A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows
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 is defined to be a smooth function that is positive in one region and negative in the other. The implicit surface represents the current position of an interface. This interface is advected by a vector field that is, in case of two-phase flows, the solution of the Navier–Stokes equations. The elementary advection equation for interface evolution is: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.
Section snippets
Governing equation
In the present work, the three-dimensional flow field of two non-miscible laminar incompressible fluids is calculated. The two fluids are denoted respectively by (+) and (−) and have distinct viscosity and density (ρ+, μ+) and (ρ−, μ−). Fig. 1 shows an illustration of a configuration with two fluids.
The solution in both phases, denoted as phase (+) and phase (−), are obtained simultaneously. The non-dimensional equations are given by the incompressible Navier–Stokes equations:
Numerical method
This section describes our numerical implementation for the computation of two-phase incompressible flows. As we have chosen to work with standard finite elements (for its ease of treatment of complex geometries), the computation of incompressible flows will involve two sources of potential numerical instabilities. One source is due to the presence of convective terms in the momentum equations. The other potential source of instability may be due to an inappropriate combination of interpolation
Numerical results and discussion
In this section, the coupling of the incompressible two-phase flow solver and the interface solver described in the previous section is tested and applied to several 2D and 3D two-phase flows problems. As we are only working with tetrahedral meshes, we use for the computation of the 2D flows a 2D mesh that is extruded over a thin layer in the third dimension.
Our methods were implemented in Standard C++, compiled with GNU g++ v3.3, and run on one CPU of a 2.4 GHz Intel(R) Pentium(R) 4 with 512 KB
Conclusion
A unified approach for the numerical simulation of three-dimensional two-phase flows has been presented. The approach relies on an implicit stabilized finite element approximation for the Navier–Stokes equations and discontinuous Galerkin method for the level set method (DG-LSM).
Such a combination of those two numerical methods results in a simple and effective algorithm that allows to simulate diverse flow regimes (ranging from stokes flow to highly convective flows), presenting also large
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2022, Computers and Mathematics with ApplicationsCitation Excerpt :DG methods are well-suited for level set advection [21,27] or interface tracking in multiphase flows [20,45] due to their low numerical dissipation achieved by the use of local high order polynomial approximations. On the other hand, level set reinitialization is generally solved with a more robust, lower-order finite volume scheme [10] or is totally avoided using geometric distancing based on neighbor search algorithms [26,28] even though the other parts of the solver benefit high-order DG discretizations. This is probably because high-order DG methods for the HJ equation, like many other high-order methods, produce oscillations when the approximation space is inadequate to resolve the main features of the exact solution.
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