ROTATION IMPLEMENTATION OF A CIRCULAR ... - CiteSeerX

J. Appl. Math. & Computing Vol. 22(2006), No. 1 - 2, pp. 67 - 82

Website: http://jamc.net

ROTATION IMPLEMENTATION OF A CIRCULAR CYLINDER IN INCOMPRESSIBLE FLOW VIA STAGGERED GRID APPROACH MINGQING XIAO∗ , YUAN LIN, JAMES H. MYATT, R. CHRIS CAMPHOUSE AND SIVA S. BANDA

Abstract. In this paper, we present a finite difference method for the implementation of the rotation of a circular cylinder in the incompressible flow field by solving the two-dimensional unsteady Navier-Stokes equations. The approach is to use staggered grid method so that the accuracy and order of convergence of the associated algorithms can be maintained. The proposed method is easy to be implemented and is effective. A set of simulations for the flow dynamics is provided to show the computational results. AMS Mathematics Subject Classification : 35R10, 39A10, 93C20. Key words and phrases : Navier-Stokes equations, incompressible fluid, finite difference, staggered grid.

1. Introduction Flow past a circular cylinder is one of the fluid systems that researchers have made a significant progress toward obtaining a more fundamental understanding of the mechanism of instability that leads to turbulence. Circular cylinder wakes are typical of a class of shear flows where the instability of velocity at different spatial positions along the path of its streamwise development changes in a significant and a complicated way. In the near-wake region where there is a large velocity deficit, the local flow is typically absolutely unstable, and in the far-wake region where the velocity deficit is much small, the local flow is almost always convectively unstable. Received February 2, 2005. Revised June 6, 2005. ∗ Corresponding author. Research supported in part by the Summer Faculty Fellowship Program from Air Force Office of Scientific Research, 2001 and 2002, and in part by Southern Illinois University Faculty Seed Grant, 2002. c 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

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The study of circular cylinder wake flow has received more and more attention in recent years. On one hand it appears in a wide range of aeronautical, civil, mechanical and chemical engineering applications (see, e.g., [4], [5], [7], [9], [12], and references therein). On the other hand, it represents a typical, archetypal, unstable flow in fluid dynamics. It is a reduction of reality to a simple model which can be treated mathematically, and which exhibits the nature of complex flow behavior. For instance, a simplified wake, created by a circular cylinder in compressible cross flow, has been used to model the energy separation and base pressure fluctuation [1]. It was recently found that studying the flow past a cylinder can help in understanding two oceanographic phenomena: separation of the Gulf Stream from the North American coastline at Cap Hatteras and the interaction of the Antarctic Circumpolar Current with topographic obstacles [15]. The system of equations governing the conservation of momentum for fluid flow is known as the Navier-Stokes equations. When combined with conservation of mass, they describes the general evolution of fluid flow. For a viscous incompressible fluid, the equations are   ∂u ρ + (u · ∇)u − µ∆u + ∇p = f (1) ∂t div u = 0, (2) where ρ > 0 is the constant density of the fluid, ν > 0 is the kinematic viscosity, and f represents volume forces that are applied to the fluid. Alternatively, equation (1) can be considered as the nondimensionalized form of the Navier-Stokes equations in which case u, p, f are nondimensionalized quantities and the kinematic viscosity µ/ρ := ν is replaced by Re−1 , Re representing the Reynolds number UL Re = , ν where U and L are the typical velocity and length used for the nondimensionalization. In this paper, we study the flow past an infinite length circular cylinder and thus consider the two-dimensional case of (1)-(2). We rewrite equations (1)-(2) in component form and let u = (u, v), p = ρp, and f = ρ(fx , fy ). Then the equations in component form read as follows: momentum equations:   ∂u ∂(u2 ) ∂(uv) ∂p 1 ∂2u ∂2u + + − = fx + 2 + (3) 2 ∂t ∂x ∂y Re ∂x ∂y ∂x   ∂p ∂v ∂(uv) ∂(v 2 ) ∂2v 1 ∂2v + + (4) + + − = fy , 2 2 ∂t ∂x ∂y Re ∂x ∂y ∂y continuity equation: ∂u ∂v + = 0. ∂x ∂y

