![Spatial discretization of a droplet in a vapor environment. Elements in the bulk region containing a smooth solution are discretized by a DG method with a local degree NDG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\textrm{DG}}$$\end{document}. Elements containing the phase boundary are subdivided into NFV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\textrm{FV}}$$\end{document} sub-cells per direction to improve the approximation of the phase boundary by the surrogate surface](https://www.researchgate.net/publication/374624648/figure/fig1/AS:11431281200513316@1697940483161/Spatial-discretization-of-a-droplet-in-a-vapor-environment-Elements-in-the-bulk-region_Q320.jpg)
![Flux calculation for two adjacent elements with degree N=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3$$\end{document} (left) and M=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=4$$\end{document} (right). The node distribution is indicated by dots in the volume and by squares on the surface. The numerical flux is computed on the solution representation of the higher polynomial degree](https://www.researchgate.net/publication/374624648/figure/fig2/AS:11431281200451201@1697940483267/Flux-calculation-for-two-adjacent-elements-with-degree-N3documentclass12ptminimal_Q320.jpg)
![Flux computation for a mixed interface with a DG element of degree N=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3$$\end{document} (left) and a FV sub-cell element with NFV=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\textrm{FV}}=4$$\end{document} sub-cells (right). DG volume nodes and FV sub-cell centers are indicated by dots. Surface nodes are indicated by squares. The numerical flux is computed on a piecewise constant FV representation FFV∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\mathcal {F}}}^*_{\textrm{FV}}$$\end{document} and projected afterwards to a DG solution FDG∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\mathcal {F}}}^*_{\textrm{DG}}$$\end{document} to provide the flux for the DG element](https://www.researchgate.net/publication/374624648/figure/fig3/AS:11431281200513317@1697940483385/Flux-computation-for-a-mixed-interface-with-a-DG-element-of-degree_Q320.jpg)
![Illustration of an exemplary two-phase Riemann problem at the phase interface (left) with the intermediate states QlG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Q}}_{l}^{{\mathcal {G}}}$$\end{document} and QvG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Q}}_{v}^{{\mathcal {G}}}$$\end{document}. The intermediate states of the Riemann problem serve as ghost states to define boundary conditions and to compute the ghost fluxes FlG(Ql,QlG)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\mathcal {F}}}_l^{{\mathcal {G}}}({\varvec{Q}}_{l},{\varvec{Q}}_{l}^{{\mathcal {G}}})$$\end{document} and FvG(QvG,Qv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\mathcal {F}}}_v^{{\mathcal {G}}}({\varvec{Q}}_{v}^{{\mathcal {G}}},{\varvec{Q}}_{v})$$\end{document} across the phase boundary (right)](https://www.researchgate.net/publication/374624648/figure/fig4/AS:11431281200451202@1697940483511/Illustration-of-an-exemplary-two-phase-Riemann-problem-at-the-phase-interface-left-with_Q320.jpg)
October 2023
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238 Reads
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6 Citations
Journal of Scientific Computing
We present an hp-adaptive discretization for a sharp interface model with a level-set ghost-fluid method to simulate compressible multiphase flows. The scheme applies an efficient p-adaptive discontinuous Galerkin (DG) operator in regions of smooth flow. Shocks and the phase interface are captured by a Finite Volume (FV) scheme on a h-refined element-local sub-grid. The resulting hp-adaptive scheme thus combines both the high order accuracy of the DG method and the robustness of the FV scheme by using p-adaptation in smooth areas and h-refinement at discontinuities, respectively. For the level-set based interface tracking, a similar hybrid DG/FV operator is employed. Both p-refinement and FV shock and interface capturing are performed at runtime and controlled by an indicator, which is based on the modal decay of the solution polynomials. In parallel simulations, the hp-adaptive discretization together with the costly interface tracking algorithm cause a significant imbalance in the processor workloads. To ensure parallel efficiency, we propose a dynamic load balancing scheme that determines the workload distribution by element-local wall time measurements and redistributes elements along a space filling curve. The parallelization strategy is supported by strong scaling tests using up to 8192 cores. The framework is applied to established benchmarks problems for inviscid, compressible multiphase flows. The results demonstrate that the hybrid adaptive discretization can efficiently and accurately handle complex multiphase flow problems involving pronounced interface deformations and merging interface contours.
![Figure 4: Time evolution of the dimensionless vertical displacement u y /h at the middle point of the panel for four Mach numbers, i.e. M ∞ = 2.05, 2.10, 2.11, 2.12. The supersonic panel flutter test case by Mouronval et al. [59] is used for the first validation of our FSI solver. A schematic of the setup is shown in Figure 3. The incoming supersonic flow is characterized by density ρ ∞ = 0.4kg/m 3 and pressure p ∞ = 13000Pa. In order to determine the approximate flutter speed, the inflow Mach numbers M ∞ are adjusted by changing inflow velocity u ∞ . The two-dimensional computational domain Ω = (0.0, 1.0) m × (0.0, 0.25) m is discretized in streamwise direction by 60 equidistant grid cells and in wall-normal direction by 10 exponentially growing cells of a stretching factor 1.15. At the inlet, the free stream flow is prescribed by means of a static Dirichlet boundary condition. The upper and outflow boundaries are set to be static supersonic outflow conditions. The bottom boundary and the panel surface are set to be static and moving slip wall boundaries, respectively. The deformation of the internal fluid mesh is based on the 'CT PS C 2 ' RBF, defined in [58], with a support radius of r = 0.2m. We apply a polynomial degree of N = 5, a Courant-Friedrichs-Lewy number of CFL = 0.9 and use the approximate Riemann solver of Roe with the entropy fix by Harten and Hyman [60] for the Euler fluxes.](https://www.researchgate.net/profile/Min-Gao-29/publication/369226406/figure/fig4/AS:11431281126756468@1678807114848/Time-evolution-of-the-dimensionless-vertical-displacement-u-y-h-at-the-middle-point-of_Q320.jpg)
![Fig. 9: Density, velocity and pressure distribution of the gas-liquid Riemann problem at the final time t = 0.2. The exact solution is compared against a numerical solution computed with a grid of 40 elements. Smooth regions are discretized by a DG scheme (blue) with N ∈ [2, 4], whereas a FV scheme (red) with N FV = 9 sub-cells per element is applied at discontinuities.](https://www.researchgate.net/profile/Pascal-Mossier/publication/366226223/figure/fig3/AS:11431281107038673@1670937106383/Density-velocity-and-pressure-distribution-of-the-gas-liquid-Riemann-problem-at-the_Q320.jpg)
