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題目： ： ： ：Computational Study of a 3D Turbulent Channel With Wavy Wall
(數值模擬紊流流經三维波形流道之流場分析)
系 系
系 系 所 所 所 所 別 別 別 別： ： ：機械與航太工程研究所 ： 學號姓名
學號姓名 學號姓名
學號姓名： ： ：M09508007 曾 郁 仁 ： 指導教授
指導教授 指導教授
指導教授： ： ：蔡 永 培 博 士 ：
中華民國 中華民國 中華民國 中華民國 九十 九十 九十 九十八 八 八 八 年 年 年 年 一 一 一 一 月 月 月 月
中文中文 中文中文摘要摘要摘要摘要 中文關鍵詞：減阻；節能；波形壁。
自然界和工程應用的紊流問題大多是受到一個或多個壁面約束的剪 切紊流問題，其中波形壁流動由於受到壁面交錯起伏彎曲面效應的作 用，成為研究壁面曲面效應對紊流影響的典型流動問題之一。複雜波形 邊壁在工程技術上有著廣泛的應用，例如：常規的運輸機和水上船隻，
其表面摩擦阻力約佔總阻力的 50%；對於水下運動的物體如潛艇，這個 比例就可達到 70%，而在長距離的管道運送中，幫浦的動力幾乎全部用 於克服表面摩擦阻力。
本研究以分析不可壓縮之三維管道具週期性波形邊界，藉以探討其 紊流流場物理特性。在數值方法的選擇上以迎風算則(UP-WIND Scheme) 直接求解奈維爾-史托克(Navier-Stokes)方程式。而時間離散上採用 LU-SSOR method 以增強數值之穩定性及加速程式之收斂。因波形管壁 具有提升曵力及減低阻力達節能之效益，本研究量測平均速度分佈，瞬 間流場圖形，紊流強度及壓力分佈等，針對三維流場做一有系統之計算 與分析，以達到對管流具波形邊界形成之紊流場有清楚的了解。
Abstract
K e y w o r d ： L o w e r d r a g , E c o n o m y e n e r g y, Wa v y w a l l .
M o s t t u r b u l e n t p r o b l e m s h a p p e n i n g t o t h e n a t u r e a n d e n g i n e e r i n g a p p l i c a t i o n s w e r e e x a c t l y t h e t u r b u l e n c e p r o b l e m f r e q u e n t l y c a u s e d b y o n e o r m o r e t u r b u l e n t s h e a r f l o w s . A m o n g t h e m , w a v y w a l l f l o w w a s a f f e c t e d b y c u r v e s u r f a c e a n d t h i s e f f e c t b e c a m e o n e o f t h e t y p i c a l p r o b l e m s i n r e s e a r c h i n g t h e e f f e c t o n t o t u r b u l e n c e f l o w t r i g g e r e d b y w a y w a l l s . C o m p l e x w a v y w a l l s w e r e b r o a d l y a p p l i e d e n g i n e e r i n g t e c h n i q u e s s u c h a s t r a n s p o r t a t i o n a i r p l a n e s a n d v e s s e l s w i t h t h e i r s u r f a c e f r i c t i o n r a t i n g 5 0 % o f t o t a l r e s i s t a n c e f o r c e . F o r m o v i n g u n d e r w a t e r s u c h a s s u b m a r i n e s , t h e s a i d s u r f a c e f r i c t i o n w i l l a m o u n t t o 7 0 % o f t o t a l f r i c t i o n . I n a d d i t i o n , i n t h e l o n g - d i s t a n c e c h a n n e l s f o r t r a n s p o r t a t i o n , a l m o s t a l l d y n a m i c s w e r e u s e d t o o v e r c o m e t h e s u r f a c e f r i c t i o n .
T h i s s t u d y w a s o p e r a t e d w i t h i n c o m p r e s s i b l e 3 - D c h a n n e l s w i t h c y c l i c w a v y b o u n d a r y t o e x p l o r e t h e p h y s i c a l p r o p e r t i e s o f t u r b u l e n c e f l o w. F o r t h e s e l e c t i o n o f n u m e r i c a l o p e r a t i o n s , t h e U P - W I N D S c h e m e w a s d i r e c t l y u s e d t o s o l v e t h e N a v i e r - S t o k e s e q u a t i o n . A s f o r t e m p o r a l d i s c r e t i z a t i o n , t h e L U - S S O R m e t h o d w a s a d o p t e d t o e n h a n c e n u m e r i c a l s t a b i l i t y a n d a c c e l e r a t e t h e e q u a t i o n c o n v e r g e n c e . B e c a u s e w a v y t u b e w a l l s w e r e f e a t u r e d w i t h i n c r e a s e d d r a g a n d r e d u c e d r e s i s t a n c e f o r c e w i t h e n e r g y s a v e d . T h i s r e s e a r c h m e a s u r e s t h e d i s t r i b u t i o n o f a v e r a g e v e l o c i t y, i n s t a n t f l o w i n g f i e l d s h a p e s , t u r b u l e n c e a n d p r e s s u r e d i s t r i b u t i o n , e t c . F u r t h e r m o r e , t h e s y s t e m a t i c c o m p u t a t i o n a n d a n a l y s i s f o r t h e 3 - D f l o w f i e l d w a s a l s o i m p l e m e n t e d . I t w a s a i m e d t o c l e a r l y u n d e r s t a n d t h e t u r b u l e n c e f i e l d s f o r m e d b y w a v y b o u n d s o f t u b e f l o w.
Acknowledgements
T h e w o r k r e p o r t e d i n t h i s m a s t e r ' s d e g r e e d e s c r i b e s m y r e s e a r c h a c t i v i t i e s a t C h u n g H u a U n i v e r s i t y d u r i n g t h e p e r i o d f r o m S e p t e m b e r 2 0 0 7 t o J a n u a r y 2 0 0 9 . I n o w w i s h t o t h a n k t h o s e w h o p r o v i d e d m e w i t h h e l p a n d e n c o u r a g e m e n t d u r i n g t h e p r e s e n t s t u d y.
