(510j) Playing (almost) safe on robustness and entropy in compressible multi-fluid dynamics with Geometry, Energy, and Entropy Compatible (GEEC) numerical schemes

Of the main numerical approaches for CFD that have been explored over the last half century, only a few have become mainstream and make the vast majority of theoretical investigations in academia and practical usage in applications. These mostly hinge on concepts of finite volume discretization, monotonicity preservation, flux upwinding, and the analysis of the associated numerical dissipation processes, more or less adapted from Godunov's original scheme. The basic building block consists in considering the discontinuities of field values at cell interfaces as perturbations whose propagation (the Riemann problem, possibly approximated) is calculated and remapped on the original cells. However, this always introduces some form of dissipation which has the advantage of removing almost all over- and undershoots, oscillations, and other artifacts, at the expense of poor behavior in isentropic conditions. The only options for recovering proper isentropic behavior are then an artificial correction of over-dissipation or an expensive crank up of the schemes' order - both options being potentially very fragile for complex non-linear or multi-fluid systems. Moreover, multi-fluid systems do not even sustain a unique solution to the Riemann problem.

However, application to what looks as "niche" problems shows that this dominant approach may not be as effective as generally accepted and has unduly benefited from a "winner-takes-all" effect. One of these problems is the simulation of isentropic flows which is actually "not-so-niche" as is of high practical interest, especially in multi-fluid systems which involve complex energy transfers.

The present contribution aims at providing some perspective on CFD numerical schemes recently designed in order to better capture isentropic flows including in multi-fluid systems. The basic principle is that isentropic flow is geometric, i.e. potential (or internal) energies only depend on fluids' densities which in turn are defined by fluid element trajectories. An isentropic numerical scheme can thus be obtained by a variational, least action principle. Corrections must be further added to enforce the important properties of energy conservation and positive dissipation. This Geometry, Energy, and Entropy Compatible approach (GEEC) is illustrated on our recently developed Arbitrary Lagrangian-Eulerian (ALE, where the mesh moves and distorts according to user's free prescriptions) compressible multi-fluid scheme [Int. J. Multiph. Flow, accepted].

The critical ingredient in the GEEC approach is the action integral discretization as it entirely defines the numerical scheme except for some residual terms of higher than the scheme's order. Careful definition of the discrete action integral ensures that the eventual GEEC compressible multi-fluid ALE scheme has the following desirable properties:

i) Arbitrary number of coupled fluids;

ii) Versatile arbitrary grid evolution (ALE), possibly calculated from a Lagrangian velocity, for instance user-predefined or provided by added on-the-fly mesh regularization algorithms;

iii) Exact conservation of masses, momentum and energy;

iv) Second-order space-and-time discretization of grid velocity, and kinetic and internal energies;

v) First-order space-and-time discretization of transport (simple upwind scheme), acceptable for weak relative displacements, weak gradients, or large meshes, that provides proof of concept to study other higher-order variational direct ALE schemes;

vi) Yet, full preservation of isentropic flows to scheme order;

vii) Single pressure closure (full relaxation) between phases;

viii) Non-standard downwind pressure gradients, dual to upwind transport, unexpected and at variance with usual finite volumes schemes.

The scheme was tested on various strenuous situations, including a supersonic crossing of eight Gaussian clouds on a shrink-then-stretch swirling ALE grid to be presented.