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RESEARCH ARTICLE | NOVEMBER 01 1999
Laminar flow past a rotating circular cylinder
Sangmo Kang; Haecheon Choi; Sangsan Lee
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Physics of Fluids 11, 3312–3321 (1999)
https://doi.org/10.1063/1.870190
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Laminar flow past a rotating circular cylinder
Sangmo Kanga) and Haecheon Choib)
National CRI Center for Turbulence and Flow Control Research, Institute of Advanced Machinery and Design, Seoul National University, Seoul 151-742, Korea
Sangsan Lee
ETRI Supercomputer Center, P.O. Box 1, Yusong, Taejeon 305-600, Korea
~Received 7 October 1998; accepted 20 July 1999!
The present study numerically investigates two-dimensional laminar flow past a circular cylinder rotating with a constant angular velocity, for the purpose of controlling vortex shedding and understanding the underlying flow mechanism. Numerical simulations are performed for flows with Re560, 100, and 160 in the range of 0 { \leqslant } \alpha { \leqslant } 2 . 5 , , where is the circumferential speed at the cylinder surface normalized by the free-stream velocity. Results show that the rotation of a cylinder can suppress vortex shedding effectively. Vortex shedding exists at low rotational speeds and completely disappears at \alpha > \alpha _ { L } , where \alpha _ { L } is the critical rotational speed which shows a logarithmic dependence on Re. The Strouhal number remains nearly constant regardless of while vortex shedding exists. With increasing , the mean lift increases linearly and the mean drag decreases, which differ significantly from those predicted by the potential flow theory. On the other hand, the amplitude of lift fluctuation stays nearly constant with increasing ( < \alpha _ { L } ) , while that of drag fluctuation increases. Further studies from the instantaneous flow fields demonstrate again that the rotation of a cylinder makes a substantial effect on the flow pattern. © 1999 American Institute of Physics. @S1070-6631~99!01211-8#
I. INTRODUCTION
Flow past a circular cylinder has been accepted as a building-block problem for understanding the vortex dynamics in bluff body wakes. At a low Reynolds number ~Re ,47!, the wake behind a ~nonrotating! cylinder comprises a steady recirculation region with two vortices symmetrically attached to the cylinder, whose size grows with increasing Reynolds number. Here, the Reynolds number Re is defined as \mathrm { R e } { = } U _ { \infty } d / \nu , where U _ { \infty } is the free-stream velocity, d the diameter of the cylinder, and the kinematic viscosity. At a Reynolds number of 47<Re,200, vortex shedding occurs in the near wake behind a cylinder due to the flow instability accompanying a large fluctuating pressure and, thus, a periodically oscillating lift force. At a higher Reynolds number ~Re>200!, the flow becomes three-dimensional and turbulent, and vortex shedding also occurs but in more complicated patterns. Such fluctuating forces induced from vortex shedding may cause structural vibrations, acoustic noise or resonance, which in some cases can trigger structure failure or enhance mixing in the wake.1 Therefore, controlling vortex shedding appropriately is critical in practical engineering environments. Many attempts have been made for controlling the wake behind a circular cylinder in recent years, especially for the purpose of suppressing vortex shedding, using passive or ~nonfeedback or feedback! active controls.
The rotation of a cylinder in a viscous uniform flow is expected to modify wake flow pattern and vortex shedding configuration, which may reduce a flow-induced oscillation or augment a lift force. The basic rationale behind the rotation effect is that as a cylinder rotates, the flow is accelerated on one side of the cylinder and decelerated on the other side. Hence, the pressure on the accelerated side becomes smaller than that on the decelerated side, resulting in a mean lift force. Such a phenomenon is referred to as the Magnus effect. As a result, the rotation significantly alters the flow pattern, and probably has an effect on a flow-induced oscillation. Most of such researches performed to date are classified largely into two categories, a rotary oscillation and a unidirectional rotation. Flow past a circular cylinder performing a rotary oscillation has been investigated by many researchers; for example, experimentally by Taneda,2 Tokumaru and Dimotakis,3 and Filler et al.4 and numerically by Wu et al.5 and Baek and Sung.6
There have also been numerous investigations of flow past a circular cylinder rotating with a constant angular velocity, which is the main concern of the present study. The flow past a circular cylinder started impulsively into rotation and translation from rest becomes transient in an early time, and then reaches either a steady or unsteady ~time-periodic! state depending on two parameters, Re and . Here, the rotational speed is defined as \alpha { = } \dot { \theta } d / ( 2 U _ { \infty } ) , where ˙ is the angular velocity of the cylinder. Ingham,7 Badr et a l . , ^ { 8 } and Ingham and Tang9 numerically investigated the rotatingcylinder flow in the laminar steady regime ~Re,47! for relatively small rotational speeds ~ <3!. They found that, although vortex shedding does not occur in the wake, the rotation delays and even inhibits the boundary-layer separation. The flow in the laminar vortex shedding regime ( 4 7 { \leqslant } \mathrm { R e } < 2 0 0 ) has also been investigated in the literature. Badr e t a l . ^ { 8 } numerically investigated unsteady flow at Re 560, 100, and 200, focusing on the flow pattern during an early time period after the impulsive rotation and translation of a cylinder. Subsequently, Tang and Ingham10 examined steady flow at \mathrm { R e } { = } 6 0 and 100 in the range of 0 { \leqslant } \alpha { \leqslant } 1 by solving time-independent governing equations, and presented the changes in the flow variables. To the best of our knowledge, there is no literature that investigates the rotation effect on the laminar flow in the fully developed stage involving vortex shedding. This has mainly motivated the present numerical simulation of flow past a constantly rotating cylinder in the laminar vortex shedding regime.
