Discretization

Convection–diffusion equation

Total variation diminishing

Applied mathematics

Mathematical optimization

Solver

Mathematics

Numerical stability

Mathematical analysis

Numerical analysis

Journal of Computational Physics

This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined staggered Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The algorithm combines the interface capturing level set method with a variable-coefficient incompressible Navier-Stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude. The discretization approach ensures second-order pointwise accuracy for both velocity and pressure with several physical boundary treatments, including velocity and traction boundary conditions. The paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver. To ensure the stability of the numerical scheme in the presence of high density and viscosity ratios, we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations. This consistency is achieved by solving an additional mass balance equation, which we approximate via a strong stability preserving Runga-Kutta time integrator and by employing the same mass flux (obtained from the mass equation) in the discrete momentum equation. The scheme uses higher-order total variation diminishing (TVD) and convection-boundedness criterion (CBC) satisfying limiter to avoid numerical fluctuations in the transported density field. The high-order bounded convective transport is done on a dimension-by-dimension basis, which makes the scheme simple to implement. We also demonstrate through several test cases that the lack of consistent mass and momentum transport in non-conservative formulations, which are commonly used in practice, or the use of non-CBC satisfying limiters can yield very large numerical error and very poor accuracy for convection-dominant high density ratio flows. Our numerical scheme also uses well-balanced surface tension and gravity force discretizations. In the hydrostatic limit, we show that the well-balanced formulation mitigates spurious flow currents and achieves discrete force-balance between the pressure gradient and surface tension or gravity.

A robust incompressible Navier-Stokes solver for high density ratio multiphase flows