h e r f o r m : These e q u a t i o n s e x p r e s s t h e w a l l s h e a r s t r e s s i n t e r m s o f 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 and t h e a m b i e n t v e l o c i t y w h i c h i s i n f l u e n c e d by t h e p r e s s u r e g r a d i e n t . I f no p r e s s u r e g r a d i e n t i s p r e s e n t e d , t h e s e c o n d t e r m on t h e r i g h t s i d e o f E q u a t i o n 2.1.5 w i l l d r o p , and i t becomes '^o de — r 3 7 ( 2 . 1 . 6 ) a The d e r i v a t i v e o f 6 w i t h r e s p e c t t o x i s v e r y s e n s i t i v e , and i n most cases t h e w a l l s h e a r s t r e s s c a n n o t be f o u n d a c c u r a t e l y b y E q u a t i o n 2.1.6. T h e r e f o r e , o t h e r a n a l y t i c a p p r o a c h e s t o f i n d t h e w a l l s h e a r s t r e s s a r e needed. T h e r e a r e two ways, w h i c h a r e b a s e d on v e l o c i t y p r o f i l e d i s t r i -b u t i o n , t o f i n d t h e w a l l s h e a r s t r e s s on a smooth p l a t e i n a t u r -b u l e n tf l o w : one i s b a s e d on t h e power v e l o c i t y d i s t r i b u t i o n and t h e o t h e r i s
b a s e d on t h e 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 . The power v e l o c i t y d i s t r i b u t i o n i s shown a s :
1_ u
f o r R e y n o l d s n u m b e r s , Re^^ , b e t w e e n 5 x 10^ and 10^, And t h e f o l l o w i n g r e l a t i o n f o r a c i r c u l a r p i p e , b y N i k u r a d s e , when m o d i f i e d f o r t h e smooth s u r f a c e i n a t u r b u l e n t f l o w c a s e , i s 1 - \ « 0.0225 ^ P U / " a « , 7 ( 2 . 1 . 8 ) w h e r e v i s t h e k i n e m a t i c v i s c o s i t y , ^ . By u s i n g E q u a t i o n s 2.1.2 and 2.1.6, E q u a t i o n 2.1.8 can be w r i t t e n as 1 h § • 0.0225 j ^ ] ^ . ( 2 . 1 . 9 )
The d e r i v i n g p r o c e d u r e s a r e o m i t t e d h e r e and can be f o u n d f r o m S c h l i c h t i n g ( 1 9 6 8 ) t h a t 1 /U X " 5 6 ( x ) - 0.036 X ( 2 . 1 . 1 0 ) and ^ / U X " 5 Cf' - 0-0576 - S - j ( 2 . 1 . 1 1 ) w h e r e t h e c^' i s t h e l o c a l d r a g c o e f f i c i e n t w h i c h i s d e f i n e d as ( 2 . 1 . 1 2 ) % , U / E q u a t i o n s 2.1.10 and 2.1.11 a p p l y o n l y f o r a smooth p l a t e i n a t u r b u l e n t b o u n d a r y l a y e r and a r e v a l i d on t h e a s s u m p t i o n t h a t t h e b o u n d a r y l a y e r i s t u r b u l e n t f r o m t h e l e a d i n g edge o n w a r d . Th© l o c a l d r a g c o e f f i c i e n t e q u a t i o n b a s e d on t h e power v e l o c i t y d i s t r i b u t i o n i s r e s t r i c t e d t o U 6/v < 10^ . The l o c a l d r a g
c o e f f i c i e n t deduced f r o m t h e 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 l e a d s t o a f a i r l y cumbersome s e t o f e q u a t i o n s . However, H. S c h l i c h t i n g f o u n d an e m p i r i c a l e q u a t i o n w h i c h f i t s w e l l w i t h l a b o r a t o r y e x p e r i m e n t a t i o n , i . e . , C f ' = (2 l o g Re^ - 0 . 6 5 ) " ^ - ^ ( 2 . 1 . 1 3 ) w h e r e U^x/v , t h e R e y n o l d s number, i s b a s e d on t h e l o n g i t u d i n a l d i s t a n c e x . No r e s t r i c t i o n i s p o s e d on E q u a t i o n 2.1.13, The f o l l o w i n g a r e a number o f r e c o g n i z e d l o c a l d r a g c o e f f i c i e n t e q u a t i o n s f o r a smooth p l a t e i n a t u r b u l e n t b o u n d a r y l a y e r , F. S c h u l t z -Granow ( 1 9 4 1 ) f o u n d t h a t t h e r e s i s t a n c e f o r m u l a f r o m h i s e x p e r i m e n t s was C f ' = 0.370 ( l o g Re^)-2-S84 ( 2 . 