Ocean Engineering 94 (2015) 3 6 - 5 0
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j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g
Influence of wall proximity on flow around two tandem
circular cylinders
X.K. Wang^'* J.-X. Zhang ^ Z. Hao^ B. Zhou^ S.K. Tan^
" Maritime Researcix Centre, Nanyang Technological University, 639798, Singapore"School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China ' College of Logistics Engineering, Shanghai Maritime University, Shanghai 201306, China
CrossMark
A R T I C L E I N F O A B S T R A C T
Article history: Received 3 August 2013 Accepted 30 November 2014 Available online 16 December 2014 Keywords:
Tandem cylinders Wall proximity Vortex shedding Drag and lift coefficients Spectrum of lift coefficient Particle image velocimetry (PIV)
A n e x p e r i m e n t a l s t u d y w a s c o n d u c t e d to investigate the f l o w a r o u n d t w o t a n d e m c y l i n d e r s placed near and parallel to a p l a n e w a l l . T h e R e y n o l d s n u m b e r b a s e d o n the c y l i n d e r d i a m e t e r ( D ) w a s 6 3 0 0 . T h e c y l i n d e r c e n t r e t o c e n t r e s p a c i n g ratio (Z.*=£,/D) w a s varied f r o m 1.5 to 6, a n d t h e g a p h e i g h t t o -c y l i n d e r - d i a m e t e r ratio (G* = G / D ) f r o m 0.15 to 2. T h e f l o w fields w e r e m e a s u r e d u s i n g Parti-cle Image V e l o c i m e t r y (PIV), i n c o n j u n c t i o n w i t h m e a s u r e m e n t s of fluid d y n a m i c forces ( d r a g a n d lift) o n the d o w n s t r e a m c y l i n d e r u s i n g load cell. T h e f l o w strongly d e p e n d s on the c o m b i n e d v a l u e of G* a n d I * . W i t h reference to G * , the f l o w could be classified as v o r t e x - s h e d d i n g s u p p r e s s i o n r e g i m e ( G * < 0 . 3 ) , i n t e r m e d i a t e - g a p r e g i m e ( 0 . 3 < G * < 1 ) w h e r e v o r t e x s h e d d i n g o'ccurs but is i n f l u e n c e d b y w a l l proximity, a n d large-gap r e g i m e ( C * > 1) w h e r e the w a l l i n f l u e n c e b e c o m e s negligible. Similarly, three categories c a n be identified as a f u n c t i o n o f i * , n a m e l y , e x t e n d e d - b o d y r e g i m e 1 < £ , * < 2 , r e a t t a c h m e n t r e g i m e at 2 <Z,* < 4 , a n d i m p i n g i n g r e g i m e at L * > 4 . V a r i a t i o n s of d y n a m i c d r a g a n d lift coefficients, spectra, S t r o u h a l n u m b e r s , a n d R e y n o l d s s h e a r stress are also p r e s e n t e d to c h a r a c t e r i z e the d i f f e r e n t flow r e g i m e s in the G * - L * plane.
© 2014 E l s e v i e r Ltd. A l l rights r e s e r v e d .
1. Introduction
T h e i n t e r f e r e n c e o f f l o w a r o u n d t w o c i r c u l a r c y l i n d e r s is o f b o t h a c a d e m i c i n t e r e s t a n d p r a c t i c a l i m p o r t a n c e , see S u m n e r ( 2 0 1 0 ) f o r a c o m p r e h e n s i v e r e v i e w . A m o n g t h e m a n y possible a r r a n g e m e n t s o f t h e t w o cylinders to be positioned i n relative to the f l o w direction, t h e t a n d e m c o n f i g u r a t i o n has b e e n extensively studied. This t y p e o f interference, r e f e n e d to as 'wake i n t e r f e r e n c e ' by Z d r v k o v i c h (1987), is a f u n c t i o n o f the inter-cylinder distance (expressed as the r a t i o b e t w e e n t h e centre-to-centi-e spacing and the c y l i n d e r diameter, L * = L / D , t h e r e a f t e r abbreviated as the spacing ratio). Z d r a v k o v i c h (1987) p r o p o s e d t h a t the f l o w can be classified i n t o t h r e e basic types: ( i ) single b l u f f - b o d y regime at small L* (1 < L * < 1.2~1.8), w h e r e p e r i o d i c v o n K a r m a n v o r t e x s h e d d i n g is observed o n l y i n the w a k e o f the d o w n s t r e a m cylinder; ( i i ) read:achment regime at m o d e r a t e L* (1.2 —1.8 < L* < 3 . 4 ~ 3 . 8 ) , w h e r e t h e shear layers e m a n a t i n g f r o m the u p s t r e a m cylinder read:ach o n t o the sutface o f t h e d o w n s t r e a m cylinder; ( i i i ) i m p i n g i n g r e g i m e at large L* (L* > 3.4 ~ 3.8), w h e r e v o n K a r m a n vortices are shed f r o m t h e u p s t r e a m c y l i n d e r and p e r i o d i -cally i m p i n g e o n the d o w n s t r e a m cylinder. Z h o u a n d Y i u ( 2 0 0 6 ) s h o w e d t h a t t h e r e a t t a c h m e n t r e g i m e (2 < L* < 5) can be f u r t h e r
* Corresponding author. Tel.: + 6 5 6790S619; fax: -1-65 6790S620. E-maii address: cxkwang@ntu.edu.sg (X.K. Wang).
http://dx.doi.Org/10.1016/j.oceaneng.201411.01S 0 0 2 9 - 8 0 1 8 / © 2014 Elsevier Ltd. All rights reserved.
s u b - d i v i d e d i n t o t w o d i s t i n c t categories, f o r w h i c h t h e r e a t t a c h m e n t is o n t h e rear and leading surfaces o f the d o w n s t r e a m cylinder, respectively (see Fig. 1). The exact values o f L* t o delineate the boundaries b e t w e e n d i f f e r e n t regimes d e p e n d o n t h e value o f Reynolds n u m b e r (Carmo et al., 2010) a n d f r e e - s t r e a m t u r b u l e n c e i n t e n s i t y ( L j u n g k r o n a et al., 1991). The critical spacing ratio ( ! * „ ) , at w h i c h periodic v o r t e x s h e d d i n g begins t o occur f r o m the u p s t r e a m cylinder, varies f r o m L * c r = 3 t o 5 i n t h e literature (e.g., Lee et al., 2 0 0 9 ) . Correspondingly, the f l u i d forces o n t h e cylinders w o u l d experience a discontinuous ' j u m p ' at a b o u t L*a ( Z d r a v k o v i c h and Pridden, 1977). Moreover, X u and Z h o u ( 2 0 0 4 ) s h o w e d t h a t t h e v o r t e x s h e d d i n g frequency is d e p e n d e n t o n Reynolds n u m b e r over the range Re = 8 0 0 - 4 . 2 x l O ' ' (Ke=UD/v, w h e r e u is t h e k i n e m a t i c viscosity o f f l u i d ) .
On t h e o t h e r h a n d , t h e r e are a n u m b e r o f e n g i n e e r i n g practices i n w h i c h c y l i n d r i c a l structures are placed near a p l a n e w a l l , s u c h as s u b m a r i n e pipelines, risers a n d cables o n seabed. To date, m a n y researchers have e x a m i n e d t h e i n f l u e n c e o f w a l l p r o x i m i t y o n a single c y l i n d e r w i t h t h e cross-section o f e i t h e r c i r c u l a r (e.g., B e a r m a n a n d Z d r a v k o v i c h , 1978; Lei et al., 1999; Price e t al., 2 0 0 2 ; D i p a n k a r a n d Sengupta, 2 0 0 5 ; Nishino e t al., 2 0 0 7 ; W a n g a n d Tan, 2 0 0 8 a ; L i n et al., 2 0 0 9 ; Sarkar and Sarkar, 2010; O n g et al., 2012; W a n g e t al., 2013), square (e.g., W a n g a n d Tan 2 0 0 8 b ; M a h i r , 2 0 0 9 ) o r r e c t a n g u l a r (e.g., M a i t i , 2 0 1 2 ; M a i t i a n d Bhad:, 2014). The n e a r b y w a l l affects n o t o n l y the d y n a m i c pressure a n d forces o n t h e
X.K. Wang et al. / Ocean Engineenng 94 (2015) 36-50 37
cylinder, b u t also t h e w a k e p a t t e r n and f l o w - i n d u c e d v i b r a t i o n s . The r a t i o b e t w e e n t h e gap h e i g h t to the c y l i n d e r d i a m e t e r ( G * = G / D , abbreviated hereafter as t h e gap r a t i o ) is f o u n d t o be the p r e d o m i n a n t p a r a m e t e r The i m p e r m e a b i l i t y o f the w a l l poses an i r r o t a t i o n a l c o n s t r a i n t t o the w a k e d e v e l o p m e n t , r e s u l t i n g i n suppression o f the classical v o n K a r m a n v o r t e x s h e d d i n g t h a t is i n absolute i n s t a b i l i t y ( H u e r r e and M o n k e w i t z , 1990) b e l o w a c r i t i c a l gap r a t i o (G*cr). As sketched i n Fig. 2(a), w h e n G* < G\r, t h e w a k e is steady w i t h a l o n g r e c i r c u l a t i o n r e g i o n : w h i l e t h e gap f l o w keeps attached o n t h e w a l l , t h e u p p e r shear layer e m a n a t i n g f r o m t h e c y l i n d e r e x h i b i t s as elongated K e l v i n H e l m h o l t z (KH) t y p e o f r o l l -ups ( i n convective i n s t a b i l i t y ) . However, w h e n G* > G'cr (Fig. 2(b)), the gap f l o w is s t r o n g e n o u g h to d e t a c h u p w a r d f r o m t h e w a l l ( o r u p w a s h ) , a n d t o i n t e r a c t w i t h t h e u p p e r shear layer t o f o r m discrete vortices. I t s h o u l d be n o t e d t h a t t h e v o r t e x s h e d d i n g is a s y m m e t r i c about t h e h o r i z o n t a l w a k e centerline; also, there is a c o u p l i n g b e t w e e n t h e l o w e r shear layer and the w a l l b o u n d a r y layer, as reflected b y the p h e n o m e n o n t h a t each anticlocl<wise v o r t e x is a c c o m p a n i e d by a s m a l l cloclcwise v o r t e x i n t h e near w a l l r e g i o n . The v a l u e o f G * c r « 0.3 s l i g h t l y varies w i t h Re and thickness o f t h e w a l l b o u n d a r y layer (e.g., Buresti a n d Lanciotti, 1992; Price et al., 2002).
