Numerical dissipation of upwind schemes in low Mach flow

Abstract

SUMMARYThis paper presents a modified Roe scheme for the simulation of multicomponent compressible flows with low Mach features. This modification reduces the excess dissipation of kinetic energy in Godunov-type methods at low Mach. The modification is shown to work effectively to Ma = 0.0002 using a single-mode Kelvin-Helmholtz instability as a test case, and reproduces the correct Ma 2 incompressible pressure scaling. Computational expense is negligible.

Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number

“…In other words, under sufficient condition (28) and when the initial conditions are wellprepared, PDE (26) does not create any spurious wave. At the opposite, the first point of Theorem 2.2 is not really important since estimate (27) does not mean that qðt P 0Þ is almost in the well-prepared subspace E. This first point is only mentioned to underline the importance of the invariance of E. Moreover, we underline the fact that we do not impose that the energy EðtÞ is a constant or a decreasing function (nevertheless, Eðt P 0Þ is bounded since (26) is supposed to be well-posed).…”

“…In that case, estimate (50) induced by any 1D well-prepared initial condition should be replaced by the estimate p i ðs ac Þ ¼ p à þOðMDxÞ (see below the 2D case for the details). In fact, the formal analysis in [4,8] cannot be satisfactory because, as in [6,10], they use 1D arguments (an 1D argument is also used in [26]: see estimate (11) in [26]). As a consequence, any divergence-free field is trivial and it is impossible to clearly understand the origin of the inaccuracy of Godunov type schemes at low Mach number.…”

“…Proof of Theorem 2.2: Estimate (27) is a direct consequence of the well-posedness of PDE (26). Let us now define qðt; xÞ solution of (26) with the initial condition qðt ¼ 0; xÞ :¼ q 0 À Pq 0 .…”

“…Moreover, the previous formal analysis (partly inspired from [6,10]) does not take into account the space dimension since it is an 1D analysis (1D arguments are also used in [4,8,26]): we will see in the sequel that the space dimension is essential to analyse Godunov type schemes at low Mach number from a formal or theoretical point of view.…”

Turbulent mixing in spherical implosions

“…This low Mach number behaviour in TURMOIL is better than many Godunov methods. However, work on improving the low Mach number behaviour of Godunov methods [8] may make them well suited for calculating compressible turbulent flow.…”