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A Preconditioned Implicit Free-Surface Capture Scheme for Large Density Ratio on Tetrahedral Grids

Published online by Cambridge University Press:  20 August 2015

Xin Lv ,
Qingping Zou ,
D.E. Reeve  and
Yong Zhao
Xin Lv*
Affiliation:
International Center for Numerical Methods in Engineering (CIMNE)-Singapore, 51 Science Park Road, #04-20, The Aries Singapore Science Park II, 117586, Singapore
Qingping Zou*
Affiliation:
Department of Civil and Environmental Engineering, University of Maine, Orono, Maine 04469, USA
D.E. Reeve*
Affiliation:
Center for Coastal Dynamics and Engineering, School of Engineering, University of Plymouth, Devon PL4 8AA, United Kingdom
Yong Zhao*
Affiliation:
College of Engineering, Alfaisal University, Al Maathar Road P.O. Box 50927, Riyadh 11533, Kingdom of Saudi Arabia
*
Corresponding author.Email:lvxin@cimne-sgp.com
Email:qingping.zou@maine.edu
Email:dominic.reeve@plymouth.ac.uk
Email:zyong@alfaisal.edu
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Abstract

We present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids. The current scheme improves our recently reported method [10] in several aspects. Specifically, we modified the original eigensystem by applying a preconditioning matrix so that the new eigensystem is virtually independent of density ratio, which is typically large for practical two-phase problems. Further, we replaced the explicit multi-stage Runge-Kutta method by a fully implicit Euler integration scheme for the Navier-Stokes (NS) solver and the Volume of Fluids (VOF) equation is now solved with a second order Crank-Nicolson implicit scheme to reduce the numerical diffusion effect. The preconditioned restarted Generalized Minimal RESidual method (GMRES) is then employed to solve the resulting linear system. The validation studies show that with these modifications, the method has improved stability and accuracy when dealing with large density ratio two-phase problems.

