Commun. Comput. Phys., 16 (2014), pp. 1181-1200.


A New Approach for Error Reduction in the Volume Penalization Method

Wakana Iwakami 1*, Yuzuru Yatagai 2, Nozomu Hatakeyama 3, Yuji Hattori 4

1 Yukawa Institute for Theoretical Physics, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan; Advanced Research Institute for Science & Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan.
2 Department of Applied Information Sciences, Graduate School of Information Sciences, Tohoku University, 6-3-09 Aoba, Aramaki-aza, Aoba-ku, Sendai, Miyagi 980-8579, Japan.
3 NICHe, Tohoku University, 6-6-10 Aoba, Aramaki-aza, Aoba-ku, Sendai, Miyagi 980-8579, Japan.
4 Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi, 980-8577, Japan.

Received 22 May 2013; Accepted (in revised version) 7 May 2014
Available online 29 August 2014
doi:10.4208/cicp.220513.070514a

Abstract

A new approach for reducing error of the volume penalization method is proposed. The mask function is modified by shifting the interface between solid and fluid by $\sqrt{\nu\eta}$ toward the fluid region, where $\nu$ and $\eta$ are the viscosity and the permeability, respectively. The shift length $\sqrt{\nu\eta}$ is derived from the analytical solution of the one-dimensional diffusion equation with a penalization term. The effect of the error reduction is verified numerically for the one-dimensional diffusion equation, Burgers' equation, and the two-dimensional Navier-Stokes equations. The results show that the numerical error is reduced except in the vicinity of the interface showing overall second-order accuracy, while it converges to a non-zero constant value as the number of grid points increases for the original mask function. However, the new approach is effective when the grid resolution is sufficiently high so that the boundary layer, whose width is proportional to $\sqrt{\nu\eta}$, is resolved. Hence, the approach should be used when an appropriate combination of $\nu$ and $\eta$ is chosen with a given numerical grid.

AMS subject classifications: 60-08

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Key words: Volume penalization method, immersed boundary method, compact scheme, error reduction.

*Corresponding author.
Email: wakana@heap.phys.waseda.ac.jp (W. Iwakami), yuzuru@dragon.ifs.tohoku.ac.jp (Y. Yatagai), hatakeyama@aki.niche.tohoku.ac.jp (N. Hatakeyama), hattori@fmail.ifs.tohoku.ac.jp (Y. Hattori)
 

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