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Suction
Engineering Turbulence Modelling and Experiments 3 W. Rodi and G. Bergeles (Editors) © 1996 Elsevier Science B.V. All rights reserved.
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Direct Numerical Simulation of Transitional Backward-Facing Step Flow Manipulated by Oscillating Blowing/Suction G u n t e r Bàrwolff
a
, H a n s Wengle * a n d H a n s g e o r g Jeggle b
c
H e r m a n n - F o t t i n g e r - I n s t i t u t fur T h e r m o - u n d F l u i d d y n a m i k , Technische U n i v e r s i t à t Berlin, D-10623 Berlin, G e r m a n y
a
I n s t i t u t fur S t r ô m u n g s m e c h a n i k u. A e r o d y n a m i k , L R T / W E 7, U n i v e r s i t à t der B u n d e s w e h r M u n c h e n , D-85577 N e u b i b e r g , G e r m a n y ,
b
F a c h b e r e i c h 3, M a t h e m a t i k , F u n k t i o n a l a n a l y s i s u. N u m e r i s c h e M a t h e m a t i k , Technische U n i v e r s i t à t Berlin, D-10623 Berlin, G e r m a n y
c
Direct n u m e r i c a l s i m u l a t i o n of t r a n s i t i o n a l flow over a backward-facing s t e p h a s been carried o u t for t h r e e cases of t i m e - p e r i o d i c b l o w i n g / s u c t i o n forcing t h r o u g h a n a r r o w crosswind slot close t o t h e edge of t h e s t e p . In a g r e e m e n t w i t h c o r r e s p o n d i n g e x p e r i m e n t s a significant r e d u c t i o n in t h e m e a n recirculation l e n g t h could b e achieved by l o w - a m p l i t u d e e x c i t a t i o n . T h e t r a n s i t i o n t o t u r b u l e n c e in t h e shear layer, b o u n d i n g t h e recirculation region, is significantly affected by t h e r e a t t a c h m e n t process, a n d by t h e m a n i p u l a t i o n with a time-periodic disturbance. 1.
Introduction
T h e u n d e r s t a n d i n g of flow s e p a r a t i o n a n d r e a t t a c h m e n t is i m p o r t a n t for t h e design of engineering a p p l i c a t i o n s . O n t h e one h a n d , t h e presence of recirculation a n d t u r b u l e n c e m a y help, e.g. t o e n h a n c e t h e m i x i n g of different a d m i x t u r e s in a flow. O n t h e o t h e r h a n d , t h e presence of s e p a r a t e d flow regimes, e.g. in i n t e r n a l flows, m a y cause loss of t h e available energy. T h e r e is a n u r g e n t need for t h e i m p r o v e m e n t of technological processes involving t u r b u l e n t flows, a n d it h a s been shown in recent t i m e s t h a t t h e r e is a possibility of c h a n g i n g a n d controlling t h e n a t u r a l s t r u c t u r e of t u r b u l e n c e t o achieve a c e r t a i n desired b e h a v i o u r of t h e flow, such as d r a g r e d u c t i o n , m i n i m i z a t i o n / m a x i m i z a t i o n of m i x i n g , or t h e r e d u c t i o n of s e p a r a t i o n . To c a r r y o u t such m a n i p u l a t i o n s successfully, a d e e p e r u n d e r s t a n d i n g of t h e u n d e r l a y i n g d y n a m i c s of t h e basic s t r u c t u r e s in t u r b u l e n t flow is required, which can b e o b t a i n e d by investigating t h e t h r e e - d i m e n s i o n a l a n d t i m e - d e p e n d e n t flow fields. T h e r e are two basic simulation concepts available t o c a l c u l a t e t h e t h r e e - d i m e n s i o n a l a n d t i m e - d e p e n d e n t s t r u c t u r e of a t u r b u l e n t flow. O n t h e one h a n d , t h e so-called Direct Numerical Simulation (DNS) requires t o resolve all t h e relevant scales in a t u r b u l e n t flow, b u t its r a n g e of a p p l i c a t i o n is limited t o relatively small R e y n o l d s n u m b e r s (often t o o small from a p r a c t i c a l engineering p o i n t of view). O n t h e o t h e r h a n d , t h e so-called Large*To whom correspondence should be sent; e-mail: [email protected]
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Eddy Simulation (LES) is c a p a b l e t o deliver directly t h e s p a t i a l a n d t e m p o r a l b e h a v i o u r of a t least t h e large-scale s t r u c t u r e s of high R e y n o l d s n u m b e r flows, a n d only t h e effects of t h e small-scale m o t i o n s which c a n n o t b e resolved on a given c o m p u t a t i o n a l m e s h need t o b e m o d e l l e d w i t h a so-called subgrid-scale m o d e l . A m o n g t h e flow geometries used for t h e studies of s e p a r a t e d flows, t h e m o s t frequently selected one is t h e backward-facing s t e p : t h e g e o m e t r y is simple, t h e s e p a r a t i o n p o i n t is fixed a t t h e s t e p , t h e s t r e a m l i n e s a p p r o a c h i n g t h e s t e p are n e a r l y parallel t o t h e wall. T h e r e is only one m a j o r s e p a r a t i o n region w i t h m e a n r e a t t a c h m e n t a t a b o u t six t o eight s t e p heights (for R e y n o l d s n u m b e r s above 1000). For turbulent boundary layer inflow, a n d a s t e p height R e y n o l d s n u m b e r of 5100, t h e r e is a D N S available from Le, Moin a n d K i m [1], a n d a c o r r e s p o n d i n g L E S from Akselvoll a n d Moin [2]. For m u c h higher R e y n o l d s n u m b e r s , a n d t u r b u l e n t b o u n d a r y layer inflow, or t u r b u l e n t channel inflow, L E S solutions are available from K o b a y a s h i a n d Togashi [3] for R e = 4 6 0 0 0 , a n d from A r n a l a n d Friedrich [5] for R e = 1 6 5 0 0 0 . T h e case of laminar boundary layer inflow is of p a r t i c u l a r interest b e c a u s e of t h e fact t h a t t h e free shear layer u n d e r g o e s t r a n s i t i o n t o t u r b u l e n c e a n d , over a relatively long d i s t a n c e , t h i s shear layer is s e p a r a t i n g a t u r b u l e n t recirculation zone below ( b o u n d e d by rigid walls) from a n o n - t u r b u l e n t region above. T h e r e is is s o m e i n f o r m a t i o n available on t h e onset of t h r e e - d i m e n s i o n a l i t y a n d early t r a n s i t i o n from a D N S investigation of K a i k t i s et al. [6] for R e = 8 0 0 . L E S solutions are available from N e t o et. al [4] for R e = 3 8 0 0 0 . T h e r e a r e e x p e r i m e n t a l investigations beeing carried o u t , a t a R e y n o l d s n u m b e r of 3000, by t h e research g r o u p of Prof. H. Fernholz (Technical University of B e r l i n ) . For t h i s R e y n o l d s n u m b e r t h e r e is a chance of c a r r y i n g o u t successfully a D N S . T h e r e are recent reviews on t h e subject of m a n a g e m e n t a n d control of l a m i n a r / t u r b u l e n t flows by Bushnell a n d M c G i n l e y [7] a n d by Fiedler a n d Fernholz [8]. R e l a t e d t o t h e topic of t h i s p a p e r , t h e r e a r e r e p o r t s on m o r e recent e x p e r i m e n t a l work by K i y a et al. [9] on a forced recirculation b u b b l e a t a b l u n t cylinder, by H a s a n [10] a n d by H a s a n a n d K h a n [11] on t h e m a n i p u l a t i o n of t h e free shear layer after a b a c k w a r d facing s t e p . Using direct n u m e r i c a l s i m u l a t i o n of a m a n i p u l a t e d p l a n e channel flow, Choi et al. [12] showed t h a t a 25 % d r a g r e d u c t i o n can b e achieved by i m p l e m e n t i n g a p r o p e r control loop for t h e m a n i p u l a t i o n s applied. T h e two m a i n objectives of t h i s s t u d y are: (a) t o c a r r y o u t a direct n u m e r i c a l simu lation of a b a c k w a r d facing s t e p flow, w i t h l a m i n a r inflow, a t r a n s i t i o n a l free s h e a r layer b o u n d i n g t h e ( t u r b u l e n t ) recirculation zone, a n d a relaxing t u r b u l e n t b o u n d a r y layer af t e r t h e r e a t t a c h m e n t zone, a n d (b) t o m a n i p u l a t e t h i s flow by b l o w i n g / s u c t i o n t h r o u g h a crosswind n a r r o w slot close t o t h e edge of t h e s t e p .
