Mass Conservative and Energy Stable Finite Difference Methods for the Quasi-incompressible Navier-Stokes-Cahn-Hilliard system: Primitive Variable and Projection-Type Schemes

摘要

In this paper we describe two fully mass conservative, energy stable, finitedifference methods on a staggered grid for the quasi-incompressibleNavier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressiblefluid flow with variable density and viscosity. Both methods, namely theprimitive method (finite difference method in the primitive variableformulation) and the projection method (finite difference method in aprojection-type formulation), are so designed that the mass of the binary fluidis preserved, and the energy of the system equations is always non-increasingin time at the fully discrete level. We also present an efficient, practicalnonlinear multigrid method - comprised of a standard FAS method for theCahn-Hilliard equation, and a method based on the Vanka-type smoothing strategyfor the Navier-Stokes equation - for solving these equations. We test thescheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylorinstability. Quantitative comparisons are made with existing analyticalsolutions or previous numerical results that validate the accuracy of ournumerical schemes. Moreover, in all cases, mass of the single component and thebinary fluid was conserved up to 10 to -8 and energy decreases in time.