Abstract
We consider methods for the 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 appearing in porous media problems. We show that by solving the Navier–Stokes equation for the momentum variable instead of the velocity the corresponding saddle point problem, arising at each time step, no special treatment of the pressure variable is required and leads to an efficient preconditioning of the arising block matrix. This study consists of two parts, of which this paper constitutes Part I. Here we present the algorithm, compare it with a broadly used projectiontype method and illustrate some advantages and disadvantages of both techniques via analysis and numerical experiments. In addition we also include test results for a method, based on coupling of the Navier–Stokes equations with a phase-field model, where the variable density function is handled in a different way.
Highlights
Variable density problems arise in many complex fluid flow processes of current interest, and have been studied intensively via numerical simulations
We have studied various aspects of the numerical solution of the variable density Navier–Stokes equations – discretization, operator splitting and linearization, and the interplay between the related errors
We have considered preconditioning techniques, suitable for the arising linear systems, in order to enable fast and robust numerical simulations of the underlying flow phenomena
Summary
Variable density problems arise in many complex fluid flow processes of current interest, and have been studied intensively via numerical simulations. Important variable density problems arise in laminar flows and the work, presented in this article, deals primarily with the numerical solution of such models. The main idea of the projection methods is first to compute a velocity field without taking into account incompressibility, and perform a pressure correction, which is a projection back to the subspace of solenoidal (divergence-free) vector fields In those schemes, both types of time discretizations, explicit and implicit have been utilized. A major part of the numerical simulation of nonstationary flow models is the ability to efficiently solve the nonlinear time-dependent variable density, variable-viscosity Navier–Stokes (NS) equations. We conclude the paper with some remarks, such as the possibility to couple with elasticity equations when solving porous media problems
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