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Section: New Results

Low Mach number flows simulations issues

Our activity for developing schemes suitable for the simulation of low Mach number flows considers the two main techniques developed initially for dealing with either zero Mach number flows (pressure-velocity coupling) or compressible flows (density based approach). For both approaches, we concentrated this year on the specific difficulties related to unsteady flows simulations. For the methodology adressing the pressure-velocity coupling with a low-order discretization technique, we introduced an inertia term in the AUSM+ -up scheme. The resulting scheme, called AUSM-IT (IT for Inertia Term), was designed as an extension of the AUSM+ -up scheme allowing for full Mach number range calculations of unsteady flows including acoustic features. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the conservation of the discrete linear acoustic energy at first order in the low Mach number limit. Its capability to properly handle low Mach number unsteady flows, that may include acoustic waves or discontinuities was numerically illustrated [7] .

As far as density based approach are concerned, an analysis of explicit RKDG schemes have been performed for unstationary acoustic waves propagating in a low Mach number flow. Classical cures of the unaccuracy of upwind schemes at low Mach number consist in using centered flux on the pressure. By a two scale asymptotic expansion of the scheme, we proved that this cure is a dead end for resolving unstationary acoustic waves, because it leads to a non dissipative scheme for the wave equations. We developed a dissipative term that can both stabilize the stationary incompressible equations, and the system of acoustic waves. The results with this new type of scheme have been presented in [8] .