H o w e v e r , l i t t l e a t t e n t i o n has b e e n p a i d t o t h e c o n f i g u r a t i o n o f t w o t a n d e m c y l i n d e r s i n p r o x i m i t y t o a plane w a l l (see Fig. 3 ) . The flow i n t e r f e r e n c e b e t w e e n t h e t w o cylinders is f u r t h e r c o m p l i -cated d u e t o t h e presence o f t h e w a l l b o u n d a r y . B h a t t a c h a r y y a a n d D h i n a k a r a n ( 2 0 0 8 ) n u m e r i c a l l y s t u d i e d the 2 - d i m e n s i o n a l ( 2 D ) flow a r o u n d t w o t a n d e m square c y l i n d e r s w i t h a l i n e a r i n c i d e n t v e l o c i t y p r o f i l e at G* = 0.5 and L * = 1 . 5 - 6 . The n o n - u n i f o r m approach flow causes difference i n the strength o f the upper a n d l o w e r shear layers. The flow can be steady u p to Re = 125 d e p e n d i n g o n t h e value of L*. M o r e recently, Harichandan and Roy (2012) simulated t h e flow around t w o near-wall t a n d e m cylinders (circular/square) at Re=:100 and 200, G * = 0 . 5 and 1, a n d L * = 2 and 5. For a g i v e n Re, t h e Strouhal numbers o f the t w o cylinders are identical, b u t t h e l i f t and drag coefficients are d i f f e r e n t .
As described above, there is l i m i t e d i n f o r m a t i o n available o n t h e flow a r o u n d t w o t a n d e m cylinders i n p r o x i m i t y to a w a l l b o u n d a r y . T w o aspects need a t t e n t i o n . Firstiy, t h e o n l y t w o p u b l i s h e d studies, namely, D h i n a k a r a n ( 2 0 0 8 ) a n d H a r i c h a n d a n and Roy (2012), w e r e c o n d u c t e d at r e l a t i v e l y l o w Re ( u p to 200), t h a t is, i n t h e l a m i n a r regime. Yet, i n e n g i n e e r i n g practice t h e flow is generally i n t h e subcritical r e g i m e . Secondly, b o t h studies considered o n l y a r a t h e r l i m i t e d n u m b e r o f c o m b i n a t i o n s o f G* a n d L*, and hence a c o m p l e t e p i c t u r e i n the G*-L* plane is still unavailable. These m o t i v a t e t h e present r e l a t i v e l y systematic i n v e s t i g a t i o n f o r 0 . 1 5 < C * < 2 a n d 1.5 < L* < 7 u n d e r a c o n s t a n t Reynolds n u m b e r i n s u b c r i t i c a l r e g i m e ( R e = 6 3 0 0 ) .
2. Experimental set-up and methodology
The e x p e r i m e n t s w e r e p e r f o r m e d i n a r e - c i r c u l a t i n g o p e n c h a n n e l l o c a t e d at M a r i t i m e Research Centre, N a n y a n g T e c h n o l o -gical U n i v e r s i t y , w i t h a test s e c t i o n o f 5 m x 0.3 m x 0.45 m ( l e n g t h X w i d t h x h e i g h t ) . The c h a n n e l bed a n d t h e t w o side w a l l s
o f t h e test section w e r e m a d e o f glass t o a l l o w f o r o p t i c a l access. The f r e e - s t r e a m v e l o c i t y w a s u n i f o r m t o w i t h i n 1.5% across t h e test section, a n d the t u r b u l e n c e i n t e n s i t y i n t h e f r e e stream w a s b e l o w 2%.
Fig. 3 shows a sketch o f t h e t w o t a n d e m c y l i n d e r s placed near a n d p a r a l l e l t o a p l a n e w a l l . The c y l i n d e r m o d e l s w e r e m a d e o f s m o o t h , t r a n s p a r e n t acrylic r o d w i t h an o u t e r d i a m e t e r o f D = 1 5 m m . D u r i n g the e x p e r i m e n t s , the f r e e - s t r e a m v e l o c i t y was k e p t c o n s t a n t at L / = 0 . 4 2 m / s ( R e = 6 3 0 0 ) . The a p p r o a c h b o u n d a r y layer was f u l l y d e v e l o p e d w i t h a thickness o f 5 = 7 m m ( - 0.5D). The c y l i n d e r s ' c e n t r e - t o - c e n t r e spacing w a s v a r i e d as L = 2 2 . 5 , 30, 4 5 , 60, 75, 9 0 a n d 105 m m (L* = 1.5-7), a n d t h e gap h e i g h t G = 2 . 2 5 , 6, 9 , 1 2 , 21 a n d 3 0 m m ( C * = 0 . 1 5 - 2 ) . T h e r e f o r e , t o t a l l y 4 2 cases w e r e c o n s i d e r e d i n t h e p r e s e n t study.
The s p a n [b) o f t h e c y l i n d e r s w a s 2 0 0 m m , l e a d i n g t o a n aspect r a t i o (AR) o f i / D = 1 3 . 3 . This v a l u e w a s c o n s i d e r e d t o be large e n o u g h ( A R > 1 0 a c c o r d i n g to p r e v i o u s finding, f o r e x a m p l e . L a m a n d Z o u , 2010) t o ensure a n o m i n a l l y 2 D flow i n t h e near w a k e . T h e r e f o r e , t h e v e l o c i t y m e a s u r e m e n t s w i t h Particle Image V e l o c i -m e t r y (PIV) w e r e p e r f o r -m e d i n t h e -m i d - s p a n p l a n e . The o r i g i n o f t h e c o o r d i n a t e s y s t e m was located at t h e c e n t e r o f t h e u p s t r e a m c y l i n d e r , w i t h x, y a n d z d e n o t i n g t h e s t r e a m w i s e , transverse a n d spanwise d i r e c t i o n s , respectively. The p o s i t i v e d r a g a n d l i f t forces are i n t h e x- a n d y - d i r e c t i o n s , respectively.
Velocity measurements w e r e p e r f o m e d u s i n g a digital PIV system (LaVision m o d e l ) . The flow field was i l l u m i n a t e d w i t h a double cavity Nd:YAG laser l i g h t sheet at 532 n m w a v e l e n g t h ( L i t r o n model, p o w e r ~ 1 3 5 mJ per pulse, d u r a t i o n ~ 5 ns). S p h e r i c e l ® 110P8 h o l l o w glass spheres ( n e u t r a l l y b u o y a n t w i t h a m e a n diameter o f 13 p m ) w e r e seeded i n t h e flow as Q-acer particles. The images w e r e
Upper shear layer (elongated, K-H type)
Gap flow
b Upper shear layer (broken, Kérman type)
L
Upwash
<ï*5
Fig. 2. Schematic of the flow around a near-wall single cylinder: (a) vortex-sheddlng-suppression regime at small gap ratio; and (b) vortex-shedding regime at moderate gap ratio. Proposed based on the flow measurement results in Wang and Tan (2008a).
1 D - 2 D 2D-3D
Kcnltuclimcnt oii rcur surface
3 D - 5 0
Rcntlachnicni on leading surface
> 5 D
• E x l e n d s d - b o d y ' R e g l m a • R e a t t a c h m e n f R e g i m e ' I m p i n g i n o ' R e g i m o
3S X.K Wang et al / Ocean Engineering 94 (2015) 36-50
Upstream cylinder Downstream cylinder
Fig. 3. Schematic of the flow around two near-wall tandem cylinders.
recorded u s i n g a 1 2 - b i t CCD camera w i t l i a r e s o l u t i o n o f 1600 X 1200 pixels. LaVision Davis s o f t w a r e ( V e r s i o n 7.2) was used to process t h e particle images and d e t e r m i n e t h e v e l o c i t y vectors. Particle d i s p l a c e m e n t was calculated u s i n g the f a s t - F o u r i e r - t r a n s f o i m (FFT) based c r o s s - c o n e l a t i o n a l g o r i t h m w i t h standard Gaussian sub-p i x e l f i t s t r u c t u r e d as an iterative m u l t i - g r i d m e t h o d . The sub-processing p r o c e d u r e i n c l u d e d t w o passes, s t a r t i n g w i t h a g r i d size o f 64 x 6 4 pixels, s t e p p i n g d o w n to 3 2 x 32 p k e l s o v e r l a p p i n g b y 50%, w h i c h r e s u l t e d i n a set o f 7 5 0 0 vectors (100 x 7 5 ) f o r a t y p i c a l f i e l d . I n b e t w e e n the t w o passes, t h e vector maps w e r e f i l t e r e d b y using a 3 x 3 m e d i a n f i l t e r i n order to remove possible outliers. T h e n u m b e r o f particles i n a 3 2 x 3 2 p i x e l w i n d o w was o f t h e o r d e r o f 1 0 ~ 1 5 to y i e l d s t r o n g correlations. The f i e l d o f v i e w was set at 190 m m x 143 m m , t h e r e f o r e t h e spatial resolution was 1.9 m m x 1.9 m m (i.e., 0.13D x 0.13D). For each case, a series o f 1050 instantaneous f l o w fields was a c q u i r e d at the s a m p l i n g f r e q u e n c y o f 1 5 f i z ( o r 7 0 s recordings), i n o r d e r t o achieve a reasonably statistical convergence o f the measured quantities, such as Reynolds shear stress. The u n c e i t a i n t y i n t h e instantaneous velocities (it a n d v ) was estimated to be a b o u t 3.5% f o r the present setup. The instantaneous spanwise v o r t i c i t y (cOz = A v / A x - A u / A y ) w a s calculated u s i n g t h e least squares e x t r a p o l a t i o n scheme. The u n c e i t a i n t y i n Wz was estimated to be a b o u t 10% based o n t h e m e t h o d proposed b y Fouras a n d Soria (1998).
