TY - JOUR
T1 - Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system
T2 - Primitive variable and projection-type schemes
AU - Guo, Z.
AU - Lin, P.
AU - Lowengrub, J.
AU - Wise, S. M.
N1 - ZG gratefully acknowledgespartial support from the 150th Anniversary Postdoctoral Mobility Grant of London Mathematical Society (PMG14-15 09). JL and ZG gratefully acknowledge partial support from National Science Foundation Grants NSF-DMS-1719960, NSF-DMS-1522775 and the National Institute of Health grant P50GM76516 for the Center of Excellence in Systems Biology at the University of California, Irvine. PL is partially supported by the National Natural Science Foundation of China (No. 91430106) and the Fundamental Research Funds for the Central Universities (No. 06500073). SMW gratefully acknowledges support from a grant from the National Science Foundation (NSF-DMS 1418692) and partial support from the Mathematics Department at the University of California, Irvine through NSF Grants DMS-1522775 and DMR-150733.
PY - 2017/11/1
Y1 - 2017/11/1
N2 - In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn–Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier–Stokes equation – for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh–Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10−8 and energy decreases in time.
AB - In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn–Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier–Stokes equation – for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh–Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10−8 and energy decreases in time.
KW - Binary fluid flow
KW - Energy stability
KW - Multigrid
KW - Phase-field method
KW - Staggered finite differences
KW - Variable density
U2 - 10.1016/j.cma.2017.08.011
DO - 10.1016/j.cma.2017.08.011
M3 - Article
AN - SCOPUS:85028800508
SN - 0045-7825
VL - 326
SP - 144
EP - 174
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
ER -
