Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T12:21:17.075Z Has data issue: false hasContentIssue false
  • English
  • Français

An Iterative Two-Fluid Pressure Solver Based on the Immersed Interface Method

Published online by Cambridge University Press:  20 August 2015

Sheng Xu
Sheng Xu*
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, TX 75275-0156, USA
*
*Corresponding author.Email:sxu@smu.edu
Get access

Abstract

An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm. The iteration is constructed by introducing an unsteady term in the pressure Poisson equation. In each iteration step, a Helmholtz equation is solved on the Cartesian grid using FFT. The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across two-fluid interfaces. This solver can also be used to solve Poisson equations on irregular domains.

Keywords

76T9935J0565N06Poisson solverthe immersed interface methodtwo-fluid flowjump conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 528 - 543
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]Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), 12–26.Google Scholar
[2]Huang, H. and Li, Z., Convergence analysis of the immersed interface method, IMA J. Numer. Anal., 19 (1999), 583–608.CrossRefGoogle Scholar
[3]Ito, K., Lai, M.-C. and Li, Z., A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains, J. Comput. Phys., 228 (2009), 2616–2628.Google Scholar
[4]LeVeque, R. J. 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
[5]LeVeque, R. J. 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
[6]Li, Z. and Ito, K., The immersed interface method –numerical solutions of PDEs involving interfaces and irregular domains, SIAM Frontiers in Applied Mathematics, 33 (2006), ISBN: 0-89971-609-8.Google Scholar
[7]Li, Z., Wan, X., Ito, K. and Lubkin, S. R., An augmented approach for the pressure boundary condition in a Stokes flow, Commun. Comput. Phys., 1 (2006), 874–885.Google Scholar
[8]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
[9]Mittal, R. and Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37 (2005), 239–261.Google Scholar
[10]Peaceman, D. and Rachford, H., The numerical solution of elliptic and parabolic differential equations, J. SIAM, 3 (1955), 28–41.Google Scholar
[11]Tryggvason, G.et al., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169 (2001), 708–759.Google Scholar
[12]Xu, S. and Wang, Z. J., Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation, SIAM J. Sci. Comput., 27 (2006), 1948–1980.Google Scholar
[13]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
[14]Xu, S. and Wang, Z. J., A 3D immersed interface method for fluid-solid interaction, Comput. Methods Appl. Mech. Engrg., 197 (2008), 2068–2086.CrossRefGoogle Scholar
[15]Xu, S., The immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow, J. Comput. Phys., 227 (2008), 5045–5071.Google Scholar
[16]Xu, S., Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation, Discrete and Continuous Dynamical Systems, Supplement 2009 (2009), 838–845.Google Scholar