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Let Ω be the region exterior to a circular disk with radius r and center at (xc , yc ). Then the control of rotation is acting on the circumference: (u, v) = ω(t) (−(y − yc ), x − xc )

on ∂Ω,

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where ω(t) is the angular speed (possibly time-dependent) of the cylinder. When ω ≡ 0, (6) is the standard Dirichlet boundary condition. The setting of (6) follows the convention, that is, if ω > 0 the rotation is counter clockwise, and if ω < 0 the rotation is clockwise. In most literatures, boundary conditions of partial differential equations, such as Dirichlet condition, Neumann conditions, etc., usually do not contain dynamical actions. The appearance of dynamical actions (called controlling actions) on the boundary complicates the dynamics of the system, and also the numerical approach. Increased attention has been devoted to the development of techniques capable of enhancing our ability to control unsteady flows in a wide variety of configurations and applications. Controlling the flow around these configurations can lead to greatly improved efficiency and performance of the system. It is well-known from the experiments that applying the boundary control by a rotation of the flow around a circular cylinder can lead to 30 to 60% drag reduction, compared to the fixed cylinder configuration, for Reynolds numbers in the range of 200 to 1000 [12]. The significant drag reduction by rotating the circular cylinder is due, in part, to its bluntness which makes for strong interaction with outer flow, and to its continuous surface, on which the point of separation is free to move from one position to another. In this paper, we presents a numerical approach for the implementation of the rotation movement of a cylinder in the incompressible flow via a staggered grid method. The proposed approach is new and easy to be implemented. Specifically, we classify the boundary of the circular cylinder into different types of boundary cells, and the rotation is implemented in these cells in different ways based on the cell’s characterizations. The approach does not require an extensive refinement of grids, which is costly. We provide simulations of two types of rotation, one-directional rotation and harmonic rotation, which are of interest in applications.

2. Numerical approach Our numerical calculation is made over a collection of uniformly space discrete grid points. When the central differences are used for the incompressible NavierStokes equations, pressure oscillations can possibly occur if all three unknown functions u, v and p, are evaluated at the same grid points [3]. There are two ways suggested for fixing the problem. One is to utilize the upwind differences, and the other is to use a staggered grid. The advantage of using the staggered grid is that a central difference approach can still be maintained an thus a better

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accuracy can be achieved. A staggered grid is illustrated in Fig.1. The pressures are calculated at the cell center, the horizontal velocities u are evaluated in the midpoints of the vertical cell edges, and the vertical velocities are computed in the midpoints of horizontal cell edges. Cell (i, j) occupies the spatial region [(i − 1)∆x, i∆x] × [(j − 1)∆y, j∆y], and the corresponding index (i, j) is assigned to the pressure at the cell center as well as to the u-velocity at the right edge and the v-velocity at the upper edge of this cell. The key feature is that the pressures and velocities are evaluated at different grid points so that the nonphysical oscillations due to the numerical computation can be avoided.

Figure 1. Staggered grid cells 2.1. Discretization of the Navier-Stokes equations Let the rectangular region be Ω = [0, a] × [0, b] on which we introduce a grid. The grid is divided into imax cells of equal size in the x-direction and jmax cells in the y-direction. We denote ∆x =

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Then using the central difference scheme, we have ∂ 2 u ui+1,j − 2ui,j + ui−1,j = ∂x2 i,j (∆x)2 ∂ 2 u ui,j+1 − 2ui,j + ui,j−1 = ∂y 2 i,j (∆y)2

M-Q. Xiao, Y. Lin, J. H. Myatt, R. C. Camphouse and S. S. Banda

∂ 2 v ∂x2 i,j ∂ 2 v ∂y 2 i,j ∂p ∂x i,j ∂p ∂y i,j

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vi+1,j − 2vi,j + vi−1,j (∆x)2 vi,j+1 − 2vi,j + vi,j−1 = (∆y)2 pi+1,j − pi,j = ∆x pi,j+1 − pi,j . = ∆y =

Considering that the convective terms in the momentum equations can become dominant at high Reynolds numbers, following [6], we use a mixture of the central differences and the donor-cell discretization. In each cell (i, j), we set  2  2 ! ∂(u2 ) 1 ui,j + ui+1,j ui−1,j + ui,j − = ∂x i,j ∆x 2 2  1 |ui,j + ui+1,j | (ui,j − ui+1,j ) +γ ∆x 2 2 |ui−1,j + ui,j | (ui−1,j − ui,j )  − ; 2 2 1  (vi,j + vi+1,j ) (ui,j + ui,j+1 ) ∂(uv) = ∂y i,j ∆y 2 2 (vi,j−1 + vi+1,j−1 ) (ui,j−1 + ui,j )  − 2 2 1  |vi,j + vi+1,j | (ui,j − ui,j+1 ) +γ ∆y 2 2 |vi,j−1 + vi+1,j−1 | (ui,j−1 − ui,j )  − ; 2 2 1  (ui,j + ui,j+1 ) (vi,j + vi+1,j ) ∂(uv) = ∂x i,j ∆x 2 2 (ui−1,j + ui−1,j+1 ) (vi−1,j + vi,j )  − 2 2 1  |ui,j + ui,j+1 | (vi,j − ui+1,j ) +γ ∆x 2 2 |ui−1,j + ui−1,j+1 | (vi−1,j − vi,j )  − ; 2 2   2 !  2 ∂(v 2 ) 1 vi,j + vi,j+1 vi,j−1 + vi,j − = ∂y i,j ∆y 2 2  1 |vi,j + vi,j+1 | (vi,j − ui,j+1 ) +γ ∆y 2 2 −

