Commun. Comput. Phys., 12 (2012), pp. 544-576.


A Full Eulerian Fluid-Membrane Coupling Method with a Smoothed Volume-of-Fluid Approach

Satoshi Ii 1*, Xiaobo Gong 2, Kazuyasu Sugiyama 1, Jinbiao Wu 3, Huaxiong Huang 4, Shu Takagi 5

1 Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656, Japan.
2 Department of Engineering Mechanics, NAOCE, Shanghai Jiaotong University, Shanghai 200240, China.
3 LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China.
4 Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada.
5 Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656, Japan; and Computational Science Research Program, RIKEN, 2-1 Hirosawa Wako, Saitama, 351-0198, Japan.

Received 14 December 2010; Accepted (in revised version) 11 August 2011
Available online 20 February 2012
doi:10.4208/cicp.141210.110811s

Abstract

A novel full Eulerian fluid-elastic membrane coupling method on the fixed Cartesian coordinate mesh is proposed within the framework of the volume-of-fluid approach. The present method is based on a full Eulerian fluid-(bulk) structure coupling solver (Sugiyama et al., J. Comput. Phys., 230 (2011) 596-627), with the bulk structure replaced by elastic membranes. In this study, a closed membrane is considered, and it is described by a volume-of-fluid or volume-fraction information generally called VOF function. A smoothed indicator (or characteristic) function is introduced as a phase indicator which results in a smoothed VOF function. This smoothed VOF function uses a smoothed delta function, and it enables a membrane singular force to be incorporated into a mixture momentum equation. In order to deal with a membrane deformation on the Eulerian mesh, a deformation tensor is introduced and updated within a compactly supported region near the interface. Both the neo-Hookean and the Skalak models are employed in the numerical simulations. A smoothed (and less dissipative) interface capturing method is employed for the advection of the VOF function and the quantities defined on the membrane. The stability restriction due to membrane stiffness is relaxed by using a quasi-implicit approach. The present method is validated by using the spherical membrane deformation problems, and is applied to a pressure-driven flow with the biconcave membrane capsules (red blood cells).

AMS subject classifications: 74F10, 76M20, 74L15

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Key words: Fluid-membrane interaction, Eulerian formulation, Cartesian mesh, volume-of-fluid, quasi-implicit approach, red blood cell.

*Corresponding author.
Email: sii@fel.t.u-tokyo.ac.jp (S. Ii), x.gong@sjtu.edu.cn (X. Gong), sugiyama@fel.t.u-tokyo.ac.jp (K. Sugiyama), jwu@math.pku.edu.cn (J. Wu), hhuang@yorku.ca (H. Huang), takagi@mech.t.u-tokyo.ac.jp (S. Takagi)
 

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