Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method

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1 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method Roberto Rojas 1*, Kosuke Hayashi 1, Takeshi Seta and Akio Tomiyama 1 1 Graduate School of Engineering, Kobe University 1-1, Rokkodai, Nada, Kobe, , Japan Graduate School of Science and Engineering, Toyama University 3190, Gofuku, Toyama, , Japan Received: 3 August 01, Accepted: 1 November 01 Abstract The applicability of the immersed boundary-finite difference lattice Boltzmann method (IB-FDLBM) to high Reynolds number flows about a circular cylinder is examined. Two-dimensional simulations of flows past a stationary circular cylinder are carried out for a wide range of the Reynolds number, Re, i.e., 1 Re An immersed boundary-lattice Boltzmann method (IB-LBM) is also used for comparison. Then free-falling circular cylinders are simulated to demonstrate the feasibility of predicting moving particles at high Reynolds numbers. The main conclusions obtained are as follows: (1) steady and unsteady flows about a stationary cylinder are well predicted with IB-LBM and IB-FDLBM, provided that the spatial resolution is high enough to satisfy the conditions of numerical stability, () high spatial resolution is required for stable IB-LBM simulation of high Reynolds number flows, (3) IB- FDLBM can stably simulate flows at very high Reynolds numbers without increasing the spatial resolution, (4) IB-FDLBM gives reasonable predictions of the drag coefficient for 1 Re , and (5) IB-FDLBM gives accurate predictions for the motion of free-falling cylinders at intermediate Reynolds numbers. 7 Keywords: Immersed boundary method, Finite difference lattice Boltzmann method, High Reynolds number flow, Drag, Flows about cylinders NOMENCLATURE A negative viscosity term c lattice speed, c = 1. c S sound speed c discrete particle velocity, c = (c x, c y ) C D drag coefficient C L lift coefficient D particle diameter f particle velocity distribution function f eq equilibrium distribution function f q frequency of vortex shedding F external force F D drag force F L lift force F x component in the x-direction of F(X L,t) component in the y-direction of F(X L,t) F y * Corresponding Author: address: rojas@cfrg.scitec.kobe-u.ac.jp

2 8 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method g particle velocity distribution function including an additional collision term G i direct forcing term for f i m sign function N m number of Lagrangian points at immersed boundary p pressure Q number of discrete particle velocity R particle radius Re Reynolds number St Strouhal number t time T time scaled with U 0 and D u fluid velocity, u = (u, v) u P velocity of solid body U 0 free stream velocity W weighting function x Eulerian coordinates, x = (x, y) X L Coordinates of Lagrangian point, X L = (X L, Y L ) X P Coordinates of particle P, X P = (X P, Y P ) δ smoothed-delta function δ h one-dimensional smoothed-delta function S area segment of solid body t time step size T time interval scaled with U 0 and D V computational cell volume x lattice width in the x direction y lattice width in the y direction ν kinematic viscosity ρ density φ relaxation time for single-relaxation time model Ω domain Subscript i direction of discrete particle velocity I,J indexes of lattice point L index of Lagrangian node Superscript n discrete time T transpose 1. INTRODUCTION Fluid-solid interaction (FSI) problems are encountered in a wide variety of natural phenomena and engineering applications. The immersed-boundary method (IBM) has become a promising numerical tool for dealing with FSI including moving solid bodies. Several methods combining IBM and Lattice Boltzmann method (IB-LBM) have been proposed for predicting liquid-solid twophase flows [1-6]. These IB-LBMs, however, have some restrictions, e.g., (1) the lattice spacing and the time step size cannot be independently altered since the Lattice Boltzmann equation (LBE) requires a discrete particle to move from a lattice point to its neighbor during one time step, and () simulation of high Reynolds number flows with a low relaxation time is apt to be unstable so that the spatial resolution must be increased to perform stable computation at high Reynolds numbers [7,8]. Although several extensions of LBE have been made to overcome the numerical instability [9-11], these methods still have the former restriction. In our previous study [1], we therefore combined an immersed boundary method [13] and a finite difference lattice Boltzmann method (IB-FDLBM) [14] to remove the two restrictions. Using finite difference schemes to solve the discrete Boltzmann equation allows us to alter the time step size and the lattice spacing independently. In addition, the discrete Boltzmann equation includes an Journal of Computational Multiphase Flows