2. P r o b l e m specification 2.1. Governing equations and solution strategy For a direct n u m e r i c a l s i m u l a t i o n (DNS) t h e conservation e q u a t i o n s for m a s s a n d m o m e n t u m m u s t b e solved w i t h o u t any a d d i t i o n a l a s s u m p t i o n s or m o d e l l i n g r e l a t e d t o t h e effects of t u r b u l e n t m o t i o n , i.e. t h e original Navier-Stokes e q u a t i o n s m u s t b e discretized in space a n d t i m e such t h a t all t h e relevant scales in a t u r b u l e n t flow a r e resolved. For e x a m p l e , referring t o s p a t i a l scales, t h e size of t h e c o m p u t a t i o n a l d o m a i n m u s t b e large
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e n o u g h t o a c c o m o d a t e t h e largest t u r b u l e n t scales (which is d e t e r m i n e d by t h e scale of t h e a p p a r a t u s ) , a n d t h e grid s p a c i n g m u s t b e sufficiently fine t o resolve t h e d i s s i p a t i o n lenght scale (which is of t h e o r d e r of t h e K o l m o g o r o v scale η = (u/e) ^). T h e n , t h e r a t i o of t h e two scales (cubed) provides a n e s t i m a t e for t h e t o t a l n u m b e r of grid p o i n t s required. Less s t r i n g e n t e s t i m a t i o n s consider, for e x a m p l e , t w o - p o i n t correlations a n d (one-dimensional) energy s p e c t r a t o j u d g e w h e t h e r a d e q u a t e resolution c a n b e achieved. 1
T h e direct result from a D N S is t h e t i m e - d e p e n d e n t a n d t h r e e - d i m e n s i o n a l velocity a n d pressure field. B y t i m e - a v e r a g i n g of t h e i n s t a n t a n e o u s fields ( a n d m a k i n g use of available h o m o g e n e o u s directions) t h e m e a n fields c a n b e o b t a i n e d . A s soon as t h e m e a n fields have reached s t a b l e (i.e. t i m e - i n d e p e n d e n t ) values, t h e f l u c t u a t i n g fields c a n b e evaluated. Finally, from these fluctuating fields, t h e r o o t - m e a n - s q u a r e values (e.g. for velocity, vorticity, a n d pressure fluctuations) c a n b e c a l c u l a t e d , as well as t h e R e y n o l d s stresses a n d o t h e r higher-order s t a t i s t i c s .
2.2. Flow configuration, computational grid, and b o u n d a r y conditions T h e flow configuration selected is a b a c k w a r d facing s t e p (see figure 1) w i t h a n e x p a n s i o n r a t i o of 1.2 (defined as t h e r a t i o b e t w e e n t h e vertical d i m e n s i o n after a n d before t h e s t e p ) . If all l e n g t h scales a r e n o r m a l i z e d by t h e s t e p height h, t h e d i m e n s i o n s of t h e c o m p u t a t i o n a l d o m a i n a r e ( L , L , L ) = ( 1 7 . 6 , 6 . 0 , 6 . 0 ) w i t h a n e n t r y l e n g t h of 5 reference l e n g t h s . Here, χ is t h e m a i n flow direction, y is t h e l a t e r a l direction, ζ is t h e vertical direction, a n d x / h = 0 c o r r e s p o n d s t o t h e position of t h e s t e p . (Note t h a t in figure 1 n o t t h e whole c o m p u t a t i o n a l d o m a i n is visible). T h e l e n g t h of t h e e n t r y section was sufficiently long t h a t , s t a r t i n g from a n uniform velocity profile a t t h e inlet section (at x / h = -5.0), a t t h e edge of t h e s t e p ( x / h = 0 ) a Blasius profile is developed w i t h a b o u n d a r y layer thickness of a b o u t δ/h = 0.2. T h i s s t a r t i n g c o n d i t i o n for a l a m i n a r inflow, a n d t h e R e y n o l d s n u m b e r of Reh — U h/v = 3000 (based on s t e p height h a n d t h e inlet free s t r e a m velocity U ) have b e e n selected in a c c o r d a n c e w i t h c u r r e n t e x p e r i m e n t a l investigations carried o u t by t h e g r o u p of Prof. H. Fernholz a t t h e H e r m a n n - F o t t i n g e r - I n s t i t u t ( T U Berlin, G e r m a n y ) . A n o slip b o u n d a r y c o n d i t i o n is used along t h e lower wall (with t h e s t e p ) , a n d a slip con dition is used along t h e u p p e r b o u n d a r y of t h e c o m p u t a t i o n a l d o m a i n . 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 a r e used in t h e l a t e r a l direction a n d , for t h e results p r e s e n t e d here, n o r m a l gra dients of flow variables have b e e n set t o zero a t t h e outflow cross section a n d , in a d d i t i o n , a buffer section of l e n g t h 2.7h h a s b e e n a d d e d t o t h e original t o t a l l e n g t h (at x / h = 1 5 . 0 ) of t h e c o m p u t a t i o n a l d o m a i n t o a c c o u n t for u p s t r e a m d i s t o r t i o n s of t h e flow by t h e simple outflow b o u n d a r y c o n d i t i o n (for t h e c u r r e n t s i m u l a t i o n s we i m p o s e a m o r e a p p r o p r i a t e convective outflow b o u n d a r y c o n d i t i o n ) . T h e flow is m a n i p u l a t e d by periodic b l o w i n g / s u c t i o n t h r o u g h a crosswind slot a t t h e edge of t h e s t e p . T h e oscillating j e t is inclined u n d e r 45° a n d its exit velocity c a n b e described by Vj = A · sin(2nft), w i t h a frequency f a n d a n a m p l i t u d e A as t h e two m a i n p a r a m e t e r s of t h e controlled d i s t u r b a t i o n . T h e m a n i p u l a t i o n is i m p l e m e n t e d via t i m e d e p e n d e n t b o u n d a r y c o n d i t i o n s for t h e h o r i z o n t a l (u) a n d vertical (w) velocity c o m p o n e n t s on two n e i g h b o u r i n g crosswind rows of grid p o i n t s close t o t h e edge of t h e s t e p . x
y
z
0
0
et
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2.3. Numerical solution m e t h o d T h e Navier-Stokes e q u a t i o n s for an incompressible fluid a r e solved numerically on a staggered grid using second-order finite-differencing in t i m e a n d space (explicit leapfrog for t i m e discretization, c e n t r a l differencing for convection a n d t i m e - l a g g e d diffusion t e r m s ) . T h e p r o b l e m of pressure-velocity coupling is solved iteratively by a pressure (and veloc ity) correction m e t h o d , requiring t h e dimensionless divergence t o b e below 1 0 ~ after every t i m e s t e p a t all grid p o i n t s . To resolve all t h e relevant scales of t h e flow we used (N , N , N ) = ( 5 1 2 , 1 2 8 , 1 6 0 ) , i.e. a b o u t 11 million grid p o i n t s , in t h e s t r e a m w i s e (x), t h e l a t e r a l (y), a n d t h e vertical (z) direction. At t h e inlet section, t h e s t r e a m w i s e grid spacing s t a r t s w i t h a value of AX = δχ/h = 0.1 which t h e n slowly decreases t o AX = 0.04 a t t h e edge of t h e s t e p (at x / h = 0 . 0 ) a n d t h e n r e m a i n s c o n s t a n t a t t h i s value t h r o u g h o u t t h e c o m p u t a t i o n a l d o m a i n . In t h e vertical direction t h e grid s p a c i n g is c o n s t a n t a t a value of AZ = 0.01563 from Z = 0 . 0 t o Z = 1 . 5 , a n d t h e n slowly increases t o AZ = 0.18 a t Z = 6 . 0 . In t h e l a t e r a l direction we used a n e q u i d i s t a n t grid s p a c i n g of AY = 0.046875. M e a s u r e d in u n i t s of t h e viscous l e n g t h (using a wall shear velocity of u = 0.055 from X = 1 5 . 0 ) t h e d i s t a n c e of t h e first grid p o i n t t o a rigid wall is a b o u t AZ = 1.1, t h e l a t e r a l resolution is a b o u t AY = 7.0, a n d in t h e l o n g i t u d i n a l direction t h e resolution is a b o u t AX+ = 6.0. 3
x
y
z
+
+
+