A p i e z o e l e c t r i c l o a d cell (Kistler M o d e l 9317B) w a s used to d i r e c t l y m e a s u r e t h e fluid d y n a m i c forces o n t h e d o w n s t r e a m cylinder, d r a g ( F D ) a n d l i f t [FL). The o u t p u t signal w a s c a p t u r e d w i t h a N a t i o n a l I n s t r u m e n t s A / D card at a s a m p l i n g rate o f 100 Hz ( a t least 1 o r d e r o f m a g n i t u d e greater t h a n t h e v o r t e x s h e d d i n g f r e q u e n c y , w h i c h w a s a b o u t 5 - 6 H z ) . The d u r a t i o n o f r e c o r d i n g f o r each case w a s a b o u t 2 0 0 s, w h i c h c o r r e s p o n d e d to a b o u t 1 0 0 0 cycles o f v o i t e x s h e d d i n g a n d was s u f f i c i e n t l y l o n g a c c o r d i n g to t h e c r i t e r i o n p r o p o s e d b y Sakamoto et al. (1987). T h e d i m e n s i o n l e s s s h e d d i n g f r e q u e n c y was expressed as S t r o u h a l n u m b e r ( S t = / D / U ) , w h e r e ƒ is t h e f r e q u e n c y d e t e r m i n e d f r o m spectral analysis o f t h e fluctuating l i f t c o e f f i c i e n t u s i n g p o w e r spectral d e n s i t y (PSD) f u n -c t i o n . Also, t h e m e a n and r o o t - m e a n - s q u a r e (RMS) values o f fluid d y n a m i c d r a g a n d l i f t c o e f f i c i e n t s {CD = 2FD/plfiDb a n d CL^2FI/ pU'^Db) w e r e calculated, w h e r e p is t h e fluid density. T h r o u g h a n u m b e r o f repeated m e a s u r e m e n t s o n a single c y l i n d e r , t h e uncer-t a i n uncer-t y i n uncer-t h e m e a n d r a g w a s d e uncer-t e r m i n e d uncer-t o be w i uncer-t h i n 1%. The dauncer-ta f o r a f r e e - s t a n d i n g (isolated) single c y l i n d e r m e a s u r e d a t t h e same Reynolds n u m b e r ( R e = 6 3 0 0 ) served as t h e b e n c h m a r k reference: Coo = l . l , CD'O = 0 . 0 5 5 , Cid = 0 . 0 7 5 and S t o = 0 . 2 ( w h e r e t h e subscript 0 denotes t h e i s o l a t e d c y l i n d e r ) .
3. Results and discussion
3.1. Near-wall single cylinderT h e effects o f w a l l p r o x i m i t y o n a single c y l i n d e r has b e e n e x a m i n e d i n t h i s s e c t i o n . Fig. 4 s h o w s t h e v a r i a t i o n o f t h e d y n a m i c f o r c e c o e f f i c i e n t s ( C D , Q , Ch a n d Ci) v e r s u s G*, t o g e t h e r w i t h t h e d a t a r e p o r i i e d b y Roshko e t al. ( 1 9 7 5 ) a n d L e i et al. ( 1 9 9 9 ) . As s h o w n i n Fig. 4 ( a ) , a p r o m i n e n t f e a t u r e is t h a t t h e m o s t d r a m a t i c change o f Co occurs f r o m s m a l l - t o i n t e r m e d i a t e - g a p r a t i o s (e.g., G * < 0 . 7 5 ) ; w h e n G * > 1 , b y c o n t r a s t , i t r e m a i n s a p p r o x i m a t e l y c o n s t a n t at CD « 1 . 1 ( a s y m p t o t i c a l l y a p p r o a c h i n g t h e v a l u e f o r a n i s o l a t e d c y l i n d e r ) , i n d i c a t i n g t h a t t h e w a l l e f f e c t s b e c o m e n e g l i -g i b l e . A s i m i l a r t r e n d is f o u n d f o r t h e RMS c o e f f i c i e n t s ( C ó a n d Cl) as s h o w n i n Fig. 4 ( c ) . H o w e v e r , i t is n o t e d t h a t t h e a s y m p t o t i c values are c o n s i d e r a b l y l o w e r t h a n t h o s e r e p o r t e d i n Roshko e t al. ( 1 9 7 5 ) a n d L e i et a l . ( 1 9 9 9 ) , n a m e l y , Co « 1 . 1 versus 1.3 a n d Ci 0.075 v e r s u s 0.6. T h e d i s c r e p a n c y is l i k e l y a t t r i b u t e d t o t h e d i f f e r e n c e i n m e a s u r e m e n t t e c h n i q u e s a n d o n c o m i n g flow c o n d i -t i o n s ( s u c h as Re a n d &l-t;5). N o -t e -t h e d a -t a i n Roshko e-t al. ( 1 9 7 5 ) a n d Lei et a l . ( 1 9 9 9 ) w e r e based o n pressure d i s t r i b u t i o n a r o u n d t h e c y l i n d e r circumference f o r a n e l e m e n t a l slice ( r e f e r r e d to as sectional force b y N o r b e r g ( 2 0 0 3 ) ) , w h i l e the present study measured the total force o n t h e w h o l e span o f t h e cylinder, w h i c h always has a l o w e r
m a g n i d i d e due t o t h e end effects ( W e s t a n d A p e l t , 1997). I n the p r e s e n t study, t h e thickness o f b o u n d a r y layers d e v e l o p e d o n the side w a l l s w h e r e the t w o ends o f the c y l i n d e r w e r e attached was about 0.5D, so the l e n g t h o f t h e c y l i n d e r subjected t o e n d effects was about I D (or 75% o f the t o t a l span). Therefore, the d i f f e r e n c e b e t w e e n the measured t o t a l force a n d t h e ideal sectional force w o u l d be less t h a n 10%. I n fact, t h e present results are i n good a g r e e m e n t w i t h t h e p u b l i s h e d data o n a single^cylinder u s i n g s i m i l a r measure-m e n t technique ( l o a d cell), such as Co « 1 . 1 8 6 a n d Ci « 0.089 i n L a measure-m e t al. ( 2 0 0 3 ) f o r R e = 4 . 8 x W, a n d C/ « 0.08 i n Tadrist e t a l . (1990) f o r R e = 7 0 0 0 . As depicted i n Fig. 4 ( b ) , the c y l i n d e r experiences a p o s i t i v e m e a n l i f t ( Q > 0 ) at s m a l l - t o intermediate-G*, suggesting t h a t t h e c y l i n d e r is pushed u p w a r d fi'om t h e w a l l . The m e a n l i f t c o e f f i c i e n t has a m a x i m u m o f Q x 0.3 at the smallest gap ratio ( G * = 0 . 1 5 ) , and t h e r e a f t e r decreases m o n o t o n i c a l l y u n t i l r e a c h i n g the a s y m p t o t i c value o f C i , = 0 a t G * > 1.