3 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama 9 additional collision term [15] which works as a negative viscosity in the macroscopic level and improves the numerical stability. IB-FDLBM gives good predictions for particles at intermediate Reynolds numbers such as steady flows past a stationary circular cylinder, a single particle falling in a liquid, and interaction of two falling particles. However the applicability of IB-FDLBM to particles at high Reynolds numbers has not been examined yet. Numerical simulations of high Reynolds number flows about circular cylinders are, therefore, carried out in this study using IB-FDLBM. Flows past a stationary circular cylinder are simulated for a wide range of Reynolds number, Re, i.e. 1 Re Predicted drag coefficients and Strouhal numbers are compared with available experimental and numerical data. Calculations using IB-LBM [6] are also carried out for comparison. Free-falling cylinders at intermediate and high Reynolds numbers are also simulated to demonstrate the applicability of the method to liquid-solid two-phase flows.. NUMERICAL METHODS IB-LBM proposed by Kang & Hassan [6] is used for comparing the applicability of IB-LBM to high Reynolds number flows with that of IB-FDLBM. The outlines of the two methods are described in the following..1 Immersed Boundary-Lattice Boltzmann Method The lattice Boltzmann equation is given by [6,15] 1 eq t fi( x+ ci t, t + t) = fi( x, t) [ fi( x,) t fi (,)] x t + G i i t t t G i t φ [ ( x+ c, + ) + ( x, )] (1) where f i is the particle velocity distribution function in the ith direction, f i eq the equilibrium distribution function, x the position vector, t the time, c i the discrete particle velocity, t the time step size, φ the relaxation time and G i the direct-forcing term acting on the fluid in the vicinity of immersed boundaries to impose the boundary condition. The macroscopic fluid density, ρ, and velocity, u, are given by ρ= f i ρu= Q 1 i= 0 Q 1 cifi i= 0 () (3) where Q is the number of discrete velocities. The two-dimensional nine-velocity (DQ9) model is used in this study for the discrete velocity. The equilibrium distribution function of this model is given by eq i i f W (4) i = iρ ( c u) ( c u) 3u u 4 c c c where W i is the weighting function and c the lattice velocity (c = 1). The values of c i and W i are summarized in Table 1, where c x and c y are the components of c i in the Cartesian coordinates (x, y). Applying the Chapman-Enskog expansion to Eq. (1) recovers the macroscopic conservation equations for incompressible Newtonian fluids with second-order accuracy: u = 0 (5) u + = p F u u + u + u T ν [ ( ) ] + t ρ ρ (6) where the superscript T denotes the transpose, F the external force acting on the fluid at the Volume 5 Number 1 013