Flow at a higher Reynolds number ( \mathrm { R e } { \geqslant } 2 0 0 ) is more difficult to numerically simulate because of its intrinsic three-dimensionality and complexity. Tokumaru and Dimotakis11 experimentally investigated flow past a circular cylinder with both net steady rotation and forced oscillations at \mathrm { R e } { \approx } 1 0 ^ { 3 } by suggesting the so-called virtual vortex method. They found that a higher cylinder aspect ratio yields a higher maximum lift coefficient in the case of steady rotation and the addition of forced rotary oscillations to the steady rotation increases or decreases the lift coefficient. Most investigations of unsteady flow at a high Reynolds number are limited to an early transient period because the flow at a later time becomes too complex to analyze. Badr and \mathrm { D e n n i s } ^ { 1 2 } numerically investigated unsteady flow for \mathbf { R e } { \geqslant } 2 0 0 , while Coutanceau and \mathrm { M e n a r d } ^ { 1 3 } experimentally investigated the same flow ~ 50.5 and 1! for comparison. Subsequently, Badr et al.14 carried out a comparative study of both the time evolutions of two-dimensional unsteady flow obtained numerically and experimentally for 1 0 ^ { 3 } { \leqslant } \mathrm { R e } { \leqslant } 1 0 ^ { 4 } and 0 . 5 { \leqslant } \alpha { \leqslant } 3 . They found that vortex shedding completely disappears for \alpha > \alpha _ { L } , where the critical value \alpha _ { L } is almost independent of Re and is about 2. Chen et al.15 investigated the same unsteady flow at { \mathrm { R e } } { = } 2 0 0 for \alpha { \leqslant } 3 . 2 5 , and reported that shedding of more than one vortex occurs even for \alpha { = } 3 . 2 5 , which is in contrast to the previous result of Badr et al.14 In company with these conflicting observations, there have been a few systematic researches about \alpha _ { L } and its dependence on Re when a cylinder rotates at a constant angular velocity. Chang and Chern16 and Chen et al.15 suggested different basic modes of vortex shedding depending on Re and , respectively, by identifying the behavior of stagnation point and by characterizing the asymptotic stability of the flow. Recently, Hu et al.17 systematically studied the Hopf bifurcation describing the transition from a steady to timeperiodic solution through a Galerkin method and a system theory. They obtained the transition curve on a plane of Re and and reported that the rotation may delay the onset of vortex shedding and decrease the vortex-shedding frequency.
The objectives of the present study are to numerically investigate the effect of the rotation on flow past a constantly rotating circular cylinder in the laminar vortex shedding regime ( 4 7 { \leqslant } \mathrm { R e } < 2 0 0 ) and to understand the underlying mechanism of vortex shedding suppression. This is accomplished by presenting the Strouhal number, mean flow quantities at the cylinder surface, lift and drag coefficients, and instantaneous flow fields in the fully developed stage, at various Re and . The two-dimensional unsteady Navier–Stokes equations are solved using a fully-implicit, fractional-step method in time18 and a second-order central difference scheme in space. The numerical simulations are performed for flows at Re540, 60, 100, and 160 in the range of 0 { \leqslant } \alpha { \leqslant } 2 . 5 . Note that, in case of a stationary cylinder, the flow is steady at \mathrm { R e } { = } 4 0 and unsteady ~time-periodic! at Re 560, 100, and 160. All the numerical simulations are continued until the flow reaches a fully developed state, where all the flow characteristics are analyzed. The paper is organized as follows: Sec. II explains details of the numerical method. Subsequently, in Sec. III, we discuss the results from the simulations. Finally, a summary is presented in Sec. IV.
II. NUMERICAL METHOD
The two-dimensional unsteady governing equations for an incompressible flow can be written as
\frac {\partial u _ {i}}{\partial t} + \frac {\partial (u _ {i} u _ {j})}{\partial x _ {j}} = - \frac {\partial p}{\partial x _ {i}} + \frac {1}{\mathrm{Re}} \frac {\partial^ {2} u _ {i}}{\partial x _ {j} \partial x _ {j}}, \tag {1}
\frac {\partial u _ {i}}{\partial x _ {i}} = 0, \tag {2}
where x _ { i } are the Cartesian coordinates and u _ { i } are the corresponding velocity components. All variables are nondimensionalized by the cylinder diameter d and the free-stream velocity U _ { \infty } . Notice that the notation sets ( u , v ) and ( x , y ) are used interchangeably with ( u _ { 1 } , u _ { 2 } ) and ( x _ { 1 } , x _ { 2 } ) , respectively, in this paper.
Equations ~1! and ~2! are transformed to, by introducing generalized coordinates \eta ^ { i } and volume fluxes q ^ { i } across the faces of the cell,
\frac {\partial q ^ {i}}{\partial t} + N ^ {i} (\mathbf {q}) = - G ^ {i} (p) + L _ {1} ^ {i} (\mathbf {q}) + L _ {2} ^ {i} (\mathbf {q}), \tag {3}
D ^ {i} q ^ {i} = \frac {1}{\mathbf {J}} \left(\frac {\partial q ^ {1}}{\partial \eta^ {1}} + \frac {\partial q ^ {2}}{\partial \eta^ {2}}\right) = 0, \tag {4}
where { \mathbf { q } } = ( q ^ { 1 } , q ^ { 2 } ) , Ni is the convection term, G ^ { i } ( p ) is the pressure gradient term, L _ { 1 } ^ { i } and L _ { 2 } ^ { i } are the diffusion terms without and with cross-derivatives, respectively, and D ^ { i } is the divergence operator. More details are described in Choi et a l . ^ { 1 8 } The transformed equations ~3! and ~4! are integrated in time using a fully-implicit, fractional-step method,18 composed of four-step time splitting. All terms in ~3! including cross-derivative diffusion terms are advanced with the Crank–Nicolson method in time and are resolved with a second-order central difference scheme in space. A Newton method is used to solve the discretized nonlinear equation.
