Abstract
In the present study, a new immersed boundary technique is proposed for the simulation of two-dimensional viscous incompressible flow interacting with moving solid boundary. The numerical integration is based on a second-order fraction step method under the staggered grid spatial framework. Base on the direct momentum forcing on the Cartesian grid, a “Solid-body-forcing” procedure is used to keep a suitable velocity field in the solid domain. Five different test problems are simulated using the present technique (flows over an asymmetrically placed in a channel, in-line oscillating cylinder and transverse oscillation cylinder in a free stream, in-line oscillating cylinder in a fluid at rest, two cylinders moving with respect to each other). Two forcing strategies, including extrapolation (Scheme 1) and interpolation (Scheme 2) are used in the stationary boundary problems and get good results. However, the two forcing strategies can not predict the flow field adequately for moving boundary problems. It can be observed that server oscillations of the predicted lift and drag coefficients occur when using the interpolation or extrapolation procedures. In order to quantify oscillations, “Fourier Series Expansion” was used to analyze the predicted lift and drag coefficients. It provides a criterion to judge the difference between Fourier series forms and original ones. A “Solid-body-forcing” procedure can modify the velocity field inside the boundary when it moves, and satisfying result can be observed when the forcing strategy is using a combination of interpolation and Solid-body-forcing procedure. Besides, a simple interaction between two cylinders with different paths can also be predicted by the present method, indicating the usability of the present method for various moving solid boundary.

Abstract
In the present study, a new immersed boundary technique is proposed for the simulation of two-dimensional viscous incompressible flow interacting with moving solid boundary. The numerical integration is based on a second-order fraction step method under the staggered grid spatial framework. Base on the direct momentum forcing on the Cartesian grid, a “Solid-body-forcing” procedure is used to keep a suitable velocity field in the solid domain. Five different test problems are simulated using the present technique (flows over an asymmetrically placed in a channel, in-line oscillating cylinder and transverse oscillation cylinder in a free stream, in-line oscillating cylinder in a fluid at rest, two cylinders moving with respect to each other). Two forcing strategies, including extrapolation (Scheme 1) and interpolation (Scheme 2) are used in the stationary boundary problems and get good results. However, the two forcing strategies can not predict the flow field adequately for moving boundary problems. It can be observed that server oscillations of the predicted lift and drag coefficients occur when using the interpolation or extrapolation procedures. In order to quantify oscillations, “Fourier Series Expansion” was used to analyze the predicted lift and drag coefficients. It provides a criterion to judge the difference between Fourier series forms and original ones. A “Solid-body-forcing” procedure can modify the velocity field inside the boundary when it moves, and satisfying result can be observed when the forcing strategy is using a combination of interpolation and Solid-body-forcing procedure. Besides, a simple interaction between two cylinders with different paths can also be predicted by the present method, indicating the usability of the present method for various moving solid boundary.

Reference:

