Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T08:58:32.645Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 August 2015

Satoshi Ii ,
Xiaobo Gong ,
Kazuyasu Sugiyama ,
Jinbiao Wu ,
Huaxiong Huang  and
Shu Takagi
Satoshi Ii*
Affiliation:
Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656, Japan
Xiaobo Gong*
Affiliation:
Department of Engineering Mechanics, NAOCE, Shanghai Jiaotong University, Shanghai 200240, China
Kazuyasu Sugiyama*
Affiliation:
Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656, Japan
Jinbiao Wu*
Affiliation:
LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China
Huaxiong Huang*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada
Shu Takagi*
Affiliation:
Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656, Japan Computational Science Research Program, RIKEN, 2-1 Hirosawa Wako, Saitama, 351-0198, Japan
*
Corresponding author.Email:sii@fel.t.u-tokyo.ac.jp
Email:x.gong@sjtu.edu.en
Email:sugiyama@fel.t.u-tokyo.ac.jp
Email:jwu@math.pku.edu.cn
Email:hhuang@yorku.ca
Email:takagi@mech.t.u-tokyo.ac.jp
Get access

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 consid-ered, 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).

Keywords

74F1076M2074L15Fluid-membrane interactionEulerian formulationCartesian meshvolume-of-fluidquasi-implicit approachred blood cell
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 544 - 576
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barthés-Biesel, D. and Rallison, J.M., The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid. Mech., 113 (1981) 251–267.Google Scholar
[2]Barth, T.J. and Frederickson, P.O., High-Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper, 90–0013 (1990).Google Scholar
[3]Brackbill, J.U., Kothe, D.B., Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100 (1992) 335–354.Google Scholar
[4]Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng., 99(1992) 235–394.Google Scholar
[5]Ceniceros, H.D., Fisher, J.E., Roma, A.M., Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method, J. Comput. Phys., 228 (2009) 7137–7158.CrossRefGoogle Scholar
[6]Cottet, G.H. and Maitre, E., A level set method for fluid-structure interactions with immersed surfaces, Math. Model. Meth. Appl. Sci., 16 (2006) 415–438.Google Scholar
[7]Cottet, G.H., Maitre, E., Milcent, T., Eulerian formulation and level set models for incompressible fluid-structure interaction, Math. Modeling and Numer. Anal., 42 (2008) 471–492.Google Scholar
[8]Cummins, S.J., Francois, M.M., Kothe, D.B., Estimating curvature from volume fractions, Comput. Structures, 83 (2005) 425–434.Google Scholar
[9]Dunne, T., An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaptation, Int. J. Numer. Meth. Fluids, 51 (2006) 1017–1039.Google Scholar
[10]Eggleton, C.D. and Popel, A.S., Large deformation of red blood cell ghosts in a simple shear flow, Phys. Fluids, 10 (1998) 2182–2189.Google Scholar
[11]Evans, E., Fung, Y., Improved measurement of the erythrocyte geometry, Microvasc Res., 4 (1972) 335–347.Google Scholar
[12]Fauci, L., Peskin, C.S., A computational model of aquatic animal locomotion, J. Comput. Phys., 77 (1988) 86–108.Google Scholar
[13]Ferdowsi, P.A., Bussmann, M., Second-order accurate normals from height functions, J. Comput. Phys., 227 (2008) 9293–9302.Google Scholar
[14]Fogelson, A.L., A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 56 (1984) 111–134.Google Scholar
[15]Fogelson, A.L., Continuum models of platelet aggregation: formulation and mechanical properties, SIAM J. Appl. Math., 52 (1992) 1089–1110.Google Scholar
[16]Francois, M.M., Cummins, S.J., Dendy, E.D., Kothe, D.B., Sicilian, J.M., Williams, M.W., A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys., 213 (2006) 141–173.Google Scholar
[17]Gaehtgens, P., Dührssen, C. and Albrecht, K.H., Motion, deformation, and interaction of blood cells and plasma during flow through narrow capillary tubes. Blood Cells, 6 (1980) 799–817.Google Scholar
[18]Glimm, J., Grove, J.W., Li, X.L., Shyue, K.M., Zeng, Y.N. and Zhang, Q., Three-dimensional front tracking, SIAM J. Sci. Comp., 19 (1998) 703–727.Google Scholar
[19]Gong, X., Sugiyama, K., Takagi, S., Matsumoto, Y., The deformation behavior of multiple red blood cells in a capillary vessel, J. Biomech. Eng., 131 (2009) 074504.Google Scholar
[20]Harlow, F.H., Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (1965) 2182–2189.Google Scholar
[21]Hou, T.Y., Shi, Z., Removing the stiffnessof elastic force from the immersed boundary method for 2D Stokes equations, J. Comput. Phys., 227 (2008) 9138–9169.Google Scholar
[22]Hou, T.Y., Shi, Z., An efficient semi-implicit immersed boundary method for the Navier-Stokes equations, J. Comput. Phys. 227 (2008) 8968–8991.Google Scholar
[23]Hirt, C.W., Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39 (1981) 201–225.Google Scholar
[24]Ii, S. and Xiao, F., A simplified interface capturing scheme using a continuous function, Proc. of SNA+MC 2010, (2010).Google Scholar
[25]Ii, S., Sugiyama, K., Takeuchi, S., Takagi, S. and Matsumoto, Y., Animplicit full Eulerian method for the fluid-structure interaction problem, Int. J. Numer. Meth. Fluids, 65 (2011) 150–165.Google Scholar