2.4. Performance of c o m p u t e r program on a parallel c o m p u t e r T h e c o m p u t e r code used h a s been developed originally for o p t i m a l use on a vector c o m p u t e r . T h e results presented in t h i s p a p e r have been c o m p u t e d on a massively parallel processing s y s t e m ( C R A Y T 3 D ) a n d d o m a i n d e c o m p o s i t i o n h a s b e e n used t o parallelize t h e code. O n t h e given a r c h i t e c t u r e , a t w o - d i m e n s i o n a l d o m a i n d e c o m p o s i t i o n (in t h e horizontal p l a n e of t h e c o m p u t a t i o n a l d o m a i n ) was t h e easiest way t o a d o p t t h e given p r o b l e m size t o t h e available n u m b e r of processors. T h e original i t e r a t i v e ( p o i n t - b y - p o i n t ) p r e s s u r e / v e l o c i t y correction p r o c e d u r e involved a r e c u r r e n c y leading t o a conflict a t t h e b o u n d a r i e s of t h e s u b d o m a i n s . To e l i m i n a t e t h e p r o b l e m t h e c o m p u t a t i o n of t h e pressure is d o n e in two passes (first for o d d cells, t h e n for even cells). I n t e r c h a n g e of b o u n d a r y values between s u b d o m a i n s is d o n e b e t w e e n t h e two i t e r a t i o n sweeps using " explicit s h a r e d m e m o r y message p a s s i n g " . T h i s t y p e of c o m m u n i c a t i o n t a k e s into a c c o u n t t h e global a d d r e s s space of t h e T 3 D . T h e modified c o m p u t e r p r o g r a m is a factor of 8.5 faster t h a n a single processor C R A Y Y - M P a n d needs a b o u t 1.2 · 1 0 ~ C P U - s e c p e r t i m e s t e p p e r grid p o i n t on a C R A Y T 3 D w i t h 64 P E ' s . A b o u t 1.4 G B y t e of m e m o r y is needed t o i n t e g r a t e t h e basic flow variables in t i m e a n d t o collect t h e s t a t i s t i c s of first a n d second order. W i t h a dimensionless t i m e s t e p of 0.002 a b o u t 150000 t i m e s t e p s are required t o reach 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 of t h e flow field (80 reference t i m e s ) a n d t o collect t h e c o m p l e t e s t a t i s t i c s over a b o u t 220 reference t i m e s . T h i s leads t o a n overall c o m p u t i n g t i m e of a b o u t 500 C P U - h o u r s on a T 3 D (64 P E ' s ) for t h e reference case w i t h o u t m a n i p u l a t i o n . A d d i t i o n a l cases w i t h m a n i p u l a t i o n ( s t a r t i n g from t h e s t e a d y s t a t e of t h e reference case) a n d using t h e s a m e t i m e s t e p (which is still a factor of 10 below t h e n u m e r i c a l s t a b i l i t y limit) a b o u t 100 C P U - h o u r s are required t o collect a b o u t 100 t i m e - u n c o r r e l a t e d s a m p l e s of t h e d i s t u r b e d flow, which is t h e m i n i m u m for sufficiently s t a b l e first-order s t a t i s t i c s (using in a d d i t i o n t h e h o m o g e n e o u s l a t e r a l direction of t h e p r o b l e m t o i m p r o v e t h e s t a t i s t i c s ) . 6
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3. Discussion of results and conclusions I n s t a n t a n e o u s flow field: F r o m figure 1 ( i n s t a n t a n e o u s velocity field) a n d figure 2 ( i n s t a n t a n e o u s vorticity field) it b e c o m e s evident t h a t t h e m o s t i n t e r e s t i n g a n d t h e m o s t active flow regimes a r e t h e free shear layer u n d e r g o i n g t r a n s i t i o n t o t u r b u l e n c e a n d t h e r e a t t a c h m e n t zone. S t r o n g h o r i z o n t a l velocity fluctuations are c r e a t e d close t o t h e wall by t h e s p l a s h i n g d o w n of t h e s h e a r layer o n t o t h e b o t t o m p l a t e . T h e vertical (w) velocity shows ringlike p a t t e r n s (e.g. a t X = 6 . 5 , Y = 3 . 0 ) w i t h upwelling flow, s u r r o u n d i n g t h e inner a r e a w i t h d o w n flow (which h a s also been observed in t h e e x p e r i m e n t s ) . T h e i n s t a n t a n e o u s spanwise vorticity c o n t o u r s (ω ) show t h e large scale roll-up of t h e s h e a r layer which is s t r o n g l y influenced by t h e r e a t t a c h m e n t of t h e flow, c a u s i n g smaller scale s t r u c t u r e s of t h e vorticity. M e a n f l o w field: T h e m e a n vertical velocity profiles of t h e l o n g i t u d i n a l velocity com p o n e n t (averaged in t i m e a n d over t h e l a t e r a l h o m o g e n e o u s direction) i n d i c a t e t h e rela tively low intensity of t h e m e a n flow in t h e recirculation region, see figure 3 a n d figure 4. D o w n s t r e a m of t h e m e a n r e a t t a c h m e n t line t h e profiles develop t o t h e typical s h a p e of a t u r b u l e n t b o u n d a r y layer. However, t h e vertical profiles of t h e m e a n s t r e a m w i s e velocity c o m p o n e n t , n o r m a l i z e d in wall c o o r d i n a t e s , show a t a d o w n s t r e a m l o c a t i o n of x / h = 1 5 . 0 t h a t t h e flow is still far from b e i n g recovered t o a fully developed t u r b u l e n t b o u n d a r y layer (see figure 5 ). υ
M a n i p u l a t i o n o f t h e f l o w : T h e flow h a s been d i s t u r b e d by a n oscillating (blow i n g / s u c t i o n ) l a t e r a l wall j e t close t o t h e edge of t h e s t e p . Always u s i n g a frequency of f = 5 0 Hz ( t h e m o s t successful frequency found in t h e e x p e r i m e n t a l i n v e s t i g a t i o n ) , we car ried o u t t h r e e different s i m u l a t i o n s w i t h a m p l i t u d e s A = 1 0 ~ , A = 1 0 ~ , a n d A = 1 0 " of t h e t i m e - p e r i o d i c forcing causing a significant r e d u c t i o n of t h e l e n g t h of t h e recirculation zone. T h i s b e c o m e s evident from a c o m p a r i s o n of one of t h e m a n i p u l a t e d cases ( A = 0 . 0 1 ) w i t h t h e u n m a n i p u l a t e d reference case ( A = 0 . 0 ) , see figure 3 a n d figure 4. To d e t e r m i n e m o r e a c c u r a t e l y t h e m e a n recirculation l e n g t h , profiles of t h e m e a n s t r e a m w i s e velocity c o m p o n e n t along a wall-parallel line t h r o u g h t h e first grid p o i n t n e x t t o t h e wall have b e e n d r a w n in figure 6. If t h e zero-crossing of t h e s e l o n g i t u d i n a l profiles is t a k e n t o b e t h e p o s i t i o n of t h e m e a n recirculation l e n g t h , it b e c o m e s obvious from figure 6 t h a t a significant r e d u c t i o n can b e achieved: w i t h d i s t u r b a t i o n a m p l i t u d e A = 1 0 ~ a r e d u c t i o n t o 84 %, w i t h A = 1 0 ~ a r e d u c t i o n t o 66 %, a n d w i t h A = 1 0 a r e d u c t i o n t o a b o u t 57 % of t h e value of t h e reference case w i t h o u t e x c i t a t i o n . C o m p a r i s o n w i t h e x p e r i m e n t : T h e e x p e r i m e n t a l investigation in [14] shows a 30 % r e d u c t i o n of t h e m e a n recirculation l e n g t h a p p l y i n g a similar t i m e - p e r i o d i c forcing in t h e frequency r a n g e b e t w e e n 40 Hz a n d 60 Hz. T h i s c o r r e s p o n d s t o t h e numerically s i m u l a t e d case w i t h f = 5 0 Hz a n d A — 1 0 ~ . T h e m e a s u r e d m e a n recirculation l e n g t h of t h e u n d i s t u r b e d reference case, Χητ /h = 6.4, differs from t h e c o r r e s p o n d i n g value, x /h — 7A, in t h e n u m e r i c a l s i m u l a t i o n . Small differences b e t w e e n e x p e r i m e n t a n d n u m e r i c a l s i m u l a t i o n m a y explain t h i s d e v i a t i o n of a b o u t 16 %, e.g. t h e r e is a s m a l l second s t e p a t t h e edge of t h e s t e p m o d e l used in t h e e x p e r i m e n t , a n d t h e r e is a t u r b u l e n c e intensity of a b o u t 0.35 % in t h e i n c o m i n g flow of t h e e x p e r i m e n t . A m o r e d e t a i l e d c o m p a r i s o n between e x p e r i m e n t a n d n u m e r i c a l s i m u l a t i o n is one of t h e objectives of t h e c u r r e n t investigation. 3
2
1
3
2
- 1
2
r
224
C o n c l u s i o n s : T h e first series of direct n u m e r i c a l s i m u l a t i o n s of m a n i p u l a t e d t r a n s i t i o n a l backward-facing s t e p flow showed t h a t a significant r e d u c t i o n of t h e m e a n recircu lation zone can b e o b t a i n e d by a l o w - a m p l i t u d e t i m e - p e r i o d i c b l o w i n g / s u c t i o n e x c i t a t i o n t h r o u g h a n a r r o w crosswind slot a t t h e edge of t h e s t e p . T h e flow fields are very c o m p l e x showing t h e large-scale roll-up of t h e free shear layer, t h e t r a n s i t i o n t o t h r e e - d i m e n s i o n a l s t r u c t u r e s , t h e s t r o n g effects of t h e r e a t t a c h m e n t on t h e t r a n s i t i o n , a n d t h e very slow recovering after t h e r e a t t a c h m e n t zone t o a fully developed t u r b u l e n t b o u n d a r y layer. F r o m a first e v a l u a t i o n of t h e d a t a sets it can b e concluded t h a t t h e m a j o r effect of t h e forcing is a n acceleration of t h e t r a n s i t i o n process. In t h e c u r r e n t a n d future work t h e e v a l u a t i o n of t h e flow fields will b e c o n t i n u e d t o e x t r a c t m o r e i n f o r m a t i o n a b o u t t h e s t a tistical p r o p e r t i e s a n d t h e d y n a m i c s of t h e d o m i n a n t flow s t r u c t u r e s . T h i s will b e d o n e in s t r o n g c o o p e r a t i o n w i t h t h e c u r r e n t e x p e r i m e n t a l work t o utilize t h e a d v a n t a g e s of b o t h m e t h o d s of investigation.