4 30 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method immersed boundaries to satisfy the boundary condition and p the pressure given by p = ρc /3, which gives the speed of sound, c S = c / 3. The kinematic viscosity, ν, is given by c t 1 ν = φ 3 (7) Since t and c are equal to 1, Eq. (7) means φ = 3ν + 1/. Solution of Eq. (1) is split into the following four steps [6]: (I) First-forcing step t fi'( x, t) = fi( x, t) + G i( x, t ) (8) (II) Collision step 1 eq fi"( x, t) = fi'( x,) t [ fi'(,) x t fi '( x, t)] φ (9) (III) Second-forcing step t fi'''( x, t) = fi"( x, t) + G i( x, t ) (10) (IV) Streaming step f ( x+ c t, t + t) = f '''( x, t) i i i (11) where f, f and f are temporary distribution functions. Fluid motion is computed at Eulerian grid points using Eqs. (8)-(11), whereas solid boundaries immersed in a flow field are represented using Lagrangian points. The Eulerian coordinates at the grid point IJ and the coordinates of the Lth Lagrangian point are denoted by x IJ = (x I, y J ) and X L = (X L, Y L ), respectively. The external force, F(X L, t), acting on the fluid to impose the no-slip boundary condition is given by [6] u FX (, t) = ρ L ( X, t) u( X, t) t P L L (1) where u P (X L, t) is the velocity at the Lth Lagrangian point, and u(x L, t) is the fluid velocity interpolated by using ux ( L, t) = ux ( IJ, t) δ( xij XL) V xij Ω (13) Table 1 Discrete particle velocity and weighting coefficients in the DQ9 model i c x 0 c 0 -c 0 c -c -c c c y 0 0 c 0 -c c c -c -c W i 4/9 1/9 1/9 1/9 1/9 1/36 1/36 1/36 1/36 Journal of Computational Multiphase Flows

5 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama 31 where δ is the smoothed-delta function, V the volume of a computational cell and Ω the domain in which δ 0. The delta function is given by xi XL yj YL δ( xij XL ) = δh δh x y (14) where x and y are the lattice spacings in the x and y directions, respectively, and δ h is the onedimensional smoothed-delta function. There are several smoothed-delta functions such as fourpoint stencil piecewise, four-point cosine or three-point delta functions. It has been pointed out that the following two-point stencil delta function [16] gives better predictions than the other wider stencil functions [6,17]: δ h () r = 1 r for r 1 0 for r > 1 (15) Therefore Eq. (15) is used in the following simulations. The force calculated using Eq. (1) is distributed onto the Eulerian grid points by using the delta function: N m F( xij, t) = F( XL, t) δ( xij XL ) S x L = 1 (16) where N m is the number of Lagrangian points and S the area segment of a solid body, e.g., S = πr/n m for a circular cylinder of radius R and unit length. The discrete forcing term is given by G ( x, t) = 3Wc F( x, t) i IJ i i IJ (17) The necessary and sufficient condition for numerical stability is [18]: φ> 1 (18) As an example, let us consider a flow past a two-dimensional circular cylinder. The Reynolds number, Re, is defined by Re = U 0 D/ν, where U 0 is the free stream velocity and D the cylinder diameter, which is equal to the number of lattice spacings assigned to the cylinder diameter when x = 1. The velocity is also bounded by U 0 /c S < 0.3 [18] for satisfying the condition of incompressibility. The viscosity must be decreased to increase Re at given U 0 and D. As can be understood from Eq. (7), this reduction makes φ approaching to 1/ and the computation becomes unstable as ν decreases. Hence a large D, i.e., high spatial resolution, is required to satisfy Eq. (18) at high Reynolds numbers.. Immersed Boundary-Finite Difference Lattice Boltzmann Method Cao et al. [8] proposed to solve the discrete Boltzmann equation instead of Eq. (1) by using finite difference schemes. The discrete Boltzmann equation in IB-FDLBM is given by [1,14] fi + = 1 g f t eq c f + i i i i Gi φ ( ) (19) where g i is the distribution function including an additional collision term for numerical stability: Volume 5 Number 1 013