The flow geometry and coordinate system along with boundary conditions are shown in Fig. 1. Here, notation sets ( r , \theta ) and ( u _ { r } , u _ { \theta } ) are also introduced to address, respectively, the radial and azimuthal directions and their corresponding velocity components. In this study, we use an O-type mesh, rather than a C-type mesh which may not be appropriate in the case of ~oscillatory! rotation of a cylinder.
FIG. 1. Flow geometry and coordinate system along with boundary conditions.
In the C-type mesh in case of rotation, the nonzero circumferential velocity of a cylinder, unless particularly considered, makes the solution singular at the base point, causing a fatal spurious effect on the flow field. The O-type mesh used in this study is created following Beaudan and \mathrm { M o i n } . ^ { 1 9 } For the computational domain with 0 . 5 { \leqslant } r { < } R _ { D } and 0 ^ { \circ } { \leqslant } \theta < 3 6 0 ^ { \circ } , the grid points r _ { m } ~ ( m { = } 1 , 2 , . . . , M ) in the radial direction are
r _ {m} = 0. 5 + (R _ {D} - 0. 5) \frac {1 - s ^ {(m - 1)}}{1 - s ^ {M - 1}}, \tag {5}
where R _ { D } is the radius of the computational domain, s the stretching factor and M the number of the radial grid points. The grid points \theta _ { n } \ ( n { = } 1 , 2 , . . . , N ) in the azimuthal direction are more clustered in the wake region than in the potential region, where N is the number of the azimuthal grid points. The grid boundary, or envelope, of the wake region ( x _ { b } ( h ) , y _ { b } ( h ) ) is specified analytically as
\left\{ \begin{array}{c} x _ {b} (h) = \left(R _ {D} \cos \theta_ {0} - \cos \theta_ {i} + G\right) h ^ {2} - G h + \cos \theta_ {i}, \\ y _ {b} (h) = \left(R _ {D} \sin \theta_ {0} - \sin \theta_ {i} + G \tan \theta_ {i}\right) \frac {e ^ {b h} - 1 - b h}{e ^ {b} - 1 - b} \\ - G h \tan \theta_ {i} + \sin \theta_ {i}, \end{array} \right. \tag {6}
where h is a continuous parameter and \theta _ { i } and \theta _ { 0 } are the azimuthal positions of the grid boundary at r = 0 . 5 and R _ { D } , respectively. Within the wake region, the azimuthal grid points are equispaced. In this study, the grid points ( r _ { m } , \theta _ { n } ) are generated with s = 1 . 0 1 6 , G = - 5 , b = 1 0 , \theta _ { i } = 8 5 ^ { \circ } , and \theta _ { 0 } = 3 0 ^ { \circ } .
The condition of a constant circumferential velocity, u _ { r } { } = 0 and u _ { \theta } { = } \alpha , is imposed on the cylinder surface. The farfield boundary is divided into inflow and outflow boundaries, that is x { \leqslant } 0 and x { > } 0 , respectively. The former boundary has a Dirichlet boundary condition, u = 1 and \scriptstyle { V = 0 } , while the latter boundary has a convective outflow condition, \partial u _ { i } / \partial t + c \partial u _ { i } / \partial x = 0 , ^ { 2 0 } where c is the space-averaged streamwise ~x-direction! exit velocity.
In the present study, a parameter set of R _ { D } { = } 5 0 , M { \times } N = 2 4 1 \times 2 4 1 and \Delta t = 0 . 0 2 has been chosen to solve the governing equations on the O-type mesh. The computational time step \Delta t = 0 . 0 2 corresponds to a maximum \mathrm { C F L } { = } 1 . 5 { - } 4.5. To confirm the chosen parameter values, parametric studies at Re5100 and 51 have been performed and the typical results are presented in Table I. The studies are accomplished by successively varying one parameter while keeping the others unchanged. The relative errors in the table show that the results obtained with the chosen parameter values are well converged with respect to the domain size, and spatial and temporal resolutions. The appropriateness of the boundary condition u = 1 and \scriptstyle { V = 0 } imposed on the farfield boundary ( x { \leqslant } 0 ) has also been examined because there may be an induced flow due to the cylinder rotation even at large distances away from the cylinder. This is accomplished by replacing the uniform flow on the far-field boundary (x \leqslant 0 ) with the potential flow involving the rotation effect as follows:
\left\{ \begin{array}{l} u = 1 - \frac {\cos 2 \theta}{4 R _ {D} ^ {2}} - \frac {\alpha \sin \theta}{2 R _ {D}}, \\ v = - \frac {\sin 2 \theta}{4 R _ {D} ^ {2}} + \frac {\alpha \cos \theta}{2 R _ {D}}. \end{array} \right. \tag {7}
In the case of \alpha { = } 2 . 5 ~the largest investigated in this paper! at Re5100, the inclusion of the rotation effect in the far-field boundary condition changed the lift coefficient, surface vorticity and surface pressure by less than 0.5%. This proves the applicability of the uniform inflow boundary condition at sufficiently large distances from the rotating cylinder.
