A Numerical Study of Flow Past a Rotationally Oscillating ...staff.ustc.edu.cn/~xlu/download/JFS_1996b.pdf · Journal of Fluids and Structures (1996) 10 , 829 – 849 A NUMERICAL

Journal of Fluids and Structures (1996) 10 , 829 – 849

A NUMERICAL STUDY OF FLOW PAST A ROTATIONALLY OSCILLATING CIRCULAR

CYLINDER

X . -Y . L U

Department of Modern Mechanics , Uni y ersity of Science and Technology of China Hefei , Anhui 2 3 0 0 2 6 , P .R . China

AND

J . S A T O

Department of Aeronautics and Astronautics , Faculty of Engineering Uni y ersity of Tokyo , Tokyo 1 1 3 , Japan

(Received 15 November 1995 and in revised form 5 June 1996)

Vortex shedding from a rotationally oscillating circular cylinder in a uniform flow is studied by numerical solutions of the two-dimensional incompressible Navier-Stokes equations using primitive variables . To demonstrate the viability and accuracy of the method , we calculate the approach flow past a rotating cylinder with constant angular velocity . For approach flow past a rotationally oscillating cylinder , the object of the study is to examine the ef fect of oscillating rotation on the flow structure . In the present study , some basic patterns of vortex shedding can be identified according to our calculated results and are in good agreement with available experiments . In addition , the influence of the oscillating frequency and amplitude on the forces acting on the cylinder is also investigated .

÷ 1996 Academic Press Limited

1 . INTRODUCTION

T H E U N S T E A D Y F L O W P A S T A B L U F F B O D Y has received a great deal of attention , owing mainly to its theoretical and practical significance . For a rotationally oscillating circular cylinder in a uniform flow , the three parameters governing the development of the flow are the Reynolds number , defined by : Re 5 2 UR / … , where U is the uniform freestream velocity , … is the kinematic viscosity and R is the radius of the circular cylinder ; the rotating amplitude , defined by the ratio of maximum circumferential velocity of the cylinder surface and U , a 5 v R / U , where v is the maximum angular velocity about cylinder axis ; and the oscillating frequency f e , or f e / f o , where f o is vortex-shedding frequency for flow past the cylinder without rotational oscillation .

To describe the problem , earlier studies of an oscillating flow past a circular cylinder have used the vorticity / stream-function approach , e . g ., Zhang et al . (1993) , Wang & Dalton (1991) , Justesen (1991) , and Lecointe & Piquet (1989) ; and the velocity / pressure description , e . g ., Lu & Dalton (1996) , Hall & Grif fin (1993) , Tamura et al . (1988) , Chilukuri (1987) and Hurlbut et al . (1982) . But , only a comparatively smaller number of researchers have investigated numerically the ef fects of rotational oscillation of the cylinder . Wu et al . (1989) have studied the wake features of a cylinder oscillating rotationally about its axis for Re 5 300 to 500 at oscillating frequencies at or near the Karman vortex frequency . Additionally , a number of researchers , e . g ., Chen

0889 – 9746 / 96 / 080829 1 21 $25 . 00 ÷ 1996 Academic Press Limited

X . -Y . LU AND J . SATO 830

et al . (1993) , Badr & Dennis (1985) and Badr et al . (1989) have studied computationally single-direction rotary motions of a circular cylinder in a uniform flow . A fairly full account of an unsteady flow past a rotating cylinder is given by Badr et al . (1990) .

Experimental studies of an approach flow past a rotationally oscillating circular cylinder were conducted by several researchers . Taneda (1978) studied the ef fects of rotational oscillation at Re 5 30 to 300 , and indicated that , at very high oscillating frequencies and magnitudes , the vortex-shedding process could be nearly eliminated . Tokumaru et al . (1991) have shown that rotational oscillation at very large magnitudes can produce significant reduction in drag on the cylinder . Also , Tokumaru et al . (1993) have investigated the mean lift variation of a circular cylinder executing both steady rotation and rotary oscillation with a net rotation rate in a uniform flow . Filler et al . (1991) have studied experimentally the frequency response of the shear layers separating from a circular cylinder subjected to small-amplitude rotational oscillations for Re 5 250 to 1200 .