|vi,j−1 + vi,j | (vi,j−1 − vi,j )  , 2 2

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where γ is a parameter, which in simulations is chosen to satisfy   ui,j ∆t vi,j ∆t , . 1 ≥ γ ≥ max i,j ∆x ∆y

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2.2. Boundary values for the domain Figure 2 shows a 2-D domain Ω with a circular inside. The boundary values of u and v are required to solve the discretization of the momentum equation. These velocity values are based on the boundary conditions. In this paper, no-slip condition is applied to the top and the bottom boundaries, and inflow/outflow condition are used on the left-handed/right-handed sides, respectively. 40 30 20 10 20

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Figure 2. Grid of the domain Ω (1) No-slip condition (top and bottom) This implies that the continuous velocities should be exactly zero in both horizontal and vertical directions. Hence, in the staggered grid, we set the velocity values on the top and the bottom to be zero, i.e. u0,j = 0,

uimax ,j = 0,

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Since the u-values and the v-values do not lie on the vertical boundaries and horizontal boundaries respectively, the zero boundary values is imposed as follows, vr =

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Thus on these boundaries, the no-slip condition is implemented by setting v0,j = −v1,j ,

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(2) Outflow condition (right-handed side boundary) In case of outflow boundary condition, the velocity remains unchanged in the direction normal to the boundary. Therefore, it is implemented

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by setting u0,j = u1,j , v0,j = v1,j ,

uimax ,j = uimax −1,j , vimax +1,j = vimax ,j ,

j = 1, ..., jmax , j = 1, ..., jmax ,

ui,0 = ui,1 , vi,0 = vi,1 ,

ui,jmax +1 = ui,jmax , vi,jmax = vi,jmax −1 ,

i = 1, ..., imax , i = 1, ..., imax .

(3) Inflow condition (left-handed side boundary) In this case, the velocities are given in the direction normal to the boundaries. On the direction tangential to the boundary, we average the values on either side of the boundary. So the formulae for the boundary cells are set up similarly as those in the outflow condition. A simulation of flow contours without rotating the cylinder is provided in Fig. 4, where Re = 200. From the simulation, one can see that vortices alternatively leave the top and the bottom of the cylinder with a definite frequency, which is called vortex shedding. The vortexes trail behind the cylinder in two rows, known as K´ am´ an vortex street. The frequency in this case is f ≈ 0.625. It is well-known that the von K´ arm´ an vortex street induces a strong unsteady transverse periodic force on the cylinder which can lead to cylinder oscillations with large amplitudes. Such oscillations can cause structural damage and structure failure in applications. Thus one of methods for controlling this type of vibrations is to rotate the cylinder in order to change the frequency of the oscillation.

boundary

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Figure 3. Treatment of Boundary Values 3. Implementation of rotations 3.1. One direction rotation In this section, we consider the case in which the cylinder is rotated in one

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direction. Recall that the relation between the angular speed ω and the speed on the circumference of the circular cylinder is governed by (u, v)

= ω(t) (−(y − yc ), x − xc )