To further validate the choice of numerical method and mesh, numerical simulations at the identical conditions used in previous publications have also been carried out and the results are presented in Figs. 2 and 3. Figure 2~a! shows the variation of St with Re at \scriptstyle { \alpha = 0 } . , compared with the results of Park et a l . ^ { 2 1 } and Williamson’s correlations,22 while Fig. 2~b! shows the variation of C _ { L } with at { \mathrm { R e } } { = } 2 0 . , compared with the results of Badr et a l . ^ { 8 } and Ingham and \mathrm { T a n g . } ^ { 9 } Here, the Strouhal number St is defined as { \mathrm { S t } } { = } f d / U _ { \infty } , where f is the vortex-shedding frequency. The lift and drag coefficients, C _ { L } and C _ { D } , are defined respectively as C _ { L } { = } 2 L / ( \rho U _ { \infty } ^ { 2 } d ) and C _ { D } { = } 2 D / ( \rho U _ { \infty } ^ { 2 } d ) , where L and D are the lift and drag forces per unit cylinder length, respectively. It is clearly shown that the present results are in excellent agreement with the previous ones. Figure 3 shows time developments of velocity profiles in an early time after the impulsive rotation and translation of a cylinder at Re5200 and \alpha { = } 0 . 5 , compared with the experimental measurements of Coutanceau and Me´nard.13 The results, v-velocity along \theta { = } 0 ^ { \circ } and u-velocity along \theta { = } 9 0 ^ { \circ } , are not only in excellent agreement with the experimental measurements, but also with the computational results of Badr and \mathrm { D e n n i s } ^ { 1 2 } ~not shown here!. The comparisons made in Figs. 2 and 3 indicate the appropriateness of the numerical method and mesh used in this study.
TABLE I. Parametric studies at \mathbf { R e } { = } 1 0 0 and \alpha { = } 1 . 0 . Here, the relative errors ~%! with respect to the result from R _ { D } = 5 0 , M \times N = 2 4 1 \times 2 4 1 and \Delta t = 0 . 0 2 are parenthesized. \Delta r _ { b } and \Delta \theta _ { b } denote the relative magnitudes of grid-spacings at the base point. St is the Strouhal number, and C _ { L } and C _ { D } are the lift and drag coefficients, respectively. The overbar denotes the time averaging and the prime indicates the amplitude of fluctuation @see Eq. ~12!#.
| R_DM×N | \Delta r_bΔ \theta_b | \Delta t | St | \bar{C}_L | \bar{C}_D | C'_L | C'_D |
| 50 | 1 | 0.02 | 0.1655 | -2.4881 | 1.1040 | 0.3631 | 0.0993 |
| 241×241 | 1 | ||||||
| 100 | 0.99 | 0.02 | 0.1650 | -2.4833 | 1.0979 | 0.3603 | 0.0988 |
| 285×241 | 1 | (0.30) | (0.19) | (0.55) | (0.77) | (0.50) | |
| 50 | 0.38 | 0.02 | 0.1656 | -2.5027 | 1.0993 | 0.3576 | 0.0980 |
| 301×301 | 0.80 | (0.06) | (0.59) | (0.43) | (1.51) | (1.31) | |
| 50 | 1 | 0.01 | 0.1656 | -2.4881 | 1.1039 | 0.3630 | 0.0991 |
| 241×241 | 1 | (0.06) | (0.00) | (0.01) | (0.03) | (0.20) |
FIG. 2. ~a! St vs Re at \alpha { = } 0 ; \bigcirc , the present study; L, Park et al. ~Ref. 21! ~6413241, C grid!; ¯, Williamson’s correlations ~Ref. 22!. ~b! C _ { L } vs at \ R e { = } 2 0 \colon { \mathcal { O } } , the present study; – • h• – , Badr et al. ~Ref. 8 ) \ - \textcircled { < } - , Ingham and Tang ~Ref. 9!.
After verifying the numerical method, we have conducted numerical simulations of flow past a constantly rotating cylinder at Re540, 60, 100, and 160 by successively increasing from 0 to 2.5. Since the fully developed flow is independent of initial conditions ~Fig. 4!, all the simulations may be started with arbitrary initial conditions only if the fully developed flow fields are to be analyzed.
III. RESULTS
A. Strouhal number
In the case of no rotation ~ 50!, the flow at \mathrm { R e } { = } 4 0 is steady, while the flows at Re560, 100, and 160 are timeperiodic on account of vortex shedding ~it was shown by Park et a l . ^ { 2 1 } and Fey et a l . ^ { 2 3 } that vortex shedding occurs at Re>47!. Simulations show that vortex shedding exists at low rotational speeds and then completely disappears at > \alpha _ { L } , where the critical rotational speed \alpha _ { L } depends on the Reynolds number, for example, \alpha _ { L } { \approx } 1 . 4 , 1.8, and 1.9 for
FIG. 3. Time developments of velocity profiles at Re5200 and 50.5. ~a! v-velocity along 50°; ~b! u-velocity along 590°. —, the present study; s, h, n, and L, Coutanceau and Me´nard ~Ref. 13!.
Re560, 100, and 160, respectively. No vortex shedding develops regardless of in the case of Re540. Variation of the Strouhal number with respect to while vortex shedding exists is shown in Fig. 5. Rotation of a cylinder does not significantly alter St in the range of \alpha { < } \alpha _ { L } . More specifically, the Strouhal number stays nearly constant at low \alpha ^ { \prime } \mathrm { s } , , decreases slightly as approaches \alpha _ { L } , and sharply reduces to zero at \alpha > \alpha _ { L } . The present result supports the assumption of Badr et al.8 that St is more or less independent of ~St '0.14 and 0.16, respectively, for \mathrm { R e } { = } 6 0 and 100, consistent with the present result!, but contradicts the result of Hu et a l . ^ { 1 7 } that St decreases steadily with increasing as shown in Fig. 5. However, the result of Hu et al. seems to be inaccurate, which shows a poor agreement with those of Park e t a l . ^ { 2 \dot { 1 } } and Williamson22 at Re560 and 50.