F i r s t a n d f o r e m o s t , I w i s h t o e x p r e s s m y s i n c e r e g r a t i t u d e t o m y a d v i s o r P r o f e s s o r D r. Y. P. Ts a i f o r t h i s v a l u a b l e a d v i c e a n d e n c o u r a g e m e n t d u r i n g t h e c o u r s e o f t h i s s t u d y. Wi t h o u t h i s p a t i e n c e a n d c a r e f u l g u i d a n c e t h i s w o r k w o u l d n o t b e a c c o m p l i s h e d . I a m i n d e b t e d t o t h e m e m b e r s o f m y o r a l d e f e n c e c o m m i t t e e , D r. B . J . Ts a i , D r. R . Z . H u a n g a n d D r. C . C . N i e n f o r t h e i r c o n s t r u c t i v e c r i t i c i s m . B e s i d e s , m y s i n c e r e t h a n k s a l s o g o t o t h e s c h o l a r s h i p s s u p p o r t e d b y t h e C F D l a b p a r t n e r, J a m e s p . O y a n , C . C . C h o u a n d Y. R . L i n . A n d M 1 0 1 l a b p a r t n e r, C . H . L e e e t a l .
F i n a l l y, a n d m o s t i m p o r t a n t l y, I w o u l d l i k e t o d e d i c a t e t h i s w o r k t o C i n d y f o r h e r s u p p o r t . I w i s h t o t h a n k m y p a r e n t s f o r a l l t h e i r e n c o u r a g e m e n t . T h e y g a v e m u c h m o r e t h a n t h e y e x p e c t e d , b u t t h e y c o n t r i b u t e d m o r e t h a n t h e y c a n i m a g i n e .
Contents
中文摘要· · · 1
A b s t r a c t · · · 2
A c k n o w l e d g e m e n t s · · · 3
C o n t e n t s · · · 4
F i g u r e c o n t e n t s · · · 5
1 . I n t r o d u c t i o n · · · 8
2. Numerical method · · · 11
2 . 1 F l o w c o n f i g u r a t i o n · · · 11 2 . 2 G o v e r n i n g e q u a t i o n · · · 1 2 2 . 3 U p w i n d s c h e m e · · · 1 5 2 . 4 Te m p o r a l D i s c r e t i z a t i o n · · · 1 6 2 . 5 C o m p u t a t i o n a l G r i d · · · 1 8 2 . 6 I n i t i a l c o n d i t i o n a n d b o u n d a r y c o n d i t i o n · · · 2 0 2 . 6 . 1 I n i t i a l c o n d i t i o n · · · 2 0 2 . 6 . 2 B o u n d a r y c o n d i t i o n · · · 2 0 2 . 6 . 3 B o u n d a r y c o n d i t i o n o n t o p o f b a s e a n d b e l o w · · · 2 0 3 . R e s u l t s · · · 2 1 3 . 1 T h e A n a l y s i s o n a f l o w s c a u s e d b y w a v y w a l l s · · · 2 1 3 . 2 T h e C o m p a r i s o n o f P h y s i c a l S p a c e · · · 2 2 3 . 4 C o n c l u s i o n s · · · 2 5 R e f e r e n c e s · · · 2 6 F i g u r e · · · 2 7
Figure contents
Fig.2-1 Computational domain of flow over a wavy wall · · · 12
Fig.2-2 Computational Grid resolutions · · · 20
Fig.3-1 Computational Grid· · · 27
Fig.3-2 Streamlines for channel flow over wavy wall at Re=3460· · · 27
Fig.3-3 Instantaneous velocity vectors at Re=3460 · · · 28
Fig.3-4 Mean streamwise velocity contours at Re=3460 · · · 28
Fig.3-5 Mean vertical velocity contours at Re=3460 · · · 29
Fig.3-6 Comparison the turbulence intensities also agree reasonably well with Cherukat et al.[2] · · · 30
Fig.3-7 Shows the speed contours from Nakagawa et al.[3] for varying x/λ+ from 0.0 to 1.0 · · · 31
Fig.3-8 Shows the speed contours for Reynolds number is 3460, x/λ= 0.05 · · · 31
Fig.3-9 Mean velocity profiles at crest and trough streamline locations · · · 32
Fig.3-10 Computational domain of flow over a single wavy wall · · · 34
Fig.3-11 Computational domain of flow over a bilateral wavy wall · · · 34
Fig.3-12 The computational grid of flat boundary oscillation A=0mm, the wave length λ+ =50.8mm · · · 34
Fig.3-13 The computational grid of bilateral boundary oscillation A=2.54mm, the wave length λ+ =50.8mm · · · 35
Fig.3-14 The computational grid of single boundary oscillation A=2.54mm, the wave length λ+ =50.8mm · · · 35
Fig.3-15 Distributions of instantaneous streamwise velocity contours for flat boundary in the X-Y planes, amplitude A=0mm · · · 36
Fig.3-16 Distributions of instantaneous streamwise velocity contours for single wavy wall in the X-Y planes, amplitude A=2.54mm · · · 37
Fig.3-17 Distributions of instantaneous streamwise velocity contours for bilateral wavy wall in the X-Y planes, amplitude A=2.54mm · · · 38
Fig.3-18 Distributions of instantaneous streamwise velocity contours for flat boundary in the Y-Z planes, amplitude A=0mm· · · 39
Fig.3-19 Distributions of instantaneous streamwise velocity contours for single wavy wall in the Y-Z planes, amplitude A=2.54mm· · · 39
Fig.3-20 Distributions of instantaneous streamwise velocity contours for bilateral wavy wall in the Y-Z planes, amplitude A=2.54mm· · · 39
boundary in the X-Y planes, amplitude A=0mm · · · 40 Fig.3-22 Distributions of instantaneous cross-stream velocity contours for single
wavy wall in the X-Y planes, amplitude A=2.54mm · · · 41 Fig.3-23 Distributions of instantaneous cross-stream velocity contours for bilateral
wavy wall in the X-Y planes, amplitude A=2.54mm · · · 42 Fig.3-24 Distributions of instantaneous cross-stream velocity contours for flat
boundary in the Y-Z plane, amplitude A=0mm· · · 43 Fig.3-25 Distributions of instantaneous cross-stream velocity contours for single
wavy wall in the Y-Z plane, amplitude A=2.54mm· · · 43 Fig.3-26 Distributions of instantaneous cross-stream velocity contours for bilateral
wavy wall in the Y-Z plane, amplitude A=2.54mm· · · 43 Fig.3-27 Distributions of instantaneous spanwise velocity contours for flat boundary
in the Y-Z planes, amplitude A=0mm · · · 44 Fig.3-28 Distributions of instantaneous spanwise velocity contours for single wavy