[1.] Ye T, Mittal R, Udaykumar HS, Shyy W. An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J Comp Phys 1999;156:209.
[2.] LeVeque RJ, Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 1994;31:1019.
[3.] LeVeque RJ, Li Z. Immersed interface method for Stokes flow with elastic boundaries or surface tension, SIAM J Sci Comput 1997;18:709.
[4.] Calhoun D. A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions. J Comp Phys 2002;176:231.
[5.] Li Z, Lai M-C. The immersed interface method for the Navier–Stokes equations with singular forces. J Comp Phys 2001;171:822.
[6.] Li Z. An overview of the immersed interface method and its applications. Taiwanese J Math 2003;7:1.
[7.] Peskin CS. Flow patterns around heart valves: a numerical method. J Comp Phys 1972;10:252.
[8.] Peskin CS. The fluid dynamics of heart valves: Experimental, theoritiacal and computational methods. Annual Review of Fluid Mechanics 1982;14:235
[9.] Beyer RP, LeVeque RL. Analysis of a one-dimensional model for the immersed boundary method. SIAM J Numer Anal 1992;29:332.
[10.] Lai M-C, Peskin CS. An immersed boundary method with formal second order accuracy and reduced numerical viscosity. J Comp Phys 2000;160:705.
[11.] Goldstein D, Handler R, Sirovich L. Modeling a no-slip flow with an external force field. J Comp Phys 1993;105:354.
[12.] Goldstein D, Hadler R, Sirovich L. Direct numerical simulation of turbulent flow over a modeled riblet covered surface. J Fluid Mech 1995;302:333.
[13.] Lima E Silva ALF, Silveira-Neto A, Damasceno JJR. Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method. J Comp Phys 2003;189:351.
[14.] Saiki EM, Biringen S. Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method. J Comp Phys 1996;123:450.
[15.] Mittal R, Iaccarino G. Immersed boundary methods. Annual Review of Fluid Mechanics 2005;37:239-261.
[16.] Mohd-Yusof J. Combined immersed boundary/B-Spline method for simulations of flows in complex geometries in complex geometries CTR annual research briefs. NASA Ames/Stanford University;1997.
[17.] Verzicco R, Mohd-Yosuf J, Orlandi P, Haworth D. LES in complex geometries using boundary body forces. AIAA Journal 2000;38:427-433.
[18.] Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary methods for three dimensional complex flow simulations. J Comp Phys 2000;161:30.
[19.] Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Computer and Fluids 2004;33:375-404.
[20.] Tseng YH, Ferziger JH. A ghost-cell immersed boundary method for flow in complex geometry. J Comp Phys 2003;192:593.
[21.] Kim J, Kim D, Choi H. An immersed-boundary finite-volume method for simulations of flow in complex geometries. J Comp Phys 2001;171:132.
[22.] Su SW, Lai MC, Lin CA. A simple immersed boundary technique for simulating complex flows with rigid boundary. Computers and Fluids 2007;36:313-324.
[23.] Yang J, Balaras E. An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries. J Comp Phys 2006;215:12-24.
[24.] Tyagi M, Acharya S. Large eddy simulation of turbulent flows in complex and moving rigid geometries using the immersed boundary method. J Numer Meth Fluids 2005;48:691-722.
[25.] Udaykumar HS, Mittal R, Shyy W. Computation of solid–liquid phase fronts in the sharp interface limit on fixed grids, J Comp Phys 1999;153:535-574.
[26.] Udaykumar HS, Mittal R, Rampunggoon P, Khanna A. A sharp interface Cartesian grid method for simulating flows with complex moving boundaries. J Comp Phys 2001;174:345-380.
[27.] Marella S, Krishnan S, Liu H, Udaykumar HS. Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations. J Comp Phys 2005;210:1-31.
[28.] Kim D, Choi H. Immersed boundary method for flow around an arbitrarily moving body. J Comp Phys 2006;212:662-680.
[29.] Xu S, Wang ZJ. An immersed interface method for simulating the interaction of a fluid with moving boundaries. J Comp Phys 2006;216:454-493.
[30.] Harlow FH and Welsh JE. Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys Fluids 1965;8:2181-2189.
[31.] Choi H and Moin P. Effiects of the Computational Time Step on Numerical Solutions of turbulent flow, J Comp Phys 1994;113:1.
[32.] Chorine AJ. Numerical solution of the Navier-stokes equations. Math Comp 1968;22:745.
[33.] Van den Vorst HA, Sonneveld P. CGSTAB, a more smoothly converging variant of CGS. Technical Report 90-50, Delft University of Technology 1990.
[34.] Schafer M, Turek S. Flow simulation with High-Performance Computer II, Notes in Numerical Fluid Mechanics 1996.
[35.] Tanida Y, Okajima A, Watanabe Y. Stability of a circular cylinder oscillating in uniform flow or in a wake. J Fluid Mech 1973;61:769.
[36.] Griffin OM, Ramberg SE. Vortex shedding from a cylinder vibrating in line with an incident uniform flow. J Fluid Mech 1976;75:257.
[37.] Hurlbut SE, Spaulding ML, White FM. Numerical solution for laminar two dimensional flow about a cylinder oscillating in a uniform stream. Trans ASME, J Fluids Eng 1982;104:214.
[38.] Dutsch H, Durst F, Becker S, Lienhart H. Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers. J Fluid Mech 1998;360:249-271.
[39.] Russell D, Wang ZJ. A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow. J Comp Phys 2003;191:177-205.