A cut finite element method for incompressible two-phase Navier–Stokes flows

Abstract

We present a space-time Cut Finite Element Method (CutFEM) for the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. The numerical method is able to accurately capture the strong discontinuity in the pressure and the weak discontinuity in the velocity field across evolving interfaces without re-meshing processes or regularization of the problem. We combine the strategy proposed in [P. Hansbo, M. G. Larson, S. Zahedi, Appl. Numer. Math. 85 (2014), 90-114] for the Stokes equations with a stationary interface and the space-time strategy presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016), . We also propose a strategy for computing high order approximations of the surface tension force by computing a stabilized mean curvature vector. The presented space-time CutFEM uses a fixed mesh but includes stabilization terms that control the condition number of the resulting system matrix independently of the position of the interface, ensure stability and a convenient implementation of the space-time method based on quadrature in time. Numerical experiments in two and three space dimensions show that the numerical method is able to accurately capture the discontinuities in the pressure and the velocity field across evolving interfaces without requiring the mesh to be conformed to the interface and with good stability properties.

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

“…and a moving interface with x = 0. We assume a 1 − x (t), a 2 − x (t) are non-zero, and have the same sign at any fixed time t. In the following we define a space-time CutFEM with discontinuous elements in both space and time following [9,15,32]. We emphasise that we do not explicitly construct a space-time domain in R d+1 as is done in for example [29].…”

“…In CutFEM the physical domain is embedded into a computational domain equipped with a quasi-uniform mesh. Elements that have an intersection with the domain of interest define the active mesh and associated to that is a finite dimensional function space and a weak form that together define the numerical scheme [9,14,21,31]. Interface and boundary conditions are typically imposed weakly.…”

“…A second result is a space-time CutFEM for the case of a moving interface. We use a framework similar to that proposed in [9,15,32], but here we use discontinuous elements both in space and time. The interface condition is imposed weakly as above, with the same restriction on penalties imposed by conservation.…”

Efficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methods

“…Application areas of harmonic weighting include multimaterial Stokes problems [52], Helmholtz problems in which the weighting depends on sound speed [55], as well as incompressible two-phase flow and fluid structure interaction [47]. In cut cell finite element methods, the weighting strategy is sometimes adapted to account not only for differing viscosity coefficients, but also for the measure of the cut element (and in the case of penalty parameters, also the measure of the cut face), as carefully analyzed by Annavarapu et al [6] (see also [10; 50]); applications of this idea include Stokes problems [33] and two-phase incompressible flow [30], the latter work also suggesting that the weights could take into account the viscosityto-density ratio of the two fluids. Methods which weight based on viscosity as well as cut element size have recently been adapted to handle extreme cases of these combinations by Gürkan and Massing [32].…”

Pore-scale gas–water flow in rock: Visualization experiment and simulation

“…In recent years, with the improvement of pore-scale flow theories and computer performance, the numerical simulation of pore-scale flow in rock has developed rapidly [15][16][17][18][19]. At present, the commonly used simulation methods include the Lattice Boltzmann Method (hereafter referred to as "the LBM") [23][24][25][26][27] and the Navier-Stokes equation based numerical simulation method [28][29][30][31][32][33][34][35]. Both methods are based on the pore network characteristics and used to simulate the pore-scale fluid flow in rock.…”