6 3 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method g A f f f eq = φ ( ) i i i i (0) where A is a parameter, which works as a negative viscosity in the macroscopic level. The macroscopic fluid density, velocity and the equilibrium distribution function are given by Eqs. (), (3) and (4), respectively. The macroscopic conservation equations for incompressible Newtonian fluids, Eqs. (5) and (6), are also recovered from Eq. (19), whereas the kinematic viscosity is given by c ν = A (1) 3 ( φ ) The lattice velocity, c, is equal to 1 and A is 1/ in the present simulations. Equation (1) therefore implies φ = 3ν + 1/. Treatment of immersed boundaries is similar to that explained in the previous section, whereas the external force is evaluated by [13] u FX (, t) = ρ L ( X, t) u( X, t) t P L L () The second-order Runge-Kutta scheme is used for the time integration of Eq. (19): f n+ 1 / i n t 1 eq = fi + ci gi ( fi fi ) + Gi φ n+ 1 n 1 eq fi = fi + t ci gi fi fi + Gi φ ( ) n n+ 1 / (3) (4) where the superscript n is the discrete time, i.e. t = n t. Effects of difference schemes for the advection term, i.e. the second-order central difference scheme, first- and second-order upwind schemes, on planar Couette flows have been examined. The third-order upwind scheme is used in this study because it gives the most accurate prediction: the first-order derivative of g i in the x direction is given by g = + + m g x y g x y g x y g i i( I m, J) 6 i( I m, J) 3 i( I, J) i( xi+ m, yj) x 6 x (5) where m is the sign function: m = 1 if c x > 0, otherwise m = 1. The calculation for the y direction is similar. The condition of numerical stability is given by φ> t (6) Let us again consider the flow past a cylinder. The condition for incompressibility, U 0 /c S < 0.3, must be satisfied in IB-FDLBM as well as in IB-LBM. High spatial resolution is required in IB- LBM to increase Re since the reduction of ν makes φ approaching to 1/. On the other hand, IB- FDLBM can make the relaxation time high enough to satisfy the stability condition, Eq. (6), even at low ν by increasing A. Therefore, IB-FDLBM may be able to simulate high Reynolds number flows without increasing the spatial resolution. Journal of Computational Multiphase Flows

7 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama FLOW PAST A STATIONARY CIRCULAR CYLINDER 3.1 Numerical conditions Flow past a stationary circular cylinder is calculated to examine the applicability of IB-LBM and IB-FDLBM to a wide range of Re. Two-dimensional simulations are carried out for 1 Re Table summarizes drag coefficients, C D, and Strouhal numbers, St, at Re = 100, 1000 and predicted by using two-dimensional simulations by finite-difference and finite element methods [19-1,6,7]. The predictions at Re = 100 agree well with the measured data [-4]. Mansy et al. [5] showed that small-scale three-dimensional disturbances appear in the near wake region above Re = 160~00. The two-dimensional simulation, of course, cannot reproduce the three-dimensional flow structure. Therefore the predictions at Re = 1000 and [0,1,6,7] differ from the measured data. However these two-dimensional simulations give more or less the same C D and St, and little depend on Re. This is because the separation point of flow at the cylinder surface is not altered at Re = 1000 and Thus, the drag at high Reynolds numbers mainly consists of form drag and the contribution of three-dimensional structure to the drag coefficient may become weaker and weaker as Re increases. The computational domain is shown in Fig. 1. The dimensions of the domain are 50D and 40D in the x and y directions. The cylinder is located at (0D, 0D) and D/ x = 40. The computational grid is uniform and x = y = 1. The number of Lagrangian points is 16 and these points are evenly distributed along the cylinder surface. The left boundary is uniform inflow. The right, top and bottom boundaries are continuous outflow. The value of A is 0.5 in all the IB-FDLBM simulations. The time steps are 1.0 and 0.5 in IB-LBM and IB-FDLBM simulations, respectively. Table Comparison of C D and St of flow past a stationary circular cylinder for 10 Re 10 4 Re C D St FDM, D* [19] FDM, D [0] FDM, D [1] Measured [,3,4] FDM, D [0] FDM, D [1] FEM, D** [6] FEM, D [7] Measured [,3,4] FDM, D [1] FEM, D [7] Measured [,3,4] *FDM: finite difference method; **FEM: finite element method Volume 5 Number 1 013