1 . 1 4 ) N i k u r a d s e ( 1 9 4 2 ) a l s o c o n d u c t e d a v e r y c o m p r e h e n s i v e s e r i e s o f e x p e r i -ments on f l a t p l a t e s . He f o u n d t h e l o c a l d r a g c o e f f i c i e n t f o r m u l a as Cf- = 0.02296 (Re^)-°-^5^ . ( 2 . 1 , 1 5 ) L u d w i e g and T i l l m a n (1950) p r o p o s e d an a n a l y t i c method f o r t h e c a l c u l a -t i o n o f l o c a l d r a g c o e f f i c i e n -t o f a f l a -t p l a -t e , w i -t h o r w i -t h o u -t p r e s s u r e g r a d i e n t on i t , i . e . , - 0 . 6 7 8 H f V l ° - 2 6 8 C f ' » 0.246 x 10" ( 2 . 1 . 1 6 ) T h r o u g h o u t t h e s t u d y o f a smooth b o u n d a r y i n t h i s w o r k , t h e
v i r t u a l o r i g i n was e s t i m a t e d f o r t h e c a l c u l a t i o n o f Re^ b y t h e method
o f R u b e s i n ( 1 9 5 1 ) , P h y s i c a l l y , t h e t u r b u l e n t b o u n d a r y c a n n o t s t a r t
w i t h z e r o b o u n d a r y l a y e r t h i c k n e s s , and t h e e f f e c t i v e s t a r t i n g p o i n t i s
c a l l e d t h e v i r t u a l o r i g i n . The m e t h o d o f e s t i m a t i n g t h e v i r t u a l o r i g i n
t o g e t h e r w i t h t h e e x p e r i m e n t a l r e s u l t s w i l l be d i s c u s s e d i n d e t a i l i n
C h a p t e r 4. H i e l o c a l d r a g c o e f f i c i e n t E q u a t i o n s 2.1.6, 2.1.12, 2.1.14,
and 2.1.16 were u s e d i n t h i s w o r k . P a r t i c u l a r l y , E q u a t i o n 2.1.6 w i l l
be d i s c u s s e d a t f u r t h e r l e n g t h i n C h a p t e r 4.
The r o u g h p l a t e i s more common i n e n g i n e e r i n g p r o b l e m s t h a n
t h e smooth p l a t e and d e s e r v e s more a t t e n t i o n . I f t h e r e l a t i v e r o u g h
-ness - J — i s c o n s i d e r e d , t h e r e l a t i v e r o u g h n e s s w i l l d e c r e a s e a l o n g t h e p l a t e . The h i s t h e r o u g h n e s s h e i g h t w h i c h r e m a i n s c o n s t a n t f o r u n i -f o r m r o u g h p l a t e , w h i l e 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 c r e a s e s downs t r e a m . T h i downs c i r c u m downs t a n c e caudownsedowns t h e f r o n t o f t h e p l a t e t o b e h a v e d i f -f e r e n t l y -f r o m t h e r e a r w a r d p o r t i o n as -f a r as t h e i n -f l u e n c e o -f r o u g h n e s s on d r a g i s c o n c e r n e d . The c o m p l e t e l y r o u g h f l o w i s o v e r t h e f o r w a r d p o r t i o n , f o l l o w e d by t h e t r a n s i t i o n r e g i o n and, e v e n t u a l l y , t h e r o u g h
p l a t e may become h y d r a u l i c a l l y smooth i f i t i s s u f f i c i e n t l y l o n g . The
f o r e s t and b r u s h y c a n o p i e s d i d n o t have a h y d r a u l i c a l l y smooth r e g i o n
i n t h i s w o r k . The d i m e n s i o n l e s s r o u g h n e s s p a r a m e t e r - y — i s u s e d t o d e f i n e t h e l i m i t b e t w e e n c h a r a c t e r c a t e g o r i e s as f o l l o w s : u^k c o m p l e t e l y r o u g h : ^ ^ > 70 u^k t r a n s i t i o n : 5 < < 70 ( 2 . 1 . 1 7 ) h y d r a u l i c a l l y smooth: — — < 5 w h e r e kg a t h e e q u i v a l e n t s a n d r o u g h n e s s . Based on t h e 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 , P r a n d t l and S c h l i c h t i n g p r e s e n t e d t h e f o l l o w i n g e q u a t i o n :