where (x, y) ∈ ∂Ω

The challenge for the implementation is that the obstacle in the numerical approach is a polygon instead of a circular disk. Here the polygon is used to approximate the circular disk. For the Dirichlet boundary condition, the numerical implementation does not present any difficulty. However, if the cylinder is rotated, the numerical approach becomes complicated and subtle, since the fluid flow dynamics in which a circular disk is rotated is very different from the one in which a polygon is rotated. The usual solution is to increase grid refinement. But this would result in a significant cost in computation time. Thus it is necessary to develop a technique for the approximation of the rotation of a circular cylinder without the requirement of an extensive refinement of grids near the cylinder. In order to capture the dynamics of the system, in which the circular cylinder is rotated, it is necessary to take a close look on those cells near the boundary of the cylinder. We first divide all cells into two types: fluid cells and cylinder cells. Fluid cell is a cell which lies completely or mostly in the complement of Ω, and cylinder cell is a cell which lies completely or mostly in Ω. Figure 5 illustrates the classification of fluid cells and cylinder cells. The cylinder cells are those gray cells, and the fluid cells are those white cells. Next we consider those cylinder cells (called boundary cells) that are directly adjacent to the fluid cells. Based on cells’ locations related to Ω, we further classify them into eight different types of groups: north, northwest, west, southwest, south, southeast, east, and northeast, as illustrated in Figure 5. Now we show how to set those boundary cell values when the cylinder is rotated. To start, let us consider the northern cell (with index (i, j)) as shown in Figure 5. We shall view the velocity (ui,j , vi,j ) at this cell to be the velocity on the boundary of the disk. Recall that the control of rotation is acting on the circumference: (u, v) = k(t) (−(y − yc ), x − xc ) on ∂Ω, and (xc , yc ) is the coordinates of the circular center in the domain. Let xi = i∆x and yi = j∆y. Now we set vi,j = k(t)(xi − xc ) since vi,j is the v-value on the boundary of the disk. For the u-value, we enforce it by averaging the values on either side of the boundary (fluid cell and cylinder cell): ui−1,j + ui−1,j+1 = −k(t)(yj − yc ) 2 ui,j + ui,j+1 = −k(t)(yj − yc ). 2

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Figure 5. The circular in the domain The other cell’s setting is similar and we provide the detail as follows: (1) Northern Cells vi,j = k(t)(xi − xc ) ui,j = −ui,j+1 − 2k(t)(yj − yc ) ui−1,j = −ui−1,j+1 − 2k(t)(yj − yc ) (2) Northwest Cells vi,j = k(t)(xi − xc ) ui−1,j = −k(t)(yj − yc ) vi,j−1 = −vi−1,j−1 + 2k(t)(xi − xc ) ui,j = −ui,j+1 − 2k(yj − yc ) (3) Western Cells ui−1,j = −k(t)(yj − yc ) vi,j = −vi−1,j + 2k(t)(xi − xc ) vi,j−1 = −vi−1,j−1 − 2k(t)(xi − xc ) (4) Southwest Cells vi,j−1 = k(t)(xi − xc ) ui−1,j = −k(t)(yj − yc ) vi,j = −vi−1,j + 2k(t)(xi − xc ) ui,j = −ui,j−1 − 2k(t)(yj − yc )

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(5) Southern Cells vi,j−1 = k(t)(xi − xc ) ui,j = −ui,j−1 − 2k(t)(yj − yc ) ui−1,j = −ui−1,j−1 − 2k(t)(yj − yc ) (6) Southeast Cells vi,j−1 = k(t)(xi − xc ) ui,j = −k(t)(yj − yc ) vi,j = −vi+1,j + 2k(t)(xi − xc ) ui−1,j = −ui−1,j−1 − 2k(t)(yj − yc ) (7) Eastern Cells ui,j = −k(t)(yj − yc ) vi,j = −vi+1,j + 2k(t)(xi − xc ) vi,j−1 = −vi+1,j−1 − 2k(t)(xi − xc ) (8) Northeast Cells vi,j = k(t)(xi − xc ) ui,j = −k(t)(yj − yc ) vi,j−1 = −vi+1,j−1 + 2k(t)(xi − xc ) ui−1,j = −ui−1,j+1 − 2k(t)(yj − yc ) The classification of these eight groups of boundary cells is independent of the grid refinement. In the coding, flags have been set up to indicate the corresponding cell computations. Fig. 6 shows the flow contours when Re = 200 at several time steps, where the angular speed k is set to be 6. From the simulation, one can see that the flow is regularized, compared to the case without a rotation. The simulation outcome is consistent with our analysis results discussed in [16]. 3.2. Harmonic rotations Harmonic rotation is also called oscillatory rotation as those presented in the experiment conducted by Tokumaru and Dimotakis [12]. When the forcing is set to be sinusoidal, there are two degrees of freedom, namely the frequency fe and the amplitude k of the angular velocity. The forcing strouhal number is defined as dfe Sf = . (9) Umean The forcing angular velocity in this paper is set to be ω(t) = k sin(2πSf t)

(10)