wall in the Y-Z planes, amplitude A=2.54mm· · · 45 Fig.3-29 Distributions of instantaneous spanwise velocity contours for bilateral wavy
wall in the Y-Z planes, amplitude A=2.54mm· · · 46 Fig.3-30 Distributions of instantaneous spanwise velocity contours for flat boundary
in the Y-Z plane, amplitude A=0mm · · · 47 Fig.3-31 Distributions of instantaneous spanwise velocity contours for single wavy
wall in the Y-Z plane, amplitude A=2.54mm· · · 47 Fig.3-32 Distributions of instantaneous spanwise velocity contours for bilateral wavy
wall in the Y-Z plane, amplitude A=2.54mm· · · 47 Fig.3-33 Three-dimensional instantaneous velocity chart for flat boundary in the
mean streamwise and normal velocity · · · 48 Fig.3-34 Three-dimensional instantaneous velocity chart for single wavy wall in the
mean streamwise and normal velocity · · · 48 Fig.3-35 Three-dimensional instantaneous velocity chart for bilateral wavy wall in
the mean streamwise and normal velocity · · · 49 Fig.3-36 Three-dimensional instantaneous velocity chart for flat boundary in the
cross-stream direction · · · 49 Fig.3-37 Three-dimensional instantaneous velocity chart for single wavy wall in the
cross-stream direction · · · 50 Fig.3-38 Three-dimensional instantaneous velocity chart for bilateral wavy wall in
the cross-stream direction · · · 50
Fig.3-39 Three-dimensional instantaneous velocity chart for flat boundary in the
spanwise direction · · · 51 Fig.3-40 Three-dimensional instantaneous velocity chart for single wavy wall in the
spanwise direction · · · 51 Fig.3-41 Three-dimensional instantaneous velocity chart for bilateral wavy wall in
the spanwise direction· · · 52 Fig.3-42 Distributions of instantaneous streamwise velocity for flat boundary, single
wavy wall and bilateral wavy wall in the X-Y planes · · · 53 Fig.3-43 Distributions of instantaneous cross-stream velocity for flat boundary, single
wavy wall and bilateral wavy wall in the X-Y planes · · · 54 Fig.3-44 Distributions of instantaneous spanwise velocity for flat boundary, single
wavy wall and bilateral wavy wall in the X-Y planes · · · 55 Fig.3-45 Pressure curve from the studies for Reynolds number is 3460 · · · 56
1 . I n t ro d u c t i o n
T h e w a y t o r e d u c e r e s i s t a n c e c a u s e d b y w a v y w a l l s w a s q u i t e d i f f e r e n t f r o m o t h e r c o n d i t i o n s . R e s e a r c h e r s h o w e d l i t t l e k n o w l e d g e a b o u t i t s i n t r i n s i c p r o p e r t i e s . T h e b l i n d n e s s o n q u a n t i t a t i v e a n a l y s i s a n d o p t i m a l s u r f a c e d e s i g n s t o r e d u c e r e s i s t a n c e w a s u n a v o i d a b l e . R e c e n t l y, t h e r e s e a r c h i n g e f f o r t w a s m a i n l y c o n c e n t r a t e d o n t h e d i s c l o s u r e a n d c o n t r o l o f f l o w i n g s t r u c t u r e s . I t w a s p e o p l e ’s e x p e c t a t i o n t h a t t h e e x p l o r a t i o n o n w a v y w a l l s c o u l d b e i m p r o v e d t h e o r e t i c a l l y w i t h t h e a p p l i c a t i o n s t o e n g i n e e r i n g t e c h n i q u e s t o r e d u c e r e s i s t a n c e c a u s e d b y w a v y w a l l s . I n t h i s s t u d y, C u i e t a l [ 2 ] a n d C h e r u k a t e t a l [ 3 ] p r o p o s e d t h e e x p e r i m e n t a l d a t a f o r t h e c o m p a r i s o n o f f l o w i n g s t r u c t u r a l c h a n g e s , w a l l p r e s s u r e g r a d i e n t s a n d f a i r l y c o m p l e x r e s i s t a n c e c h a n g e s c a u s e d b y w a l l s u r f a c e . O b v i o u s l y, t h e c a p a b i l i t y f o r w a v y w a l l s t o r e d u c e r e s i s t a n c e w a s b e c a u s e t h e e x i s t e n c e o f w a v y w a l l s c o u l d i m p r o v e p a r t i a l t u r b u l e n c e s t r u c t u r e n e a r b y w a l l s . To r e a c h t h e b r e a k t h r o u g h i m p r o v e m e n t t o r e d u c e r e s i s t a n c e c a u s e d b y w a l l s , t h e f i r s t s t e p h a d t o b e t a k e n i n r e s e a r c h i n g t h e f l o w i n g c h a r a c t e r i s t i c s n e a r b y w a v y w a l l s .
E x p e r i m e n t a l r e s e a r c h o f f l o w o v e r t h e w a v y w a l l w a s p e r f o r m e d a s e a r l y a s 1 9 3 2 . T h e t h e o r y o f t h e s u r f a c e w a l l v a r i a t i o n h a s b e e n d e v e l o p e d . Z i l k e r a n d H a n r a t t y [ 4 ] , Z i l k e r e t a l . [ 5 ] c o n t i n u e d t o c o n d u c t s t u d i e s i n c l u d i n g t h e e x p e r i m e n t s o f g a u g i n g w a l l s u r f a c e p r e s s u r e , m e a s u r i n g m e a n v e l o c i t y o f s t r e a m a n d t h e w a l l s u r f a c e s h e a r i n g s t r e s s a n d a n a l y z i n g t h e m o b i l e c h a r a c t e r i s t i c s . I n 1 9 8 5 P a t e l [ 6 ] h a s i m p r o v e d s i m u l a t i o n f o r s e v e r a l k i n d s o f n e a r w a l l w i t h l o w R e y n o l d s n u m b e r k−ε m o d e l . P a t e l a n d C h e n [ 7 ] h a v e u s e d d o u b l e - d e c k e d m o d e l t o s o l v e t u r b u l e n t c o m p l e x s t a l l e d f l o w. S i n c e t h i s m o d e l s a v e s t h e g r i d s , t h e r e f o r e i t c a n e f f e c t i v e l y s a v e t h e c o m p u t a t i o n s p a c e a n d t i m e a n d e n h a n c e s t h e c o m p u t a t i o n f e a s i b i l i t y.