Fig. 5(a) s h o w s t h e time h i s t o r i e s o f d y n a m i c l i f t c o e f f i c i e n t ( C L ) o n t h e c y l i n d e r at d i f f e r e n t gap r a t i o s . As G * increases, t h e signal changes f r o m a c h a o t i c p a t t e r n at G * = 0.15 a n d 0.25, t o a p e r i o d i c p a t t e r n at G * > 0.4 w i t h a m u c h h i g h e r m a g n i t u d e o f fluctuation. T h e c h a o t i c p a t t e r n at s m a l l - G * is d u e t o t h e cessation o f p e r i o d i c
X.K. Wang et al / Ocean Engineering 94 (2015) 36-50 39 Present Lei etal. (1999) Roshko et al. (1975) s h e d d i n g becomes s t r o n g e r w i t h i n t h i s range (a s i m i l a r t r e n d is i n f e r r e d f r o m t h e v e l o c i t y d a t a i n W a n g a n d T a n ( 2 0 0 8 a ) ) . A t G*=0.15 or 0.25, o n t h e o t h e r h a n d , a w e a k peak is d i s c e r n i b l e a t a r e l a t i v e l y h i g h f r e q u e n c y , t h a t is, S t » 0.85, w h i c h c o r r e s p o n d s t o t h e K - H roU-ups i n shear layer i n s t a b i l i t y . R a j a g o p a l a n a n d A n t o n i a ( 2 0 0 5 ) p r o p o s e d an e m p i r i c a l r e l a t i o n s h i p b e t w e e n t h e shear layer i n s t a b i l i t y f r e q u e n c y ( f d ) a n d t h e v o r t e x s h e d d i n g f r e q u e n c y ( f y ) as a f u n c t i o n o f Re f o r a n i s o l a t e d c y l i n d e r , n a m e l y , / s i / / , / = 0 . 0 2 9 x [^g0.65_ j j ^ g p r e d i c t e d v a l u e u s i n g this e q u a t i o n f o r R e = 6 3 0 0 is
f^^/fy=8.55, w h i c h is a b o u t t w i c e t h e m e a s u r e d v a l u e o f 0.85/ 0 . 2 = 4 . 2 5 . H o w e v e r i t is noted t h a t the w o r k i n g f l u i d i n Rajagopalan and A n t o n i a ( 2 0 0 5 ) is air (vs. w a t e r i n the present study), w h i c h w o u l d result i n relatively t h i n n e r shear layers and hence h i g h e r ƒ j , , since/si is inversely p r o p o r t i o n a l t o the shear layer thickness (Gerrard, 1967). Fig. 5(b) also shows t h a t at G* = 0.4 (corresponding to onset o f periodic v o r t e x shedding), the s p e c t m m exhibits co-existence o f b o t h peaks ( S t « 0.2 and 0.85), albeit rather weak, i m p l y i n g a t r a n s i t i o n / c o m p e t i -tion b e t w e e n the t w o types o f instability ( v o n K a r m a n vs. K-H).
0.7 0.6 0,5 0.4 -I O 0.3 0.2 0,1 0.0 0.1 -0.03 — B — Present ^ ^ l e i et al. (1999) Roshko et al. (1975) 1 ' ) ' 1 1 I 1 1 ' I ' 1 . . . . W < 7^ y » T —
Ir^--/
Ir^--/
—f—C' 1 - V - C ; . (Leietat, 1999) 0.7 0.6 0.6 - 0.4 - 0,3 0,2 0.1 0.0 0,0 0.6 1.0 1.5 2.0 2.5 3.0 G*Fig. 4. Variation of dynamic force coefficients with C for the near-wall single cylinder: (a) mean drag coefficient (Co); (b) mean lift coefficient (Cj.); and (c) RMS drag ( C d ) and lift (CÏ) coefficients. Present: Re=6300 and a=0.5D; Lei et al. (1999): R e = 1 . 3 6 x 1 0 " and 5=0.140; Roshko e t a l (1975): R e = 2 x l O " a n d « = 0 . 5 D . v o r t e x s h e d d i n g f r o m t h e c y l i n d e r (e.g.. Price e t al., 2 0 0 2 ; W a n g a n d Tan, 2 0 0 8 a ) . Also, t h e c r i t i c a l gap r a t i o (G*cr) is b e t w e e n 0.25 a n d 0.4, i n accordance w i t h t h e r e p o r t e d value o f G * c r « 0.3 i n t h e l i t e r a t u r e . T h e p e r i o d i c i t y o f l i f t signals is r e f l e c t e d i n t h e c o r r e -s p o n d i n g -spectra -s h o w n i n Fig. 5 ( b ) . For G* > 0.4, each -s p e c t r u m displays one obvious p e a k at S t « 0.2 ( s i m i l a r t o t h e case o f an i s o l a t e d c y l i n d e r ) , w i t h t h e m a g n i t u d e o f t h e peak p r o g r e s s i v e l y i n c r e a s i n g w i t h G* u n t i l G* = 0.8. This i n d i c a t e s t h a t v o r t e x
3.2. Near-wall tandem cylinders
3.2.1. Instantaneous flow patterns around the cylinders
Fig. 6 shows a representative snapshot o f t h e instantaneous v o r t i c i t y fields a r o u n d the n e a r - w a l l t a n d e m cylinders at selected gap ratios (G*=0.15, 0.4, 0.6 and 1.4) and spacing ratios ( L * = 2 , 3 and 5). It is obvious t h a t t h e flow p a t t e m s d e p e n d o n b o t h G* a n d L*. Periodic v o i t e x s h e d d i n g f r o m b o t h cylinders is suppressed w h e n G* is small. For G* = 0 . 1 5 (1^'^ r o w ) , t h e upper shear layer e m a n a t i n g f r o m the u p s t r e a m c y l i n d e r is K - H t y p e o f roU-ups ( e l o n g a t e d i n t h e streamwise d i r e c t i o n w i t h negative v o r t i c i t y ) , w h i c h pass over (or overshoot) the d o w n s t r e a m c y l i n d e r However, the l o w e r shear layer, w h i c h is evident i n the spacing b e t w e e n the cylinders, is rather w e a k i n m a g n i t u d e a n d s m a l l i n size. For G''' = 0.4 (2""^ r o w ) , b o t h shear layers are still largely i n K - H type at small spacing ratios (e.g., L * = 2 or 3). A t L * = 5 , o n tiie other hand, they begin to display as relatively large, discrete 'patches' o f v o i t i c i t y b e h i n d the d o w n s t r e a m cylinder, i n d i -cative o f the occurrence o f vortex shedding. However, n o vortex shedding is observed f r o m t h e upsb-eam cylinder: w h i l e t h e upper shear layer is s t i l l i n K - H type, t h e l o w e r shear layer e i t h e r reattaches steadily o n t h e leading surface o f the d o w n s t r e a m c y l i n d e r at L ' * = 2 and 3, or dissipates around x / D 4 at L* = 5. In a d d i t i o n , flow-induced separation is f o u n d i n t h e near w a l l region, similar t o the case o f t h e near-wall single cylinder s h o w n i n Fig. 2(b). W h e n C* increases to 0.6 (3'''' r o w ) , the w a l l effects still exist, b u t to a lesser degree. A t this gap ratio, periodic vortex shedding is always observed f r o m the d o w n -sti-eam cylinder, as w e l l as f r o m the upstream c y l i n d e r at wide-spacing ratios (e.g., L* = 5). A t G'' = 1.4 ( 4 * r o w ) , the w a l l effects become almost negligible such that t h e flow is similar to the free-standing case. Obviously, the three cases, L*=2, 3 and 5, belong t o the 'extended-body', 'reattachment and ' i m p i n g i n g ' regimes, respectively, as s h o w n i n Fig. 1. The critical spacing ratio is L*cr ~ 4.5, w h i c h is also consistent w i t h t h e reported values o f L * a - = 3 - 5 i n t h e literature.
A closer e x a m i n a t i o n o f the i n s t a n t a n e o u s v o r t i c i t y fields i n d i -cates t h a t t h e effects o f w a l l p r o x i m i t y c a n n o t s i m p l y be described as i n h i b i t i n g v o r t e x s h e d d i n g f r o m the c y l i n d e r s ; i n s t e a d , i t plays a c o m p l e x role i n a f f e c t i n g t h e shear layer d e v e l o p m e n t a n d i n t e r a c -t i o n . Take -t h e case o f L* = 2 a-t d i f f e r e n -t gap ra-tios ( l e f -t c o l u m n i n Fig. 6 ) as an e x a m p l e . A t G'* = 1.4, t h e flow is i n e x t e n d e d - b o d y r e g i m e , and t h e t w o shear layers separated f r o m t h e u p s t r e a m c y l i n d e r are k e p t n e a r l y h o r i z o n t a l l y a n d w r a p a r o u n d the d o w n -s t r e a m c y l i n d e r A t i n t e r m e d i a t e gap ratio-s (G* = 0.4 a n d 0.6), h o w e v e r , t h e l o w e r shear layer is b r o k e n i n t o t w o segments. The one i n b e t w e e n t h e t w o c y l i n d e r s deflects u p w a r d a n d reattaches o n t h e l e a d i n g surface o f t h e d o w n s t r e a m c y l i n d e r . S i m i l a r shear layer d e f l e c t i o n a n d r e a t t a c h m e n t are e v i d e n t f o r L * = 3 a t i n t e r -m e d i a t e gap ratios.
40 X.K. Wang et al. / Ocean Engineering 94 (2015) 36-50
a
0,4 0.2 0,0 0.4 0,2 0,0 0,4 0,2 0,0 c 'o g> 0.4 0,2 0,0 0,4ro
% 0,2 0,0 0.4 0.2 O.D 0.4 0.2 0.0 : 1 1 ' 1 1 — 1 1 1 1 -1 ' G * = 0 . -1 5 1 . 1 . 1 . 1 G* = 0.25 : 1 . 1 , 1 . 1 . ' G* = 0.4 1 . 1 . 1 1 G* = 0.5h
fl 1 . 1 , 1 1 q * = 0.6 1 . 1 , 1 . 1 . • G * = 0 , 8 ji Jjl
fl 1 - G * = 1 1 4 6t
(s)
10 0.01 \-0,00 0.01 0.00 0,01- r — ' r
G* = 0.151,
S l = 0 , 8 6 I I 1 — G* = 0.25 0.00 F O 001 co ^ 0,00 0,04 0,02 0.00 0.04 0.02 0.00 0.04 0,02 0,00 S t = 0 , S 5 G* = 0.4 Sl=0,85 G* = 0.5 S l = 0 , 2 G* = 0.6 G* = 0.8 S l = 0 . 2 G * = 1 0.0 0.2 0.4 0.6 S t 0.8Fig. 5. (a) Time histories of dynamic lift coefficient (Cj.) on the near-wall single cylinder at different gap ratios; and (b) corresponding spectra based on power spectral density (PSD) function. L * = 2 L * = 3 L * = 5 Ö -0.4
(j!:mU
" öt
1.4Q
SI
• r r • \ . V '' f l
0 . ) »
r r - I —1 — I — 1— r 1 6 x/D —I r-10 O - 7— 1 — 1 — 1— r 2 A —I r -10 O - I —1— I —1— I — I —I —t — r x/DFig. 6. A representative snapshot of the instantaneous normalized vorticity (lu* = wD/U) fields around the near-wall tandem cylinders at i ' = 2 , 3 and 5 and G*=0.15, 0.4, 0.6 and 1.4. Positive: solid red lines; negative: dashed blue lines. Cut-off value |<w''|,^i„ = l ; contour interval=0.5. (For interpretation ofthe references to color in this figure, the reader is referred to the web version of this article.)