8 34 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method 50D Uniform inflow at U 0 y (0D,0D) Cylinder 40 x for D N m =16 40D (0,0) x Continuous outflow Figure 1. Computational domain for flows past a circular cylinder The drag, F D, and lift forces, F L, are calculated by N m FD = Fx( X L, t) S x L = 1 N m FL = Fy( X L, t) S x L = 1 (7) (8) where F(X L, t) = (F x (X L, t), F y (X L, t)). The drag and lift coefficients, C D and C L, are defined by C C D L FD = 1 ρud 0 FL = 1 ρud 0 (9) (30) The Strouhal number is defined by St = fd q U 0 (31) where f q is the frequency of vortex shedding, which is calculated from the time evolution of the lift coefficient. 3. Results of IB-LBM The drag coefficients predicted by IB-LBM are plotted against the Reynolds number in Fig.. Good agreement between measured [] and predicted C D is obtained up to Re = 100. For 500 < Re < 000, the predicted drag coefficients differ from the measured data. However, the drag coefficient of flows at Re = 1000 agrees well with other predictions obtained using the twodimensional simulations with FDM and FEM [0, 1, 6, 7]. When Re > 000, numerical instability arises and converged solutions are not obtained with the spatial resolution, D/ x = 40. In the stable computation at Re = 000, φ = To keep φ = also at Re = 500, the spatial resolution must be increased to D/ x = 50. Likewise, a spatial resolution must be D/ x = 000 at Journal of Computational Multiphase Flows

9 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama 35 Re = Thus the computational costs drastically increases with Re. Figure 3 shows the predicted Strouhal number. At Re = 100, the prediction agrees well with the experimental data [3]. The predicted Strouhal numbers for Re 500 differ from the experimental data [3, 4], though the Strouhal number at Re = 1000 is similar to those predicted with the other methods Measured (Wieselsberger, 19) FDM (Lei et al., 000) FDM (Matsumiya et al., 1993) IB-LBM (present) Unstable C D Figure. Drag coefficient of a stationary cylinder (IB-LBM) Measured (Roshko, 1954) Measured (Schewe, 1983) FDM (Lei et al., 000) FDM (Matsumiya et al., 1993) IB-LBM (present) Unstable St Figure 3. Strouhal number of a stationary cylinder (IB-LBM) 3.3 Results of IB-FDLBM Figure 4 shows the drag coefficient predicted by using IB-FDLBM. Good agreements are obtained up to Re = 100. Similar to the result of IB-LBM, the predicted drag coefficients slightly differ from the experimental data at Re = 500 and However the predicted drag coefficient at Re = 1000 agree well with the numerical predictions [0, 1, 6, 7]. The flows at the higher Reynolds numbers, Re = and , are successfully simulated without any numerical instability and the predicted drag coefficients agree well with the measured data. The predicted Strouhal number is larger than the measured data as shown in Fig. 5. Three-dimensional simulation should be done for obtaining better prediction of St in a turbulent flow. Volume 5 Number 1 013

10 36 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method The flows about the cylinder in different regimes are shown in Figs Creeping flow is obtained at Re = 1. Two stationary eddies are formed behind the cylinder at Re = 40. These flow patterns are in good agreement with experimental observations [8]. Figure 7 shows the streamlines of the flow at Re = 100 at two instants. The flow is unsteady and periodic vortex shedding (Karman vortex street) takes place behind the cylinder. The streamlines of flows at Re = show irregular vortex shedding. In reality, flows at this Reynolds number possess more vortical structures with three-dimensional fluctuations [9], which are, of course, not obtained with the present twodimensional simulation Measured (Wieselsberger, 19) FDM (Lei et al., 000) FDM (Matsumiya et al., 1993) IB-FDLBM (present) C D Re Figure 4. Drag coefficient of a stationary cylinder (IB-FDLBM) Measured (Roshko, 1954) Measured (Schewe, 1983) FDM (Lei et al., 000) FDM (Matsumiya et al., 1993) IB-FDLBM (present) St Re Figure 5. Strouhal number of a stationary cylinder (IB-FDLBM) Journal of Computational Multiphase Flows

