2018/03/19 by Tormod Landet, Landet, Tormod, Kent‐André Mardal +3 · 1 citation
Engineering · #Computational Fluid Dynamics and Aerodynamics #Advanced Numerical Methods in Computational Mathematics #Fluid Dynamics and Turbulent Flows
paper · pdf · doi:10.48550/arxiv.1803.06976
Solving the Navier-Stokes equations when the density field contains a large\nsharp discontinuity---such as a water/air free surface---is numerically\nchallenging. Convective instabilities cause Gibbs oscillations which quickly\ndestroy the solution. We investigate the use of slope limiters for the velocity\nfield to overcome this problem in a way that does not compromise on the mass\nconservation properties. The equations are discretised using the interior\npenalty discontinuous Galerkin finite element method that is divergence free to\nmachine precision.\n A slope limiter made specifically for exactly divergence free (solenoidal)\nfields is presented and used to illustrated the difficulties in obtaining\nconvectively stable fields that are also exactly solenoidal. The lessons\nlearned from this are applied in constructing a simpler method based on the use\nof an existing scalar slope limiter applied to each velocity component.\n We show by numerical examples how both presented slope limiting methods are\nvastly superior to the naive non-limited method. The methods can solve\ndifficult two-phase problems with high density ratios and high Reynolds\nnumbers---typical for marine and offshore water/air simulations---in a way that\nconserves mass and stops unbounded energy growth caused by the Gibbs\nphenomenon.\n