P a t e l e t a l . [ 8 ] c a l c u l a t i o n o f t h e w a v y w a l l t u r b u l e n t f l o w w a s c o n d u c t e d o n t w o k i n d s o f w a v e h e i g h t r a t i o s i n 1 9 9 1 . T h e s i m u l a t e d s t r e a m l i n e s p i c t u r e i n t h e n e a r w a l l v o r t e x a r e a a n d t h e c r o s s s e c t i o n v e l o c i t y d i s t r i b u t i o n w e r e i n a g o o d a g r e e m e n t 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 . C o m p a r i s o n o f f r i c t i o n c o e f f i c i e n t c u r v e s w i t h p r e s s u r e c o e f f i c i e n t s w a s a l s o m a d e b e t w e e n t h e w a v y w a l l s u r f a c e a n d f l a t s u r f a c e . T h e e f f e c t s o f t h e p r o f i l e o n f l o w s i n t h e n e a r w a l l a r e a w e r e a n a l y z e d . F e r r i r a a n d L o p e s [ 9 ] c a r r i e d o n t h e m u l t i - g r o u p w i n d t u n n e l e x p e r i m e n t s o n t h e u n i m o d u l a r f l o w s o f s i n u s o i d a l w a v y w a l l
f l o w f i e l d . T h e l o w R e y n o l d s n u m b e r k - ε m o d e l w i t h t h e c o n t r o l v o l u m e t r i c m e t h o d w a s u s e d t o c o m p u t e w a v y w a l l n e a r z o n e s t a l l e d f l o w f o r v a r i o u s w a v e h e i g h t r a t i o s (A/λ . M o n t a l b a n o a n d ) M c C r e a d y [ 1 0 ] u s e d t h e w a v e s t a b i l i t y t h e o r y a n d a d d e d s m a l l p e r t u r b a t i o n q u a n t i t y t o O r r - S o m m e r f e l d e q u a t i o n t o d e v e l o p t h e r e l a t i o n b e t w e e n t h e l a m i n a r w a l l s u r f a c e p r e s s u r e a n d t h e s h e a r s t r e s s e s .
A i r i a u a n d G i o v a n n i n i [ 11 ] u t i l i z e d s t a t i s t i c a l s i m u l a t i o n t o o b t a i n t h e s t r e a m f u n c t i o n a n d v o r t e x c h a r t s a t d i f f e r e n t t i m e f o r t h e s i n u s o i d a l w a v y w a l l . A l s o t h e a v e r a g e s t r e a m f u n c t i o n a n d t h e a v e r a g e w a l l s u r f a c e p r e s s u r e c o e f f i c i e n t c u r v e s w e r e c a l c u l a t e d u s i n g t h e t i m e a v e r a g e m e t h o d . Wi t h t h e s e t o o l s v o r t i c i t y a n d v a r i a t i o n o f t h e p r e s s u r e g r a d i e n t a l o n g t h e w a l l s u r f a c e w e r e a n a l y z e d . M a l a m a t a r i s 1 a n d B o n t o z o g l o u [ 1 2 ] t h e d i m e n s i o n l e s s N a v i e r – S t o k e s e q u a t i o n s a r e s o l v e d i n t h e w h o l e r a n g e o f t h e l a m i n a r f l o w r e g i m e . N u m e r i c a l p r e d i c t i o n s a r e c o m p a r e d w i t h a v a i l a b l e e x p e r i m e n t a l d a t a f o r v e r y l o w R e y n o l d s n u m b e r s . T h e e m p h a s i s i n t h e d i s c u s s i o n o f r e s u l t s i s g i v e n i n t h e p r e s e n t a t i o n o f f r e e s u r f a c e p r o f i l e s , s t r e a m l i n e s , v e l o c i t y, a n d p r e s s u r e d i s t r i b u t i o n s a l o n g t h e f r e e s u r f a c e a n d t h e w a l l . T h e i n t e r a c t i o n o f t h e d i m e n s i o n l e s s n u m b e r s o f t h e f l o w i s s t u d i e d , c r i t e r i a f o r f l o w r e v e r s a l a r e e s t a b l i s h e d , a n d a r e s o n a n c e p h e n o m e n o n a t h i g h R e y n o l d s n u m b e r s i s i n v e s t i g a t e d . B o e r s m a [ 1 3 ] . T h e e v o l u t i o n i n s p a c e a n d t i m e o f p a r t i c l e s a r e r e l e a s e d i n t h i s f l o w w i l l b e e x a m i n e d . I t w i l l b e s h o w n t h a t s m a l l w a v e s o n t h e c h a n n e l b o t t o m c a n g e n e r a t e l a r g e l o n g i t u d i n a l v o r t i c e s s i m i l a r t o L a n g m u i r v o r t i c e s t h a t a r e o b s e r v e d i n f l o w s w i t h w a v e s a t t h e f r e e - s u r f a c e . T h e s i m u l a t i o n r e s u l t s s h o w t h a t t h e c o n c e n t r a t i o n o f t h e p a r t i c l e s i s m a x i m a l o n t h e d o w n s t r e a m s i d e o f t h e w a v e c r e s t . N a k a g a w a e t a l . [ 1 4 ] , M e a s u r e m e n t s o f t u r b u l e n c e w i t h l a s e r D o p p l e r v e l o c i m e t r y ( L D V ) a r e c o m p a r e d f o r t u r b u l e n t f l o w o v e r a f l a t s u r f a c e a n d a s u r f a c e w i t h s i n u s o i d a l w a v e s o f s m a l l w a v e l e n g t h . T h e w a v y b o u n d a r y w a s h i g h l y r o u g h i n t h a t t h e f l o w s e p a r a t e d . T h e R e y n o l d s n u m b e r b a s e d o n t h e h a l f - h e i g h t o f t h e c h a n n e l a n d t h e b u l k v e l o c i t y w a s 4 6 , 0 0 0 . T h e w a v e l e n g t h w a s 5 m m a n d t h e h e i g h t t o w a v e l e n g t h r a t i o w a s 0 . 1 . T h e r o o t - m e a n - s q u a r e s o f t h e v e l o c i t y f l u c t u a t i o n s a r e a p p r o x i m a t e l y e q u a l i f n o r m a l i z e d w i t h