ÏÏ; = k ^" r ) * h (2.1.18) w h e r e i s c a l l e d t h e r o u g h n e s s f u n c t i o n and i s a c o n s t a n t 8.5 u n d e r c o m p l e t e l y r o u g h c o n d i t i o n s . I n o r d e r t o f i n d t h e r e l a t i o n b e t w e e n t h e e q u i v a l e n t sand r o u g h n e s s , k^ , and t h e p h y s i c a l h e i g h t o f t h e r o u g h -n e s s , h , S c h l i c h t i -n g p e r f o r m e d e x p e r i m e -n t s f o r a l a r g e -number o f r o u g h n e s s e s a r r a n g e d on a f l a t p l a t e and d e t e r m i n e d t h e c o n s t a n t i n t h e u n i v e r s a l e q u a t i o n , TT- =• T '•nf^] + B^ . ^ F - ' M h M «2 • ( 2 . 1 . 1 9 ) On c o m p a r i n g E q u a t i o n s 2.1.18 and 2.1.19, t h e e q u i v a l e n t sand r o u g h n e s s can be o b t a i n e d f r o m t h e r e s u l t i n g e q u a t i o n : 1 / ^ F ^ " ( i r - - B2 . ( 2 . 1 . 2 0 ) Paeshke ( 1 9 3 7 ) d e m o n s t r a t e d f r o m e x p e r i m e n t t h a t E q u a t i o n 2.1.19 can be a p p l i e d t o t h e m o t i o n o f a n a t u r a l w i n d o v e r s u r f a c e s w h i c h w e r e c o v e r e d w i t h d i f f e r e n t k i n d s o f v e g e t a t i o n . He f o u n d t h a t B = 5 when t h e p h y s i c a l h e i g h t o f t h e v e g e t a b l e g r o w t h h i s u s e d . I n a c c o r d a n c e w i t h E q u a t i o n 2.1.20 t h i s i s t h e same as t a k i n g t h e e q u i v a l e n t sand r o u g h n e s s t o be k^ = 4 h The e q u i v a l e n t sand r o u g h n e s s i s u s e f u l f o r t h e r o u g h p l a t e s t u d y , f o r t h e r e a r e c h a r t s b a s e d on c^' , k^ , Re^ e x i s t i n g i n S c h l i c h t i n g ( 1 9 6 8 ) . The c^' o f t h e e x p e r i m e n t a l c a n o p i e s i n t h i s
w o r k can a l s o be a p p r o x i m a t e d by t h e method m e n t i o n e d above. However,
f o r a d e t a i l e d i n s i d e l o o k i n t o t h e r o u g h n e s s and t h e l o c a l d r a g
coe f f i c i coe n t o f t h coe coe x p coe r i m coe n t a l c a n o p i coe s , t h coe t o l l o w i n g mcoethod i s r coe
Many i n v e s t i g a t o r s u s e d t h e 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
and o m i t t e d t h e r o u g h n e s s f u n c t i o n , B , t o f i n d t h e i r a e r o d y n a m i c
r o u g h n e s s z and s h e a r v e l o c i t y u^, f r o m t h e e q u a t i o n
w h e r e u ^ and z^ a r e two unknowns. Two s e t s o f u and y d a t a
f r o m t h e same 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 h a v e t o be u s e d t o s o l v e
E q u a t i o n 2.1.2 f o r u^, and z^ . The v a l u e o f u and y , h o w e v e r ,
s h o u l d be t a k e n a t a c e r t a i n d i s t a n c e away f r o m t h e r o u g h s u r f a c e f o r
t h e 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 i s n o t p r e s e n t a d j a c e n t t o t h e
r o u g h w u r f a c e . Deacon ( 1 9 5 3 ) d i d a s t u d y b y u s i n g E q u a t i o n 1.1.2 f o r
n a t u r a l s u r f a c e s w i t h o u t v e g e t a t i o n s u c h as s e a , d e s e r t s and snow s u r
-f a c e , and -f o r n a t u r a l s u r -f a c e s w i t h low v e g e t a t i o n s u c h as mown g r a s s