B. Mean flow quantities
The mean pressure coefficient and vorticity around the cylinder surface for cases with various combinations of the rotational speed and Reynolds number are shown in Figs. 6 and 7. In these figures, the effect of on the mean flow quantities, denoted by the solid line, is examined by fixing Re at \mathbf { R e } { = } 1 0 0 , while the effect of Re, denoted by the dashed line, is done by fixing at 51.0. The pressure distributions obtained from the potential flow theory are also plotted for¯ comparison. The mean flow quantity, denoted by \overline { { ( ) } } , is computed by taking the temporal average of the flow quantities over one complete time cycle after the flow becomes fully developed.
FIG. 4. Variation of C _ { L } according to the initial condition at t = 0 \ \mathrm { ( R e = } 1 0 0 and \alpha { = } 1 . 0 ) ; ~ ( { \mathrm { a } } ) ~ - 2 0 { \leqslant } t { \leqslant } 7 0 and ~b! 7 0 { \leqslant } t { \leqslant } 1 0 0 . - , started impulsively into rotation and translation from rest; –•–, started from steady flow at \alpha { = } 2 . 0 ; \mathrm { ~ -- ~ } , started from time-periodic flow at \scriptstyle \alpha = 0 . . This figure shows that the flow becomes fully time-periodic after a long enough time, irrespective of the initial conditions.
Figure 6 shows the mean pressure coefficients, defined as \overline { { C } } _ { p } = 2 ( \bar { p } - p _ { \infty } ) / \rho U _ { \infty } ^ { 2 } , where p _ { \infty } is the free-stream pressure, around the cylinder surface with increasing in the range of 0 { \leqslant } \alpha { \leqslant } 2 . 5 at \mathrm { R e } { = } 1 0 0 . . As expected, the mean pressure for \scriptstyle { \alpha = 0 } is symmetric about \theta { = } 1 8 0 ^ { \circ } ~stagnation point!, leading to a zero mean lift. As increases, the flow becomes asymmetric and at the same time the pressure on the lower ~or the accelerated flow! side of the cylinder ~ '270°! decreases, resulting in a negative ~downward! mean lift. In addition, the stagnation point adhered to the cylinder surface moves in the reverse direction of rotation with increasing and then departs from the cylinder surface when \alpha { \gtrsim } 0 . 5 . Such observations are also found for all the other Reynolds numbers \mathrm { R e } { = } 4 0 , 6 0 , and 160. In Fig. 6 the results are also compared with those from the potential flow theory, given as
FIG. 5. Variation of St according to at Re560, 100, and 160: \mathrm { O , R e = 1 6 0 ; } \scriptstyle \prod , \mathrm { R e } = 1 0 0 ; \Delta , \mathrm { R e } = 6 0 ; \emptyset , Hu et al. ~Ref. 17! ~Re560!.
FIG. 6. Mean pressure coefficients around the cylinder surface at \alpha { = } 0 , 0 . 5 , , 1.0, 1.5, 2.0, and 2.5 for Re5100, compared with those from the potential flow theory for \scriptstyle { \alpha = 0 } and 1 . 0 ; - , the present study; - \cdot - . the potential theory. Also shown are the mean pressure coefficients at \mathrm { R e } { = } 6 0 and 160 for \alpha { = } 1 . 0 , , denoted by – – –.
\bar {C} _ {p} = 1 - 4 \sin^ {2} \theta + 4 \alpha \sin \theta - \alpha^ {2}, \tag {8}
for cases with \scriptstyle { \alpha = 0 } and 1.0. The comparison reveals that there is a big discrepancy between the potential and real viscous flows. The mean pressure coefficients at \mathrm { R e } { = } 6 0 and 160 for \alpha { = } 1 . 0 are also shown in Fig. 6. The results indicate that a change of the Reynolds number has a negligible effect on the pressure coefficient at the cylinder surface.
Figure 7 shows the mean vorticities around the cylinder surface for the same conditions as in Fig. 6. In this study, the mean surface vorticity is computed from \overline { { \omega } } = 2 \alpha + \partial \overline { { u } } _ { \theta } / \partial r in the ( r , \theta ) coordinate. In the case of \alpha { = } 0 , , the vorticity has the positive and negative peak values, respectively, at \theta { \approx } 2 3 0 ^ { \circ } and \theta \approx 1 3 0 ^ { \circ } , while it is nearly zero in the wake. With increasing , the vorticity increases in its magnitude, significantly deviating from zero in the wake region. The local peak in the wake region ( 3 0 0 ^ { \circ } < \theta < 3 6 0 ^ { \circ } ) also increases with the peak point moving in the direction of rotation as increases, and then becomes of a maximum ~negative! value when *1.5. On the other hand, increasing Re involves an overall growth of the vorticity magnitude around the cylinder surface. The shear stress ~or friction coefficient! around the cylinder surface, defined as \overline { { \tau } } _ { r \theta } = ( - 2 \alpha + \partial \overline { { u } } _ { \theta } / \partial r ) / \mathrm { R e } , can be explained from the surface vorticity shown in Fig. 7. Now that { \partial \overline { { u } } _ { \theta } } / { \partial r } is much larger than 2 for all the rotational speeds investigated, the distribution of the surface shear stress is very similar to that of the surface vorticity.
