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ICCM 2019
9th-13th July, Singapore, Singapore An implicit second order immersed boundary method An implicit second order immersed boundary method for viscous fluid flow simulation is proposed. The method is based on SIMPLE approach [1] for solution of fluid flow equation and IBM [2] for considering of fluid and solid body interaction. Due to rectangular staggered grid the mass conservation law is satisfied exactly while assuming constant velocity components along the mesh edges. The classical 5-point difference and the central differences are used for Laplace operator and convective terms respectively. Combined with one layer of fictitious cells it provides the second order of approximation of the boundary conditions and fluid flow equations. In contrast to the widely used methods [3-5] with Adams- Bashforth method used for approximation of convective terms Crank-Nicolson approach is used to increase the stability. Also the iteration procedure is used instead of Runge-Kutta method for time integration. It increases the method stability and guarantee that boundary conditions at the walls and immersed boundary are satisfied at each time step for the final velocity distribution after the pressure correction. The pressure term in the momentum equation is taken from "n" and "n+1" steps to provide the second order approximation. Due to using the iteration procedure it is possible to apply simple direct forcing instead of multi-direct forcing [4] to reduce the computation time. The proposed version of the IBM is written without any additional force terms introduced into fluid flow equations. Due to that there is no need to care about approximation of such source terms in time and the error that is introduced by the immersed boundary is caused by approximation in space only. The proposed method has been verified on the problems with analytical solution: Taylor-Green and Lamb-Oseen vortices with Reynolds number varied from 1 to 100. To estimate the influence of immersed boundary on the method convergence the solid body rotated with prescribed speed is placed into the center of the computational domain. The results of simulations of stationary and non-stationary flow around the cylinder were compared with the results of other authors.This research was supported by the Russian Science Foundation (project No. 17-71-20139). References [1] Patankar S. (1980) Numerical Heat Transfer and Fluid Flow, Taylor & Francis. [2] Peskin C. S. (1972) Flow patterns around heart valves: a numerical method J. Comput. Phys. 10. 252-271 [3] Uhlmann M. (2005) An immersed boundary method with direct forcing for the simulation of particulate flows J. Comp. Phys. 209(2). 448-476. [4] Breugem W.-P. (2012) A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows J. Comp. Phys. 231. 4469-4498. [5] Kempe T., Lennartz M., Schwarz S., Frohlich J. (2015) Imposing the freeslip condition with a continuous forcing immersed boundary method J. Comp. Phys. 282 183-209 |