Error analysis of energy-conservative BDF2-FE scheme for the 2D Navier-Stokes equations with variable density

In this paper, we present an error estimate of a second-order linearized finite element (FE) method for the 2D Navier-Stokes equations with variable density. In order to get error estimates, we first introduce an equivalent form of the original system. Later, we propose a general BDF2-FE method for solving this equivalent form, where the Taylor-Hood FE space is used for discretizing the Navier-Stokes equations and conforming FE space is used for discretizing density equation. We show that our scheme ensures discrete energy dissipation. Under the assumption of sufficient smoothness of strong solutions, an error estimate is presented for our numerical scheme for variable density incompressible flow in two dimensions. Finally, some numerical examples are provided to confirm our theoretical results.Comment: 22 pages, 1 figure

Parallel Multiphase Navier-Stokes Solver

We study and implement methods to solve the variable density Navier-Stokes equations. More specifically, we study the transport equation with the level set method and the momentum equation using two methods: the projection method and the artificial compressibility method. This is done with the aim of numerically simulating multiphase fluid flow in gravity oil-water-gas separator vessels. The result of the implementation is the parallel Aspen software framework based on the massively parallel deal.II . For the transport equation, we briefly discuss the theory behind it and several techniques to stabilize it, especially the graph laplacian artificial viscosity with higher order elements. Also, we introduce the level set method to model the multiphase flow and study ways to maintain a sharp surface in between phases. For the momentum equation, we give an overview of the two methods and discuss a new projection method with variable time stepping that is second order in time. Then we discuss the new third order in time artificial compressiblity method and present variable density version of it. We also provide a stability proof for the discrete implicit variable density artificial compressibility method. For all the methods we introduce, we conduct numerical experiments for verification, convergence rates, as well as realistic models

Approximation Techniques for Incompressible Flows with Heterogeneous Properties

We study approximation techniques for incompressible flows with heterogeneous properties. Speci cally, we study two types of phenomena. The first is the flow of a viscous incompressible fluid through a rigid porous medium, where the permeability of the medium depends on the pressure. The second is the ow of a viscous incompressible fluid with variable density. The heterogeneity is the permeability and the density, respectively. For the first problem, we propose a finite element discretization and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence is exponential, we propose a splitting scheme which involves solving only two linear systems. For the second problem, we introduce a fractional time-stepping scheme which, as opposed to other existing techniques, requires only the solution of a Poisson equation for the determination of the pressure. This simpli cation greatly reduces the computational cost. We prove the stability of first and second order schemes, and provide error estimates for first order schemes. For all the introduced discretization schemes we present numerical experiments, which illustrate their performance on model problems, as well as on realistic ones