11 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama (a) Re = 1 37 (b) Re = 40 Fig. 6 Streamlines of steady flows about a circular cylinder at intermediate Re (a) Dimensionless time U0 t/d = (b) U0 t/d = 1.5 Fig. 7 Streamlines about circular cylinder at Re = 100. (b) U0 t/d = 1.5 (a) Dimensionless time U0 t/d = Figure 8. Streamlines about circular cylinder at Re = CIRCULAR CYLINDERS FALLING FREELY AT INTERMEDIATE AND HIGH REYNOLDS NUMBERS The motion of a free-falling circular cylinder is simulated to examine the applicability of IBFDLBM to moving boundary problems. It is known that the Strouhal number of a free-falling cylinder is lower than that of a stationary cylinder at the same Reynolds number [30], that is, the vortex shedding is retarded due to the cylinder motion. Namkoong et al. [30] carried out simulations of free-falling cylinders at intermediate Reynolds numbers (Re < 188) using a finite element method and the mechanism of the St reduction has been described as follows: the cylinder tends to move toward the low-pressure side, where a vortex is generated and then separates, so that the pressure on this side recovers with the retardation of vortex separation. Simulations of freefalling cylinders at intermediate Re are first carried out using IB-FDLBM to examine whether or not IB-FDLBM can reproduce the St reduction. The numerical setup is similar to that shown in Fig. 1, whereas the domain size is the same as that used in Namkoong et al., i.e., 70D and 100D in the x and y directions. The initial cylinder position is (0D, 50D). The computational grid is uniform near the cylinder, whereas it is non-uniform in the other region. The dimensions of the uniform grid region are 5D and D in the x and y directions, and the spatial resolution of this region is D = 40 x ( x = y = 1). The number of Lagrangian points is 16. Uniform flow in the x direction enters from the left boundary and crosses the cylinder. First the cylinder is fixed at the initial position until the Volume 5 Number 1 013

12 38 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method flow reaches a quasi-steady state. Then, the constraint on cylinder motion is removed and the translation and rotation of the cylinder are calculated by solving the equations of motion. The solution algorithm of the equation of motion can be found in [1]. The acceleration of gravity directing toward x is added and its value is set so as to balance the gravitational force with the drag force. Hence the motion of the free-falling cylinder is observed in the frame of reference moving with the cylinder velocity. The Reynolds numbers simulated are 96, 119, 138 and 156. Figure 9 shows a comparison of the Strouhal numbers of stationary and free-falling cylinders. The predicted Strouhal numbers of the stationary cylinder agree very well with those measured by Roshko [3]. The Strouhal number of the free-falling cylinder takes lower values than that of the stationary cylinder and agrees well with the predictions by Namkoong et al. Then, a circular cylinder falling at a high Reynolds number is simulated. The numerical conditions are similar to those in section 3: the domain size is 50D and 40D in the x and y directions, the initial cylinder position is (0D, 0D), D/ x = 40 and the number of Lagrangian points is 16. The Reynolds number is The cylinder oscillates due to the vortex shedding. The amplitude of the oscillation is not constant because the flow fluctuation at this Re is irregular. The averaged drag coefficient is approximately 1., which is the same as that of the stationary cylinder. The predicted Strouhal number of the stationary cylinder is 0.34 while St for the free-falling cylinder Stationary (Measured by Roshko, 1961) Freely falling (Namkoong et al., 008) Stationary (Present) Freely falling (Present) St Re Figure 9. Strouhal numbers of stationary and free-falling cylinders The processes of vortex shedding for the stationary cylinder and for the free-falling cylinder are shown in Figs. 10 (a) and (b), respectively. The cross symbol represents the initial position of cylinder center. The time interval, T, scaled with U 0 and D is 1.5. In the case of the stationary cylinder, the negative vortex (clockwise vortex represented with blue color) behind the cylinder is elongated in the streamwise direction at the dimensionless time T and is shed downstream when the positive vortex (counter-clockwise vortex represented with red color) is formed (T + T). The positive vortex is elongated (T + T) and is shed when the new negative vortex is formed (T + 3 T). Then the next shedding cycle starts (T + 4 T). Thus the period of vortex shedding is about 3 T. The vortex-shedding process for the free-falling cylinder is similar to that of the stationary cylinder, whereas the cylinder motion affects the vortex motion, i.e., each vortex is more elongated compared with that of the stationary cylinder. The period of vortex-shedding process is also changed to about 4 T. Journal of Computational Multiphase Flows

