Third-order less oscillatory and less diffusive compact stencil-based upwind schemes, and their applications to incompressible flows and free surface flows

Abstract

We propose novel third-order less oscillatory and less diffusive compact stencil-based upwind schemes for the approximation of the continuity equation. The proposed schemes are based on the constrained interpolation profile-conservative semi-Lagrangian schemes. An important feature of the proposed schemes is that the interpolation functions are constructed using only variables within one upwind cell (a cell average and two boundary values). Furthermore, the proposed schemes have third-order accuracy and are also less oscillatory, less diffusive, and fully conservative. The proposed schemes are validated through various benchmark problems and comparisons with experiments of two droplets collision/separation and droplet splashing. The numerical results have shown that the proposed schemes have third-order accuracy for smooth solution, and capture discontinuities and smooth solutions simultaneously without numerical oscillations. The proposed schemes can capture the secondary vorticity of lid-driven cavity flow of Re = 7500 with a Cartesian grid of 64 × 64. The numerical results of two droplets collision/separation of We = 40 show that the proposed schemes can reproduce droplets collision/separation with quite coarse grids. These numerical results of droplet splashing have demonstrated that proposed schemes can reduce numerical diffusions well against existing schemes and robust.