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Figure 6. Contour (Re = 200) at various time steps. Cylinder is rotated in counter clockwise with angular speed k = 6. Thus, the magnitude of the velocity varies with a period of 1/Sf . Both maximum angular speed k and the forcing strouhal number are read from an external parameter file. Similar to the previous section, we implement the harmonic rotation via the following eight different types of cells:

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Figure 7. Vorticity contour for harmonic rotation when Re = 200. The sequence plots represent about one period of the flow dynamics. The angular speed k = 6, and the forcing frequency fe = 1.5. (1) Northern Cells vi,j = ω(t)(xi − xc ) ui,j = −ui,j+1 − 2ω(t)(yj − yc ) ui−1,j = −ui−1,j+1 − 2ω(t)(yj − yc )

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(2) Northwest Cells vi,j = ω(t)(xi − xc ) ui−1,j = −ω(t)(yj − yc ) vi,j−1 = −vi−1,j−1 + 2ω(t)(xi − xc ) ui,j = −ui,j+1 − 2ω(t)(yj − yc ) (3) Western Cells ui−1,j = −ω(t)(yj − yc ) vi,j = −vi−1,j + 2ω(t)(xi − xc ) vi,j−1 = −vi−1,j−1 − 2ω(t)(xi − xc ) (4) Southwest Cells vi,j−1 = ω(t)(xi − xc ) ui−1,j = −ω(t)(yj − yc ) vi,j = −vi−1,j + 2ω(t)(xi − xc ) ui,j = −ui,j−1 − 2ω(t)(yj − yc ) (5) Southern Cells vi,j−1 = ω(t)(xi − xc ) ui,j = −ui,j−1 − 2ω(t)(yj − yc ) ui−1,j = −ui−1,j−1 − 2ω(t)(yj − yc ) (6) Southeast Cells vi,j−1 = ω(t)(xi − xc ) ui,j = −ω(t)(yj − yc ) vi,j = −vi+1,j + 2ω(t)(xi − xc ) ui−1,j = −ui−1,j−1 − 2ω(t)(yj − yc ) (7) Eastern Cells ui,j = −ω(t)(yj − yc ) vi,j = −vi+1,j + 2ω(t)(xi − xc ) vi,j−1 = −vi+1,j−1 − 2ω(t)(xi − xc ) (8) Northeast Cells vi,j = ω(t)(xi − xc ) ui,j = −ω(t)(yj − yc ) vi,j−1 = −vi+1,j−1 + 2ω(t)(xi − xc ) ui−1,j = −ui−1,j+1 − 2ω(t)(yj − yc )

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In simulation, we set Sf = 0.75, k = 6, and Re = 200. The period of the flow is about 1.33 and the dynamics is dominated by the forcing frequency. This is consistent with the experiment results presented in [12], and a finite element computation shown in [8]. Summary In this paper, we present a simple approach of the finite difference for the implementation of a circular cylinder rotation in the incompressible flow. When the proposed method is applied, the accuracy and the convergence of the algorithm will be maintained, while an extensive refinement of grids does not required. Such an approach can be used to establish feedback loop to control vortex shedding. The further study will be reported elsewhere. References 1. J. R. Ackerman, J. P. Gostelow, A. Rona and W. E. Carscallen, Energy separation and base pressure in the wake of a circular cylinder, Proc. of the 1st Flow Control Conference of AIAA, AIAA 2002-2075, St. Louis, Missouri, 24 - 27 Jun 2002. 2. G. Addington, J. Hall and J. Myatt, Reduced order modeling applied to reactive flow control, Proc. of the 1st Flow Control Conference of AIAA, AIAA 2002-4807, St. Louis, Missouri, 24 - 27 Jun 2002. 3. J. D. Anderson, Computational Fluid Dynamics, McGraw-Hill, Inc. 1995. 4. B. Pier, On the frequency selection of finite-amplitude vortex shedding in the cylinder wake, J. Fluid Mech. 458 (2002), 407-417. 5. E. A. Gillies, Low-dimensional control of the circular cylinder wake, J. Fluid Mech. 371 (1998), 157-178. 6. M. Griebel, T. Dornseifer, and T. Neunhoeffer, Numerical Simulation in Fluid Dynamics, SIAM, Philadelphia, 1998. 7. W. R. Graham, J. Peraire and K. Y. Tang, Optimal control of vortex shedding using low order models Part I: open-loop model development, Int. J. for Num. Methods in Engr. 44(7) (1997), 973-990. 8. J. He, M. Chevalier, R. Glowinski, R. Metcalfe, A. Nordlander and J. Periaux, Drag reduction by active control for flow past cylinders, Computational mathematics driven by industrial problems, (Martina Franca, 1999), 287–363, Lecture Notes in Math., 1739, Springer, Berlin, 2000. 9. X. Ma and G. E. Karniadakis, A low-dimensional model for simulating three-dimensional cylinder flow, J. Fluid Mech. 458 (2002), 181-190. 10. M. Schumm, E. Berger and P. A. Monkewitz, Self-excited oscillations in the wake of twodimensional bluff bodies and their control, J. Fluid Mech. 271 (1994), 17-53. 11. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, New York, Springer, 1997 12. P. T. Tokumaru and P. E. Dimotakis, Rotary oscillation control of a cylinder wake, J. Fluid Mech. 224 (1991), 77-90. 13. M. Tome and S. McKee, GENSMAC: A computational marker and cell method for free surface flows in general domains, J. Comput. Phys. 110 (1994), 171-186. 14. K. Roussopoulos, Feedback control of vortex shedding at low Reynolds numbers, J. Fluid Mech. 248 (1993), 267-296. 15. Claire E. Tansley and David P. Marshall, Flow past a cylinder on a β plane, with application to Gulf Stream Separation and the antarctic circumpolar current, J. of Physical Oceanography, 31 (2001), 3274-3283.

