A mimetic numerical scheme for multi-fluid flows with thermodynamic and geometric compatibility on an arbitrarily moving grid. (November 2020) - Vazquez-Gonzalez, Thibaud; Llor, Antoine; Fochesato, Christophe
Highlights: A compressible multi-fluid ALE scheme is built using a least action principle. Thermodynamic consistency is thus ensured, with explicit internal energy PDEs. Novel non-standard downwind discrete pressure gradients appear in the process. The scheme preserves masses, momentum, and energy exactly for any number of fluids. Future schemes will benefit from this proof of concept. Graphical abstract: Abstract: Simulating transient and compressible multi-fluid flows in extreme situations such as Inertial Confinement Fusion is especially challenging because of numerous and sometimes conflicting constraints : large number of fluids, both isentropic and strongly shocked compressible evolution, highly variable, stiff or contrasted equations of state, large heat sources, large deformations, and transport over large distances. Models and schemes for such flows all share a common non-dissipative "backbone" structure of per-fluid mass, momentum, and energy evolution-and-transport equations, coupled through pressure terms. A novel and efficient multi-fluid numerical scheme for discretizing the backbone equations over a moving grid (ALE or Arbitrary Lagrangian–Eulerian) is here generated through a "Geometry, Energy, and Entropy Compatible" mimicking procedure [Eur. J. Mech. B – Fluids 67, 494 (2017)]. Starting from the discretized density fields, energy fields, and transport operators, the procedure yields the discrete evolution equations in a practically univocal way. With arbitrarily moving grids, number of fluids, contrasts of volume fractions and equations of state, the resulting scheme is fully conservative in masses, momentum, and energy, preserves isentropic behavior to the scheme order, and ensures per-fluid thermodynamic consistency. Noticeably, optimal isentropic behavior is obtained thanks to a non-standard downwind form of pressure gradient. Multi-fluid numerical test cases—including Sod's shock tube, Ransom's faucet, and a nine-fluids crossing test—are performed in two-dimensions using deliberately strenuous grid motion strategies. The results confirm the expected properties and illustrate the robustness, stability and versatility of the scheme at finite resolution, though it is not intended to be used as is. The scheme is a foundation block to be complemented with (system-dependent and ubiquitous) physical dissipation terms which provide the necessary damping of divergingly unstable modes at vanishing wavelengths.