A sharp-interface discontinuous Galerkin method for simulation of two-phase flow of real gases based on implicit shock tracking

Abstract

We present a high-order, sharp-interface method for simulation of two-phase flow of real gases using implicit shock tracking. The method is based on a phase-field formulation of two-phase, compressible, inviscid flow with a trivial mixture model. Implicit shock tracking is a high-order, optimization-based discontinuous Galerkin method that automatically aligns mesh faces with non-smooth flow features to represent them perfectly with inter-element jumps. It is used to accurately approximate shocks and rarefactions without stabilization and converge the phase-field solution to a sharp interface one by aligning mesh faces with the material interface. Time-dependent problems are formulated as steady problems in a space-time domain where complex wave interactions (e.g., intersections and reflections) manifest as space-time triplet points. The space-time formulation avoids complex re-meshing and solution transfer that would be required to track moving waves with mesh faces using the method of lines. The approach is applied to several two-phase flow Riemann problems involving gases with ideal, stiffened gas, and Becker-Kistiakowsky-Wilson (BKW) equations of state, including a spherically symmetric underwater explosion problem. In all cases, the method aligns element faces with all shocks (including secondary shocks that form at time t > 0), rarefactions, and material interfaces, and accurately resolves the flow field on coarse space-time grids.

Figures (13)

• Figure 1: Shock-agnostic space-time mesh and first-order solution (density) used to initialize HOIST solver (top), and converged HOIST solution (density) with (middle) and without (bottom) mesh edges.
• Figure 2: Shock-agnostic space-time mesh and first-order solution (phase) used to initialize HOIST solver (top), and converged HOIST solution (phase) with (middle) and without (bottom) mesh edges.
• Figure 3: Two-phase HOIST solution (density) for single-phase, ideal gas Sod shock tube problem using one slab (bottom) and the mesh and solution used to initialize the HOIST solver (top).
• Figure 4: Left: Slice of two-phase HOIST solution (density) (\ref{['line:density_sod_multi']}) to the single-phase, ideal gas Sod shock tube problem at the final time, and the corresponding analytical solution (\ref{['line:density_sod_multi_anyl']}). Right: Mesh convergence of the two-phase HOIST method ($p=2$) applied to the single-phase Sod shock tube measured using $L^1$ error at time $t=0.1$. Legend (right): HOIST solution error (\ref{['line:err_multi_sod_conv']}) and optimal convergence rate ($p+1$) (\ref{['line:err_multi_sod_conv_ideal']}).
• Figure 5: Two-phase HOIST SQP iterations (density) for two-phase, ideal gas Sod shock tube problem using one slab. SQP iterations $n = 0, 25, 50, 75, 100$ (top-to-bottom).

Theorems & Definitions (2)

• Remark 1
• Remark 2