X.K Wang et al / Ocean Engineering 94 (2015) 36-50 41
Based o n the PIV m e a s u r e m e n t results, a m a p o f f l o w patterns a r o u n d t h e n e a r - w a l l t a n d e m cylinders i n t h e G*-L* plane is p r o p o s e d i n Fig. 7. W i t h reference t o L*, i t can be r o u g h l y d i v i d e d i n t o t h r e e basic types o f spacing, t h a t is, close (1 < L* < 2), m o d e r a t e (2 < I * < 4 ) a n d w i d e (L* > 4 ) , w h i c h are r o u g h l y e q u i v a l e n t t o the 'extended-body', ' r e a t t a c h m e n t ' a n d ' i m p i n g i n g ' regimes, respec-tively, f o r f r e e - s t a n d i n g t a n d e m cylinders. Similarly, t h e f l o w can be b r o a d l y classified as a f u n c t i o n o f G*: ( i ) large-gap r e g i m e ( a p p r o x . > 1), t h e f l o w a n d v o r t e x s h e d d i n g characteristic are s i m i l a r t o the f r e e - s t a n d i n g case; ( i i ) i n t e r m e d i a t e - g a p r e g i m e ( a p p r o x . 0.3 < 1 ) ,
w h e r e p e r i o d i c v o r t e x s h e d d i n g occurs, b u t t h e s t r e n g t h o f v o r t e x s h e d d i n g reduces w i t h decreasing G*; a n d ( i i i ) small-gap r e g i m e (approx. G* < 0.3), w h e r e p e r i o d i c v o r t e x s h e d d i n g is c o m p l e t e l y suppressed.
3.2.2. Forces, lift spectra and Strouhal numbers on the downstream cylinder
This section presents t h e time h i s t o r i e s o f f l u c t u a t i n g l i f t o n t h e d o w n s t r e a m c y l i n d e r a n d t h e c o r r e s p o n d i n g spectra f o r d i f f e r e n t I n c r e a s i n g p i t c h ratio (L*) o Oi CQ <Q OJ Ó' C l o s e - s p a c i n g M o d e r a t e - s p a c i n g
d
-r^
y^
- ^ 0 W i d e - s p a c i n gO
I QX).
T3Q
S>
ozxiu a
TJ
Fig. 7. Overview of flow patterns for the near-wall two tandem cylinders as a function of both C and i * .
b 0,2 0.0 -0,2 0.2 0,0 -0,2 ^ 0,2
S
0,0lg
-0,28
0.2 ^ 0-0 g -0.2 § 0.2 • § 0,0 -0,2 0,2 0, -0,2 0.2 0.0 -0.2 -I 1 I -L* = 4 L * = 5 L* = 60 ^ii-t'Hf^^Af^]^^^^
'4
L* = 7 4 6t(s)
1 0 0.01I-0.00 0.01 0.00 0,01 0,00 0.01 0.00 0.01 0.00 0,01 0,00 0.01 0.00 ' 1 1 1 ' 1 ' 1 ' L* = •] 5 SI = 0.86 1 , 1 . 1 , f . L'* ~ 2 st = 0,86 " . 1 . 1 . 1 . 1 . L* = 3 I . I . I , Sl = 0,84 1 L* = 4 1 , 1 . 1 . SI = 0,84 1 | _ i _ g SI = 0.85 1 , 1 . 1 . 1 . St = 0.19 L ' = 6 " , 1 . 1 . 1 , 81 = 0,85 1 i si=o.i8 L* - 7 1 , 1 , c . st = 0,84 1 0.0 0,2 0,4 ' 0,6
St
1,04 2 X.K. Wang et al. / Ocean Engineering 94 (2015) 36-50
c o m b i n a t i o n s t o L* a n d G*. The results f o r L* = 1.5-7 at G*=0.15 are s h o w n i n Fig. 8. I t is clear t h a t t h e f l o w is i n t h e v o r t e x - s h e d d i n g s u p p r e s s i o n r e g i m e . A c c o r d i n g l y , t h e l i f t s i g n a l is i r r e g u l a r f o r a l l
s p a c i n g ratios c o n s i d e r e d . S i m i l a r t o t h e case o f t h e n e a r - w a l l single c y l i n d e r (Fig. 5), each s p e c t r u m displays a p e a k at S t « 0.85 associated w i t h K H t y p e o f roUups. I t is n o t e d t h a t f o r w i d e -0.5 0.0 -0.5 0.5 0.0 -0.5 0.5 i 0,0 O
%
O O ï ë 0,0 I I I _ l . 1 . L - 0 . 5 0 , 5 -0,5 Ulc
ro 0,5%
LL. 0,0 -0.5 0,5 0,0 -0.5 0,5 0,0 -0,5 L* = 2 L* = 3 L* = 4 L* = 5 -1 I i L L* = 6 I , I u L* = 7 4 6t(S)
1 0 0,02 0.00 | i ^ - ^ v - w A v ^ ^ v > y ^ r ~ 0,02 0,00 0,02 0,00 Q 0,02 CO 0¬ 0,00 0,02 L* = 2 st = 0.84 0.00 1 ' r L * = 1.5 st = 0,8S L* = 3 St=0,2ik
L* = 4 -L. StH0,18 O.OO 0.02-0.00 r-^"",'
0.02 L* = 5 L* = 6J0\
jé
L* = 7 0.0 0 . 2 0,4 0,6St
St = 0,B4 St = 0,85 Sl = 0,83 ...^;^V^^A,v.... Sl = 0,83 81 = 0,83 0 . 8 1.0Fig. 9. (a) Time liistories of fluctuating lift coefflcient; and (b) corresponding spectra on the downstream cylinder for different spacing ratios at G'=0.4.
c
0Ü
8
Ë CD Cro
ë
0.5 0.0 -0.5 0.5 0.0 -0.5 0.5 0,0 -0,5 0.5 0.0 -0.5 0,5 0,0 -0.5 0,5 0,0 -0,5 0,5 0,0 -0,5 ' L * = 1 . 5 ' L* = 2 L*.= 3 J . L L* = 4 L* = 6 L* = 7 0,04 0,02 0,00 0,04 0.02 0,00 0,02 0,00 0,04 W 0.02 0,00 0,05 0,00 0,05 0,00 0.05 10 L* = 3 0,00 - i 1 r L ' = 1,5 L* = 2 L* = 4 St = (),19 L* = 5 L* = 6 L* = 7 0.0 0.2t ( s )
0.4 0.6St
0,8 1,0X.K. Wang et al. / Ocean Engineering 94 (2015) 36-50 43
spacing c o n f i g u r a t i o n (L* > 4 ) , t h e r e is an a d d i t i o n a l pealc a r o u n d S t » 0 . 1 8 ( c o r r e s p o n d i n g t o tire large-scale v o n K a r m a n v o r t e x s h e d d i n g ) b u t i t is r a t h e r b r o a d - b a n d e d .
For G * = 0 . 4 (Fig. 9(a)), t h e signal g r a d u a l l y changes f r o m a chaotic p a t t e r n at L* = 1.5 a n d 2, t o a m o r e p e r i o d i c p a t t e r n a t I * >: 4. The a m p l i t u d e o f f l u c t u a t i o n increases s h a r p l y w i t h iL* f r o m close- t o m o d e r a t e - I * ( L * < 4 ) . As s h o w n i n Fig. 9 ( b ) , each spe-c t r u m a l w a y s e x h i b i t s a p e a k at St as 0 . 8 3 - 0 . 8 6 assospe-ciated w i t h t h e K-H i n s t a b i l i t y , i m p l y i n g t h a t t h e w a l l effects are s t i l l s i g n i f i c a n t . I t s h o u l d be n o t e d t h a t f r o m L * = 3 o n w a r d , a n a d d i t i o n a l p e a k at S t a : 0.18 appears as w e l l . T h e d o u b l e - p e a k character i n d i c a t e s t h e co-existence o f t w o d i f f e r e n t f l u i d d y n a m i c processes, as c o u l d be a p p r e c i a t e d f r o m Fig. 6. Located i n t h e lee o f t h e u p s t r e a m cylinder, t h e d o w n s t r e a m c y l i n d e r is s u b j e c t e d t o shear l a y e r r e a t t a c h m e n t o n its surface. A t C * = 0 . 4 , t h e shear layers are basically i n K - H t y p e o f roU-ups. H o w e v e r , w h e n L* is large e n o u g h (e.g., I * = 5), discrete 'patches' o f v o r t i c i t y are f o r m e d i n t h e w a k e of t h e d o w n s t r e a m c y l i n d e r , i n d i c a t i v e o f occurrence o f v o r t e x s h e d d i n g a t r e l a t i v e l y l o w f r e q u e n c y . Fig. 9 ( b ) s h o w s t h a t i n t h e case o f L * = 3 , t h e p e a k at St a; 0.19 is r a t h e r b r o a d - b a n d e d a n d s m a l l i n a m p l i t u d e ; w i t h f u r t h e r increase i n L*, i t becomes m o r e d i s t i n c t , suggesting t h a t v o r t e x s h e d d i n g becomes m o r e r e g u l a r and stronger.
