Numerical solution of the time-dependent Navier-Stokes equation for variable density–variable viscosity

We consider methods for numerical simulations of variable density incompressible fluids, modelled by the Navier-Stokes equations. Variable density problems arise, for instance, in interfaces between fluids of different densities in multiphase flows such as appear in porous media problems. It is shown that by solving the Navier-Stokes equation for the momentum variable instead of the velocity, the corresponding saddle point problem, which arises at each time step, becomes automatically regularized, enabling elimination of the pressure variable and leading to a, for the iterative solution, efficient preconditioning of the arising block matrix. We present also stability bounds and a second order operator splitting method. The theory is illustrated by numerical experiments. For reasons of comparison we also include test results for a method, based on coupling of the Navier-Stokes equations with a phase-field model.