In the present study , we have investigated the vortex formation and shedding in a wider regime of parameters than those previously considered for the same subject . Our oscillating rotation magnitude a ranges from 0 ? 1 to 3 ? 0 , and oscillating frequency f e / f o

takes the values 0 ? 5 , 1 ? 0 , 2 ? 0 , 3 ? 0 and 4 ? 0 at Re 5 200 , 1000 and 3000 . The primitive-variable form of the Navier-Stokes equations for incompressible flows is solved numerically by a fractional-step method (Kim & Moin 1985) . The goal of the study is to investigate the flow structure at dif ferent forcing frequencies and amplitudes . Some patterns of vortex shedding can be identified from our calculated results and are in good agreement with available experiments . In addition , the influence of the oscillating frequency and amplitude on the forces acting on the cylinder is also studied .

2 . GOVERNING EQUATIONS

A nonrotating reference frame fixed to the axis of the circular cylinder is used . In this frame , the fluid at infinity is a uniform stream with the velocity U in the x -direction , and the cylinder has an oscillatory rotation . It is supposed that the uniform translation and oscillatilng rotation of the cylinder start impulsively at the same instant , and that the flow is two-dimensional . The incompressible Navier-Stokes equations with primitive variables are used for the calculation . In dimensionless form and in polar coordinates ( r , θ ) , these equations are

Û u Û r

1 u r

1 1 r

Û y

Û θ 5 0 , (1)

Û u Û t

1 u Û u Û r

1 y Û u

r Û θ 2

y 2

r 5 2

Û p Û r

1 2

Re S = 2 u 2

2 r 2

Û y

Û θ 2

u r 2 D , (2)

Û y

Û t 1 u

Û y

Û r 1 y

Û y

r Û θ 1

u y

r 5 2

Û p r Û θ

1 2

Re S = 2 y 1

2 r 2

Û u Û θ

2 y

r 2 D , (3)

where the cylinder radius R and the approach velocity U are used as the length and velocity scales , respectively ; u and y are the dimensionless radial and circumferential velocity components , respectively , p is the dimensionless pressure , and the Laplace operator = 2 is

= 2 5 Û

2

Û r 2 1 1 r

Û

Û r 1

1 r 2

Û 2

Û θ 2 . (4)

FLOW PAST ROTATIONALLY OSCILLATING CYLINDER 831

In the present calculation , the initial velocity is taken as the 2-D potential flow over the stationary cylinder . The rotation of the cylinder enters in the boundary conditions , of which the dimensionless form is

u 5 0 , y 5 y B 5 a sin(2 π f e t ) (5)

on the body surface ( r 5 1) , and

u 5 S 1 2 1 r 2 D cos θ , y 5 2 S 1 1

1 r 2 D sin θ , for u o , 0 , (6)

Û u Û r

5 2 r 3 cos θ ,

Û y

Û r 5

2 r 3 sin θ , for u o $ 0 , (7)

at the outer boundary , where u o is the radial velocity component that is normal to the outer boundary . In the circumferential direction , we use periodic boundary conditions .

3 . NUMERICAL METHOD

We consider the Navier-Stokes equations written in the vector form ,

Û V Û t

5 2 = p 1 L ( V ) 1 N ( V ) , (8)

where V is the velocity vector ; L ( V ) and N ( V ) represent viscous and convective terms , respectively :

L ( V ) 5 2

Re = 2 V , (9)

N ( V ) 5 2 ( V ? = ) V . (10)

Using the fractional-step method (Kim & Moin 1985) , the semi-discrete form of equation (8) can be obtained by splitting it into two substeps as

V ̂ 2 V n

D t 5 N D ( V ) 1 L D ( V ) , (11)

V n 1 1 2 V ̂ D t

5 2 = p n 1 1/2 , (12)

where V ̂ is an intermediate velocity , L D ( V ) and N D ( V ) respectively denote discretized forms of L ( V ) and N ( V ) . In the splitting method , it is required that the velocity field V n 1 1 satisfy the incompressibility constraint ,

= ? V n 1 1 5 0 . (13)

Incorporating the incompressibility constraint into equation (12) , we finally arrive at a separately solvable elliptic equation for the pressure in the form

= 2 p n 1 1/2 5 1 D t

= ? V ̂ . (14)

To solve the elliptic equation (14) , a pressure boundary condition must be implemented , which is a key factor for solving the incompressible Navier-Stokes


equations . According to Gresho & Sani (1987) , the pressure condition on the body surface is given by

