Numerical simulation of rarefied supersonic flows using a fourth-order maximum-entropy moment method with interpolative closure

Abstract

Maximum-entropy moment methods allow for the modelling of gases from the continuum regime to strongly rarefied conditions. The development of approximated solutions to the entropy maximization problem has made these methods computationally affordable. In this work, we apply a fourth-order maximum-entropy moment method to the study of supersonic rarefied flows. For such conditions, we compare the maximum-entropy solutions to results obtained from the kinetic theory of gases at different Knudsen numbers. The analysis is performed for both a simplified model of a gas with a single translational degree of freedom (5-moment system) and for a typical gas with three degrees of freedom (14-moment system). The maximum-entropy method is applied to the study of the Sod shock-tube problem at various rarefaction levels, and to the simulation of two-dimensional low-collisional crossed supersonic jets. We show that, in rarefied supersonic conditions, it is important to employ accurate estimates of the wave speeds. Since analytical expressions are not presently available, we propose an approximation, valid for the 14-moment system. In these conditions, the solution of the maximum-entropy system is shown to realize large degrees of non-equilibrium and to approach the Junk subspace, yet provides a good overall accuracy and agreement with the kinetic theory. Numerical procedures for reaching second-order accurate discretizations are discussed, as well as the implementation of the 14-moment solver on Graphics Processing Units (GPUs).

Figures (23)

• Figure 1: Physical realizability boundary, Junk subspace and equilibrium in dimensionless moment space, for the 1D1V 5-moment system. Dotted parabolas represent contours of the parabolic mapping parameter, $\sigma$.
• Figure 2: Moments obtained from the numerical solution of the BGK kinetic equation. Cases (a) to (e) are obtained at increasing simulated times, that progressively approach local thermodynamic equilibrium. Arrows show the evolution of the moments with the increasing simulated time. The bottom-right panel shows the solution in the dimensionless moment space: the Junk line is shown only for reference and does not affect anyhow the kinetic solution. The physical realizability condition is automatically satisfied by the kinetic solution.
• Figure 3: Solution of the 5-moment system for the Sod shock-tube problem test case, in collisionless conditions, for $\sigma_\mathrm{lim}=10^{-4}$. Left: normalized density profile; the vertical dashed lines in the Left plot denote the main region, where the initial discontinuity breaks into various waves. Centre: dimensionless fourth-order moment; a much faster wave can be observed ($a$), followed by a further discontinuity ($b$); Right: trajectory of the system in moment space, and effect of the discontinuities. The dashed lines indicate contours at constant $\sigma$.
• Figure 4: Solution of the 5-moment system for the Sod shock-tube problem test case, in collisionless conditions, for $\sigma_\mathrm{lim}=10^{-4}$. The top panels represent the scaled density, and dimensionless heat flux and fourth-order moment. The bottom panel represents the solution in dimensionless moment space, superimposed to lines at constant $\sigma$. The point where the numerical solution crosses the Junk line is indicated with a white arrow. A red marker is also employed to help mapping the crossing region among the different plots.
• Figure 5: Solution of the 5-moment system for the Sod shock-tube problem. The red arrows indicate the evolution of the solution for increasingly longer simulated times, equivalent to the Knudsen numbers: $\mathrm{Kn}=\infty, 0.25, 0.1, 0.02$. The dashed line is obtained for $\mathrm{Kn} = 0.0001$. Regions (1), (2) and (3) indicate which parts of the initial wave structure evolve into the rarefaction fan, contact discontinuity and shock wave respectively.