Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T10:22:42.340Z Has data issue: false hasContentIssue false
  • English
  • Français

An Implementation of MAC Grid-Based IIM-Stokes Solver for Incompressible Two-Phase Flows

Published online by Cambridge University Press:  20 August 2015

Zhijun Tan ,
K. M. Lim  and
B. C. Khoo
Zhijun Tan*
Affiliation:
Guangdong Province Key Laboratory of Computational Science and School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China
K. M. Lim*
Affiliation:
Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
B. C. Khoo*
Affiliation:
Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
*
Corresponding author.Email:tzhij@mail.sysu.edu.cn
Email:mpelimkm@nus.edu.sg
Email:mpekbc@nus.edu.sg
Get access

Abstract

In this paper, a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces. The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method. The augmented variables and /or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines and are then applied to the fluid through the jump conditions. The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-type method. The numerical results show that the overall scheme is second order accurate. The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions. The proposed method avoids solution of the pressure Poisson equation, and comparisons are made to show the advantages of time savings by the present method. The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.

Keywords

65N0635R0565M12Incompressible two-phase Stokes equationstwo-phase Navier-Stokes equationsdiscontinuous viscositysingular forceimmersed interface methodUzawa methodfront tracking method
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1333 - 1362
Copyright
Copyright © Global Science Press Limited 2011

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]Adams, J., Swarztrauber, P. and Sweet, R., FISHPACK: Efficient FORTRAN subprograms for the solution of separable eliptic partial differential equations, 1999, Available on the web at http://www.scd.ucar.edu/css/software/fishpack/.Google Scholar
[2]Beyer, R. P., A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98 (1992), 145–162.CrossRefGoogle Scholar
[3]Cahouet, J. and Chabard, J.-P., Some fast 3d finite element solvers for the generalized Stokes problem, Int. J. Numer. Methods Fluids, 8 (1988), 869–895.Google Scholar
[4]Calhoun, D., A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., 176 (2002), 231–275.Google Scholar
[5]Christoph, B., Domain imbedding methods for the Stokes equations, Numer. Math., 57 (1990), 435–451.Google Scholar
[6]Dillon, R., Fauci, L., Fogelson, A. and Gaver, D., Modeling biofilm processes using the immersed boundary method, J. Comput. Phys., 129 (1996), 57–73.Google Scholar
[7]Eggleton, C. D. and Popel, A. S., Large deformation of red blood cell ghosts in a simple shear flow, Phys. Fluids, 10 (1998), 1834–1845.Google Scholar
[8]Elman, H. C., Multigrid and Krylov subspace methods for the discrete Stokes equations, Int. J. Numer. Methods Fluids, 227 (1996), 755–770.Google Scholar
[9]Elman, H. C., Preconditioners for saddle point problems arising in computational fluid dynamics, Appl. Numer. Math., 43 (2002), 75–89.Google Scholar
[10]Fauci, L.J. andPeskin, C. S., A computational model of aquatic animal locomotion, J.Comput. Phys., (1988), 85–108.Google Scholar
[11]Fogelson, A. L., A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 1 (1984), 111–134.Google Scholar
[12]Fogelson, A. L., Continuum models of platelet aggregation: formulation and mechanical properties, SIAM J. Appl. Math., 52 (1992), 1089–1110.Google Scholar
[13]Harlow, F. H. and Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (1965), 2182–2189.Google Scholar
[14]Kobelkov, G. M. and Olshanskii, M. A., Effective preconditioning of Uzawa type schemes for a generalized Stokes problem, Numer. Math., 86 (2000), 443–470.CrossRefGoogle Scholar
[15]Layton, A. T., An efficient numerical method for the two-fluid Stokes equations with a moving immersed boundary, Comput. Methods Appl. Mech. Eng., 197 (2008), 2147–2155.CrossRefGoogle Scholar
[16]Le, D. V., Khoo, B. C. and Peraire, J., Animmersed interfacemethod for viscous incompressible flows involving rigid and flexible boundaries, J. Comput. Phys., 220 (2006), 109–138.Google Scholar
[17]Lee, L. and Le Veque, R. J., An immersed interface method for incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 25 (2003), 832–856.CrossRefGoogle Scholar
[18]Le, R. J.Veque and Li, Z., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), 709–735.Google Scholar
[19]Le, R. J.Veque and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019–1044.Google Scholar
[20]Li, Z., An overview of the immersed interface method and its applications, Taiwan J. Math., 7 (2003), 1–49.Google Scholar
[21]Li, Z. and Ito, K., The immersed interface method-numerical solutions of PDEs involving interfaces and irregular domains, SIAM Frontiers Appl. Math., 33 (2006), ISBN: 0-89971-609-8.Google Scholar
[22]Li, Z., Ito, K. and Lai, M-C., An augmented approach for Stokes equations with a discontinuous viscosity and singular forces, Comput. Fluids, 36 (2007), 622–635.Google Scholar
[23]Li, Z. and Lai, M. C., The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001), 822–842.Google Scholar
[24]Li, Z. and Wang, C., A fast finite difference method for solving Navier-Stokes equations on irregular domains, Commun. Math. Sci., 1 (2003), 180–196.CrossRefGoogle Scholar
[25]Linnick, M. N. and Fasel, H. F., A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains, J. Comput. Phys., 204 (2005), 157–192.Google Scholar
[26]Oosterlee, C. W. and Lorenz, F. J. G, Multigrid methods for the Stokes system, Comput. Sci. Eng., 8 (2006), 34–43.Google Scholar
[27]Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), 220–252.CrossRefGoogle Scholar
[28]Peskin, C. S., The immersed boundary method, Acta Numerica, 11 (2002), 479–517.Google Scholar
[29]Peters, J., Reichelt, V. and Reusken, A., Fast iterative solvers for discrete Stokes equations, SIAM J. Sci. Comput., 27 (2005), 646–666.Google Scholar
[30]Russell, D. and Wang, Z. J., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191 (2003), 177–205.Google Scholar
[31]Sarin, V. and Sameh, A., An efficient iterative method for the generalized Stokes problem, SIAM J. Sci. Comput., 19 (1998), 206–226.CrossRefGoogle Scholar
[32]Shin, D. and Strikwerda, J. C., Fast solvers for finite difference approximations for the Stokes and Navier-Stokes equations, J. Australian Math. Soc., 38 (1996), 274–290.Google Scholar
[33]Stockie, J. M. and Green, S. I., Simulating the motion of flexible pulp fibres using the immersed boundary method, J. Comput. Phys., 147 (1998), 147–165.Google Scholar
[34]Stoer, J. and Bulirsch, R., Introduction to Numerical Analysis, 3rd ed., Springer-Verlag, 2002.Google Scholar
[35]Tan, Z.-J., Le, D. V., Li, Z., Lim, K. M. and Khoo, B. C., An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane, J. Comput. Phys., 227 (2008), 9955–9983.Google Scholar
[36]Tau, E. Y., Numerical solution of the steady Stokes equations, J. Comput. Phys., 99 (1992), 190–195.CrossRefGoogle Scholar
[37]Tu, C. and Peskin, C. S., Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods, SIAM J. Sci. Statist. Comput., 13 (1992), 1361–1376.Google Scholar
[38]Xu, S. and Wang, Z. J., An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. Comput. Phys., 216 (2006), 454–493.Google Scholar
[39]Xu, S. and Wang, Z. J., A 3D immersed interface method for fluid-solid interaction, Comput. Methods Appl. Mech. Eng., 197 (2008), 2068–2086.Google Scholar