s u r f a c e s . P l a t e and H i d y ( 1 9 6 7 ) u s e d E q u a t i o n 1.1.2 as w e l l t o f i n d t h e a e r o d y n a m i c r o u g h n e s s o f s m a l l w a t e r waves. One o f t h e d i f f i c u l t i e s i n t h e use o f E q u a t i o n 1.1.2 i s t h a t as -<• Q , i t does n o t a p p r o a c h t h e smooth s u r f a c e c o n d i t i o n , w h i c h i s : w h i c h i s t h e law o f t h e w a l l p r o p o s e d by L. P r a n d t l . A n o t h e r p r o b l e m i s t h a t t h e l o c a l w i n d v e l o c i t y has t o become z e r o a t t h e e l e v a t i o n o f y = i n o r d e r t o have E q u a t i o n 1.1.2 v a l i d m a t h e m a t i c a l l y , i . e . , t h e no s l i p c o n d i t i o n a t t h e s u r f a c e o f a b o u n d a r y i s s u b j e c t t o
q u e s t i o n . However, E q u a t i o n 1.1.2 has i t s m e r i t . F o r one t h i n g , u^
and i n a t u r b u l e n t f l o w a r e t h e c o u n t e r p a r t c h a r a c t e r i s t i c v e l o c i t y
and l e n g t h U and 6 i n a l a m i n a r f l o w . W i t h t h e s e two p a r a m e t e r s , ( 1 . 1 . 2 )
t h e v e l o c i t y p r o f i l e i n t u r b u l e n t f l o w does show t h e l o g a r i t h m i c
d i s t r i b u t i o n c h a r a c t e r . F o r t h e o t h e r , t h e a e r o d y n a m i c r o u g h n e s s i s
c o r r e l a t e d w i t h t h e h e i g h t o f t h e r o u g h n e s s . A l t h o u g h t h e a e r o d y n a m i c
r o u g h n e s s r e f l e c t s o n l y a q u a l i t a t i v e measure o f t h e h e i g h t o f t h e
r o u g h n e s s , t h e i s a l s o i n f l u e n c e d by u , , w h i c h i s r e l a t e d t o t h e
l o c a l s u r f a c e s t r e s - i , and hence t h e d e n s i t y o f t h e r o u g h n e s s and t h e
f l o w c o n d i t i o n .
I n t h i s w o r k , t h e h e i g h t o f s t u d i e d c a n o p i e s b e i n g r a t h e r l a r g e , t h e f l o w c h a r a c t e r can h a r d l y be shown by u s i n g E q u a t i o n 1.1.2.
Rossby and Montgomery ( 1 9 3 5 ) s u g g e s t e d t h a t f o r h i g h c r o p s , t h e equa-t i o n s h o u l d be 1 F I ' T ' I • ( 1 . 1 . 3 ) o T h i s e q u a t i o n shows t h e 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 s t a r t e d f r o m t h e e l e v a t i o n o f z e r o - p o i n t d i s p l a c e m e n t d u p w a r d and a v o i d s t h e v a r i a t i o n o f f l o w c o n d i t i o n below d . A g a i n , m a t h e m a t i c a l l y t h e l o c a l v e l o c i t y i s z e r o a t y = d * z^ , w h i c h c a n n o t be t r u e i n n a t u r e . The z e r o - p o i n t d i s p l a c e m e n t i s f o u n d f r o m e x p e r i m e n t f o r t h e d i f f e r e n t r o u g h n e s s e s p r e s e n t . F o r s i m p l i c i t y , t h e p h y s i c a l h e i g h t o f t h e r o u g h n e s s h r e p l a c e s t h e z e r o - p o i n t d i s p l a c e m e n t d i n E q u a t i o n 1.1.3. I t t h e n r e a d s : i L . = 1 „ / y - d u * k V • ( 1 - 1 . 5 ) o E q u a t i o n 1.1.5 was u s e d t o s t u d y t h e e x p e r i m e n t a l f o r e s t and b r u s h y c a n o p i e s i n t h i s w o r k . The f o l l o w i n g i s a c i t a t i o n o f e x i s t i n g s u r f a c e s h e a r s t r e s s m e a s u r i n g d e v i c e s i n n o v a t e d by v a r i o u s r e s e a r c h e r s .