Û p Û r

5 2 2

Re ( = 3 = 3 V ) ? n r 1

y 2 B

r , (15)

where n r indicates the unit vector at radial direction , and y B is given in equation (5) . In this study , a staggered grid , which is uniformly spaced in the circumferential

direction and exponentially stretched in the radial direction , is employed for the discretization of the governing equations . The transformation is given as

r 5 e π j , θ 5 π h , (16)

where 0 # j , j ̀ , 0 # h # 2 . The convective terms in equation (11) are approximated using a third-order biased upwind dif ference scheme (Kawamura & Kuwahara 1984) with all other spatial derivatives being discretized using a second-order central dif ference scheme . The time derivative in equation (11) is solved using a second-order Adams-Bashforth scheme .

The computational loop to advance the solution from one time level to the next consists of the following three substeps . First , the discretized form of equation (11) is advanced explicitly to obtain the new intermediate velocity , using a second-order Adams-Bashforth scheme . Then , using the known intermediate velocity , the pressure Poisson equation , equation (14) , is solved for the new pressure field . Finally , equation (12) is solved for the new velocity .

4 . RESULTS AND DISCUSSION

To illustrate the computational procedure , we mainly discuss the results of Re 5 1000 for f e / f 0 5 0 ? 5 , 1 ? 0 , 2 ? 0 , 3 ? 0 and 4 ? 0 , and a 5 0 ? 1 to 3 ? 0 . The number of mesh points for the calculations was 128 3 128 in the r and θ directions . The computational domain is 30 R in the radial direction , and the time step is 0 ? 0005 .

It has been determined that the computed results are independent of the time steps and the grid sizes . We compared two grid systems : ( r , θ ) 5 128 3 128 with D t 5 0 ? 0005 and ( r , θ ) 5 256 3 256 with D t 5 0 ? 00025 for the calculation of flow past the cylinder without rotational oscillation at Re 5 1000 . They produced virtually identical results as shown in Figure 1 ; the only dif ference being a phase lag for the fine grid . The phase lag

3

2

1

–1

–2

0CL

, CD

0 10 20 30 40 50 60 70 80

t

128X128, ∆τ = 0·005

256X256, ∆t = 0·00025

Figure 1 . Lift and drag coef ficients , and validation for grid resolution and time step at Re 5 1000 .


1·0

0·5

0

–0·5

–1·01 2 3 4

t = 1·0 (calc.)t = 2·0t = 3·0t = 4·0t = 5·0t = 2·0 (exp.)t = 3·0t = 4·0t = 5·0

u

r/R

(a) t = 1·0 (calc.)t = 2·0t = 3·0t = 4·0t = 5·0t = 1·0 (exp.)t = 2·0t = 4·0

3·0

2·5

2·0

1·5

1·0–0·5 0 0·5 1·0 1·5 2·0

(b)

r/R

–v Figure 2 . Velocity distribution at θ 5 0 8 and θ 5 90 8 obtained from the present computation (lines) and

experiment (symbols , Badr et al . 1990) . (a) θ 5 0 8 ; (b) θ 5 90 8 .

is attributed to the onset of the wake instability leading to vortex shedding . A similar convergence check has been done for other Reynolds numbers , Re 5 200 and 3000 (not shown here) . From Figure 1 , the Strouhal number , as determined from the lift coef ficient plot , is 0 ? 22 , which compares quite well with the experimental value of approximately 0 ? 21 .

To validate the code , some quantitative comparisons between the computational and experimental results were carried out for a rotating cylinder in the counter-clockwise direction with constant angular velocity at Re 5 1000 . Figure 2 shows a comparison of the numerically obtained velocity distribution at θ 5 0 8 and θ 5 90 8 with those obtained experimentally (Badr et al . 1990) . In Figure 3 , the trajectories of the first vortex center for the three cases a 5 0 ? 5 , 1 ? 0 and 2 ? 0 are plotted , where some experimental points (Badr et al . 1990) are also shown . All the test cases are in good agreement with experiment . Figure 4 shows our calculated streamline patterns at a 5 0 ? 5 , Re 5 1000 , which agree very well with the computed results of Badr et al . (1990) .

a = 0·5 (calc.)a = 1·0a = 2·0a = 0·5 (exp.)a = 1·0a = 2·0

2

1

0

y/R

0 1 2 3 4x/R

Figure 3 . The calculated trajectories (lines) of the first vortex center for a 5 0 ? 5 , 1 ? 0 and 2 ? 0 at Re 5 1000 . Symbols are experimental results from Badr et al . (1990) .