The r e s u l t s f o r G * = 0 . 6 are s h o w n i n Fig. 10. I n t h i s case, t h e l i f t signal becomes s i g n i f i c a n t l y m o r e p e r i o d i c t h a n t h a t o f G ' ' = 0 . 4 at t h e same spacing r a t i o . O n t h e o t h e r h a n d , t h e peak f o r t h e h i g h -f r e q u e n c y K - H r o l l - u p s becomes n e a r l y i n v i s i b l e as a t i n y h u m p , i n d i c a t i n g t h a t the w a l l e f f e c t s reduce w i t h i n c r e a s i n g G*. A l l t h e spectra except f o r those a t L*=3 a n d 4 d i s p l a y a d o m i n a n t f r e q u e n c y o f S t = 0 . 1 8 0 . 2 c o r r e s p o n d i n g t o p e r i o d i c v o r t e x s h e d d i n g . A t L * = 3 and 4 , h o w e v e r , t h e spectral peak is r a t h e r b r o a d -banded, suggestive o f w e a k e n e d v o r t e x s h e d d i n g a c t i v i t y .
A t G* > 1, t h e w a l l effects b e c o m e n e a r l y n e g l i g i b l e , see Fig. 11 f o r G* = 1.4. The h i g h - f r e q u e n c y c o m p o n e n t t h a t m a y o t h e r w i s e
e x i s t at s m a l l - a n d intermediate-G"^ disappears c o m p l e t e l y , so each s p e c t r u m is characterized b y a w e l l d e f i n e d f r e q u e n c y at S t = 0 . 1 5 0.19. Based o n t h e p r o p o s e d c l a s s i f i c a t i o n , t h e y b e l o n g to e x t e -n d e d - b o d y r e g i m e (L* = 1.5 a -n d 2 ) , r e a t t a c h m e -n t r e g i m e (L*=3 a n d 4), a n d i m p i n g i n g r e g i m e (I,* = 5 , 6 a n d 7 ) , respectively. Several f e a t u r e s can be o b s e r v e d . Firstly, i n e i t h e r e x t e n d e d - b o d y o r i m p i n g i n g r e g i m e , t h e p e a k is w e l l - d e f i n e d , w h i l e i n r e a t t a c h m e n t r e g i m e ( I * = 3 a n d 4 ) i t is s o m e w h a t b r o a d - b a n d e d . Secondly, t h e p e r i o d i c i t y o f l i f t signal does n o t v a r y m o n o t o n i c a l l y w i t h L*; instead, i t first achieves a m i n i m u m at L ' * = 3 ( o n s e t o f r e a t t a c h -m e n t r e g i -m e ) a n d t h e n a -m a x i -m u -m at L'* = 5 ( o n s e t o f i -m p i n g i n g 0.24 w 0.22 -] 0.20 0.18 0.15 0.14 - G * = 1.4 - G * = 2
- Free-standing (Xu and Zhou, 2004) -Free-standing (igarashi, 1981)
— I 1 1 —
12 14 16
Fig. 12. Variation of Strouhal number (St) with V at large gap ratios, together with published data on two tandem cylinders under free-standing conditions.
Ü
8
cn g ro =i o 0.5 0.0 -0.5 0,5 0.0 -0,5 0,5 0,0 -0.5 0.5 0.0 -0.5 0.5 0.0 -0.5 0.5 0.0 -0.5 0.5 0.0 -0.5 : ' ' ' L* = 1.5' ' ' : 1 . 1 , 1 , 1 ,-L -L* =
2 : f . 1 : L* =c
I . l , 1Miliiii
- 1 ' , n 1 1 1 1 .' p ' 11 ' 1 °Ï , ' '
1 ' 11
ii | f i |
M
1 1imi4
i
4 6t
(s)
10 0.1 0.0 0,1 0.0 0.1 0.0 Q 0.1 CL 0.0 0.2 0.1 0.0 0.2 0.1 0,0 0.2 0,1 0,0 0,0 S I = 0,186 SI = 0,18 - A y St = 0.164 , *
st=,ö,1 0,2 L * = 1,5L* =
2L* =
3 L* = 4 L* = 5L* =
6 L * = 7 0.4 0,6St
0,8 1.044 X.K. Wang et al. / Ocean Engineering 94 (2015) 36-50
r e g i m e ) , as i n d i c a t e d b y t h e a m p l i t u d e o f l i f t f l u c t u a t i o n o r t h e m a g n i t u d e o f spectral peak. These observations i n d i c a t e t h a t t h e d o w n s t r e a m c y l i n d e r is i n f l u e n c e d b y t h e a d v e c t i o n a n d i m p i n g e -m e n t o f v o r t i c e s shed f r o -m t h e u p s t r e a -m c y l i n d e r . A t L* > 5, t h e peak c o n t i n u e s to decrease i n m a g n i t u d e w i t h i n c r e a s i n g L*, suggesting t h a t t h e i n t e r f e r e n c e b e t w e e n t h e t w o c y l i n d e r s is r e d u c i n g .
Fig. 11(b) s h o w s t h a t t h e values o f St f o r t h e spectral p e a k i n i m p i n g i n g r e g i m e (L* > 5) m a i n t a i n a p p r o x i m a t e l y c o n s t a n t . For L * < 4 , o n t h e o t h e r h a n d , St first d r o p s w i t h L* f r o m 0.185 at L * = 1 . 5 t o a m i n i m u m o f 0.15 at L * = 3 , a n d t h e n recovers t o 0.17 at L * = 4 . The i n i t i a l d e c l i n e o f St w i t h L* at s m a l l - t o m o d e r a t e - L * seems t o be a n i n h e r e n t f e a t u r e w h e n t h e w a l l p r o x i m i t y effects are n e g l i g i b l e , since a s i m i l a r t r e n d is f o u n d f o r G* = 2 as w e l l as f o r f r e e - s t a n d i n g t a n d e m c y l i n d e r s p u b l i s h e d i n t h e l i t e r a t u r e (e.g..
Igarashi 1 9 8 1 ; X u a n d Z h o u , 2 0 0 4 ) , as s h o w n i n Fig. 12. This is l i k e l y a t t r i b u t e d t o t h e fact t h a t a n increase i n L* a l l o w s t h e shear layers e m a n a t e d f r o m t h e u p s t r e a m c y l i n d e r to g r o w t h i c k e r u p o n r e a c h i n g t h e surface o f t h e d o w n s t r e a m c y l i n d e r . A c c o r d i n g l y , St decreases p r o g r e s s i v e l y w i t h £*, because a t h i c k e r shear layer leads t o a l o w e r v o r t e x s h e d d i n g f r e q u e n c y f r o m a c y l i n d e r (Roshko, 1954). H o w e v e r , t h i s t r e n d c a n n o t be sustained w i t h f u r t h e r i n -crease i n L* since t h e flow w o u l d change i n t o i m p i n g i n g r e g i m e at L * c r « 4, f o r w h i c h t h e shear layers e m a n a t e d f r o m t h e u p s t r e a m c y l i n d e r w i l l r o l l u p i n t o discrete v o r t i c e s i n b e t w e e n t h e t w o c y l i n d e r s a n d are n o l o n g e r d i r e c t l y c o n n e c t e d w i t h t h e v o r t e x f o r m a t i o n f r o m t h e d o w n s t r e a m c y l i n d e r . The f o r c e data i n d i c a t e t h a t t h e w a l l p r o x i m i t y t e n d s t o i n h i b i t p e r i o d i c v o r t e x s h e d d i n g f r o m t h e c y l i n d e r s . F u r t h e r m o r e , spectral analysis w a s a p p l i e d t o t h e v e l o c i t y data to i l l u s t r a t e t h e p e r i o d i c 0 . 0 4 0 . 0 0 0 . 0 0 0.05 Q CO 0 . 0 0 ° - 0 . 1 0 0 . 1 0 0 . 0 5 0 . 0 0 0 . 1 0 0.05 0 . 0 0 0 . 0 2 h 0 . 0 2 \ -
G* = 0.4
k
c
0.05 \-G* = 0.15
Upper
Lower
G* = 0,6
^ e
G* = 0.8
G*' = 1.4
. fG* = 2
0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5St
0 . 2 0 0 . 2 5Fig. 13. Spectra of transverse velocity (v) in between the two cylinders for the case of L * = 5 at: (a) G*=0.15, (b) 0.4, (c) 0.6, (d) 0.8, (e) 1.4 and (f) 2. The velocity signals are retrieved from: (x,y)=(2.5D, - 0 . 5 D ) , i.e., in the lower shear layer; and (x,y)=(2.5D,0.5D), i.e., in the upper shear layer.