REFERENCES 1.
Le, H., Moin, P. a n d K i m , J. (1993): Direct n u m e r i c a l s i m u l a t i o n of t u r b u l e n t flow over a backward-facing s t e p , 9 t h S y m p . T u r b u l e n t S h e a r Flows, A u g u s t 16-18, 1993, Kyoto, J a p a n 2. Akselvoll, K. a n d Moin, P. (1993): Large e d d y s i m u l a t i o n of a b a c k w a r d facing s t e p flow, E n g i n e e r i n g T u r b u l e n c e Modelling a n d E x p e r i m e n t s 2, W . R o d i a n d F . Martelli ( E d i t o r s ) , Elsevier Science P u b l i s h e r s 3. K o b a y a s h i , T . a n d Togashi, S. (1993): C o m p a r i s o n of t u r b u l e n c e m o d e l s applied t o backward-facing s t e p flow by L E S d a t a base, E n g i n e e r i n g T u r b u l e n c e Modelling a n d E x p e r i m e n t s 2, W . R o d i a n d F . Martelli ( E d i t o r s ) , Elsevier Science P u b l i s h e r s 4. N e t o , A.S., G r a n d , D., M e t a i s , O. a n d Lesieur, M. (1991): L a r g e - e d d y s i m u l a t i o n of t h e t u r b u l e n t flow in t h e d o w n s t r e a m region of a backward-facing s t e p , Physical Review L e t t e r s 66, p p . 2320-2323 5. A r n a l , M. a n d Friedrich, R. (1992): Large-eddy s i m u l a t i o n of a t u r b u l e n t flow w i t h s e p a r a t i o n , T u r b u l e n t S h e a r Flows 8, Springer-Verlag 6. K a i k t i s , L., K a r n i a d a k i s , G . E . a n d Orszag, S.A. (1991): O n s e t of three-dimensionality, equilibria, a n d early t r a n s i t i o n in flow over a backward-facing s t e p , J. F l u i d Mech. 231, p p . 501-528 7. Bushnell, D . M. a n d McGinley, C. (1989): T u r b u l e n c e control in wall flows, A n n . Rev. F l u i d Mech. 2 1 , p p . 1-20 8. Fiedler, H. E. a n d Fernholz, H. H. (1990): O n m a n a g e m e n t a n d control of t u r b u l e n t s h e a r flow, P r o g . Aerospace Sci. 27, p p . 305-387 9. Kiya, M., Shimizu, M., Mochizuki, O. a n d Ido, Y. (1993): T h e forced t u r b u l e n t s e p a r a t i o n b u b b l e , 9 t h S y m p . T u r b u l e n t S h e a r Flows, A u g u s t 16-18, K y o t o , J a p a n 10. H a s a n , Μ. A. Z. (1992): T h e flow over a backward-facing s t e p u n d e r controlled dist u r b a t i o n : l a m i n a r s e p a r a t i o n , J. F l u i d Mech. 238, p p . 73-96 11. H a s a n , Μ. A. Z. a n d K h a n , A. S. (1992): O n t h e s t a b i l i t y characteristics of a r e a t t a c h i n g shear layer w i t h n o n l a m i n a r s e p a r a t i o n , Int. J. H e a t a n d F l u i d Flow 1 3 / 3 , p p . 224-231 12. Choi, H., Moin, P. a n d K i m , J. (1994): Active t u r b u l e n c e control for d r a g r e d u c t i o n
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in w a l l - b o u n d e d flows, J. F l u i d Mech. 262, p p . 75-110 13. Bàrwolff, G., Ketelsen, K. a n d Thiele, F . (1995): P a r a l l e l i z a t i o n of a finite-volume Navier-Stokes solver on a T 3 D massively parallel s y s t e m , P r o c e e d i n g s of 6 t h Inter n a t i o n a l S y m p o s i u m on C o m p u t a t i o n a l F l u i d D y n a m i c s , S e p t e m b e r 5-9, 1995, Lake Tahoe, Nevada, USA 14. H u p p e r t z , A. (1994): Beeinflussung der S t r ô m u n g h i n t e r einer r u c k w a r t s g e w a n d t e n Stufe d u r c h d r e i d i m e n s i o n a l e A n r e g u n g , D i p l o m a - T h e s i s (in g e r m a n ) , H e r m a n n F o t t i n g e r - I n s t i t u t fur T h e r m o - u n d F l u i d d y n a m i k , T U Berlin
F i g u r e 1. Isolines of i n s t a n t a n e o u s velocity field : above: U ( X , Z ) ; m i n = - 0 . 5 , m a x = + 0 . 5 , i n c r e m e n t s . 1, in m i d p l a n e below: U ( X , Y ) , V ( X , Y ) a n d W ( X , Y ) ; m i n — 0 . 1 , m a x = + 0 . 1 , i n c r e m e n t = 0.02, close t o t h e b o t t o m wall (U a n d V a t Z = 0 . 0 0 8 , W a t Z = 0 . 0 2 3 ) full lines: positive values, d a s h e d lines: n e g a t i v e values
F i g u r e 2. Isolines of i n s t a n t a n e o u s vorticity field : ω (Χ,Ζ), ω (Χ,Ζ) a n d ω (Χ, Ζ)] m i n = - 5 . 0 , m a x = + 5 . 0 , i n c r e m e n t a l . 0 , in m i d p l a n e , d a s h e d lines: negative values, full lines: positive values χ
ζ
υ
MPPPPPPPIII 10
MMPPPIIIII 10
F i g u r e 3. Vertical profiles of m e a n s t r e a m w i s e velocity < U >: above: A = 0 . 0 below: A = 0 . 0 1
F i g u r e 4. Isolines of m e a n s t r e a m w i s e a n d m e a n vertical velocity c o m p o n e n t : above: reference case w i t h o u t m a n i p u l a t i o n ( A = 0 . 0 ) below: m a n i p u l a t e d case w i t h A = 0 . 0 1 left: m e a n s t r e a m w i s e velocity < U(X,Z) >; min=-0.2, m a x = + 1 . 0 right: m e a n vertical velocity < W(X,Z) >;min=-0.02,max=+0.02,iner.=0.004 full lines: positive values, d a s h e d lines: negative values
F i g u r e 5. M e a n s t r e a m w i s e velocity profile in wall c o o r d i n a t e s a t X = x / h = 1 5 . 0 ( A = 0 . 0 )
228
10
I I I I I I I I
CM I
Ο
A=0.0 A=0.001 A=0.01 A=0.1
V IM ' I /I -10
10
X F i g u r e 6. E v a l u a t i o n of m e a n recirculation l e n g t h x /h: from m e a n s t r e a m w i s e velocity d i s t r i b u t i o n along t h e b o t t o m wall a t height Z = 1.1 (first grid p o i n t n e x t t o wall) for t h e reference case ( w i t h o u t m a n i p u l a t i o n , A = 0 . 0 ) a n d t h r e e cases w i t h different m a n i p u l a t i o n a m p l i t u d e s : A = 0 . 0 0 1 , 0.01, a n d 0.1 r
+
15
219
Direct Numerical Simulation of Transitional Backward-Facing Step Flow Manipulated by Oscillating Blowing/Suction G u n t e r Bàrwolff
a
, H a n s Wengle * a n d H a n s g e o r g Jeggle b
c
H e r m a n n - F o t t i n g e r - I n s t i t u t fur T h e r m o - u n d F l u i d d y n a m i k , Technische U n i v e r s i t à t Berlin, D-10623 Berlin, G e r m a n y
a
I n s t i t u t fur S t r ô m u n g s m e c h a n i k u. A e r o d y n a m i k , L R T / W E 7, U n i v e r s i t à t der B u n d e s w e h r M u n c h e n , D-85577 N e u b i b e r g , G e r m a n y ,
b
F a c h b e r e i c h 3, M a t h e m a t i k , F u n k t i o n a l a n a l y s i s u. N u m e r i s c h e M a t h e m a t i k , Technische U n i v e r s i t à t Berlin, D-10623 Berlin, G e r m a n y
c
Direct n u m e r i c a l s i m u l a t i o n of t r a n s i t i o n a l flow over a backward-facing s t e p h a s been carried o u t for t h r e e cases of t i m e - p e r i o d i c b l o w i n g / s u c t i o n forcing t h r o u g h a n a r r o w crosswind slot close t o t h e edge of t h e s t e p . In a g r e e m e n t w i t h c o r r e s p o n d i n g e x p e r i m e n t s a significant r e d u c t i o n in t h e m e a n recirculation l e n g t h could b e achieved by l o w - a m p l i t u d e e x c i t a t i o n . T h e t r a n s i t i o n t o t u r b u l e n c e in t h e shear layer, b o u n d i n g t h e recirculation region, is significantly affected by t h e r e a t t a c h m e n t process, a n d by t h e m a n i p u l a t i o n with a time-periodic disturbance. 1.