S c h u l t z - G r u n w ( 1 9 4 0 ) s u c c e e d e d i n d i r e c t measurement by means o f a
m e c h a n i c a l b a l a n c e , see F i g u r e 2 . 1 . 1 , b u t t h e b a l a n c e was n o t easy t o
use f o r i t needs a l a r g e amount o f i n s t r u m e n t a l a c c e s s o r i e s . Page and
F a l k n e r ( 1 9 3 0 ) u s e d t h e p r e s s u r e o r i f i c e a t t h e p o i n t o f t h e w a l l w h e r e
t h e s h e a r i n g s t r e s s i s t o be measured. A p p r o x i m a t e l y 1/20 mm above
t h e o r i f i c e was a s h a r p k n i f e edge, a l s o c a l l e d a S t a n t o n t u b e . The
p o r t i o n o f t h e v e l o c i t y n e a r t h e w a l l i s t h e n dammed up b e t w e e n k n i f e
edge and w a l l . The p r e s s u r e r i s e s b e l o w t h e k n i f e edge w i t h r e s p e c t
t o t h e u n d i s t u r b e d s t a t i c p r e s s u r e gave Fago and F a l k n e r a measure f o r
t h e w a l l s h e a r i n g s t r e s s , s i n c e t h e v e l o c i t y d i s t r i b u t i o n i n t h e w a l l p r o x i m i t y i s d e f i n i t e l y c o r r e l a t e d t o t h e s h e a r i n g s t r e s s , However, b e c a u s e o f t h e d i f f i c u l t y i n h a n d l i n g and o f t h e e x t r e m e l y s e n s i t i v e t e s t p r o b e , t h e m e t h o d was n o t e a s y t o u s e , H, L u d w i e g ( 1 9 5 0 ) u s e d a s m a l l h e a t t r a n s f e r e l e m e n t w h i c h was b u i l t i n t o t h e w a l l and had a h i g h e r t e m p e r a t u r e t h a n t h e f l o w . He f o u n d a d e f i n i t e r e l a t i o n s h i p b e t w e e n s u r f a c e s h e a r s t r e s s and h e a t t r a n s f e r as f o l l o w s : 1 Nu - 0.807 ( J i ) t j ( 2 . 1 , 2 2 ) where Nu • N u s s e l t number I • t h e l e n g t h o f t h e h e a t e l e m e n t o - t h e t h e r m a l d i f f u s i v i t y , | ^ . The d e r i v a t i o n s t e p s a r e o m i t t e d h e r e b u t can be f o u n d i n h i s p a p e r . The h e a t t r a n s f e r e l e m e n t i s h e a t e d by a s m a l l e l e c t r i c h e a t e r . A warm b o u n d a r y l a y e r , s t a r t i n g f r o m t h e f o r w a r d edge o f t h e e l e m e n t , i s f o r m e d . The h e a t v o l u m e Q o f t h e e l e m e n t can be c a l c u l a t e d f r o m
t h e h e a t i n p u t t o t h e e l e m e n t a f t e r t h e warm b o u n d a r y l a y e r r e a c h e s t h e
s t e a d y s t a t e . The t e m p e r a t u r e s a t t h e w a l l and a t t h e edge o f t h e warm
b o u n d a r y l a y e r can be d e t e c t e d b y a t h e r m o c o u p l e . By k n o w i n g 9 and - T^) , t h e N u s s e l t n i m b e r can be o b t a i n e d f r o m w h e r e Nu = ^ = n r r r ^ - l T T ( 2 . 1 . 2 3 ) ^ o " a = b t ( ? -T ) ( 2 . 1 . 2 4 ) O 00 b = t h e w i d t h o f t h e h e a t e l e m e n t . The t h e r m a l d i f f u s i v i t y , a , can a l s o be o b t a i n e d w i t h t h e i n f o r m a t i o n
(J^-TJ