X.K. Wang et al. / Ocean Engineering 94 (2015) 36-50 4 5
n a t u r e o f t h e f l o w up to ƒ = 7 , 5 Hz (i.e., half the PIV s a m p l i n g rate o f 15 Hz), or St Kl 0.27. Fig. 13 shows the velocity spertra between the t w o cylinders f o r the case o f L * = 5 at difFerent gap ratios. The velocity signals are rehieved f r o m ( x , y ) = ( 2 . 5 D , - 0 . 5 D ) a n d (2.5D, - 0 . 5 D ) , w h i c h are respectively located i n the l o w e r and upper shear layers. A t G*=0.15, there is no peak over the measurement range, whereas at G * > 0 . 4 , each spech'um begins to exhibit a d o m i n a n t peak at St=0.170.2 ( w h i c h is i n accordance w i t l i the l o w f r e q u e n c y c o m p o -n e -n t i -n l i f t spectra). This c o -n f i r m s the validity o f usi-ng l i f t sig-nal as a-n indicator o f vortex shedding process. The peak m a g n i t u d e increases w i t h G* particularly over the range o f G * < 0 . 8 , suggesting t h a t the effects o f w a l l p r o x i m i t y are decreasing a n d the strength o f v o r t e x shedding becomes sti'onger (similar conclusion is i n f e r r e d f r o m the l i f t specti-a). It is n o t e w o r t h y that f o r a given G*, the peak i n the upper shear layer (denoted b y red line) is generally higher i n m a g n i t u d e t h a n that i n the l o w e r shear layer (denoted b y black line) f o r G* < 1.4, indicating flow a s y m m e t r y about the w a k e centeriine. A t large enou gh gap ratios (e.g., G * = 2 ) , the t w o specfi-a almost coincide w i t h each o t h e r
Fig. 14 presents t h e v a r i a t i o n s o f t h e m e a n d r a g ( C D ) a n d l i f t ( C L ) , RMS d r a g ( C ó ) a n d l i f t {Q) c o e f f l c i e n t s o n t h e d o w n s t r e a m c y l i n d e r as a f u n c t i o n o f L* f o r d i f f e r e n t gap ratios, t o g e t h e r w i t h t h e c o r r e s p o n d i n g values o f t h e isolated single c y l i n d e r f o r c o m -p a r i s o n . Located i n t h e lee o f t h e u -p s t r e a m c y l i n d e r , t h e m e a n d r a g
c o e f f i c i e n t o n t h e d o w n s t r e a m c y l i n d e r (Co) r e m a i n s c o n s i s t e n t l y l o w e r t h a n t h a t o f t h e i s o l a t e d c y l i n d e r As s h o w n i n Fig. 14(a), C D increases m o n o t o n i c a l l y w i t h L* f o r a l l gap ratios considered, b u t
2.8 2.4 2.0 1.6 1.2 0.4 O No vortex shedding • Vortex shedding o • o o o o o o
Region of broad-banded peiitcs in lift spectra
Fig.
15.
Critical gap and spacing ratios (C*cr and l * c r ) for onset of vortex shedding from the dovi^nstream cylinder.ICJ 1.2 1.0 0.6 0.2 4 0.0 4 -0.2 - • - G ' = 0.15 —c— 0.4 - A - - 0.6 - V — 0.8 — 1.4 2 i.ïiflilifil '•mfilf f v l i i i t k i -I U 0.30 0.25 0.20 0,10 0,05 0.00 0.05
Fig. 14. Variation of dynamic force coefflcients on the downstream cylinder with L * at different gap ratios: (a) mean drag coefficient CCD); (b) mean lift coefflcient ( C J ; (c) R M S drag coefficient (Có); and (d) R M S lift coefficient (Q).
46 X.K Wans a/. / Ocean Engineenng 94 (2015) 36-50
at d i f f e r e n t rates (as r e f l e c t e d b y slope o f t h e curves). A t G*=0.15, CD is a b o u t 0.1 a n d increases s l i g h t l y w i t h L*. As C * increases t o 2, t h e l o w e n d o f each c u r v e ( a t L* = 1.5) decreases u n t i l r e a c h i n g a m i n i m u m o f ( C D ) n , i n = - 0 . 1 3 , w h e r e a s t h e h i g h e n d (at L* = 7) c o n t i n u e s t o rise u p t o (CD)max=0-84. T h e r e f o r e , w h e n G* is r e l a t i v e l y l a r g e (G* > 0.8), t h e d o w n s t r e a m c y l i n d e r experiences a d r a g i n v e r s e ( f r o m n e g a t i v e t o p o s i t i v e ) w i t h i n t h e range o f 2 ^ L * < 3 . T h i s is a w e l l - k n o w n p h e n o m e n o n f o r f r e e - s t a n d i n g t a n d e m c y l i n d e r s , f o r instance, S u m n e r e t a l . ( 2 0 0 5 ) r e p o r t e d a m i n i m u m o f (CD)n,in= - 0 . 5 5 at L * = 1 . 1 2 5 . T h e n e g a t i v e ( a t t r a c t i v e ) d r a g a t small-L* is d u e t o the f a c t t h a t t h e d o w n s t r e a m c y l i n d e r is c o m p l e t e l y e n w r a p p e d b y t h e shear layers f r o m t h e u p s t r e a m c y l i n d e r , a n d h e n c e experiences a n e g a t i v e pressure. T h e m e a n l i f t c o e f f i c i e n t (CL), as s h o w n i n Fig. 14(b), o n t h e o t h e r h a n d , varies s i g n i f i c a n t l y w i t h b o t h G* a n d L*. S i m i l a r t o t h e n e a r - w a l l single c y l i n d e r , t h e d o w n s t r e a m c y l i n d e r g e n e r a l l y experiences a p o s i t i v e 1.4 1.2 1.0 I U 0.4
0.2 - Near-wall single cylinder (Present)
-Near-wall single cylinder (Roshko etal., 1975) - Downstream cylinder (L* = 7) I'o 1.0 0,8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0,6 -0.8 0,5 1.0 1.5 2.0 2.5 3.0 Present (G* = 3)
Harimi and Sagfiafian (2012) - V - Z d r a v k o v i c h and Pridden (1977)
— \ ' 1 1 1 ' 1 ' r
Fig. 16. Variation ofthe mean drag coefficient (Co) on the dowfnstream cylinder with: (a) G* at large spacing ratio ( L ' = 7 ) , and comparison with published data (Roshko et al., 1975) and the present measurement data on a near-wall single cylinder; and (b) V at large spacing ratio (G*=3), and comparison with published data under free-standmg conditions (Zdravkovich and Pridden, 1977; Harimi and Saghafian, 2012).
L * = 2 L * = 3 L * = 5
m.DIU
X.K. Wang et al. / Ocean Engineering 94 (2015) 36-50 47
m e a n l i f t ( Q > 0 ) , b u t i t is c o n s i d e r a b l y s m a l l e r i n a m p l i t u d e (CL i « 0 . 0 5 - 0 , 2 ) . The s i g n i f i c a n t v a r i a t i o n o f CL w i t h G* a n d L* is p r o b a b l y d u e to t h e f o l l o w i n g t w o reasons. The f i r s t is t h e possible m i s a l i g n m e n t o f t h e models, since i t is e x t r e m e l y d i f f i c u l t , i f n o t i m p o s s i b l e , t o achieve a p e r f e c t a l i g n m e n t e x p e r i m e n t a l l y . Sec-ondly, t h e f l o w i n t h e gap m a y keep steadily r e a t t a c h e d o n t h e surface o f t h e d o w n s t r e a m c y l i n d e r , b u t m a y also s p o n t a n e o u s l y f l i p - f l o p v e r t i c a l l y so t h a t t h e r e a t t a c h m e n t p o i n t varies i r r e g u l a r l y over t i m e , w h i c h is i n a n a l o g y t o t h e f l o w b e t w e e n t w o s i d e b y -side c y l i n d e r s at s m a l l gap r a t i o s ( S u m n e r et al., 1999). Figs. 14 (c) a n d 14(d) s h o w t h a t t h e v a r i a t i o n t r e n d s o f Cb a n d Cl w i t h I * are s i m i l a r , a l t h o u g h t h e l a t t e r has a m u c h h i g h e r ( 3 ~ 4 t i m e s ) m a g n i t u d e . I n g e n e r a l , CÓ a n d Cl d i s p l a y d i f f e r e n t v a r i a t i o n t r e n d s w i t h respect t o L* d e p e n d i n g o n t h e v a l u e o f G*, w h i c h is c o n s i s t e n t w i t h t h e PlV data. A t G*=0.15 w h e n v o r t e x s h e d d i n g is suppressed, Cl a n d Cl increase s t e a d i l y w i t h L* over t h e m e a s u r e m e n t range. I n i n t e r m e d i a t e - g a p r e g i m e ( G * = 0 . 4 , 0.6 a n d 0.8), t h e y increase r a p i d l y w i t h L* u n t i l r e a c h i n g a m a x i m u m at L * = 5 (i.e., o n s e t o f i m p i n g i n g r e g i m e ) . A s i m i l a r
c o n v e x shape is f o u n d f o r G * = 1 . 4 a n d 2, w h e r e t h e t w o curves a l m o s t coincide, i m p l y i n g d i m i n i s h i n g e f f e c t s o f w a l l p r o x i m i t y .