[26]Ii, S., Sugiyama, K., Takeuchi, S., Takagi, S., Matsumoto, Y. and Xiao, F., An interface capturing method with a continuous function: the THINC method with multi-dimensional reconstruction, J. Comput. Phys., 231 (2012) 2328–2358.Google Scholar
[27]Jiang, G.S., Shu, C.W., Efficient implementation of WENO schemes, J. Comput. Phys., 126 (1996) 202–228.Google Scholar
[28]Kajishima, T., Conservation properties of finite difference method for convection, Trans. Jpn. Soc. Mech. Eng. B, 60 (1994) 2058–2063 (in Japanese).Google Scholar
[29]Lai, M.C. and Peskin, C.S., An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity, J. Comput. Phys., 160 (2000) 705–719.CrossRefGoogle Scholar
[30]LeVeque, R.J., Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numerical Analysis, 31 (1994) 1019–1044.Google Scholar
[31]Le, D.V., Khoo, B.C., Peraire, J., An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, J. Comput. Phys., 220 (2006) 109–138.CrossRefGoogle Scholar
[32]Le, D.V., White, J., Peraire, J., Lim, K.M. and Khoo, B.C., An implicit immersed boundary method for three-dimensional fluid-membrane interactions, J. Comput. Phys., 228 (2009) 8427–8445.Google Scholar
[33]Li, Z., Lai, M.C., The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001) 822–842.Google Scholar
[34]Liu, C., Walkington, N.J., An Eulerian description of fluids containing visco-elastic particles, Arch. Rational Mech. Anal., 159 (2001) 229–252.CrossRefGoogle Scholar
[35]López, J., Hernández, J., On reducing interface curvature computation errors in the height function technique, J. Comput. Phys., 229 (2010) 4855–4868.Google Scholar
[36]Mori, Y., Peskin, C.S., Implicit second-order immersed boundary methods with boundary mass, Comput. Methods Appl. Mech. Eng., 197 (2008) 2049–2067.CrossRefGoogle Scholar
[37]Nagano, N., Sugiyama, K., Takeuchi, S., Ii, S., Takagi, S. and Matsumoto, Y., Full-Eulerian Finite-Difference Simulation of Fluid Flow in Hyperelastic Wavy Channel, Journal of Fluid Science and Technology, 5 (2010) 475–490.Google Scholar
[38]Okazawa, S., Kashiyama, K., Kaneko, Y., Eulerian formulation using stabilized finite element method for large deformation solid dynamics, Int. J. Numer. Meth. in Engng., 72 (2007) 1544–1559.CrossRefGoogle Scholar
[39]Osher, S.J., Sethian, J.A., Fronts propagating with curvature dependent speed. Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988) 12–49.Google Scholar
[40]Peskin, C.S., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10 (1972) 252–271.CrossRefGoogle Scholar
[41]Peskin, C.S., The immersed boundary method, Acta Numerica, 11 (2002) 479–517.Google Scholar
[42]Pilliod, J.E. Jr., Puckett, E.G., Second-order accurate volume-of-fluid algorithms for tracking material interfaces, J. Comput. Phys., 199 (2004) 465–502.Google Scholar
[43]Pozrikidis, C., Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow, J. Fluid. Mech., 297 (1995) 123–152.Google Scholar
[44]Pozrikidis, C., Effect of bending stiffness on the deformation of liquid capsules in simple shear flow, J. Fluid. Mech., 440 (2001) 269–291.Google Scholar
[45]Skalak, R., Tözeren, A., Zarda, P.R. and Chien, S., Strain energy function of red blood cell membranes, Biophys. J., 13 (1973) 245–264.Google Scholar
[46]Stewart, P.A., Lay, N., Sussman, M. and Ohta, M., An Improved Sharp Interface Method for Viscoelastic and Viscous Two-Phase Flows, J. Sci. Comp., 35 (2008) 43–61.Google Scholar
[47]Sussman, M., Smereka, P., Osher, S., A level set approach for computing solutions to incompressible two-phase flows, J. Comput. Phys., 114 (1994) 146–159.Google Scholar
[48]Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S. and Matsumoto, Y., Full Eulerian simulations of biconcave neo-Hookean particles in a Poiseuille flow, Comput. Mech., 46 (2010) 147–157.Google Scholar
[49]Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S., Matsumoto, Y., A full Eulerian finite difference approach for solving fluid-structure coupling problems, J. Comput. Phys., 230 (2011) 596–627.Google Scholar
[50]Taylor, G.I., The deformation of emulsions in definable fields of flows, Proc. R. Soc. Lond. A, 146 (1934) 501–523.Google Scholar
[51]Tryggvason, G., Bunner, B.B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., Jan, Y.J., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169 (2001) 708–759.Google Scholar
[52]Unverdi, S.O., Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100 (1992) 25–37.Google Scholar
[53]Hoogstraten, P.A.A Van, Slaats, P.M.A, Baaijens, F.P.T., A Eulerian approach to the finite element modeling of neo-Hookean rubber material, Appl. Sci. Res., 48 (1991) 193–210.Google Scholar
[54]Xiao, F., Yabe, T., Computation of complex flows containing rheological bodies, Computational Fluid Dynamics J., 8 (1999) 43–49.Google Scholar
[55]Xiao, F., Honma, Y., Kono, K., A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Meth. Fluids, 48 (2005) 1023–1040.Google Scholar
[56]Yokoi, K., Efficient implementation of THINC scheme: A simple and practical smoothed VOF algorithm, J. Comput. Phys., 226 (2007) 1985–2002.Google Scholar
[57]Youngs, D.L., Time-dependent multi-material flow with large fluid distortion, in Morton, K.W. and Baines, M.J. (eds), Numerical Methods for Fluid Dynamics, Academic, New York, 1982, 273–285.Google Scholar
[58]Youngs, D.L., An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code, Technical Report, 44/92/35, AWRE, 1984.Google Scholar