. TTius. t h e s u r f a c e s h e a r s t r e s s can be c a l c u ¬ l a t e d f r o m E q u a t i o n 2.1.22. A n o t h e r use o f E q u a t i o n 2.1.22 i s t o d e t e r -a i n e -a r e l -a t i o n s h i p b e t w e e n . Nu -and -a b y -a c -a l i b r -a t i o n me-as- meas-u r e m e n t w h e r e s meas-u r f a c e s h e a r s t r e s s e s a r e known. W i t h t h e s u c c e s s o f L u d w i e g ' s s u r f a c e s h e a r m e a s u r i n g d e v i c e , L u d w i e g and T i l l m a n ( 1 9 5 0 ) h a d t h e c o n t r i b u t i o n o f t h e a n a l y t i c Equa-t i o n 2.1.16 f o r Equa-t h e c a l c u l a Equa-t i o n o f Equa-t h e l o c a l d r a g c o e f f i c i e n Equa-t , w i Equa-t h o r w i t h o u t p r e s s u r e g r a d i e n t on a smooth b o u n d a r y i n a t u r b u l e n t f l o w . S m i t h and W a l k e r ( 1 9 5 9 ) u s e d a f l o a t i n g e l e m e n t s k i n - f r i c t i o n b a l a n c e t o measure t h e l o c a l s u r f a c e - s h e a r f o r a t u r b u l e n t f l o w on a f l a t smooth p l a t e h a v i n g z e r o - p r e s s u r e g r a d i e n t . I T j e i r d a t a , v e r i f i e d b y u s i n g a c a l i b r a t e d t o t a l head t u b e l o c a t e d on t h e s u r f a c e o f t h e t e s t w a l l , a g r e e d w e l l w i t h t h e measurements o f S c h u l t z - G r u n o w . I n t h i s w o r k , a s h e a r p l a t e was u s e d t o measure t h e s u r f a c es h e a r s t r e s s o f a smooth b o u n d a r y and o f e x p e r i m e n t a l c a n o p i e s .
o f t h e o b j e c t on t h e s h e a r p l a t e . D e t a i l s o f t h e p l a t e a r e g i v t n i n C h a p t e r 3.
2-2 The Momentum I n t e g r a l f o r A p p r o x i m a t e T r e e Drag C a l c u l a t i o n
I n F i g u r e 2.2.1 b y t a k i n g a c o n t r o l volume a r o u n d a t r e e and
c o n s i d e r i n g t h e law o f c o n s e r v a t i o n o f momentum, t h e sum o f t h e f o r c e
a c t i n g i s e q u a l t o t h e change i n momentum t h r o u g h t h e b o u n d a r i e s .
S i n c e t h e p r e s s u r e i s c o n s t a n t , t h e momentum e q u a t i o n , as a p p l i e d t o
t h e c o n t r o l volume shown i n F i g u r e 2 . 2 . 1 , can be s t a t e d a s :
u? dy d2 U. d y dz ( 2 . 2 . 1 ) U,
•rr-r-r
T-^—r-r-TT-T
X « l t 1 F i g u r e 2.2.1 C o n t r o l volume a r o u n d a t r e ewhere Y = t h e v e r t i c a l d i s t a n c e w h i c h i s l a r g e r t h a n 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 a t x = 1 and x = i + 1 Z B t h e l a t e r a l d i s t a n c e w h i c h c o v e r s t h e t r e e wake i n f l u e n c e zone a t x = i + 1 u^ • t h e l o c a l w i n d v e l o c i t y a t x » i " i * l ° t h e l o c a l w i n d v e l o c i t y a t x « i + 1 f p - t h e t o t a l drag f o r c e a c t i n g on t h e t r e e fjy • t h e t o t a l s u r f a c e s h e a r f o r i <^ x <^ i + 1 and 0 <_ Z <^ z The d r a g f o r c e on t h e t r e e i s f a r l a r g e r t h a n t h e s u r f a c e s h e a r on t h e w a l l . The f ^ i s t h u s n e g l e c t e d and t h e t r e e d r a g i s , u s i n g t h e n u a e r l c a l r e p r e s e n t a t i o n : m m " a f " i + l - " i ^ * ^ " i - " i + P AyAz) ( 2 . 2 . 2 )
The above method can be u s e d t o f i n d the t o t a l d r a g f o r c e o f
a number o f t r e e s , p r o v i d e d t h a t t h e c o n t r o l volume i s l a r g e enough t o
c o v e r a l l t r e e s i n i t . T h i s method was u s e d on one o f t h e o r c h a r d
canopy c a s e s . The name o f t h e o r c h a r d canopy u s e d h e r e i n d i c a t e s t h a t
t r e e s a r e e q u a l l y s p a c e d i n b o t h l o n g i t u d i n a l and l a t e r a l d i r e c t i o n s .
