Numerical Solution of the Time-Dependent Navier–stokes Equation for Variable Density–variable Viscosity. Part I

Abstract

PostprintThis is the accepted version of a paper published in Mathematical Modelling and Analysis. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination. AbstractWe 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 projection-type 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.The theory including stability bounds and a second order splitting method is dealt with in Part II of the study.

Preconditioners for Two-Phase Incompressible Navier--Stokes Flow

“…Further complicating matters is the ASGS stabilisation used and the dynamic two-phase nature of the simulations. We have implemented an approach which treats F −1 as a block preconditioned GMRES iteration where the sub-blocks are approximated using multigrid methods and this has shown signs of being effective both in our work and others [26]; see also results in [2]. Unfortunately, maintaining solver robustness throughout an entire multi-physics simulation remains an issue.…”

“…The details of the weak formulation and finite element discretisation are known and are found in, for example, [23]; we summarise the resulting finite element problem here. We assume appropriate, though not necessarily inf-sup stable, finite element spaces for velocity and pressure given by V h ⊂ H 1 (Ω) 2 and Q h ⊂ p ∈ L 2 (Ω) : Ω p = 0 (9) respectively. Then we wish to find u h ∈ V h and p h ∈ Q h such that…”

“…However, in some important cases this restriction on the CFL number is not strictly necessary. 2 Thus, in the second simulation, we use a fixed time-step of ∆t = 0.01. In this case, the CFL number is larger than one for much of the simulation, reaching a maximum of 20.5 and typically being above 2.5.…”

Scheme for Evolutionary Navier-Stokes-Fourier System with Temperature Dependent Material Properties Based on Spectral/hp Elements

“…where v = [u, v, w] T is the velocity vector (by setting w = const. = 0 we restrict to 2D problem), p is a variable related to the thermodynamic pressure, 1 T denotes the temperature, D = 1 2 ∇v + (∇v) T is the symmetric part of the rate of strain 1 We call thermodynamic pressure the variable acting in the equation of state, e.g. p = ρRT for ideal gas.…”

“…Table 1 Chosen results concerning equation systems with variable material parameters Eq. type ∇ • v μ κ ρ [1] nonst. 0 μ(ρ) const.…”