Introduction
T h e u n d e r s t a n d i n g of flow s e p a r a t i o n a n d r e a t t a c h m e n t is i m p o r t a n t for t h e design of engineering a p p l i c a t i o n s . O n t h e one h a n d , t h e presence of recirculation a n d t u r b u l e n c e m a y help, e.g. t o e n h a n c e t h e m i x i n g of different a d m i x t u r e s in a flow. O n t h e o t h e r h a n d , t h e presence of s e p a r a t e d flow regimes, e.g. in i n t e r n a l flows, m a y cause loss of t h e available energy. T h e r e is a n u r g e n t need for t h e i m p r o v e m e n t of technological processes involving t u r b u l e n t flows, a n d it h a s been shown in recent t i m e s t h a t t h e r e is a possibility of c h a n g i n g a n d controlling t h e n a t u r a l s t r u c t u r e of t u r b u l e n c e t o achieve a c e r t a i n desired b e h a v i o u r of t h e flow, such as d r a g r e d u c t i o n , m i n i m i z a t i o n / m a x i m i z a t i o n of m i x i n g , or t h e r e d u c t i o n of s e p a r a t i o n . To c a r r y o u t such m a n i p u l a t i o n s successfully, a d e e p e r u n d e r s t a n d i n g of t h e u n d e r l a y i n g d y n a m i c s of t h e basic s t r u c t u r e s in t u r b u l e n t flow is required, which can b e o b t a i n e d by investigating t h e t h r e e - d i m e n s i o n a l a n d t i m e - d e p e n d e n t flow fields. T h e r e are two basic simulation concepts available t o c a l c u l a t e t h e t h r e e - d i m e n s i o n a l a n d t i m e - d e p e n d e n t s t r u c t u r e of a t u r b u l e n t flow. O n t h e one h a n d , t h e so-called Direct Numerical Simulation (DNS) requires t o resolve all t h e relevant scales in a t u r b u l e n t flow, b u t its r a n g e of a p p l i c a t i o n is limited t o relatively small R e y n o l d s n u m b e r s (often t o o small from a p r a c t i c a l engineering p o i n t of view). O n t h e o t h e r h a n d , t h e so-called Large*To whom correspondence should be sent; e-mail: [email protected]
220
Eddy Simulation (LES) is c a p a b l e t o deliver directly t h e s p a t i a l a n d t e m p o r a l b e h a v i o u r of a t least t h e large-scale s t r u c t u r e s of high R e y n o l d s n u m b e r flows, a n d only t h e effects of t h e small-scale m o t i o n s which c a n n o t b e resolved on a given c o m p u t a t i o n a l m e s h need t o b e m o d e l l e d w i t h a so-called subgrid-scale m o d e l . A m o n g t h e flow geometries used for t h e studies of s e p a r a t e d flows, t h e m o s t frequently selected one is t h e backward-facing s t e p : t h e g e o m e t r y is simple, t h e s e p a r a t i o n p o i n t is fixed a t t h e s t e p , t h e s t r e a m l i n e s a p p r o a c h i n g t h e s t e p are n e a r l y parallel t o t h e wall. T h e r e is only one m a j o r s e p a r a t i o n region w i t h m e a n r e a t t a c h m e n t a t a b o u t six t o eight s t e p heights (for R e y n o l d s n u m b e r s above 1000). For turbulent boundary layer inflow, a n d a s t e p height R e y n o l d s n u m b e r of 5100, t h e r e is a D N S available from Le, Moin a n d K i m [1], a n d a c o r r e s p o n d i n g L E S from Akselvoll a n d Moin [2]. For m u c h higher R e y n o l d s n u m b e r s , a n d t u r b u l e n t b o u n d a r y layer inflow, or t u r b u l e n t channel inflow, L E S solutions are available from K o b a y a s h i a n d Togashi [3] for R e = 4 6 0 0 0 , a n d from A r n a l a n d Friedrich [5] for R e = 1 6 5 0 0 0 . T h e case of laminar boundary layer inflow is of p a r t i c u l a r interest b e c a u s e of t h e fact t h a t t h e free shear layer u n d e r g o e s t r a n s i t i o n t o t u r b u l e n c e a n d , over a relatively long d i s t a n c e , t h i s shear layer is s e p a r a t i n g a t u r b u l e n t recirculation zone below ( b o u n d e d by rigid walls) from a n o n - t u r b u l e n t region above. T h e r e is is s o m e i n f o r m a t i o n available on t h e onset of t h r e e - d i m e n s i o n a l i t y a n d early t r a n s i t i o n from a D N S investigation of K a i k t i s et al. [6] for R e = 8 0 0 . L E S solutions are available from N e t o et. al [4] for R e = 3 8 0 0 0 . T h e r e a r e e x p e r i m e n t a l investigations beeing carried o u t , a t a R e y n o l d s n u m b e r of 3000, by t h e research g r o u p of Prof. H. Fernholz (Technical University of B e r l i n ) . For t h i s R e y n o l d s n u m b e r t h e r e is a chance of c a r r y i n g o u t successfully a D N S . T h e r e are recent reviews on t h e subject of m a n a g e m e n t a n d control of l a m i n a r / t u r b u l e n t flows by Bushnell a n d M c G i n l e y [7] a n d by Fiedler a n d Fernholz [8]. R e l a t e d t o t h e topic of t h i s p a p e r , t h e r e a r e r e p o r t s on m o r e recent e x p e r i m e n t a l work by K i y a et al. [9] on a forced recirculation b u b b l e a t a b l u n t cylinder, by H a s a n [10] a n d by H a s a n a n d K h a n [11] on t h e m a n i p u l a t i o n of t h e free shear layer after a b a c k w a r d facing s t e p . Using direct n u m e r i c a l s i m u l a t i o n of a m a n i p u l a t e d p l a n e channel flow, Choi et al. [12] showed t h a t a 25 % d r a g r e d u c t i o n can b e achieved by i m p l e m e n t i n g a p r o p e r control loop for t h e m a n i p u l a t i o n s applied. T h e two m a i n objectives of t h i s s t u d y are: (a) t o c a r r y o u t a direct n u m e r i c a l simu lation of a b a c k w a r d facing s t e p flow, w i t h l a m i n a r inflow, a t r a n s i t i o n a l free s h e a r layer b o u n d i n g t h e ( t u r b u l e n t ) recirculation zone, a n d a relaxing t u r b u l e n t b o u n d a r y layer af t e r t h e r e a t t a c h m e n t zone, a n d (b) t o m a n i p u l a t e t h i s flow by b l o w i n g / s u c t i o n t h r o u g h a crosswind n a r r o w slot close t o t h e edge of t h e s t e p .