The t r e e a r r a y i n t h e l o n g i t u d i n a l d i r e c t i o n i s c a l l e d a " c o l u m n " and
i n t h e l a t e r a l d i r e c t i o n i s c a l l e d a " r o w , " w h e r e t h e l o n g i t u d i n a l d i
-r e c t i o n i s p a -r a l l e l t o t h e c e n t e -r l i n e o f t h e w i n d t u n n e l . The case
computed h e r e was a one c o l u m n , t w e n t y row o r c h a r d canopy w i t h 25.4 cm
t r e e s p a c i n g . The a m b i e n t w i n d v e l o c i t y was 915 cm/sec. A CDC 6400
2.3 S i n g l e Roughness Element The p r o b l e m o f d e t e r m i n i n g t h e d r a g f o r c e on a s i n g l e r o u g h n e s s e l e m e n t , w h i c h i s m o u n t e d on a f l a t p l a t e , i s c o m p l i c a t e d . The d r a g depends on t h e c h a r a c t e r i s t i c s o f t h e b o u n d a r y - l a y e r f l o w as w e l l as on t h e g e o m e t r y o f t h e r o u g h n e s s e l e m e n t . A t h e o r e t i c a l s o l u t i o n t o t h e p r o b l e m i s n o t f e a s i b l e a t p r e s e n t ; t h e r e f o r e , a n a l y s e s must be b a s e d on e x p e r i m e n t a l d a t a . K. W i e g h a r d t ( 1 9 5 3 ) and W. T i l l m a n n ( 1 9 5 3 ) c a r r i e d o u t a l a r g *
number o f measurements on r o u g h n e s s i n G o e t t i n g e n L a b o r a t o r y . The w a l l
f r i c t i o n d r a g was m e a s u r e d on a 50 cm by 30 cm r e c t a n g u l a r t e s t p l a t e w i t h a b a l a n c e . The d r a g on t h e t e s t p l a t e w i t h a s i n g l e r o u g h n e s s e l e m e n t on i t gave an i n c r e a s e i n d r a g . A f t e r s u b t r a c t i n g t h e d r a g o f t h e t e s t p l a t e w i t h o u t t h e r o u g h n e s s e l e m e n t , t h e d i f f e r e n c e was t h e n p r e s u m e d t o be t h e d r a g f o r c e o f t h e r o u g h n e s s e l e m e n t , d e n o t e d b y A f ^ . W i e g h a r d t u s e d t h e f o l l o w i n g d e f i n i t i o n f o r t h e d r a g c o e f f i c i e n t o f t h e r o u g h n e s s e l e m e n t s : A f . A (2.3.1) qA where Cp • t h e d r a g c o e f f i c i e n t o f t h e r o u g h n e s s e l e m e n t A f p a t h e d r a g f o r c e o f t h e r o u g h n e s s e l e m e n t A - t h e l a r g e s t f r o n t a l a r e a o f t h e r o u g h n e s s e l e m e n t p e r p e n d i c u l a r t o t h e d i r e c t i o n o f f l o w q a t h e s t a g n a t i o n p r e s s u r e a v e r a g e d o v e r t h e h e i g h t o f t h e r o u g h n e s s e l e m e n t w h i c h i s
- 1 1
q = ^ 2 P ( y ) dy . ( 2 . 3 . 2 )
Jo
H e r e , a q u e s t i o n i s r a i s e d because t h e f l o w p a t t e r n b e h i n d t h e r o u g h n e s s
e l e m e n t changes t h e d r a g measurement o f t h e f l a t p l a t e , and hence t h e