s h e a r s t r e s s . C a l c u l a t i o n s w i t h a d i r e c t n u m e r i c a l s i m u l a t i o n ( D N S ) a r e u s e d t o s h o w t h a t t h e f l u i d i n t e r a c t s w i t h t h e w a l l i n q u i t e d i f f e r e n t w a y s f o r f l a t a n d w a v y s u r f a c e s . T h e y s h o w s i m i l a r i t y i n t h a t l a r g e q u a d r a n t 2 e v e n t s i n t h e o u t e r f l o w, f o r b o t h c a s e s , a r e a s s o c i a t e d w i t h p l u m e s t h a t e m e r g e f r o m t h e w a l l r e g i o n a n d e x t e n d o v e r l a r g e d i s t a n c e s . M e a s u r e m e n t s o f s k e w n e s s o f t h e s t r e a m w i s e a n d w a l l - n o r m a l v e l o c i t y f l u c t u a t i o n s a n d q u a d r a n t a n a l y s e s o f t h e R e y n o l d s s h e a r s t r e s s e s a r e q u a l i t a t i v e l y s i m i l a r f o r f l a t a n d w a v y s u r f a c e s . H o w e v e r, t h e s k e w n e s s m a g n i t u d e s a n d t h e r a t i o o f t h e q u a d r a n t 2 t o q u a d r a n t 4 c o n t r i b u t i o n s a r e l a r g e r f o r t h e w a v y s u r f a c e . T h u s , t h e r e i s e v i d e n c e t h a t t u r b u l e n t s t r u c t u r e s a r e u n i v e r s a l i n t h e o u t e r f l o w a n d f o r q u a n t i t a t i v e d i f f e r e n c e s i n t h e s t a t i s t i c s t h a t r e f l e c t d i f f e r e n c e s i n t h e w a y i n w h i c h t h e f l u i d i n t e r a c t s w i t h t h e w a l l . Z i l k e r e t a l . [ 1 5 ] m e a s u r e m e n t s o f t h e s h e a r - s t r e s s v a r i a t i o n a l o n g a n d t h e v e l o c i t y p r o f i l e s a b o v e a s o l i d w a v y w a l l b o u n d i n g a t u r b u l e n t f l o w a r e p r e s e n t e d f o r w a v e s w i t h h e i g h t - t o - l e n g t h r a t i o s o f 2 a / λ = 0 · 0 3 1 2 a n d 0 · 0 5 . T h e s e a r e c o m p a r e d w i t h p r e v i o u s m e a s u r e m e n t s o f t h e w a l l s h e a r s t r e s s r e p o r t e d b y T h o r s n e s s ( 1 9 7 5 ) a n d b y M o r r i s r o e ( 1 9 7 0 ) f o r 2 a / λ = 0 · 0 1 2 . T h e i n v e s t i g a t i o n c o v e r e d a r a n g e o f c o n d i t i o n s f r o m t h o s e f o r w h i c h a l i n e a r b e h a v i o u r i s o b s e r v e d t o t h o s e f o r w h i c h a s e p a r a t e d f l o w i s j u s t b e i n g i n i t i a t e d .
A b o v e m e n t i o n e d r e s e a r c h e s o n w a v y w a l l s w e r e m a i n l y f o c u s e d o n s i n e w a v y w a l l s . H o w e v e r, t h e r e w e r e f e w s t u d i e s o n z i g z a g w a v y w a l l s ( t r i a n g u l a r w a v y w a l l s ) a n d r e c t a n g u l a r w a v y w a l l s b e c a u s e t h e p h y s i c a l n u m e r i c a l m e a s u r i n g o n w a v e t i p s f o r b o t h z i g z a g a n d r e c t a n g u l a r w a v y w a l l s w a s m o r e d i f f i c u l t . M . C . G o o d a n d P. N . J o u b e r ( 1 9 6 8 ) [ 1 7 ] u s e d t o i m p l e m e n t e x p e r i m e n t s o n s q u a r e w a v e s ( L / H = 1 . 0 ) a n d t h i n w a v e s ( L / H = 0 . 1 ) w i t h t h e c r o s s - s e c t i o n a l v e l o c i t y ) a n d p r e s s u r e c o e f f i c i e n t o f w a l l s u r f a c e c o n c l u d e d . P a s t r e s e a r c h e s u s e d t o d i s c l o s e t h e f l o w i n g c h a r a c t e r i s t i c s c a u s e d b y w a v y w a l l s t o a c e r t a i n d e g r e e ( e x p e r i m e n t a n d c o m p u t a t i o n ) , s u c h a s t h e t y p e s o f f l o w i n g f i e l d s , a r e a s o f s w i r l z o n e s , d i s t r i b u t i o n o f s p a n w i s e v e l o c i t y, p r e s s u r e o f w a l l s u r f a c e a n d s h e a r s t r e s s , e t c .
2 . N u m e r i c a l m e t h o d 2 . 1 F l o w c o n f i g u r a t i o n
T h e p a r a m e t e r s o f t h e s i m u l a t i o n a r e t h e s a m e a s u s e d b y C h e r u k a t e t a l . [ 3 ] . I n t h e i r n u m e r i c a l , t h e a m p l i t u d e a n d t h e w a v e l e n g t h o f t h e w a v y w a l l w e r e 2 . 5 4 m m a n d 5 0 . 8 m m , r e s p e c t i v e l y.