In what follows , we first observe typical vortex patterns for selected values of the oscillating frequency ( f e / f o ) and amplitude ( a ) at Re 5 1000 . Then , we investigate the influence of the flow structure on the forces acting on the cylinder .

4 . 1 . V O R T E X P A T T E R N S A N D F O R C E C O E F F I C I E N T S A T Re 5 1000

4 . 1 . 1 . f e / f o 5 0 ? 5 ; a 5 0 . 5 , 1 ? 0 , 2 ? 0 , 3 ? 0

Figure 5 shows the vorticity contours during a half oscillating cycle for a 5 0 ? 5 . At this subharmonic oscillating frequency , f e / f o 5 0 ? 5 , two opposite vortices are shed from each side of the cylinder over the half-cycle (i . e ., T e / 2 , where T e is the rotational oscillating period) . In this case , a synchronized vortex-shedding mode occurs , and the period of the vortex shedding is T e / 2 , or T o , where T o is the vortex-shedding period for flow past the cylinder without rotational oscillation . As the oscillating amplitude increases , e . g ., a 5 1 ? 0 , and 2 ? 0 , nonsynchronized vortex patterns (not shown here) are formed in the near wake . But , if the amplitude increases further , the vortex-shedding changes to a synchronized mode again . Figure 6 shows the vortex shedding at a 5 3 ? 0 in one oscillating cycle , which locks on to the cylinder oscillation . In the near wake , three vortices of the same sign per half-cycle are shed from the same side of the cylinder . The next two vortices may be attributed to an additional separation caused by the most recently generated vortex .

The force coef ficients are shown in Figure 7 for a 5 0 ? 5 and 3 ? 0 . The value of the time-averaged drag coef ficient and the r . m . s . value of the lift coef ficient for a 5 3 ? 0 are larger than that for a 5 0 ? 5 , because the transverse scale of the wake at a 5 3 ? 0 is wider than at a 5 0 ? 5 . We can see how the lift coef ficients vary with the vortex- shedding frequency , and drag coef ficients vary with twice the vortex-shedding frequency . As shown in Figures 6 and 7 , the vortex-shedding frequency is 2 f e (i . e ., f o ) at a 5 0 ? 5 , and f e at a 5 3 ? 0 , respectively .

4 . 1 . 2 . f e / f o 5 1 ? 0 ; a 5 1 ? 0 , 2 ? 0 , 3 ? 0

At this oscillating frequency , all the cases we have calculated from a 5 0 ? 1 to 3 ? 0 are the synchronized vortex modes locked to the cylinder oscillations . Here , we only show the vortex patterns for a 5 1 ? 0 , 2 ? 0 and 3 ? 0 in Figures 8 – 10 . At a 5 1 ? 0 , the vortex formation is similar to the classical Karman vortex street ; two opposite-sign vortices are shed from the two sides of the cylinder in one oscillating cycle . At a 5 2 ? 0 and 3 ? 0 , however , the flow patterns shown in Figures 9 and 10 , indicate that two vortices of the same sign are shed in a half-cycle from the same side of the cylinder . The first vortex is induced by the opposite-sign vortex , i . e ., the most recently generated one from the other side of the cylinder . We also calculated the vortex patterns at frequencies of f e / f o 5 0 ? 9 and 1 ? 1 . The vortex modes are similar to those for f e / f o 5 1 ? 0 at the same oscillating amplitudes .

Figure 11 shows the variation of the force coef ficients at a 5 1 ? 0 and 3 ? 0 . The lift coef ficients vary at the vortex-shedding frequency , and the drag coef ficients oscillate at twice the shedding frequency . Figure 11 also indicates that the value of time-averaged drag and the r . m . s . value of the lift coef ficients for a 5 3 ? 0 are larger than for a 5 1 ? 0 .

4 . 1 . 3 . f e / f o 5 2 ? 0 ; a 5 2 ? 0 , 0 ? 5 , 1 ? 0 , 2 ? 0

Figure 12 shows the vortex patterns for a 5 0 ? 2 over two oscillating cycles (i . e ., 2 T e ) . Two vortices with opposite signs are shed from either side of the cylinder in two

FLOW PAST ROTATIONALLY OSCILLATING CYLINDER 835(a

)(b

)

(d)

(c)

Fig

ure

4 . C

ompu

ted

stre

amlin

e pa

tter

ns o

f th

e pr

esen

t st

udy

and

Bad

r et

al .

