A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces

Abstract

Although the Discontinuous Galerkin (dg) method has seen widespread use for compressible flow problems in a single fluid with constant material properties, it has yet to be implemented in a consistent fashion for compressible multiphase flows with shocks and interfaces. Specifically, it is challenging to design a scheme that meets the following requirements: conservation, highorder accuracy in smooth regions and non-oscillatory behavior at discontinuities (in particular, material interfaces). Following the interface-capturing approach of Abgrall [1], we model flows of multiple fluid components or phases using a single equation of state with variable material properties; discontinuities in these properties correspond to interfaces. To represent compressible phenomena in solids, liquids, and gases, we present our analysis for equations of state belonging to the Mie-Grüneisen family. Within the dg framework, we propose a conservative, high-order accurate, and non-oscillatory limiting procedure, verified with simple multifluid and multiphase problems. We show analytically that two key elements are required to prevent spurious pressure oscillations at interfaces and maintain conservation: (i) the transport equation(s) describing the material properties must be solved in a non-conservative weak form, and (ii) the suitable variables must be limited (density, momentum, pressure, and appropriate properties entering the equation of state), coupled with a consistent reconstruction of the energy. Further, we introduce a physics-based discontinuity sensor to apply limiting in a solution-adaptive fashion. We verify this approach with one-and two-dimensional problems with shocks and interfaces, including high pressure and density ratios, for fluids obeying different equations of state to illustrate the robustness and versatility of the method. The algorithm is implemented on parallel graphics processing units (gpu) to achieve high speedup.

Numerical simulations of a shock interacting with successive interfaces using the Discontinuous Galerkin method: the multilayered Richtmyer–Meshkov and Rayleigh–Taylor instabilities

“…We extended this approach to the Discontinuous Galerkin method [18,19]. The Discontinuous Galerkin method [6][7][8][9][10] is a numerical method for solving partial differential equations which combines the advantages of the finite element and finite volume methods.…”

“…Additionally, a limiting procedure is required to avoid solution oscillations at flow discontinuities. We use a non-oscillatory, conservative, and high-order accurate limiting procedure based on hierarchical reconstruction, which has been suitably modified to prevent spurious pressure oscillations [19]. Solution limiting is performed gradually and hierarchically from the highest polynomial degree to the lowest to retain as much of the high-order accuracy of the method as possible.…”

“…We consider multiple adjacent ideal gases separated by singlemode perturbed interfaces. We use the high-order accurate Discontinuous Galerkin method to solve the multifluid Euler equations [18,19]. Using two-dimensional simulations, we analyze the effects of the shocks, rarefactions, and the separation distance on the mixing between the three fluids.…”

An improved unstructured WENO method for compressible multi‐fluids flow

“…A similar case can be found in Frahan [23]. In this test, a perturbation is added to the mean flow .…”

“…To investigate the order of accuracy of the present method, the two-dimensional vortex evolution problem is employed here. A similar case can be found in Frahan [23]. In this test, a perturbation is added to the mean flow .…”