Marc Henry de Frahan

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

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, high-order 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-Gruneisen 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.

Discontinuous Galerkin method for multifluid euler equations

Based on previous work extending the Discontinuous Galerkin method to multifluid flows, we analyze the performance of our numerical scheme. First, the advection of a contact discontinuity and a multifluid version of the Sod shock-tube problem are considered. We compare the oscillations generated by the non-conservative and conservative formulations of the ratio of specific heats equations, and demonstrate the non-oscillatory nature of the proposed numerical method. We compare the results for different proposed Riemann solvers. We verify the mass, momentum, and energy conservation properties of the scheme. Finally, we validate our numerical simulations against experimental results of the Richtmyer-Meshkov instability.

Investigating the Multi-layered Richtmyer-Meshkov Instability with High-Order Accurate Numerical Methods

The Richtmyer-Meshkov instability (RMI) occurs in flows where a shock interacts with a perturbed interface between two different fluids. At the interface, the interacting shock deposits baroclinic vorticity that drives the perturbation growth [1]. Conducting theoretical and experimental analysis is important to understand the physics of the RMI and its role in the context of high-energy-density physics [3], particularly inertial confinement fusion [8] and supernova collapse [6]. In these latter problems, the geometry is such that the shock interacts with multiple layers of materials. Thus, the multi-layered RMI is particularly relevant in these multi-material problems, where an accurate characterization of the level of mixing is important. Although the single interface RMI has been studied extensively in the past, through both experiments [2] and numerical simulations [5, 7], little attention has been given to the multi-layered RMI.

High-order discontinuous Galerkin methods applied to multiphase flows

In this paper, we present an application of the interface-centered binary projection method to improve the order of accuracy of the discretization of the advection terms by the Discontinuous Galerkin method. We present our multi-gpu parallel implementation and perform weak and strong scaling of our code. We apply the dg method to the multifluid Euler equations. We present numerical simulations of two multiphase problems applicable to many engineering fields. Simulations of a shock interacting with a drop of water in air are compared to experimental data. We show good agreement between the simulations and the experiments. Finally, we investigate the dynamics of a supersonic water drop impacting a wall. Large negative pressures and high tensions inside the drop resulting from the impact can lead to possible cavitation erosion of the wall.