A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows

Abstract

A numerical method for the simulation of three-dimensional incompressible two-phase flows is presented. The proposed algorithm combines an implicit pressure stabilized finite element method for the solution of incompressible two-phase flow problems with a level set method implemented with a quadrature-free discontinuous Galerkin (DG) method [E. Marchandise, J.-F. Remacle, N. Chevaugeon, A quadrature free discontinuous Galerkin method for the level set equation, Journal of Computational Physics 212 (2006) 338–357]. The use of a fast contouring algorithm [N. Chevaugeon, E. Marchandise, C. Geuzaine, J.-F. Remacle, Efficient visualization of high order finite elements, International Journal for Numerical Methods in Engineering] permits us to localize the interface accurately. By doing so, we can compute the discontinuous integrals without neither introducing an interface thickness nor reinitializing the level set.

The capability of the resulting algorithm is demonstrated with “large scale” numerical examples (free surface flows: dam break, sloshing) and “small scale” ones (two phase Poiseuille, Rayleigh–Taylor instability).

Introduction

The study of two phase flows covers a wide range of engineering and environmental flows, including small-scale bubble dynamics, wave mechanics, open channel flows, flows around a ship or structure. The main challenge for solving time-dependent two-phase flow problems in three dimensions is to provide an accurate representation of the interface that separates the two different fluids. This involves the tracking of a discontinuity in the material properties like density and viscosity.

The principal computational methods used to solve incompressible two-phase flows are the front tracking methods [3], [4], [5], [6], [7], and the front capturing methods (volume of fluid [8], [9] and level set [10], [11]).

A successful approach to deal with two phase flows, especially in the presence of topological changes, is the level set method [10]. Application of level sets in two-phase flow calculations have been extensively described by Sussman, Smereka and Osher in [11], [12], [13] and used by among others [14], [15], [16], [17]. The level set function is able to represent an arbitrary number of bubbles or drops interfaces and complex changes of topology are naturally taken into account by the method. The level set function $\varphi (\overrightarrow{x}\text{,}t)$ is defined to be a smooth function that is positive in one region and negative in the other. The implicit surface $\varphi (\overrightarrow{x}\text{,}t)=0$ represents the current position of an interface. This interface is advected by a vector field $\mathbf{u}(\overrightarrow{x}\text{,}t)$ that is, in case of two-phase flows, the solution of the Navier–Stokes equations. The elementary advection equation for interface evolution is:

$${\mathrm{\partial }}_t\varphi +\mathbf{u}·\nabla \varphi =0\text{.}$$

In [1], we have developed a high order quadrature free Runge–Kutta discontinuous Galerkin (DG) method to solve the level set equation (1) in space and time. The method was compared with classical Hamilton–Jacobi ENO/WENO methods [13], [18], [19], [20] and showed to be computationally effective and mass conservative. Besides, we showed that there was no need to reinitialize the level set.

Level sets are representing a fluid interface in an implicit manner. The main advantage of this approach is that the underlying computational mesh does not conform to the interface. Hence, discontinuous integrals have to be computed in the fluid formulation because both viscosities and densities are discontinuous in all the elements crossed by the interface. The most common approach is to define a zone of thickness 2ϵ in the vicinity of the interface (∣ϕ∣ < ϵ) and to smooth the discontinuous density and viscosity over this thickness [11], [17], [21], [22], [23]. Smoothing physical parameters in the interfacial zone may be the cause of two problems. The first one is the introduction of non-physical densities and viscosities in the smoothed region, leading to possible thermodynamical aberrations [24]. The second problem is the obligation to keep the interface thickness constant in time. For ensuring that the smoothed region has a constant thickness, one has to reinitialize the level set so that it remains a distance function. In this work, we rather adopt a discontinuous approach [25], [26] to compute the discontinuous integrals. The use of a recursive contouring algorithm [2] allows to localize the interface accurately. Consequently, we are able to compute the discontinuous integrals with a very high level of accuracy.

For the computation of the incompressible two-phase Navier–Stokes equations, various numerical methods have been developed. Among them are the projection methods [27], [28], [29], stabilized finite element methods [17], [30], [31] and artificial compressibility methods [32], [33]. A key feature of stabilized methods is that they have proved to be LBB stable and to have good convergence properties [34], [35].

In this work, we present a stabilized finite element method for computing flows in both phases and combine it with a discontinuous Galerkin level set method for computing the interface motion. The overall algorithm avoids the cost of the renormalization of the level set as well as the introduction of a non-physical interface thickness and exhibits good mass conservation properties.

The outline of this paper is as follows: we first present the governing equations in Section 2. Section 3 is devoted to the description of our computational method. We present the Navier–Stokes solver and the coupling with the discontinuous Galerkin method for the level set equation. Section 4 gives numerical examples to verify accuracy, stability and convergence properties.