FIG. 7. Mean vorticities around the cylinder surface at \alpha { = } 0 , 0 . 5 , 1 . 0 , 1 . 5 , 2.0, and 2.5 for Re5100. Also shown are the mean vorticities at Re560 and 160 for \alpha { = } 1 . 0 , , denoted by – – –.
FIG. 8. Mean lift and drag coefficients according to the rotational speed; ~a! \overline { { C } } _ { L } ; ( { \bf b } ) \ : \overline { { C } } _ { D } . \mathrm { ~ O , R e = 1 6 0 ; } \bigstar \bigstar \mathrm { , ~ } \forall \mathrm { e = 1 0 0 ; } \bigtriangleup , \mathrm { R e = 6 0 ; } \bigtriangledown , \mathrm { R e = 4 0 ; } \bigtriangleup , potential flow theory. Solid symbols ~j! are obtained from Badr et al. ~Ref. 8! at \mathbf { R e } { = } 1 0 0 . Also shown are the contributions of the pressure and friction forces; —, total; –•–, pressure; ¯, friction.
C. Lift and drag coefficients
The characteristics of lift and drag forces exerted on the cylinder rotating steadily in a viscous uniform flow is investigated in terms of the lift and drag coefficients, C _ { L } and C _ { D } . The ~total! drag and lift forces are composed of both the pressure and friction forces as follows:
C _ {L} = C _ {L p} + C _ {L f}, \quad C _ {D} = C _ {D p} + C _ {D f}. \tag {9}
Variations of the lift and drag coefficients for cases with various combinations of the rotational speed and Reynolds number are shown in Figs. 8–10.
Figure 8 shows the mean lift and drag coefficients for all the cases investigated, compared with those computed from the potential flow theory,
C _ {L} = - 2 \pi \alpha , \quad C _ {D} = 0. \tag {10}
Also shown in the same figure are the contributions of the pressure and friction forces. First, the mean lift coefficients, \bar { C } _ { L } , \bar { C } _ { L p } , and \bar { C } _ { L f } , are shown in Fig. 8~a!. In the figure, all the ~negative! lift coefficients increase linearly in proportion to for low rotational speeds and the least-square fit provides ~for <1.0!
FIG. 9. Amplitudes of the lift and drag fluctuations according to the rotational speed; ~a! C _ { L } ^ { \prime } ; ( { \bf b } ) \ C _ { D } ^ { \prime } . \bigcirc , \mathrm { R e } = 1 6 0 ; \bigcirc , \mathrm { R e } = 1 0 0 ; \bigtriangleup , \mathrm { R e } = 6 0 . Solid symbols ~j! are obtained from Badr et al. ~Ref. 8! at Re5100.
FIG. 10. Phase diagrams of C _ { L } with C _ { D } for various values of ; ~a! Re 5100; ~b! Re5160. Time advances in the clockwise direction ~on the upper branch in the case of 50!.
\bar {C} _ {L p} \sim - 2. 3 3 \alpha , \quad \bar {C} _ {L f} \sim - 0. 1 3 \alpha ,
\bar {C} _ {L} \sim - 2. 4 6 \alpha , \quad \text { for } \mathrm{Re} = 1 6 0,
\bar {C} _ {L p} \sim - 2. 3 1 \alpha , \quad \bar {C} _ {L f} \sim - 0. 1 7 \alpha ,
\bar {C} _ {L} \sim - 2. 4 8 \alpha , \quad \text { for } \mathrm{Re} = 1 0 0,
\bar {C} _ {L p} \sim - 2. 2 9 \alpha , \quad \bar {C} _ {L f} \sim - 0. 2 1 \alpha , \tag {11}
\bar {C} _ {L} \sim - 2. 5 0 \alpha , \quad \text { for } \mathrm{Re} = 6 0,
\bar {C} _ {L p} \sim - 2. 3 1 \alpha , \quad \bar {C} _ {L f} \sim - 0. 2 6 \alpha ,
\bar {C} _ {L} \sim - 2. 5 7 \alpha , \quad \text { for } \mathrm{Re} = 4 0.
The present result is in good agreement with the prediction of Badr et al. \mathrm { , } \ \bar { C } _ { L } { \sim } - 2 . 5 5 \alpha for \mathrm { R e } { = } 6 0 , but in relatively poor agreement with the predictions of Tang and Ingham,10 \bar { C } _ { L } { \sim } - 2 . 1 7 \alpha and 21.86 for { \mathrm { R e } } { = } 6 0 and 100, respectively. Such discrepancies are due to the fact that Badr et al. investigated unsteady flow by solving time-dependent governing equations as in the present study, while Tang and Ingham investigated steady flow by solving time-independent governing equations. As shown in Eq. ~11!, the lift force mostly comes from the pressure force and its contribution increases with increasing Re, for example 92%, 93%, and 95%, respectively, for \mathrm { R e } { = } 6 0 , 1 0 0 , and 160. Since a change in the Reynolds number mostly affects the friction, as implied in Figs. 6 and 7, which has a negligible contribution to the total lift force, the total lift coefficient is nearly independent of Re. It is also observed that the lift force differs significantly from the result obtained from the potential flow theory @see Fig. 8~a!#.