2. P r o b l e m specification 2.1. Governing equations and solution strategy For a direct n u m e r i c a l s i m u l a t i o n (DNS) t h e conservation e q u a t i o n s for m a s s a n d m o m e n t u m m u s t b e solved w i t h o u t any a d d i t i o n a l a s s u m p t i o n s or m o d e l l i n g r e l a t e d t o t h e effects of t u r b u l e n t m o t i o n , i.e. t h e original Navier-Stokes e q u a t i o n s m u s t b e discretized in space a n d t i m e such t h a t all t h e relevant scales in a t u r b u l e n t flow a r e resolved. For e x a m p l e , referring t o s p a t i a l scales, t h e size of t h e c o m p u t a t i o n a l d o m a i n m u s t b e large
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e n o u g h t o a c c o m o d a t e t h e largest t u r b u l e n t scales (which is d e t e r m i n e d by t h e scale of t h e a p p a r a t u s ) , a n d t h e grid s p a c i n g m u s t b e sufficiently fine t o resolve t h e d i s s i p a t i o n lenght scale (which is of t h e o r d e r of t h e K o l m o g o r o v scale η = (u/e) ^). T h e n , t h e r a t i o of t h e two scales (cubed) provides a n e s t i m a t e for t h e t o t a l n u m b e r of grid p o i n t s required. Less s t r i n g e n t e s t i m a t i o n s consider, for e x a m p l e , t w o - p o i n t correlations a n d (one-dimensional) energy s p e c t r a t o j u d g e w h e t h e r a d e q u a t e resolution c a n b e achieved. 1
T h e direct result from a D N S is t h e t i m e - d e p e n d e n t a n d t h r e e - d i m e n s i o n a l velocity a n d pressure field. B y t i m e - a v e r a g i n g of t h e i n s t a n t a n e o u s fields ( a n d m a k i n g use of available h o m o g e n e o u s directions) t h e m e a n fields c a n b e o b t a i n e d . A s soon as t h e m e a n fields have reached s t a b l e (i.e. t i m e - i n d e p e n d e n t ) values, t h e f l u c t u a t i n g fields c a n b e evaluated. Finally, from these fluctuating fields, t h e r o o t - m e a n - s q u a r e values (e.g. for velocity, vorticity, a n d pressure fluctuations) c a n b e c a l c u l a t e d , as well as t h e R e y n o l d s stresses a n d o t h e r higher-order s t a t i s t i c s .
2.2. Flow configuration, computational grid, and b o u n d a r y conditions T h e flow configuration selected is a b a c k w a r d facing s t e p (see figure 1) w i t h a n e x p a n s i o n r a t i o of 1.2 (defined as t h e r a t i o b e t w e e n t h e vertical d i m e n s i o n after a n d before t h e s t e p ) . If all l e n g t h scales a r e n o r m a l i z e d by t h e s t e p height h, t h e d i m e n s i o n s of t h e c o m p u t a t i o n a l d o m a i n a r e ( L , L , L ) = ( 1 7 . 6 , 6 . 0 , 6 . 0 ) w i t h a n e n t r y l e n g t h of 5 reference l e n g t h s . Here, χ is t h e m a i n flow direction, y is t h e l a t e r a l direction, ζ is t h e vertical direction, a n d x / h = 0 c o r r e s p o n d s t o t h e position of t h e s t e p . (Note t h a t in figure 1 n o t t h e whole c o m p u t a t i o n a l d o m a i n is visible). T h e l e n g t h of t h e e n t r y section was sufficiently long t h a t , s t a r t i n g from a n uniform velocity profile a t t h e inlet section (at x / h = -5.0), a t t h e edge of t h e s t e p ( x / h = 0 ) a Blasius profile is developed w i t h a b o u n d a r y layer thickness of a b o u t δ/h = 0.2. T h i s s t a r t i n g c o n d i t i o n for a l a m i n a r inflow, a n d t h e R e y n o l d s n u m b e r of Reh — U h/v = 3000 (based on s t e p height h a n d t h e inlet free s t r e a m velocity U ) have b e e n selected in a c c o r d a n c e w i t h c u r r e n t e x p e r i m e n t a l investigations carried o u t by t h e g r o u p of Prof. H. Fernholz a t t h e H e r m a n n - F o t t i n g e r - I n s t i t u t ( T U Berlin, G e r m a n y ) . A n o slip b o u n d a r y c o n d i t i o n is used along t h e lower wall (with t h e s t e p ) , a n d a slip con dition is used along t h e u p p e r b o u n d a r y of t h e c o m p u t a t i o n a l d o m a i n . 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 a r e used in t h e l a t e r a l direction a n d , for t h e results p r e s e n t e d here, n o r m a l gra dients of flow variables have b e e n set t o zero a t t h e outflow cross section a n d , in a d d i t i o n , a buffer section of l e n g t h 2.7h h a s b e e n a d d e d t o t h e original t o t a l l e n g t h (at x / h = 1 5 . 0 ) of t h e c o m p u t a t i o n a l d o m a i n t o a c c o u n t for u p s t r e a m d i s t o r t i o n s of t h e flow by t h e simple outflow b o u n d a r y c o n d i t i o n (for t h e c u r r e n t s i m u l a t i o n s we i m p o s e a m o r e a p p r o p r i a t e convective outflow b o u n d a r y c o n d i t i o n ) . T h e flow is m a n i p u l a t e d by periodic b l o w i n g / s u c t i o n t h r o u g h a crosswind slot a t t h e edge of t h e s t e p . T h e oscillating j e t is inclined u n d e r 45° a n d its exit velocity c a n b e described by Vj = A · sin(2nft), w i t h a frequency f a n d a n a m p l i t u d e A as t h e two m a i n p a r a m e t e r s of t h e controlled d i s t u r b a t i o n . T h e m a n i p u l a t i o n is i m p l e m e n t e d via t i m e d e p e n d e n t b o u n d a r y c o n d i t i o n s for t h e h o r i z o n t a l (u) a n d vertical (w) velocity c o m p o n e n t s on two n e i g h b o u r i n g crosswind rows of grid p o i n t s close t o t h e edge of t h e s t e p . x
y
z
0
0
et
222
2.3. Numerical solution m e t h o d T h e Navier-Stokes e q u a t i o n s for an incompressible fluid a r e solved numerically on a staggered grid using second-order finite-differencing in t i m e a n d space (explicit leapfrog for t i m e discretization, c e n t r a l differencing for convection a n d t i m e - l a g g e d diffusion t e r m s ) . T h e p r o b l e m of pressure-velocity coupling is solved iteratively by a pressure (and veloc ity) correction m e t h o d , requiring t h e dimensionless divergence t o b e below 1 0 ~ after every t i m e s t e p a t all grid p o i n t s . To resolve all t h e relevant scales of t h e flow we used (N , N , N ) = ( 5 1 2 , 1 2 8 , 1 6 0 ) , i.e. a b o u t 11 million grid p o i n t s , in t h e s t r e a m w i s e (x), t h e l a t e r a l (y), a n d t h e vertical (z) direction. At t h e inlet section, t h e s t r e a m w i s e grid spacing s t a r t s w i t h a value of AX = δχ/h = 0.1 which t h e n slowly decreases t o AX = 0.04 a t t h e edge of t h e s t e p (at x / h = 0 . 0 ) a n d t h e n r e m a i n s c o n s t a n t a t t h i s value t h r o u g h o u t t h e c o m p u t a t i o n a l d o m a i n . In t h e vertical direction t h e grid s p a c i n g is c o n s t a n t a t a value of AZ = 0.01563 from Z = 0 . 0 t o Z = 1 . 5 , a n d t h e n slowly increases t o AZ = 0.18 a t Z = 6 . 0 . In t h e l a t e r a l direction we used a n e q u i d i s t a n t grid s p a c i n g of AY = 0.046875. M e a s u r e d in u n i t s of t h e viscous l e n g t h (using a wall shear velocity of u = 0.055 from X = 1 5 . 0 ) t h e d i s t a n c e of t h e first grid p o i n t t o a rigid wall is a b o u t AZ = 1.1, t h e l a t e r a l resolution is a b o u t AY = 7.0, a n d in t h e l o n g i t u d i n a l direction t h e resolution is a b o u t AX+ = 6.0. 3
x
y
z
+
+
+
