Florian Kummer

A SIMPLE based discontinuous Galerkin solver for steady incompressible flows

In this paper we present how the well-known SIMPLE algorithm can be extended to solve the steady incompressible Navier-Stokes equations discretized by the discontinuous Galerkin method. The convective part is discretized by the local Lax-Friedrichs fluxes and the viscous part by the symmetric interior penalty method. Within the SIMPLE algorithm, the equations are solved in an iterative process. The discretized equations are linearized and an equation for the pressure is derived on the discrete level. The equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially, leading to an efficient solution procedure. The extension of the proposed scheme to the unsteady case is straightforward, where fully implicit time schemes can be used. Various test cases are carried out: the Poiseuille flow, the channel flow with constant transpiration, the Kovasznay flow, the flow into a corner and the backward-facing step flow. Using a mixed-order formulation, i.e. order k for the velocity and order k-1 for the pressure, the scheme is numerically stable for all test cases. Convergence rates of k+1 and k in the L^2-norm are observed for velocity and pressure, respectively. A study of the convergence behavior of the SIMPLE algorithm shows that no under-relaxation for the pressure is needed, which is in strong contrast to the application of the SIMPLE algorithm in the context of the finite volume method or the continuous finite element method. We conclude that the proposed scheme is efficient to solve the steady incompressible Navier-Stokes equations in the context of the discontinuous Galerkin method comprising hp-accuracy.

AN EXTENSION OF THE DISCONTINUOUS GALERKIN METHOD FOR THE SINGULAR POISSON EQUATION

We present a numerical method for solving a singular Poisson equation which solution contains jumps and kinks due to a singular right-hand side. Equations of this type may arise, e.g., within the pressure computation of incompressible multiphase flows. The method is an extension to the well-known discontinuous Galerkin (DG) method, being able to represent the jumps and kinks with subcell accuracy. In the proposed method, an ansatz function which already fulfills the jump condition is subtracted from the original problem, thereby reducing it to a standard Poisson equation without a jump. Invoking a technique that we refer to as “patching,” the construction of the ansatz function can be limited to a very narrow domain around the jump position, thus making the construction numerically cheap and easy. Under optimal conditions, the method shows a convergence order of

Patch-Recovery Filters for Curvature in Discontinuous Galerkin-Based Level-Set Methods

p+1

LEVEL SET METHOD FOR SIMULATING THE INTERFACE KINEMATICS: APPLICATION OF A DISCONTINUOUS GALERKIN METHOD

for DG polynomial degree

An extended discontinuous Galerkin framework for multiphase flows

p

A High-Order Local Discontinuous Galerkin Scheme for Viscoelastic Fluid Flow

. Still, in the worst case, a convergence order of approximately 2.4 is preserved for DG polynomial degree of 2.

Numerical investigation of laminar vortex shedding applying a discontinuous Galerkin Finite Element method

In two-phase flow simulations, a difficult issue is usually the treatment of surface tension effects. These cause a pressure jump that is proportional to the curvature of the interface separating the two fluids. Since the evaluation of the curvature incorporates second derivatives, it is prone to numerical instabilities. Within this work, the interface is described by a level-set method based on a discontinuous Galerkin discretization. In order to stabilize the evaluation of the curvature, a patch-recovery operation is employed. There are numerous ways in which this filtering operation can be applied in the whole process of curvature computation. Therefore, an extensive numerical study is performed to identify optimal settings for the patch-recovery operations with respect to computational cost and accuracy.

Highly accurate surface and volume integration on implicit domains by means of moment-fitting

A Discontinuous Galerkin (DG) Method was applied for simulating the kinematics of deforming interfaces. The level set method was used as an interface capturing method. The numerical implementations were performed in the context of the in-house code BoSSS developed at the Chair of Fluid Dynamics in Darmstadt University of Technology, see [1, 2]. As the higherorder spectral representation of the variables in the DG method, results in a precise solution to the level set advection equation, the interface kinematics could be accurately simulated without having to solve the level set re-initialization equation. But the solution exhibits an appropriate hp-convergence only if the gradient of the level set function does not have any singularity over the domain of computation. For instance, the signed-distance level set function of a circle has a singular gradient at the center of the circle. As the smoothed Heaviside and Delta functions used in the multiphase flow calculations, are commonly expressed in terms of the level set function, the level set function needs to remain signed distance in order to keep a uniform smoothing width. The signed distance property of the level set function can be recovered by solving the level set re-initialization equation. In order to obtain a solution with a monotonicity preserving behavior, a Godunovs scheme was applied for approximating the Hamiltonian of the re-initialization equation. Moreover, a notable stability improvement was achieved by adding an artificial diffusion along the characteristic lines. The solution exhibits an appropriate hpconvergence and almost no spurious movement of the interface was detected.

Simple multidimensional integration of discontinuous functions with application to level set methods

We present a framework for handling cut cells in a high order discontinuous Galerkin (DG) context. To describe the boundary between fluid phases, we use a level-set formulation. When the interface cuts a computational cell, we discretize the resulting sub-cells with the same DG method as used on standard cells. This requires a suitable quadrature procedure. Within this framework, we present a solver for the two-phase Navier-Stokes equation, a reinitialization procedure for the level-set and a solver for transport-processes on the surface.

Extended discontinuous Galerkin methods for two-phase flows: the spatial discretization

Coping with the so called high Weissenberg number problem (HWNP) is a key focus of research in computational rheology. By numerically simulating viscoelastic flow a breakdown in convergence often occurs for different computational approaches at critically high values of the Weissenberg number. This is due to two major problems concerning stability in the discretization. First, we have a mixed hyperbolic-elliptic problem weighted by a ratio parameter between retardation and relaxation time of viscoelastic fluid. Second, we have a convection-dominated convection-diffusion problem in the constitutive equations. We introduce a solver for viscoelastic Oldroyd B flow with an exclusively high-order Discontinuous Galerkin (DG) scheme for all equations using a local DG formulation in order to solve the hyperbolic constitutive equations and using a streamline upwinding formulation for the convective fluxes of the constitutive equations. The successful implementation of the local DG formulation for Newtonian fluid with appropriate fluxes containing stabilizing penalty parameters is shown in two results. First, a hk-convergence study is presented for a non-polynomial manufactured solution for the Stokes system. Second, numerical results are shown for the confined cylinder benchmark problem for Navier-Stokes flow and compared to the same flow using a symmetric interior penalty method without additional constitutive equations.