The mean drag coefficients, \bar { C } _ { D } , \ \bar { C } _ { D p } , and \bar { C } _ { D f } , are shown in Fig. 8~b!. The drag force seems to be more complicated than the lift force. It is seen that the friction drag is of the same order of magnitude as the pressure drag and thus the total drag has a relatively large dependence on Re. As Re increases, the friction and total drag coefficients decrease. As increases, the pressure drag decreases with the increase in the friction drag, resulting in the net decrease in the total drag force. Even though the total drag forces at \mathbf { R e } { = } 1 0 0 and 160 are similar, their drag decompositions are different. It is also worth noting that the pressure drag becomes negative when *2.
In view of controlling vortex shedding, the lift and drag fluctuations are of more practical importance in engineering environments. Here, the amplitudes in the fluctuation are defined, respectively, as
C _ {L} ^ {\prime} = \frac {C _ {L , \max} - C _ {L , \min}}{2}, \quad C _ {D} ^ {\prime} = \frac {C _ {D , \max} - C _ {D , \min}}{2}, \tag {12}
where the subscripts min and max denote the minimum and maximum values, respectively, in a period. The variations of C _ { L } ^ { \prime } and C _ { D } ^ { \prime } with at R e = 6 0 , 100 and 160 are shown in Fig. 9, together with those of Badr et a l . ^ { 8 } Excellent agreement is seen between the present and Badr et al.’s studies. The figure implies again that vortex shedding appears at low rotational speeds and then completrange of vortex shedding, disappears at stays nearly c \alpha > \alpha _ { L } . In and C _ { L } ^ { \prime } C _ { D } ^ { \prime } increases linearly with increasing . On the other hand, the two fluctuation amplitudes, C _ { L } ^ { \prime } and C _ { D } ^ { \prime } , increase with increasing Re.
Behaviors of the lift and drag forces presented in Figs. 8 and 9 can be represented more clearly in the form of the phase diagram, by plotting C _ { L } as a function of C _ { D } at Re = 1 0 0 and 160 as shown in Fig. 10. The closed phase diagram indicates that the flow becomes fully time-periodic. As evident in the figure, the position of a phase diagram denotes the mean lift and drag and the size denotes the corresponding amplitudes of fluctuation. The phase diagrams confirm again that the mean lift and drag forces increase and decrease, respectively, with increasing . At the same time, the amplitude of the lift fluctuation stays almost constant and that of the drag fluctuation increases in the range of vortex shedding. It is also clear that the phase diagram collapses to a point ~denoted by the asterisk! at \alpha > \alpha _ { L } , for example \alpha { = } 1 . 9 for \mathbf { R e } { = } 1 0 0 and \alpha { = } 2 . 0 for \mathrm { R e } { = } 1 6 0 . At 50, vortex shedding occurs at the frequency of C _ { L } being half that of C _ { D } , resulting in a symmetry and a crossline in the phase diagram. When the cylinder rotates, however, the flow becomes asymmetric due to the unidirectional rotation. Consequently, C _ { D } has the same frequency as C _ { L } when \alpha { > } \alpha _ { s } at occurrence of vortex shedding, for example \alpha _ { s } \approx 0 . 1 at Re = 1 0 0 . The value \alpha _ { s } tends to increase with increasing Re.
D. Instantaneous flow fields
Instantaneous flow fields are investigated to identify the underlying mechanism of the suppression of vortex shedding due to the rotation. Figure 11 shows the vorticity contours at two different times, t / T { = } 1 / 2 and 1 in one complete time cycle, with increasing at \mathbf { R e } { = } 1 0 0 , where T is the nondimensional period or a reciprocal of the Strouhal number. Here, t / T { = } 0 is the time corresponding to the maximum lift, which does not necessarily mean that t / T { = } 1 / 2 corresponds to the minimum lift. As expected, it is clearly shown that vortex shedding which develops at \scriptstyle { \alpha = 0 } also occurs at low values of \alpha , for example \alpha { < } 2 . 0 for \mathrm { R e } { = } 1 0 0 . While vortex shedding exists, the vorticity contours away from the cylinder surface are similar in overall shape, which indicates that the rotation effect is confined to the flow in the vicinity of the cylinder surface. In the near-surface flow, with increasing , the negative vorticity on the upper side of the cylinder becomes more dominant than the positive vorticity on the lower side. Subsequently, for \alpha { \geq } 2 . 0 , vortex shedding completely disappears and the flow has two stationary vorticity bubbles attached to the cylinder. As increases further, the bubbles become thinner and more inclined in the direction of rotation. In the range of \alpha { \gtrsim } 1 , the negative isovorticity lines in front of the cylinder form an acute curved tail, which grows around the cylinder in the direction of rotation with increasing .
The corresponding streamlines are shown in Fig. 12 for the same conditions. Overall, the figure shows that the rotation of a cylinder makes a substantial effect on the flow pattern. When the cylinder rotates at low rotational speeds ~ ,2.0!, two vortices are alternately shed on each side of the cylinder but the vortex shedding configuration varies according to . As increases, the vortex on the upper side in the wake becomes stronger, while that on the lower side becomes weaker. Finally, the lower vortex completely disappears when >2.0, resulting in no vortex shedding. In addition, it is valuable to compare the flow pattern to the corresponding potential flow, especially at a high rotational speed. The potential flow theory24 gives that closed streamlines surrounding the cylinder exist only when .2.0, where the stagnation point is detached from the cylinder surface. The same phenomenon can be observed in the viscous flow but for different conditions. In the present study, the flow has the closed streamlines surrounding the cylinder when \alpha { \gtrsim } 0 . 5 , where the separation point departs from the cylinder surface. The flow pattern for 52.5 is very similar to that of the potential flow.24
FIG. 11. Vorticity contours at t/T51/2 ~left-hand side! and 1 ~right-hand side! with increasing at Re5100. 5~a! 0.0, ~b! 0.5, ~c! 1.0, ~d! 1.5, ~e! 2.0, and ~f! 2.5. Contour levels are from 23 to 3 in increments of 0.2. Negative values are shown as dashed.