d r a g f o r c e d i f f e r e n c e A f ^ i s n o t t h e r e a l d r a g o f t h e r o u g h n e s s e l e m e n t . A c c o r d i n g t o P l a t e ( 1 9 6 4 ) t h e t o t a l d r a g on t h e p l a t e w i l l i n c r e a s e b e c a u s e t h e t u r b u l e n t b o u n d a r y f l o w i s d i s t u r b e d by t h e r o u g h -ness e l e m e n t . He d i d a t w o - d i m e n s i o n a l f e n c e s t u d y i n a w i n d t u n n e l and u s e d t h e f o l l o w i n g method t o d e t e r m i n e t h e d r a g f o r c e : where f j ^ = t h e i n c r e a s e i n d r a g c a u s e d by t h e f e n c e f f = t h e f e n c e d r a g f p •> t h e p l a t e d r a g f p o = t h e p l a t e d r a g u n d e r u n d i s t u r b e d b o u n d a r y l a y e r . The f p ^ can be c a l c u l a t e d f r o m w e l l - e s t a b l i s h e d t e c h n i q u e s by
S c h l i c h t i n g ( 1 9 6 0 ) and by such methods as p r e v i o u s l y m e n t i o n e d i n
sub-s e c t i o n 2.1 f o r t u r b u l e n t f l o w o v e r a sub-smooth b o u n d a r y o r be m e a sub-s u r e d , such as w i t h a s h e a r p l a t e . The f e n c e d r a g f ^ i s o b t a i n e d by m e a s u r i n g t h e p r e s s u r e i n f r o n t o f t h e f e n c e p ^ and t h e p r e s s u r e on t h e r e a r o f t h e f e n c e p^^ e x p e r i m e n t a l l y . TTie e q u a t i o n i n c a l c u l a t i n g f ^ r e a d s i-h f f = b ( p ^ - pj^)dy = ( p ^ ^ ^ - p ^ ^ ^ ) . b . h ( 2 . 3 . 4 )
where t h e p ^ ^ ^ and p^^^^ a r e t h e a v e r a g e d v a l u e s o f p ^ and p ^ ,
The f i s f r o m P where f p ( x ) - b r Cp dx • i p U / ( 2 . 3 . 5 ) J X j Cp = f r i c t i o n f a c t o r o f t h e p l a t e i n d i s t u r b e d b o u n d a r y l a y e r , Cp = f (^) , see P l a t e ( 1 9 6 5 ) . E q u a t i o n s 2.3.4 and 2.3.5 c a n a l s o be e x p r e s s e d as f f • f p ( x ) - p U / [ e ( x ) - 6 ( x p ] . ( 2 . 3 . 6 )
Equation 2.3.6 i s the f r i c t i o n drag on the p l a t e between p o i n t s x^ and X , and 6 ( x ) and 6 ( X j ) are the momentum thickness a t x and X j , r e s p e c t i v e l y . See Figure 2.3.1. Hence i t was considered a d v i s a -b l e t o measure s i n g l e t r e e alone w i t h a s p e c i a l s t r a i n gage dynamometer independent o f w a l l shear.
I n t h i s work t h e drag f o r c e and t h e drag c o e f f i c i e n t o f a s i n g l e model t r e e are s t u d i e d . The t r e e drag was o b t a i n e d by u s i n g a s t r a i n gage f o r c e dynamometer which was b u i l t on a p o r t a l gage p r i n c i p l e , Hsi and N a t h (1968). Thus, t h e drag f o r c e on a s i n g l e t r e e element was measured d i r e c t l y . The drag c o e f f i c i e n t was d e f i n e d as f o l l o w s :
S - r 4 - C2.3.7)
where
A - the f r o n t a l area od the model t r e e , and