2.4. Performance of c o m p u t e r program on a parallel c o m p u t e r T h e c o m p u t e r code used h a s been developed originally for o p t i m a l use on a vector c o m p u t e r . T h e results presented in t h i s p a p e r have been c o m p u t e d on a massively parallel processing s y s t e m ( C R A Y T 3 D ) a n d d o m a i n d e c o m p o s i t i o n h a s b e e n used t o parallelize t h e code. O n t h e given a r c h i t e c t u r e , a t w o - d i m e n s i o n a l d o m a i n d e c o m p o s i t i o n (in t h e horizontal p l a n e of t h e c o m p u t a t i o n a l d o m a i n ) was t h e easiest way t o a d o p t t h e given p r o b l e m size t o t h e available n u m b e r of processors. T h e original i t e r a t i v e ( p o i n t - b y - p o i n t ) p r e s s u r e / v e l o c i t y correction p r o c e d u r e involved a r e c u r r e n c y leading t o a conflict a t t h e b o u n d a r i e s of t h e s u b d o m a i n s . To e l i m i n a t e t h e p r o b l e m t h e c o m p u t a t i o n of t h e pressure is d o n e in two passes (first for o d d cells, t h e n for even cells). I n t e r c h a n g e of b o u n d a r y values between s u b d o m a i n s is d o n e b e t w e e n t h e two i t e r a t i o n sweeps using " explicit s h a r e d m e m o r y message p a s s i n g " . T h i s t y p e of c o m m u n i c a t i o n t a k e s into a c c o u n t t h e global a d d r e s s space of t h e T 3 D . T h e modified c o m p u t e r p r o g r a m is a factor of 8.5 faster t h a n a single processor C R A Y Y - M P a n d needs a b o u t 1.2 · 1 0 ~ C P U - s e c p e r t i m e s t e p p e r grid p o i n t on a C R A Y T 3 D w i t h 64 P E ' s . A b o u t 1.4 G B y t e of m e m o r y is needed t o i n t e g r a t e t h e basic flow variables in t i m e a n d t o collect t h e s t a t i s t i c s of first a n d second order. W i t h a dimensionless t i m e s t e p of 0.002 a b o u t 150000 t i m e s t e p s are required t o reach 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 of t h e flow field (80 reference t i m e s ) a n d t o collect t h e c o m p l e t e s t a t i s t i c s over a b o u t 220 reference t i m e s . T h i s leads t o a n overall c o m p u t i n g t i m e of a b o u t 500 C P U - h o u r s on a T 3 D (64 P E ' s ) for t h e reference case w i t h o u t m a n i p u l a t i o n . A d d i t i o n a l cases w i t h m a n i p u l a t i o n ( s t a r t i n g from t h e s t e a d y s t a t e of t h e reference case) a n d using t h e s a m e t i m e s t e p (which is still a factor of 10 below t h e n u m e r i c a l s t a b i l i t y limit) a b o u t 100 C P U - h o u r s are required t o collect a b o u t 100 t i m e - u n c o r r e l a t e d s a m p l e s of t h e d i s t u r b e d flow, which is t h e m i n i m u m for sufficiently s t a b l e first-order s t a t i s t i c s (using in a d d i t i o n t h e h o m o g e n e o u s l a t e r a l direction of t h e p r o b l e m t o i m p r o v e t h e s t a t i s t i c s ) . 6
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3. Discussion of results and conclusions I n s t a n t a n e o u s flow field: F r o m figure 1 ( i n s t a n t a n e o u s velocity field) a n d figure 2 ( i n s t a n t a n e o u s vorticity field) it b e c o m e s evident t h a t t h e m o s t i n t e r e s t i n g a n d t h e m o s t active flow regimes a r e t h e free shear layer u n d e r g o i n g t r a n s i t i o n t o t u r b u l e n c e a n d t h e r e a t t a c h m e n t zone. S t r o n g h o r i z o n t a l velocity fluctuations are c r e a t e d close t o t h e wall by t h e s p l a s h i n g d o w n of t h e s h e a r layer o n t o t h e b o t t o m p l a t e . T h e vertical (w) velocity shows ringlike p a t t e r n s (e.g. a t X = 6 . 5 , Y = 3 . 0 ) w i t h upwelling flow, s u r r o u n d i n g t h e inner a r e a w i t h d o w n flow (which h a s also been observed in t h e e x p e r i m e n t s ) . T h e i n s t a n t a n e o u s spanwise vorticity c o n t o u r s (ω ) show t h e large scale roll-up of t h e s h e a r layer which is s t r o n g l y influenced by t h e r e a t t a c h m e n t of t h e flow, c a u s i n g smaller scale s t r u c t u r e s of t h e vorticity. M e a n f l o w field: T h e m e a n vertical velocity profiles of t h e l o n g i t u d i n a l velocity com p o n e n t (averaged in t i m e a n d over t h e l a t e r a l h o m o g e n e o u s direction) i n d i c a t e t h e rela tively low intensity of t h e m e a n flow in t h e recirculation region, see figure 3 a n d figure 4. D o w n s t r e a m of t h e m e a n r e a t t a c h m e n t line t h e profiles develop t o t h e typical s h a p e of a t u r b u l e n t b o u n d a r y layer. However, t h e vertical profiles of t h e m e a n s t r e a m w i s e velocity c o m p o n e n t , n o r m a l i z e d in wall c o o r d i n a t e s , show a t a d o w n s t r e a m l o c a t i o n of x / h = 1 5 . 0 t h a t t h e flow is still far from b e i n g recovered t o a fully developed t u r b u l e n t b o u n d a r y layer (see figure 5 ). υ
M a n i p u l a t i o n o f t h e f l o w : T h e flow h a s been d i s t u r b e d by a n oscillating (blow i n g / s u c t i o n ) l a t e r a l wall j e t close t o t h e edge of t h e s t e p . Always u s i n g a frequency of f = 5 0 Hz ( t h e m o s t successful frequency found in t h e e x p e r i m e n t a l i n v e s t i g a t i o n ) , we car ried o u t t h r e e different s i m u l a t i o n s w i t h a m p l i t u d e s A = 1 0 ~ , A = 1 0 ~ , a n d A = 1 0 " of t h e t i m e - p e r i o d i c forcing causing a significant r e d u c t i o n of t h e l e n g t h of t h e recirculation zone. T h i s b e c o m e s evident from a c o m p a r i s o n of one of t h e m a n i p u l a t e d cases ( A = 0 . 0 1 ) w i t h t h e u n m a n i p u l a t e d reference case ( A = 0 . 0 ) , see figure 3 a n d figure 4. To d e t e r m i n e m o r e a c c u r a t e l y t h e m e a n recirculation l e n g t h , profiles of t h e m e a n s t r e a m w i s e velocity c o m p o n e n t along a wall-parallel line t h r o u g h t h e first grid p o i n t n e x t t o t h e wall have b e e n d r a w n in figure 6. If t h e zero-crossing of t h e s e l o n g i t u d i n a l profiles is t a k e n t o b e t h e p o s i t i o n of t h e m e a n recirculation l e n g t h , it b e c o m e s obvious from figure 6 t h a t a significant r e d u c t i o n can b e achieved: w i t h d i s t u r b a t i o n a m p l i t u d e A = 1 0 ~ a r e d u c t i o n t o 84 %, w i t h A = 1 0 ~ a r e d u c t i o n t o 66 %, a n d w i t h A = 1 0 a r e d u c t i o n t o a b o u t 57 % of t h e value of t h e reference case w i t h o u t e x c i t a t i o n . C o m p a r i s o n w i t h e x p e r i m e n t : T h e e x p e r i m e n t a l investigation in [14] shows a 30 % r e d u c t i o n of t h e m e a n recirculation l e n g t h a p p l y i n g a similar t i m e - p e r i o d i c forcing in t h e frequency r a n g e b e t w e e n 40 Hz a n d 60 Hz. T h i s c o r r e s p o n d s t o t h e numerically s i m u l a t e d case w i t h f = 5 0 Hz a n d A — 1 0 ~ . T h e m e a s u r e d m e a n recirculation l e n g t h of t h e u n d i s t u r b e d reference case, Χητ /h = 6.4, differs from t h e c o r r e s p o n d i n g value, x /h — 7A, in t h e n u m e r i c a l s i m u l a t i o n . Small differences b e t w e e n e x p e r i m e n t a n d n u m e r i c a l s i m u l a t i o n m a y explain t h i s d e v i a t i o n of a b o u t 16 %, e.g. t h e r e is a s m a l l second s t e p a t t h e edge of t h e s t e p m o d e l used in t h e e x p e r i m e n t , a n d t h e r e is a t u r b u l e n c e intensity of a b o u t 0.35 % in t h e i n c o m i n g flow of t h e e x p e r i m e n t . A m o r e d e t a i l e d c o m p a r i s o n between e x p e r i m e n t a n d n u m e r i c a l s i m u l a t i o n is one of t h e objectives of t h e c u r r e n t investigation. 3
2
1
3
2
- 1
2
r
224
C o n c l u s i o n s : T h e first series of direct n u m e r i c a l s i m u l a t i o n s of m a n i p u l a t e d t r a n s i t i o n a l backward-facing s t e p flow showed t h a t a significant r e d u c t i o n of t h e m e a n recircu lation zone can b e o b t a i n e d by a l o w - a m p l i t u d e t i m e - p e r i o d i c b l o w i n g / s u c t i o n e x c i t a t i o n t h r o u g h a n a r r o w crosswind slot a t t h e edge of t h e s t e p . T h e flow fields are very c o m p l e x showing t h e large-scale roll-up of t h e free shear layer, t h e t r a n s i t i o n t o t h r e e - d i m e n s i o n a l s t r u c t u r e s , t h e s t r o n g effects of t h e r e a t t a c h m e n t on t h e t r a n s i t i o n , a n d t h e very slow recovering after t h e r e a t t a c h m e n t zone t o a fully developed t u r b u l e n t b o u n d a r y layer. F r o m a first e v a l u a t i o n of t h e d a t a sets it can b e concluded t h a t t h e m a j o r effect of t h e forcing is a n acceleration of t h e t r a n s i t i o n process. In t h e c u r r e n t a n d future work t h e e v a l u a t i o n of t h e flow fields will b e c o n t i n u e d t o e x t r a c t m o r e i n f o r m a t i o n a b o u t t h e s t a tistical p r o p e r t i e s a n d t h e d y n a m i c s of t h e d o m i n a n t flow s t r u c t u r e s . T h i s will b e d o n e in s t r o n g c o o p e r a t i o n w i t h t h e c u r r e n t e x p e r i m e n t a l work t o utilize t h e a d v a n t a g e s of b o t h m e t h o d s of investigation.