T h e d i s t a n c e b e t w e e n t h e m e a n l o c a t i o n o f t h e w a v y s u r f a c e a n d t h e f l a t w a l l w a s 5 0 . 8 m m . A s c h e m a t i c d i a g r a m o f t h e t h r e e - d i m e n s i o n a l c o m p u t a t i o n a l d o m a i n i s s h o w n i n F i g u r e 2 - 1 . I t c o n s i s t s o f a c h a n n e l w h i c h i s u n b o u n d e d i n b o t h t h e s t r e a m w i s e ( x ) a n d s p a n w i s e ( z ) d i r e c t i o n s . T h e l o w e r w a l l h a s Nω(=4) w a v e s w i t h s i n u s o i d a l s h a p e a n d a m e a n p o s i t i o n a t t h e y = 0 p l a n e ( y i s t h e v e r t i c a l d i r e c t i o n ) . T h e f l a t w a l l i s l o c a t e d a t y = h . T h e l o c a t i o n o f t h e w a v y w a l l , y , ω
i s g i v e n b y
2 )
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w h e r e a i s t h e a m p l i t u d e o f t h e w a v ea n d ¸ i s t h e w a v e l e n g t h . T h e m e a n f l o w 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 s p r e s s u r e d r i v e n . I n t h e p r e s e n t s t u d y, w a v e l e n g t h ¸ a n d a m p l i t u d e a w e r e s e t e q u a l t o h a n d 0 . 0 5 h t o m a t c h t h e p a r a m e t e r s o f H u d s o n ’s ( 1 9 9 3 ) m e a s u r e m e n t s .
T h e f l o w i s a s s u m e d t o b e h o m o g e n e o u s i n t h e s p a n w i s e d i r e c t i o n , j u s t i f y i n g t h e u s e o f p e r i o d i c b o u n d a r y c o n d i t i o n s . T h e f l o w i s a l s o a s s u m e d t o b e p e r i o d i c i n t h e s t r e a m w i s e d i r e c t i o n . T h u s , t h e c o m p u t a t i o n a l b o x s i z e i n t h e s t r e a m w i s e (Λx=Nωλ) a n d s p a n w i s e d i r e c t i o n s( xΛ s h o u l d b e l a rg e e n o u g h t o i n c l u d e t h e l a rg e s t l e n g t h ) s c a l e o f t h e t u r b u l e n t s t r u c t u r e s . T h e e x t e n t s o f t h e c o m p u t a t i o n a l d o m a i n w e r e , r e s p e c t i v e l y, c h o s e n t o b e 4 h 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 n d 2 h i n t h e s p a n w i s e d i r e c t i o n .
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2 2
2 2
1 3
2
1 1
1 1
1 1
, ,
, ,
, , ,
, ,
, ,
, ,
( 2 - 11 )
w h e r e t h e c o m p o n e n t s o f t h e s h e a r - s t r e s s t e n s o r a r e g i v e n b y
) 2
3 ( 2
z w y v x u
xx ∂
−∂
∂
−∂
∂
= µ ∂
τ ( 2 - 1 2 a )
) 2
3 ( 2
z w x u y v
yy ∂
−∂
∂
−∂
∂
= µ ∂
τ ( 2 - 1 2 b )
) 2
3 ( 2
y v x u z w
zz ∂
−∂
∂
−∂
∂
= µ ∂
τ ( 2 - 1 2 c )
yx
xy x
v y
u τ
µ
τ =
∂ + ∂
∂
= (∂ ) ( 2 - 1 2 d )
zx
xz z
u x
w τ
µ
τ =
∂ +∂
∂
= (∂ ) ( 2 - 1 2 e )
zy
yz y
w z
v τ
µ
τ =
∂ +∂
∂
= (∂ ) ( 2 - 1 2 f )
w h e r e v i s c o s i t y c o e f f i c i e n t
1 c
1.5T c T µ = +
+
w h e r e T i s a n o n - d i m e n s i o n a l t e m p e r a t u r e ; c i s S u t h e r l a n ’s
c o n s t a n t110.4
T
∞ ;p r
t
P c k
=
µ
, g e n e r a l l y pr i s s e t a t 0 . 7 2 .2 . 3 U p w i n d s c h e m e
To i l l u s t r a t e t h e m e t h o d , c o n s i d e r t h e f o l l o w i n g o n e - d i m e n s i o n a l l i n e a r w a v e e q u a t i o n
= 0
∂ + ∂
∂
∂
x a u t
u
( 2 - 1 3 )I t d e s c r i b e s a w a v e p r o p a g a t i n g i n t h e x - d i r e c t i o n w i t h a v e l o c i t y a.
T h e p r e c e d i n g e q u a t i o n i s a l s o a m a t h e m a t i c a l m o d e l f o r o n e - d i m e n s i o n a l l i n e a r a d v e c t i o n . C o n s i d e r a t y p i c a l g r i d p o i n t i i n t h e d o m a i n . I n a o n e - d i m e n s i o n a l d o m a i n , t h e r e a r e o n l y t w o d i r e c t i o n a s s o c i a t e d w i t h p o i n t i - l e f t a n d r i g h t . I f a i s p o s i t i v e t h e l e f t s i d e i s c a l l e d u p w i n d s i d e a n d r i g h t s i d e i s t h e d o w n w i n d s i d e . S i m i l a r l y, i f a i s n e g a t i v e t h e l e f t s i d e i s c a l l e d d o w n w i n d s i d e a n d r i g h t s i d e i s t h e u p w i n d s i d e . I f t h e f i n i t e d i f f e r e n c e s c h e m e f o r t h e s p a t i a l d e r i v a t i v e ,
u x
∂ ∂ c o n t a i n s m o r e p o i n t s i n t h e u p w i n d s i d e , t h e s c h e m e i s s i m p l y a n u p w i n d s c h e m e .
