Formulation of entropy-conservative discretizations for compressible flows of thermally perfect gases

Abstract

This study proposes a novel spatial discretization procedure for the compressible Euler equations that guarantees entropy conservation at a discrete level for thermally perfect gases. The procedure is based on a locally conservative formulation, and extends the entropy-conserving schemes to the more realistic case of thermally perfect gases, while still guaranteeing preservation of both linear invariants and kinetic energy. The proposed methodology, which can also be extended to multicomponent gases and to an Asymptotically Entropy-Conservative formulation, shows advantages in terms of accuracy and robustness when compared to existing similar approaches.

Figures (5)

• Figure 1: Instantaneous visualization of the inviscid doubly periodic jet flow at $t/t_c=4$. Results have been obtained on a refined grid consisting in $N_x\times N_y=257\times 129$ evenly spaced points to guarantee a satisfactory visualization of the main flow-field structures. Here, CFL has been set to $0.5$.
• Figure 2: Inviscid doubly periodic jet flow. (a) asymptotic convergence of the novel AEC discretization. (b) error on entropy conservation.
• Figure 3: Inviscid three-dimensional Taylor--Green vortex at $\operatorname{\mathit{M}}=0.1$. (a) entropy-conservation error (b) kinetic energy evolution (c) internal energy evolution (d) kinetic energy-related pressure term evolution. For graphical clarity, data are sampled every 30 time steps for Figs. (a)-(c) and every 500 time steps for Fig. (d).
• Figure 4: Inviscid three-dimensional Taylor--Green vortex at $\operatorname{\mathit{M}}=0.1$. (a) r.m.s. temperature fluctuations. (b) r.m.s. density fluctuations. For graphical clarity, data are sampled every 30 time steps.
• Figure 5: One-dimensional Sod shock tube problem. Results obtained with the EC-TP scheme augmented with a LLF dissipation term for different numbers of spatial discretization points $N_x$.