13 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama Dimensionless time T T T + T T + T T + T T + T T + 3 T T + 3 T T + 4 T (a) Stationary cylinder T + 4 T (b) Free-falling cylinder Figure 10. Dimensionless vorticity fields about a stationary and free-falling cylinders at Re= CONCLUSION Two-dimensional simulations of flows about a circular cylinder were carried out using the immersed boundary-finite difference lattice Boltzmann method (IB-FDLBM) to investigate its applicability to high Reynolds number liquid-solid two-phase flows. For flows past a stationary Volume 5 Number 1 013

14 40 Numerical Simulation of Flows about a Stationary and a Free-Falling Cylinder Using Immersed Boundary-Finite Difference Lattice Boltzmann Method cylinder, a wide range of Reynolds number was tested, i.e., 1 Re An immersed boundary-lattice Boltzmann method (IB-LBM) was also used for comparison. The drag coefficients and Strouhal numbers obtained were compared with experimental data and available numerical predictions. Simulations of free-falling cylinders at intermediate and high Reynolds numbers were also performed. As a result, the following conclusions are obtained: (1) Steady and unsteady flows about a stationary cylinder are well predicted with IB-LBM and IB-FDLBM, provided that the spatial resolution is high enough to satisfy the conditions of numerical stability. () High spatial resolution is required for stable IB-LBM simulation of high Reynolds number flows. (3) IB-FDLBM can stably simulate flows at very high Reynolds numbers without increasing the spatial resolution. (4) IB-FDLBM gives reasonable predictions of the drag coefficient for 1 < Re < (5) IB-FDLBM gives accurate predictions for the motion of free-falling cylinders at intermediate Reynolds numbers. ACKNOWLEDGEMENTS This work has been supported by the Japan Society for the Promotion of Science (JSPS) (grantsin-aid for scientific research (B) No ). REFERENCES [1] Feng, Z.-G., Michaelides, E. E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, Journal of Computational Physics, 004, 195, [] Feng, Z.-G., Michaelides, E. E., Proteus: a direct forcing method in the simulations of particulate flows, Journal of Computational Physics, 005, 0, [3] Feng, Z.-G., Michaelides, E. E., Robust treatment of no-slip boundary condition and velocity updating for the lattice- Boltzmann simulation of particulate flows, Computers and Fluids, 009, 38, [4] Wu, J., Shu, C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications, Journal of Computational Physics, 009, 8, [5] Lin, S.-Y., Lin, C.-T., Chin, Y.-H., Tai, Y.-H., A direct-forcing pressure-based lattice Boltzmann method for solving fluid-particle interaction problems, International Journal for Numerical Methods in Fluids, 010, DOI: /fld.44. [6] Kang, S. K., Hassan, Y. A., A comparative study of direct-forcing immersed boundary-lattice Boltzmann methods for stationary complex boundaries, International Journal for Numerical Methods in Fluids, 010, DOI: /fld.304. [7] Sterling, J.D., Chen, S., Stability analysis of lattice Boltzmann methods, Journal of Computational Physics, 1996, 13, [8] Cao, N., Chen, S., Jin, S., Martinez, D., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Physical Review E, 1997, 55(1), R1-R4. [9] Lallemand, P., Luo, L.S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Physical Review E, 000, 61, [10] Karlin, I.V., Gorban, A.N., Succi, S., Boffi, V., Maximum entropy principle for lattice kinetic equations, Physical Review Letters, 1998, 81, 6-9. [11] Shu, C., Niu, X.D., Chew, Y.T., Cai, Q.D., A fractional step lattice Boltzmann method for simulating high Reynolds number flows, Mathematics and Computers in Simulation, 006, 7, [1] Rojas, R., Seta, T., Hayashi, K., Tomiyama, A., Immersed boundary-finite difference lattice Boltzmann method for liquid-solid two-phase flows, Journal of Fluid Science and Technology, 011, 6(6), [13] Dupuis, A., Chatelain, P., Koumoutsakos, P., An immersed boundary lattice-boltzmann method for the simulation of the flow past an impulsively started cylinder, Journal of Computational Physics, 008, 7, [14] Tsutahara, M., Kurita, M., Iwagami, T., A study of new finite difference lattice Boltzmann model, Transaction of Japanese Society of Mechanical Engineers, Series B, 00, 68(665), (in Japanese) [15] Chen, Y., Li, J., Introducing unsteady non-uniform source terms into the lattice Boltzmann model, International journal of Numerical Methods in Fluids, 008, 56, Journal of Computational Multiphase Flows