(199

0) a

t a

5 0 ?

5 an

d R

e 5

1000

. (a

, b)

pres

ent

com

puta

tion

; (c

, d)

resu

lts

of C

adr

et a

l . (1

990)

. (a ,

c) t

5 4 ?

0 ; (

b , d)

t 5

6 ? 0 .


(a)

(b)

(c)

(d)

Figure 5 . Vortex patterns at f e / f o 5 0 ? 5 and a 5 0 ? 5 at times : (a) 5 T e / 8 ; (b) 3 T e / 4 ; (c) 7 T e / 8 ; (d) T e .

oscillating cycles . As expected , the vortex-shedding frequency is f e / 2 (i . e ., f o ) at the smaller amplitude . At a higher oscillating amplitude , e . g ., a 5 2 ? 0 as shown in Figure 13 , vortex shedding is also synchronized with the cylinder oscillation , and the vortex-shedding frequency is f e . In the near wake , there is an alternate , out-of-phase shedding of vortices from either side of the cylinder over an oscillating cycle . The wake takes on the appearance of two parallel vortex arrays . We also calculated a 5 0 ? 5 and 1 ? 0 at this superharmonic oscillating frequency . The vortex shedding (not shown here) is that of nonsynchronized vortex modes .

Figure 14 shows the force coef ficients at a 5 0 ? 2 and 2 ? 0 . The lift coef ficient varies at twice the oscillating frequency for a 5 0 ? 2 , and at the same oscillating frequency for a 5 2 ? 0 . At this oscillating frequency , the values of the drag (time-averaged) and lift (r . m . s . ) coef ficients for a 5 2 ? 0 are smaller than that for a 5 0 ? 2 .

4 . 1 . 4 . f e / f o 5 3 ? 0 , 4 ? 0 ; a 5 2 ? 0 , 3 ? 0

Figures 15 and 16 show the vortex patterns at f e / f o 5 3 ? 0 and a 5 2 ? 0 , 3 ? 0 . At f e / f o 5 3 ? 0 and a 5 2 ? 0 , the vortex formation in the near wake is also synchronized with the cylinder oscillation , then coalesces into a structure with a lower spatial frequency some distance downstream of the cylinder . At a 5 3 ? 0 , the near-wake vortex structures lock


(b)

(c)

(d)

(a)

Figure 6 . Vortex patterns at f e / f o 5 0 ? 5 and a 5 3 ? 0 at times : (a) T e / 2 ; (b) 3 T e / 4 ; (c) 7 T e / 8 ; (d) T e .

on to the cylinder oscillation , then the vortices of the same sign coalescence , and disappear due to viscous dissipation .

Figure 17 shows the evolution of the vortex structure at f e / f o 5 4 ? 0 and a 5 2 ? 0 . Due to the higher oscillatory frequency , some small vortices are shed near the cylinder , and immediately coalesce into larger ones some distance downstream of the cylinder . In this case , vortex shedding is also synchronized with the cylinder oscillation , but the period of the downstream large vortices is 4 T e (i . e ., f o ) . This is also confirmed by the variation of the force coef ficients . These coef ficients are shown in Figure 18 for f e / f o 5 3 ? 0 , 4 ? 0 and a 5 2 ? 0 . The lift force coef ficients vary at the same oscillating frequency for f e / f o 5 3 ? 0 , and at four times the oscillating frequency for f e / f o 5 4 ? 0 .

4 . 2 . V O R T E X P A T T E R N S A T Re 5 200 A N D 3000

To discuss the ef fect of the Reynolds number on the vortex patterns , we show only the vortex patterns for Re 5 200 and 3000 at f e / f o 5 1 ? 0 and a 5 2 ? 0 in Figures 19 and 20 . Based on our computed lift coef ficient (not shown here) of flow past the cylinder without rotational oscillation , the Strouhal number is 0 ? 2 (i . e ., f o 5 0 ? 1) which


2(a)

1

0

–1

–2

CL

, C

D

8(b)

4

0

–4

–8

CL

, C

D

0 10 20 30 40 50 60 70 80 90 100t

Figure 7 . Lift and drag coef ficients at f e / f o 5 0 ? 5 : – – – , C L ; ——— , C D . (a) a 5 0 ? 5 ; (b) a 5 3 ? 0 .