T h e s c h e m e c a n b e c o m b i n e d a n d w r i t t e n i n a c o m p a c t f o r m a s
)]
( ) (
[ 1 1
1 n
i n i n
i n i n
i n
i
u a u u a u u
u
+ = −λ
+ − − + − + − ( 2 - 1 4 )w h e r e
x t
∂
= ∂
λ
( 2 - 1 5 )) 2(
1
a a
a
+ = + ( 2 - 1 6 )) 2(
1
a a
a
− = − ( 2 - 1 7 )U s i n g t h e r e l a t i o n s h i p b e t w e e n a+, a a n d a−, t h e s c h e m e a g a i n c a n b e w r i t t e n a s
) 2
2 ( )
2 ( 1 1 1 1
1 n
i n i n
i n
i n i n
i n
i
u a u u a u u u
u
+ = −λ
+ − − +λ
+ − + −( 2 - 1 8 )
2 . 4 Te m p o r a l D i s c r e t i z a t i o n
I n t h e t i m e i n t e g r a l , t h i s s t u d y u s e s i m p l i c i t e x p r e s s i o n t i m e i n t e g r a t i o n o f L U - S S O R b y Yo o n & J a m e s o n ( 1 9 8 7 ) t o t h e t w o o r d e r s p r e c i s i o n . A l s o o n l y t h e d i a g o n a l m a t r i x o f t h e c o n c e a l e d m o d e o f t h e i n v e r s e m a t r i x w a s c a l c u l a t e d t o s a v e t h e c o m p u t i n g t i m e . T h e v e c t o r i z a t i o n p r o c e s s i s q u i t e s u c c i n c t a n d e a s y t o u n d e r s t a n d .
T h e r e f o r e t h e e x p r e s s i o n m a y a p p l y o n l y i n t h e i n c o m p r e s s i b l e f l o w f i e l d .
F i r s t l y, m a k i n g l i n e a r i z a t i o n p r o c e s s i n g t o t h e n + 1 t i m e s t e p f l u x ,
∆ Ο +
∆ +
=
∆ Ο +
∆ +
=
∆ Ο +
∆ +
=
∆ Ο +
∆ +
=
∆ Ο +
∆ +
=
∆ Ο +
∆ +
=
+ + + + + +
1 2 1 2 1 2 1 2 1 2 1 2
ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
Q Q
C G G
Q Q
B F F
Q Q
A E E
Q Q
C G G
Q Q
B F F
Q Q
A E E
n v n v n v
n v n v n v
n v n v n v
n n n
n n n
n n n
( 2 - 1 9 )
w h e r e Aˆ , Bˆ , and Cˆ are J o c o b i a n m a t r i x e s o f n o n - v i s c o s i t y v a l u e f l u x e s Eˆ, Fˆ , and Gˆ; a n d Aˆv, Bˆv, and Cˆv are J o c o b i a n m a t r i x e s o f v i s c o s i t y v a l u e f l u x e sEˆv, Fˆv, and Gˆv. ∆Q) i s t h e m i c r o v a r i a b l e o f c o n s e r v a t i o n v a r i a b l e i n t h e t i m e s t e p . U s i n g t h e s i g n o f c h a r a c t e r i s t i c v a l u e s , n o n - v i s c o s i t y f l u x J a c o b i a n m a t r i x c a n b e f u r t h e r d e c o m p o s e d i n t o :
1
1 − −
− +
−
+
+ = Λ + Λ
=
i i i i i i i ii
A A R R R R
A ) ) )
( 2 - 2 0 ) w h e r e d i a g o n a l m a t r i x Λ+i i s c o m p o s e d b y n o n - n e g a t i v e c h a r a c t e r i s t i c v a l u e s o f t h e Λi a n d m a t r i x Λ−i i s c o m p o s e d b y t h e n o n - p o s i t i v e c h a r a c t e r i s t i c v a l u e s o f Λi.
U s i n g t h e i m p l i c i t E u l e r s c h e m e f o r s p a c e - t i m e d i c r e t i z a t i o n：
( )
[ ] [ ( ) ]
( )
[ ] [ ( ) ]
( )
[ ] [ ( ) ]
−
−
−
=
+
− − + −
− +
+
− − + −
− +
+
− − + −
− +
∆ + −
1 2 , 1 ,
~ 1 ~
2 , 1 ,
~
~
1 2, , 1
~ 1 ~
2, , 1
~
~
1 , 2, 1
~ 1 ~
, 2, 1
~
~ ,
ˆ, 1 , ˆ, , ,
n k j Si Gv n G
k j Si Gv G
n k j Si Fv n F
k j Si Fv F
n k j Si Ev n E
k j Si Ev t E
n k j Qi n
k j Qi k j Vi
( 2 - 2 1 )
w h e r e n i s t h e t i m e t a r g e t .
S u b s t i t u t i n g t h e l i n e a r i z e d f l u x f r o m e q u a t i o n ( 2 - 1 9 ) a n d d e c o m p o s e d J a c o b i a n m a t r i x i n t o e q . ( 2 - 2 1 ) a n d n e g l e c t i n g t h e s e c o n d a n d h i g h e r o r d e r i t e m s , a d i a g o n a l l i n e i m p l i c i t e x p r e s s i o n c a n b e o b t a i n e d .
RHS S
G G S
G G
S F F S
F F
S E E S
E E
Q S
C C Q
S C C
Q S
C C Q
S C C
Q S
B B Q
S B B
Q S
B B Q
S B B
Q S
A A Q
S A A
Q S
A A Q
S A A Q
t I V
k j v i k
j v i
k j v i k
j v i
k j v i
k j v i
n k j k i
j v i k
j k i
j v i
k j k i
j v i k
j k i
j v i
k j k i j v i k
j k i j v i
k j k i j v i k
j k i j v i
k j k i j v i
k j k i
j v i
k j k i
j v i
k j k i j v i
k j i k j i
≡
−
−
−
−
−
−
−
−
−
−
−
−
=
∆ +
−
∆ +
+
∆
−
−
∆
− +
∆ +
−
∆ +
+
∆
−
−
∆
− +
∆ +
−
∆ +
+
∆
−
−
∆
− +
∆ ∆
+ +
− +
− +
−
− + +
−
− − +
+ +
−
− + +
−
− − +
+ +
−
− + +
−
− − +
+ +
} ]
~ ) [(~ ]
~ ) {[(~
} ]
~ ) [(~ ]
~ ) {[(~
} ]
~ ) [(~ ]
~ ) {[(~
} ]
) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) [(
] ) {[(
2 , 1 2 ,
, 1 ,
2, , 1 2,
, 1
, 2, , 1
2, 1
, , 2 , 1 1 ,
, , 2 , 1 ,
1 , , 2 , 1 , ,
, 2 , 1 ,
, , ,
2 , 1 ,
1 , ,
2 , 1
, 1 , ,
2 , 1 ,
, , 2 , 1
, , ,
2, 1 ,
, , 1
2, 1
, , , 1
2, 1 ,
, , 2, 1 ,
, ,
,
) )
) )
) )
) )
) )
) )
) )
) )
) )
) ) ) )
) )
) )
) )
) )
) )
) )
) ) )
( 2 - 2 2 )
w h e r e I i s t h e u n i t m a t r i x . U s i n g v i s c o s i t y f l u x o f t h e J a c o b i a n m a t r i x i n t h e i m p l i c i t s c h e m e c a n i n c r e a s e t h e c o n v e r g i n g s p e e d .