Keywords

65E05VOFlevel setfree surfaceunstructured finite volume methodimplicit methodrestarted GMREStetrahedral grid
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 1 , January 2012 , pp. 215 - 248
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Kunza, R. F. and Boger, D. A.et al., A preconditioned Navier-Stokes method for two-phase Flows with application to cavitation prediction, Comput. Fluids, 29 (2000), 849–875.Google Scholar
[2]Barth, T. J., Analysis of implicit local linearization techniques for upwind the TVD algorithms, Twenty-fifth Aerospace Sciences Meeting, January 1987, AIAA Paper 87-0595.Google Scholar
[3]Saad, Y. and Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7(3) (1986), 856–869.CrossRefGoogle Scholar
[4]Venkatakrishnan, V. and Mavriplis, D. J., Implicit solvers for unstructured meshes, J. Comput. Phys., 105 (1993), 83–91.Google Scholar
[5]Whitaker, D. L., Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method, AIAA Paper 93-3337, 1993.Google Scholar
[6]Luo, H., Baum, J. D., Löhner, R. and Cabello, J., Implicit schemes and boundary conditions for compressible flows on unstructured meshes, AIAA Paper 94-0816, 1994.Google Scholar
[7]Barth, T. J. and Linton, S. W., An unstructured mesh Newton solver for compressible fluid flow and its parallel implementation, AIAA Paper 95-0221, 1995.Google Scholar
[8]Cuthill, E. H. and McKee, J. M., Reducing the bandwidth of sparse symmetric matrices, in: Proceedings of the 24th National Conference of the Association for Computing Machinery, 1969, 157–172.Google Scholar
[9]Nejat, A. and Ollivier-Gooch, C., Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations, J. Comput. Phys., 227 (2008), 2366–2386.Google Scholar
[10]Lv, X., Zou, Q. P., Zhao, Y. and Reeve, D. E., A novel coupled level set and volume of fluid method for sharp interface capturing on 3D tetrahedral grids, J. Comput. Phys., doi:10.1016/j.jcp.2009.12.005, 2009.Google Scholar
[11]Zhao, Y. and Tai, C. H., Higher-order characteristics-based methods for incompressible flow computation on unstructured grids, AIAA J., 39(7) (2001), 1280–1287.Google Scholar
[12]Zhao, Y., Tan, H. H. and Zhang, B. L., A high-resolution characteristics-based implicit dual time-stepping VOF method for free surface flow simulation on unstructured grids, J. Comput. Phys., 183 (2002), 233–273.CrossRefGoogle Scholar
[13]Price, W. G. and Chen, Y. G., A simulation of free surface waves for incompressible two-phase flows using a curvilinear level set formulation, Int. J. Numer. Methods Fluids, 51 (2006), 305– 330.Google Scholar
[14]Ubbink, O. and Issa, R. I., A method for capturing sharp fluid interfaces on arbitrary meshes, J. Comput. Phys., 153 (1999), 26–50.CrossRefGoogle Scholar
[15]Leonard, B. P., The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection, Comput. Meth. Appl. Mech. Eng., 88 (1991), 17–74.Google Scholar
[16]Jameson, A. and Mavriplis, D. J., Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh, AIAA J., 24 (1986), 611.Google Scholar
[17]Liou, M.-S. and Leer, B. van, Choice of implicit and explicit operators for the upwind differencing method, 26th Aerospace Sciences Meeting, Reno, Nevada, Jan 11-14, 1988.Google Scholar
[18]Luo, H., Baum, J. D. and Löhner, R., A fast, matrix-free implicit method for compressible flows on unstructured meshes, J. Comput. Phys., 146 (1998), 664–690.Google Scholar
[19]Nichols, B. D., Hirt, C. W. and Hotchkiss, R. S., SOLA-VOF: A solution algorithm for transient fluid flow with multiple free boundaries, Tech. Rep., LA-8355, Los Alamos National Laboratory, 1980.Google Scholar
[20]Hirt, C. W. and Nichols, B. D., Volume of fluid method for the dynamics of free boundaries, J. Comput. Phys., 39 (1981), 201–225.Google Scholar
[21]Soulis, J. V., Computation of two-dimensional dam-breakflood flows, Int. J. Numer. Methods Fluids, 14 (1992), 631–664.Google Scholar
[22]Martin, J. C. and Moyce, W. J., An experimental study of the collapse of liquid columns on a rigid horizontal plane, Philos. Trans. Roy. Soc. London, A244 (1952), 312–324.Google Scholar
[23]Koshizuka, S., Tamako, H. and Oka, Y., A particle method for incompressible viscous flow with fluid fragmentation, Comput. Fluid Dyn. J., 4(1) (1995), 29–46.Google Scholar
[24]Kleefsman, K. M. T., Fekken, G., Veldman, A. E. P., Iwanowski, B. and Buchner, B., A volume-of-fluid based simulation method for wave impact problems. J. Comput. Phys., 206 (2005), 363–393.Google Scholar
[25]SPH European Research Interest Community website, http://wiki.manchester.ac.uk/spheric/index.php/test2.Google Scholar
[26]Caiden, R., Fedkiw, R. P. and Anderson, C., A numerical method for two-phase flow consisting of separate compressible and incompressible regions, J. Comput. Phys., 166(1) (2001), 1–27.Google Scholar
[27]Knoll, D. A. and Keyes, D. E., Jacobian-free Newton-Krylov methods a survey of approaches and applications, J. Comput. Phys., 193 (2004), 357–397.Google Scholar
[28]Fedkiw, R. P., Aslam, T., Merriman, B. and Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152(2) (1999), 457–492.Google Scholar
[29]Liu, T. G., Khoo, B. C. and Wang, C. W., The ghost fluid method for compressible gas-water simulation, J. Comput. Phys., 204 (2005), 193–221.Google Scholar
[30]Liu, X.-D., Fedkiw, R. P. and Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J. Comput. Phys., 160 (2000), 151–178.Google Scholar
[31]Kang, M., Fedkiw, R. P., and Liu, X.-D., A boundary condition capturing method for multiphase incompressible flow, J. Sci. Comput., 15 (2000), 323–360.Google Scholar
[32]Wang, Z.-Y., Zou, Q.-P. and Reeve, D. E., Simulation of spilling breaking waves using a two phase flow CFD model, Comput. Fluids, doi:10.1016/j.compfluid.2009.06.006, 2009.CrossRefGoogle Scholar
[33]Zhang, Y.-L., Zou, Q.-P. and Greaves, D. M., Numerical simulation of free-surface flow using the level-set method with global mass correction, Int. J. Numer. Methods Fluids, doi: 10.1002/fld.2090, 2009.Google Scholar
[34]Liu, D. M. and Lin, P. Z., A numerical study of three-dimensional liquid sloshing in tanks, J. Comput. Phys., 227(8) (2008), 3921–3939.Google Scholar
[35]Clift, R., Grace, J. R. and Weber, M. E., Bubbles, Drops and Particles, Academic Press: New York, 1978.Google Scholar
[36]Chen, L., Garimella, S. V., Reizes, J. A. and Leonardi, E., The development of a bubble rising in a viscous fluid, J. Fluid Mech., 387 (1999), 61–96.Google Scholar
[37]Parolini, N., Computational Fluid Dynamics for Naval Engineering Problems, PhD Thesis, Number 3138, Ecole Polytechnique Federale de Lausanne (EPFL), 2004.Google Scholar
[38]Parolini, N. and Burman, E., A finite element level set method for viscous free-surface flows, Applied and Industrial Mathematics in Italy, Proceedings of SIMAI 2004, 417-427, World Scientific, 2005.Google Scholar
[39]Hysing, S., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S. and Tobiska, L., Quantitative benchmark computations of two-dimensional bubble-dynamics, MOX-Report No. 23/2008, 2008.Google Scholar