(a)

(b)

(c)

(c)

Figure 8 . Vortex patterns at f e / f o 5 1 ? 0 and a 5 1 ? 0 at times : (a) T e / 4 ; (b) T e / 2 ; (c) 3 T e / 4 ; (d) T e .


(a)

(b)

(c)

(d)

Figure 9 . Vortex pattens at f e / f o 5 1 ? 0 and a 5 2 ? 0 at times : (a) T e / 4 ; ( b ) T e / 2 ; (c) 3 T e / 4 ; (d) T e .

compared very well with the experimental value of approximately 0 ? 195 at Re 5 200 , and the Strouhal number is 0 ? 23 (i . e ., f o 5 0 ? 115) at Re 5 3000 which also is in good agreement with the experimental value of approximately 0 ? 22 . From the flow patterns shown in Figures 19 and 20 , the large-scale vortex structures are similar to the vortex patterns at Re 5 1000 (see Figure 9) . For other values of f e / f o and a , we also find the large vortex structures in the near wake are nearly the same for Re 5 200 , 1000 and 3000 .

4 . 3 . F O R C E C H A R A C T E R

Some of the behavior of the lift and drag coef ficients has been discussed in the previous section . In the range of our calculated parameters , the lift coef ficients vary at the same


(a)

(b)

(c)

(d)


vortex-shedding frequencies for f e / f o # 3 ? 0 ; but , for higher oscillating frequencies , the frequency of the lift variation is less than that of the vortex shedding as can be seen for f e / f o 5 4 ? 0 in Figure 18 . The large-scale vortex-shedding frequency in the near wake is f e / 4 , or f o .

Figure 21 shows the time-averaged drag coef ficient versus oscillating frequency at a 5 2 ? 0 and Re 5 1000 . The unforced value means the time-averaged drag coef ficient for approach flow without cylinder rotation . There is a significant decrease in the

2(a)

1

0

–1

–2

CL

, C

D

3(b)

2

1

–2

–3

CL

,CD

0 10 20 30 40 50 60 70 80t

0

–1


(a)

(b)

(c)

(d)

Figure 12 . Vortex patterns at f e / f o 5 2 ? 0 and a 5 0 ? 2 at times : (a) T e / 2 ; (b) T e ; (c) 3 T e / 2 ; (d) 2 T e .

(a)

(b)

(c)

(d)


2(a)

1

0

–1

–2

CL

, C

D

(b)2

1

CL

, C

D

0 10 20 30 40 50 60 70 80t

0

–1



(a)

(b)

(c)

(d)


time-averaged drag coef ficient C # D for f e / f o . 1 ; but this C # D is much higher than the unforced value for f e / f o 5 0 ? 5 . This tendency is qualitatively similar to the experimental results (Tokumaru & Dimotakis 1991) .

4 . 4 . T R A N S I T I O N A N D S E L E C T I O N O F T H E V O R T E X M O D E S

From the flow patterns shown in the previous sections , we see that the transition among dif ferent vortex modes occurs at dif ferent oscillating frequencies and amplitudes . As shown in Figures 8 – 10 for f e / f o 5 1 ? 0 and a 5 1 ? 0 , 2 ? 0 and 3 ? 0 , the transition to dif ferent vortex patterns happens gradually as the amplitude increases . Also , as the oscillating frequency , f e / f o , increases from 1 ? 0 to 4 ? 0 at a 5 2 ? 0 , several dif ferent vortex patterns occur .

In this problem , the cylinder takes both motions at the same instant , i . e ., oscillating rotation and translation . If the oscillating rotation dominates the flow , preferred vortex modes may be synchronized with the cylinder oscillation , and the vortex-shedding frequency mainly depends on the oscillating frequency . Otherwise , for the limiting condition of a 5 0 , this problem becomes a steady approach flow past a nonrotating cylinder . So , if the approach flow dominates the flow field , a preferred vortex mode may be the formation of a wake flow similar to the Karman vortex street . In the


(a)

(b)

(c)

(d)

Figure 16 . Vortex patterns at f e / f o 5 3 ? 0 and a 5 3 ? 0 at times : ( a ) T e / 4 ; (b) T e / 2 ; (c) 3 T e / 4 ; (d) T e .

nonsynchronization region , there is competition between the rotating and translating perturbations . As shown in previous sections for the vortex patterns at small rotating amplitudes , such as f e / f o 5 0 ? 5 , a 5 0 ? 5 (see Figure 5) , and f e / f o 5 2 ? 0 , a 5 0 ? 2 (see Figure 12) , the vortex patterns in the near wake are similar to the Karman vortex , and the vortex-shedding frequency is f o . As the amplitude increases , the oscillating rotation may dominate the flow , where the vortex shedding is locked on to the cylinder oscillation , and the vortex-shedding frequency becomes the forced oscillating frequency , f e .