2 . 5 C o m p u t a t i o n a l G r i d
T h e f l o w f i e l d c o n t a i n s s e v e r a l r e g i o n s , w i t h l a r g e g r a d i e n t s , w h i c h r e q u i r e c a r e f u l r e s o l u t i o n . G r i d c l u s t e r i n g i s r e q u i r e d i n t h e
v e r t i c a l d i r e c t i o n n e a r t h e w a l l s t o r e s o l v e t h e b o u n d a r y l a y e r. T h e s e p a r a t e d s h e a r l a y e r, w h i c h i s l o c a t e d o v e r t h e s e p a r a t i o n b u b b l e , a l s o n e e d s s p e c i a l a t t e n t i o n . B e c a u s e o f t h e s e r e q u i r e m e n t s , a f i n e r e s o l u t i o n s h o u l d b e m a i n t a i n e d u p t o t h e m i d d l e o f t h e c h a n n e l i n o r d e r t o r e s o l v e t h e s e p a r a t e d s h e a r l a y e r p r o p e r l y.
T h e ( x , y, z ) d i r e c t i o n w a s d i s c r e t i z e d i n t o 2 1 0 1 2 5 e l e m e n t s . T h e n u m b e r o f g r i d p o i n t s i n t h e x d i r e c t i o n w a s 1 2 5 , i n t h e y a n d z d i r e c t i o n w a s 4 1 . To r e d u c e t h e c o m p u t a t i o n a l c o s t , t h e g o v e r n i n g e q u a t i o n s w e r e f i r s t i n t e g r a t e d i n t i m e o n a v e r y c o a r s e g r i d u n t i l t h e f l o w r e a c h e d a s t a t i s t i c a l l y s t e a d y s t a t e . T h e n t h e f l o w f i e l d w a s i n t e r p o l a t e d o n t o a f i n e r g r i d t o g e t a r e s t a r t f l o w f i e l d f o r t h e s i m u l a t i o n w i t h h i g h e r r e s o l u t i o n .
T h e f o l l o w i n g r e s u l t s w e r e o b t a i n e d :
( a ) A s t r e a m w i s e r e s o l u t i o n o f a t l e a s t 1 2 5 e l e m e n t s i s r e q u i r e d ; h i g h e r r e s o l u t i o n i n t h i s d i r e c t i o n d o e s n o t c a u s e s i g n i f i c a n t c h a n g e s . ( b ) S i g n i f i c a n t c h a n g e s w e r e f o u n d i f l e s s t h a n 4 1 e l e m e n t s w e r e u s e d i n t h e v e r t i c a l d i r e c t i o n . H i g h e r r e s o l u t i o n s t h a n t h i s i m p r o v e d t h e f i r s t - o r d e r s t a t i s t i c s o n l y s l i g h t l y.
( c ) A s t h e r e s o l u t i o n s i n t h e s t r e a m w i s e a n d v e r t i c a l d i r e c t i o n s i m p r o v e , t h e s h e a r s t r e s s a l o n g t h e u p p e r w a l l i n c r e a s e s .
C o m p a r i s o n s o f s t r e a m l i n e s , ψ , o f t h e m e a n f l o w f i e l d s a r e p r e s e n t e d i n F i g u r e 2 - 2 . T h e b o u n d a r y o f t h e s e p a r a t e d r e g i o n c o r r e s p o n d s t o t h e ψ = 0 c o n t o u r. T h e s i z e o f t h e s e p a r a t i o n b u b b l e i s s e e n t o c h a n g e o n l y s l i g h t l y a s t h e r e s o l u t i o n i n c r e a s e s .
F i g . 2 - 2 C o mp a r i s o n o f t h e s t r e a ml i n e s (ψ) w i t h d i ff e r e n t r e s o l u t i o n s . ( a ) 100×20×20 e l e me n t s ; ( b ) 125×41×41 e l e me n t s ; ( c ) 251×41×41 e l e me n t s .
2 . 6 I n i t i a l c o n d i t i o n a n d b o u n d a r y c o n d i t i o n
2 . 6 . 1 I n i t i a l c o n d i t i o nT h e f l o w c o n d i t i o n s ( Re=hUb/2v=3460 ) c o n s i d e r e d , a d v e r s e p r e s s u r e g r a d i e n t s w e r e l a r g e e n o u g h t o c a u s e f l o w s e p a r a t i o n . T h e p r e s s u r e i s i n t h e s t a t i c a t m o s p h e r i c c o n d i t i o n .
2 . 6 . 2 B o u n d a r y c o n d i t i o n
B o u n d a r y c o n d i t i o n s a l o n g t h e s o l i d b o u n d a r i e s a r e n o s l i p a n d n o p e n e t r a t i o n . B o u n d a r y v a l u e o f t h e f l o w f i e l d i s a s s u m e d t o b e t h e i n i t i a l f l o w f i e l d e x i t c o n d i t i o n a n d k e p t c o n s t a n t .
2 . 6 . 3 B o u n d a r y c o n d i t i o n o n t o p o f b a s e a n d b e l o w (A)
(B)
(C)