FIG. 12. Streamlines at t/T51/2 ~left-hand side! and 1 ~right-hand side! with increasing at Re5100. 5~a! 0.0, ~b! 0.5, ~c! 1.0, ~d! 1.5, ~e! 2.0, and ~f! 2.5. Contour levels are from 20.3 to 0.3 in increments of 0.02 and from 60.3 to 61 in increments of 60.05.
Figure 13 shows the time evolution of the streamlines at Re5100 and \alpha { = } 1 . 0 . This figure indicates the way how the shed vortices are formed and convected downstream. It is seen that there are two alternate vortices over one complete time cycle with one vortex [ ^ { 6 } \mathrm { \AA } ^ { , 3 } vortex in ~e!# stronger than the other ( ^ { 6 } \mathrm { \Delta ^ { 6 } B ^ { \prime } } ) vortex! because of the unidirectional rotation, and the stronger vortex lasts longer than the other. Such an observation becomes more pronounced as the rotational speed increases.
E. Stability and bifurcation
Results have shown repeatedly that in the laminar vortex shedding regime ~47<Re,200! vortex shedding exists at low rotational speeds and completely disappears at \alpha > \alpha _ { L } . This indicates that there should be a bifurcation curve between the existence of vortex shedding and its disappearance on a two-dimensional plane of Re and . Here, such a bifurcation curve based on the onset of vortex shedding is a typical result of flow stability problem. Figure 14 shows the bifurcation curve, or borderline, between steady and unsteady ~time-periodic! regimes obtained with varying by an increment of 0.1 at several chosen Reynolds numbers. Notice that the real bifurcation curve should reside in between open circle and cross symbols in the figure. The figure shows that the \alpha _ { L } ~denoted by open circles! increases logarithmically as Re increases. The observation was also found qualitatively in Badr and Dennis,12 Coutanceau and Me´nard,13 and Badr et al., 14 who all investigated unsteady flow at high Reynolds numbers ( \mathrm { R e } { = } 2 0 0 { - } 1 0 ^ { 4 } ) and found the disappearance of vortex shedding at \alpha > \alpha _ { L } . Then, they claimed that \alpha _ { L } is nearly independent of Re and is about 2, which is apparently inconsistent with the present result but may be valid at high Reynolds numbers. Recently, Hu et al.17 found analytically the logarithmic dependence of \alpha _ { L } on Re in the range of 4 5 { \leqslant } \mathrm { R e } { \leqslant } 5 0 and <1.0 using a Galerkin method and a system theory, as also depicted in Fig. 14. Agreement between the results of the present study and Hu et al. is clearly seen. It can be concluded from Fig. 14 that the Reynolds number tends to destabilize flow past a circular cylinder, while the rotation of the cylinder tends to stabilize it.
FIG. 13. Time evolution of the streamlines at Re5100 and \alpha { = } 1 . 0 ; t / T { = } ( \mathrm { a } ) 1/8, ~b! 2/8, ~c! 3/8, ~d! 4/8, ~e! 5/8, ~f! 6/8, ~g! 7/8, and ~h! 1. Contour levels are from 20.3 to 0.3 in increments of 0.02 and from 60.3 to 61 in increments of 60.05.
IV. SUMMARY
In this paper, we have performed a numerical study of fully developed two-dimensional laminar flow past a circular cylinder rotating with a constant angular velocity for the purpose of controlling vortex shedding and understanding the underlying flow mechanism. The rotation of a circular cylinder in a viscous uniform flow may significantly modify flow patterns and, thus, reduce a flow-induced oscillation from vortex shedding. Numerical simulations were performed for flows with Re540, 60, 100, and 160 in the range of 0 { \leqslant } \alpha { \leqslant } 2 . 5 . In the case of no rotation, the flow at \mathrm { R e } { = } 4 0 is steady, while the flows at Re560, 100, and 160 are timeperiodic with vortex shedding.
Results showed that vortex shedding exists, resulting in a time-periodic flow, at low rotational speeds and completely disappears at \alpha > \alpha _ { L } , where the critical rotational speed \alpha _ { L } depends on the Reynolds number, for example \alpha _ { L } { \approx } 1 . 4 , , 1.8, and 1.9 for \mathrm { R e } { = } 6 0 . , 100, and 160, respectively. In other words, the value of \alpha _ { L } increases logarithmically as Re increases. When vortex shedding exists in the range of \leqslant \alpha _ { L } , notable changes in the flow pattern are observed. First, the Strouhal number is nearly independent of the rotational speed, but strongly dependent on the Reynolds number. Second, as the rotational speed increases, the mean lift force increases almost linearly with and the mean drag force decreases. At the same time, the amplitude of the lift fluctuation stays nearly constant and that of the drag fluctuation increases linearly with . The drag and lift fluctuations also vanish at \alpha > \alpha _ { L } . Further studies from the instantaneous flow fields demonstrate again that the rotation of a cylinder makes a substantial effect on the flow pattern.
FIG. 14. Bifurcation curve dividing into the steady and time-periodic regimes on a plane of Re and , compared to Hu et al. ~Ref. 17!. ~L!: s, vortex shedding; 3, no vortex shedding.
ACKNOWLEDGMENTS
This work was supported by National Creative Research Initiatives of the Korean Ministry of Science and Technology. The computations were performed on the CRAY YMP C90 at the Electronics and Telecommunications Research Institute. The supports are gratefully acknowledged.
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