D e p e n d i n g o n the values o f G* a n d L*, t h e shear layers emanated f r o m t h e u p s t r e a m c y l i n d e r m a y overshoot, reattach or i m p i n g e u p o n t h e d o w n s t r e a m c y l i n d e r and t h e n separate, perhaps j o i n i n g those developed o n the d o w n s t r e a m c y l i n d e r i t s e l f t o f o i m vortices a r o u n d the d o w n s t r e a m c y l i n d e r This results i n d i f f e r e n t behavi-ors o f f l u i d d y n a m i c forces o n the d o w n s t r e a m c y l i n d e r Here, a n a t t e m p t is made to i d e n t i f y the critical gap (G*cr) and spacing ( ! * „ ) ratios f o r vortex shedding based o n t h e p e r i o d i c i t y o f f l u c t u a t i n g l i f t signals and the i n t e n s i t y o f t h e peak i n l i f t spectra as s h o w n i n Figs. 8 - 1 1 ( s i m i l a r observation can be o b t a i n e d b y analysis o f t h e v e l o c i t y signal as s h o w n i n Fig. 13). The m a p f o r absence/presence o f v o r t e x shedding from the d o w n s t r e a m c y l i n d e r i n G*-L* plane is presented i n Fig. 15. The v o r t e x - s h e d d i n g suppression r e g i m e is m a i n l y located at t h e l o w e r - l e f t corner (i.e., small-G* a n d small-L*): at t h e smallest gap ratio (G* = 0.15), i t extends t h e w h o l e L* range; as G* increases, i t gradually shrinks i n w i d t h u n t i l c o m p l e t e l y disap-pears a t G* = 0.8. I n a d d i t i o n , the values o f Cl can be used to
B-B A - A S B-B G * = 0.15 0 1 ÜIU G " = 0.4 j 0 1 ÜIU G* = 0.6
F
ul
0 1 TilU G * = 0.8.2)
0 1 ÜIU G* = 1.4 / 5 7 /.V 0 ^ 1 ÏÏ/UFig. 18. Profiles of the normalized streamwise mean velocity (u/U) along two vertical lines (i.e., A-A and B-B, located behind the upstream cylinder and the downstream cylinder, respectively) for different gap and spacing ratios.
48 X.K. Wang et al / Ocean Engineering 94 (2015) 3S-50
qualitatively d e t e r m i n e t h e strength o f v o r t e x shedding. I t should be n o t e d t h a t measurements with a higher resolution i n G*-L* plane are desirable i n o r d e r t o m o r e accurately d e f i n e t h e boundaries separat-i n g t h e d separat-i f f e r e n t f l o w regseparat-imes. Furthermore, as s h o w n separat-i n Fseparat-igs. 9 - 1 1 , due to enhanced activity o f shear layer reattachment f o r L * = 3 a n d 4 i n intermediate-G* regime, the spectral peak is broad-banded i n these cases, w h i c h has been h i g h l i g h t e d by t h e shaded r e g i o n i n Fig. 15.
_ As s h o w n i n Fig. 16(a), v a r i a t i o n s o f t h e m e a n d r a g c o e f f i c i e n t ( C D ) o n t h e d o w n s t r e a m c y l i n d e r w i t h G* at t h e largest spacing r a t i o ( L * = 7 ) have b e e n c o m p a r e d w i t h the p u b l i s h e d data (Roshko e t al., 1975) a n d t h e p r e s e n t m e a s u r e m e n t o n a n e a r - w a l l single c y l i n d e r . S i m i l a r t o t h e case o f t h e single c y l i n d e r , C D i n i t i a l l y experiences a sharp increase w i t h G* b e f o r e l e v e l i n g o f f at large e n o u g h gap r a t i o s (G* s 1). H o w e v e r , a t t h e same G* i t is a p p r e -c i a b l y l o w e r t h a n t h a t o f t h e single -c y l i n d e r ( p a r t i -c u l a r w h e n G* is s m a l l ) , d u e t o t h e effects f r o m t h e u p s t r e a m c y l i n d e r . This suggests t h a t t h e spacing r a t i o o f L * = 7 is s t i l l n o t s u f f i c i e n t l y large f o r t h e t w o t a n d e m c y l i n d e r s t o be c o n s i d e r e d i n d e p e n d e n t l y . O n t h e o t h e r h a n d , as s h o w n i n Fig. 16(b), t h e v a r i a t i o n o f C D w i t h L* a t G * = 3 agrees w e l l w i t h t h e p u b l i s h e d data u n d e r f r e e - s t a n d i n g c o n d i t i o n s ( Z d r a v k o v i c h a n d P r i d d e n , 1977; H a r i m i a n d Saghafian, 2 0 1 2 ) , c o n f i r m i n g t h a t t h e gap r a t i o o f G * = 3 is large e n o u g h f o r n e g l e c t i n g t h e w a l l effects.
3.2.3. Ensemble-averaged flow pattems around the cylinders As s h o w n above, t h e v o r t e x s h e d d i n g characteristics f r o m t h e c y l i n d e r s d e p e n d o n b o t h G* a n d L*. This leads t o c o r r e s p o n d i n g v a r i a t i o n i n t h e statistical q u a n t i t i e s o f t h e f l o w , such as d i s t r i b u -tions o f m e a n v e l o c i t y vectors a n d Reynolds shear stresses.
Consistent w i t h t h e instantaneous f l o w s t r u c t u r e , t h e m e a n v e l o c i t y v e c t o r f i e l d (Fig. 17) g r a d u a l l y changes f r o m a s y m m e t r i c a l p a t t e r n s a b o u t t h e w a k e c e n t e r i i n e at s m a l l - a n d i n t e r m e d i a t e - g a p
ratios (G* < 1), t o s y m m e t r i c a l p a t t e r n s at large gap ratios (G* > 1). For a g i v e n L*, t h e r e c i r c u l a t i o n l e n g t h , d e f i n e d as t h e distance f r o m t h e c y l i n d e r base t o t h e zero m e a n s t r e a m w i s e v e l o c i t y p o i n t a l o n g t h e w a k e c e n t e r i i n e , increases w i t h G*. M e a n w h i l e , t h e gap flow is d e f l e c t e d u p w a r d i n y - d i r e c t i o n a n d r e a t t a c h o n t h e l e a d i n g face o f the d o w n s t r e a m c y l i n d e r , m o s t n o t a b l y i n t h e case o f i n t e r m e d i a t e - G * a n d m o d e r a t e - L * (e.g., G * = 0.6 a n d L * = 3 ) . This corresponds t o t h e r e g i o n o f b r o a d - b a n d e d peaks i n t h e l i f t spectra, as s h o w n i n Fig. 15.
Profiles o f t h e n o r m a l i z e d s t r e a m w i s e m e a n v e l o c i t y {u/U) a l o n g t w o v e r t i c a l lines l o c a t e d at 0.5D a f t e r t h e t r a i l i n g edges o f t h e t w o c y l i n d e r s (i.e., A - A a n d B-B) f o r d i f f e r e n t gap a n d spacing ratios are p r o v i d e d i n Fig. 18. One o b v i o u s f e a t u r e is t h a t f o r a fixed G*, t h e p r o f i l e s at d i f f e r e n t L* at A - A a l m o s t collapse, w h e r e a s t h o s e at B-B deviate f r o m each o t h e r m o r e e v i d e n t l y . T h i s indicates t h a t t h e presence o f t h e d o w n s t r e a m c y l i n d e r m a i n l y affects t h e flow b e h i n d i t . The presence o f t h e c y l i n d e r s results i n t h e v e l o c i t y d e f e c t b e h i n d each c y l i n d e r , so t h a t t h e v e l o c i t y p r o f i l e s e x h i b i t as a n "S"-shape. W h e n G* < G*cr (e.g., G* = 0.15), h o w e v e r , t h e l o w e r h a l f o f the "S"-shape is n o t o b v i o u s o r even c o m p l e t e l y disappears due t o the r a t h e r w e a k g a p flow.
F u r t h e r m o r e , t h e shear layer d e v e l o p m e n t s can be a p p r e c i a t e d f r o m the c o n t o u r s o f t h e n o r m a l i z e d Reynolds shear stress ( ï ï V / L f ^ ) i n Fig. 19. A t s m a l l - o r i n t e r m e d i a t e - G * , t h e u p p e r shear layer is b o t h s t r o n g e r i n m a g n i t u d e a n d l a r g e r i n size t h a n t h e l o w e r one. For a g i v e n L*, as G* increases f r o m 0.15 t o 1.4, t h e d i s t r i b u t i o n s o f W/U^ g r a d u a l l y b e c o m e m o r e s y m m e t r i c a b o u t t h e w a k e c e n t e r i i n e ; m e a n w h i l e , regions o f s i g n i f i c a n t uY/U^ c o n t r a c t i n t h e s t r e a m w i s e d i r e c t j o n ( t o s m a l l e r x ) , t o g e t h e r w i t h e l e v a t e d level ( o r m a g n i t u d e ) o f u'v'/U^.
S i m i l a r to t h e flow c l a s s i f i c a t i o n based o n t h e i n s t a n t a n e o u s v o r t i c i t y fields, t h e d i s t r i b u t i o n s o f uV/U^ can be d i v i d e d i n t o t h r e e d i f f e r e n t p a t t e r n s - P a t t e r n 1 : regions o f s i g n i f i c a n t u'v'/U^ are f o u n d i n t h e w a k e o f t h e d o w n s t r e a m c y l i n d e r o n l y (e.g., f o r a l l L * = 2 L * = 5
U Ö
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I—I—I—I—r- - T —I— r -4 X/D — 1— I —I -2Fig. 19. Contours ofthe normalized Reynolds shear stress (u'v'/ü^) for l ' = 2 , 3 and 5 and G*=0.15, 0.4,0.6 and 1.4. Positive: solid red lines; negative; dashed blue lines. Cut-off value luV/U^I =0.01; contour interval=0.005 (For interpretation o f t h e references to color in this figure, the reader is referred to the web version of this article.).