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16. M. Xiao, M. Novy, J. Myatt and S. Banda, A Rotary Control of the Circular Cylinder Wake: An Analytic Approach, Proc. of the 1st Flow Control Conference of AIAA, AIAA 2002-3075, St. Louis, Missouri, 24 - 27 Jun 2002. 17. M. Xiao, Y. Lin, R. Camphouse, J. Myatt and S. Banda, Numerical Simulations and Control of the Flow Past a Circular Cylinder, Proc. of the 33rd AIAA Fluid Dynamics Conference and Exhibit, AIAA-2003-4265, 2003. Mingqing Xiao received his Ph. D. in Mathematics from the University of Illinois at Urbana-Champaign in 1997. After his graduation he had been worked as a visiting research assistant professor in the University of California at Davis for two more years. He current is an associate professor at the Southern Illinois University at Carbondale. He has served on the Editorial Boards for IEEE Transactions on Automatic Control, and International Journal of Innovational Computing and Control Information since 2004. His research interests include nonlinear control, fluid flow control, and control of systems governed by partial differential equations. Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA e-mail: mxiao@@math.siu.edu Yuan Lin received his M. S. in Mathematics from Southern Illinois University at Carbondale in 2004 and M. S. in Computer Engineering from Syracuse University in 1999. With two years’ experience in the IT industry in Silicon Valley, he currently is a Ph. D. candidate in Mathematics at Southern Illinois University at Carbondale. His research interests include fluid flow control, nonlinear control and numerical optimization. Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA e-mail: ylin@@math.siu.edu James H. Myatt received his Ph. D. in Aeronautics & Astronautics in 1997 from Stanford University. He has worked in the Air Force Research Laboratory’s Air Vehicles Directorate since 1990 and is currently in the Control Design and Analysis branch. His research interests are aerodynamic flow control and flight mechanics. His current area of focus is the integration of feedback control with aerodynamic flow control. Airforce Research Laboratory, Wright-Patterson Air Force Base, Ohio, USA e-mail: James.Myatt@@wpafb.af.mil R. Chris Camphouse received his Ph. D. in Mathematics in 2001 from Virginia Polytechnic Institute and State University. After completing a one year post-doc at Virginia Tech’s Interdisciplinary Center for Applied Mathematics, Chris joined the Control Design and Analysis branch of AFRL’s Air Vehicles Directorate. His research interests include feedback control theory, reduced-order modeling, numerical methods, and partial differential equations. His current area of research involves reduced-order modeling and boundary feedback control of fluid flows. Airforce Research Laboratory, Wright-Patterson Air Force Base, Ohio, USA e-mail: Russell.Camphouse@@wpafb.af.mil Siva Banda is the Director of the Control Science Center of Excellence, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio, USA. His research interests are autonomous control of unmanned air vehicles, control of access-to-space vehicles, and aerodynamic flow control. He served on Editorial Boards for IEEE Transactions on Control Systems Technology, International Journal of Robust and Nonlinear Control, and Journal of Guidance, Control and Dynamics. He is a Fellow of AIAA, and IEEE, and is a member of the National Academy of Engineering. Airforce Research Laboratory, Wright-Patterson Air Force Base, Ohio, USA e-mail: Siva.Banda@@wpafb.af.mil