Based on our calculations , we can sketch the complete form of the vortex mode-selection diagram in the frequency-amplitude plane , as shown in Figure 22 (Karniadakis & Triantafyllou 1989) . Similar mode-selection diagrams have been found in the wake of a cylinder , e . g ., Karniadakis & Triantafyllou (1989) , Stansby (1976) . The diagram in Figure 22 gives only a qualitative summary of vortex modes in the wake ; an exact quantitative diagram is prohibitively expensive , requiring a much larger amount of computation . Furthermore , for larger amplitudes , the wake formation may change drastically .

(a)

(b)

(c)

(d)

Figure 17 . Vortex patterns at f e / f o 5 4 ? 0 and a 5 2 ? 0 at times : (a) T e ; (b) 2 T e ; (c) 3 T e ; (d) 4 T e .

1·0(a)

0·5

0CL

, C

D

1·0

(b)

0·5

–0·5

–1·0

CL

, C

D

0 10 20 30 40 50 60 70 80t

0

–0·5

Figure 18 . Lift and drag coef ficients : – – – , C L ; ——— , C D . (a) f e / f o 5 3 ? 0 , a 5 0 ? 2 ; (b) f e / f o 5 4 ? 0 , a 5 2 ? 0 .


(a)

(b)

(c)

(d)

(c)

(d)

Figure 19 . Vortex patterns for Re 5 200 at f e / f o 5 1 ? 0 and a 5 2 ? 0 at times : (a) T e / 4 ; (b) T e / 2 ; (c) 3 T e / 4 ; (d) T e .

5 . CONCLUDING REMARKS

The unsteady flow past a rotationally oscillating circular cylinder is numerically investigated at Re 5 200 , 1000 and 3000 in a range of the magnitude of rotational velocity , 0 ? 1 # a # 3 ? 0 , and oscillating frequencies 0 ? 5 # f e / f o # 4 ? 0 . This is a substan- tially wider range of parameters than for previous work on the same subject . According to our computed results , the large-scale vortex structures in the near wake are nearly the same for Re 5 200 , 1000 and 3000 . The ef fect of oscillating frequency and amplitude on the vortex formation is studied , and some basic patterns of vortex shedding are identified . The transition and selection of the vortex modes occur for dif ferent oscillating frequencies and amplitudes . The variation of the forces is clearly related to the evolution of the vortex formation in the near wake .


(a)

(b)

(c)

(d)

Figure 20 . Vortex patterns for Re 5 3000 at f e / f o 5 1 ? 0 and a 5 2 ? 0 at times : (a) T e / 4 ; (b) T e / 2 ; (c) 3 T e / 4 ; (d) T e .

3

2

1

0

Tim

e-a

vera

ge

d d

rag

co

effi

cie

nt

0 1 2 3 4 5fe /fo

Figure 21 . Time-averaged drag force coef ficient versus f e / f o for a 5 2 ? 0 and Re 5 1000 .


fe /fo

a

Lock-in boundary

Quasi-periodicity

Receptivityboundary

Lock-in

1·0

Figure 22 . Vortex mode-selection diagram .

ACKNOWLEDGEMENTS

The authors are sincerely grateful to Professor C . Dalton of the University of Houston , Professor L . X . Zhuang of the University of Science and Technology of China and Professor K . Rinoie of the University of Tokyo for their helpful comments and discussion . The first author also acknowledges the support of the Japan Society for the Promotion of Science (JSPS) during his stay at the University of Tokyo .

R EFERENCES

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A Numerical Study of Flow Past a Rotationally Oscillating ...staff.ustc.edu.cn/~xlu/download/JFS_1996b.pdf · Journal of Fluids and Structures (1996) 10 , 829 – 849 A NUMERICAL