REFERENCES 1.
Le, H., Moin, P. a n d K i m , J. (1993): Direct n u m e r i c a l s i m u l a t i o n of t u r b u l e n t flow over a backward-facing s t e p , 9 t h S y m p . T u r b u l e n t S h e a r Flows, A u g u s t 16-18, 1993, Kyoto, J a p a n 2. Akselvoll, K. a n d Moin, P. (1993): Large e d d y s i m u l a t i o n of a b a c k w a r d facing s t e p flow, E n g i n e e r i n g T u r b u l e n c e Modelling a n d E x p e r i m e n t s 2, W . R o d i a n d F . Martelli ( E d i t o r s ) , Elsevier Science P u b l i s h e r s 3. K o b a y a s h i , T . a n d Togashi, S. (1993): C o m p a r i s o n of t u r b u l e n c e m o d e l s applied t o backward-facing s t e p flow by L E S d a t a base, E n g i n e e r i n g T u r b u l e n c e Modelling a n d E x p e r i m e n t s 2, W . R o d i a n d F . Martelli ( E d i t o r s ) , Elsevier Science P u b l i s h e r s 4. N e t o , A.S., G r a n d , D., M e t a i s , O. a n d Lesieur, M. (1991): L a r g e - e d d y s i m u l a t i o n of t h e t u r b u l e n t flow in t h e d o w n s t r e a m region of a backward-facing s t e p , Physical Review L e t t e r s 66, p p . 2320-2323 5. A r n a l , M. a n d Friedrich, R. (1992): Large-eddy s i m u l a t i o n of a t u r b u l e n t flow w i t h s e p a r a t i o n , T u r b u l e n t S h e a r Flows 8, Springer-Verlag 6. K a i k t i s , L., K a r n i a d a k i s , G . E . a n d Orszag, S.A. (1991): O n s e t of three-dimensionality, equilibria, a n d early t r a n s i t i o n in flow over a backward-facing s t e p , J. F l u i d Mech. 231, p p . 501-528 7. Bushnell, D . M. a n d McGinley, C. (1989): T u r b u l e n c e control in wall flows, A n n . Rev. F l u i d Mech. 2 1 , p p . 1-20 8. Fiedler, H. E. a n d Fernholz, H. H. (1990): O n m a n a g e m e n t a n d control of t u r b u l e n t s h e a r flow, P r o g . Aerospace Sci. 27, p p . 305-387 9. Kiya, M., Shimizu, M., Mochizuki, O. a n d Ido, Y. (1993): T h e forced t u r b u l e n t s e p a r a t i o n b u b b l e , 9 t h S y m p . T u r b u l e n t S h e a r Flows, A u g u s t 16-18, K y o t o , J a p a n 10. H a s a n , Μ. A. Z. (1992): T h e flow over a backward-facing s t e p u n d e r controlled dist u r b a t i o n : l a m i n a r s e p a r a t i o n , J. F l u i d Mech. 238, p p . 73-96 11. H a s a n , Μ. A. Z. a n d K h a n , A. S. (1992): O n t h e s t a b i l i t y characteristics of a r e a t t a c h i n g shear layer w i t h n o n l a m i n a r s e p a r a t i o n , Int. J. H e a t a n d F l u i d Flow 1 3 / 3 , p p . 224-231 12. Choi, H., Moin, P. a n d K i m , J. (1994): Active t u r b u l e n c e control for d r a g r e d u c t i o n
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in w a l l - b o u n d e d flows, J. F l u i d Mech. 262, p p . 75-110 13. Bàrwolff, G., Ketelsen, K. a n d Thiele, F . (1995): P a r a l l e l i z a t i o n of a finite-volume Navier-Stokes solver on a T 3 D massively parallel s y s t e m , P r o c e e d i n g s of 6 t h Inter n a t i o n a l S y m p o s i u m on C o m p u t a t i o n a l F l u i d D y n a m i c s , S e p t e m b e r 5-9, 1995, Lake Tahoe, Nevada, USA 14. H u p p e r t z , A. (1994): Beeinflussung der S t r ô m u n g h i n t e r einer r u c k w a r t s g e w a n d t e n Stufe d u r c h d r e i d i m e n s i o n a l e A n r e g u n g , D i p l o m a - T h e s i s (in g e r m a n ) , H e r m a n n F o t t i n g e r - I n s t i t u t fur T h e r m o - u n d F l u i d d y n a m i k , T U Berlin
F i g u r e 1. Isolines of i n s t a n t a n e o u s velocity field : above: U ( X , Z ) ; m i n = - 0 . 5 , m a x = + 0 . 5 , i n c r e m e n t s . 1, in m i d p l a n e below: U ( X , Y ) , V ( X , Y ) a n d W ( X , Y ) ; m i n — 0 . 1 , m a x = + 0 . 1 , i n c r e m e n t = 0.02, close t o t h e b o t t o m wall (U a n d V a t Z = 0 . 0 0 8 , W a t Z = 0 . 0 2 3 ) full lines: positive values, d a s h e d lines: n e g a t i v e values
F i g u r e 2. Isolines of i n s t a n t a n e o u s vorticity field : ω (Χ,Ζ), ω (Χ,Ζ) a n d ω (Χ, Ζ)] m i n = - 5 . 0 , m a x = + 5 . 0 , i n c r e m e n t a l . 0 , in m i d p l a n e , d a s h e d lines: negative values, full lines: positive values χ
ζ
υ
MPPPPPPPIII 10
MMPPPIIIII 10
F i g u r e 3. Vertical profiles of m e a n s t r e a m w i s e velocity < U >: above: A = 0 . 0 below: A = 0 . 0 1
F i g u r e 4. Isolines of m e a n s t r e a m w i s e a n d m e a n vertical velocity c o m p o n e n t : above: reference case w i t h o u t m a n i p u l a t i o n ( A = 0 . 0 ) below: m a n i p u l a t e d case w i t h A = 0 . 0 1 left: m e a n s t r e a m w i s e velocity < U(X,Z) >; min=-0.2, m a x = + 1 . 0 right: m e a n vertical velocity < W(X,Z) >;min=-0.02,max=+0.02,iner.=0.004 full lines: positive values, d a s h e d lines: negative values
F i g u r e 5. M e a n s t r e a m w i s e velocity profile in wall c o o r d i n a t e s a t X = x / h = 1 5 . 0 ( A = 0 . 0 )
228
10
I I I I I I I I
CM I
Ο
A=0.0 A=0.001 A=0.01 A=0.1
V IM ' I /I -10
10
X F i g u r e 6. E v a l u a t i o n of m e a n recirculation l e n g t h x /h: from m e a n s t r e a m w i s e velocity d i s t r i b u t i o n along t h e b o t t o m wall a t height Z = 1.1 (first grid p o i n t n e x t t o wall) for t h e reference case ( w i t h o u t m a n i p u l a t i o n , A = 0 . 0 ) a n d t h r e e cases w i t h different m a n i p u l a t i o n a m p l i t u d e s : A = 0 . 0 0 1 , 0.01, a n d 0.1 r
+
15