15 Roberto Rojas, Kosuke Hayashi, Takeshi Seta and Akio Tomiyama 41 [16] Yang, X., Zhang, X., Li, Z., He, G.W., A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, Journal of Computational Physics, 009, 8, [17] Hayashi, K., Rojas, R., Seta, T., Tomiyama, A., Immersed boundary-lattice Boltzmann method using two relaxation times, Journal of Computational Multiphase Flows, 01, 4 (), [18] Chen, S., Doolen, G. D., Lattice Boltzmann method for fluid flows, Annual Review of Fluid Mechanics, 1998, 30, [19] Park, J., Kwon, K., Choi, H., Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME International Journal, 1998, 1(6), [0] Lei, C., Cheng, L., Kavanagh K., A finite difference solution of the shear flow over a circular cylinder, Ocean Engineering, 000, 7, [1] Matsumiya, H., Kieda, K., Taniguchi, N., Kobayashi, T., Numerical simulation of D flow around a circular cylinder by the third-order upwind finite difference method, Transaction of Japanese Society of Mechanical Engineers, Series B, 1993, 59(566), (in Japanese) [] Wieselsberger, C., New data on the laws of fluid resistance, National Advisory Committee for Aeronautics, 19, 84. [3] Roshko, A., On the development of turbulent wakes from vortex streets, NACA, 1954, Rep [4] Schewe, G., On the force fluctuations acting on a circular cylinder in cross flow from subcritical up to transcritical Reynolds numbers, Journal of Fluid Mechanics, 1983, 133, [5] Mansy, H., Yang, P.M., Williams, D.R., Quantitative measurements of three-dimensional structures in the wake of a circular cylinder, Journal of Fluid Mechanics, 1994, 70, [6] Mittal, S., Kumar, V., Flow-induced vibrations of a light circular cylinder at Reynolds number 10 3 to 10 4, Journal of Sound and Vibration, 001, 45(5), [7] Kondo, N., Direct third-order upwind finite element simulation of high Reynolds number flows around a circular cylinder, Journal of Wind Engineering and Industrial Aerodynamics, 1993, 46&47, [8] Bachelor, G. K., An introduction to fluid dynamics, Cambridge University Press, [9] Van Dyke, M., An album of fluid motion, 198, Parabolic Press. [30] Namkoong, K., Yoo, J. Y., Choi, H., Numerical Analysis of Two-Dimensional Motion of a Freely Falling Circular Cylinder in an Infinite Fluid, Journal of Fluid Mechanics, 008, 604, Volume 5 Number 1 013