Strong shock as a stringent test for Onsager-Burnett equations

A new set of thermodynamically consistent Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)] has recently been derived. In this work, we subject these equations to a severe test case of strong shock (Mach number = 134) for a dilute gas system composed of hard-sphere molecules. The numerical results of OBurnett equations for conserved and nonconserved variables are compared against the molecular dynamics and direct simulation Monte Carlo results available in the literature. With no tweaking of the equations in any way, we establish several fundamental aspects of OBurnett equations which other higher-order continuum theories like Burnett and Grad equations lack. In particular, evidence is put forward for smooth shock structures, the existence of heteroclinic trajectory, and positive entropy generation inside the shock at all Mach numbers. With respect to shock profiles of hydrodynamic variables, it is observed that OBurnett equations significantly improve upon the results of Navier-Stokes equations. Further comparison with regularized 13 (R13) equations at lower Mach numbers shows that OBurnett equations capture the more rarefied upstream part better than the R13 equations. These evidences suggest that the OBurnett equations are accurate and form a reliable set of higher-order transport equations. Further, it should now be possible to describe the complex structure of the shock wave correctly, even at a very large Mach number.

Figure 1

(a) Orbits in the ($u^{*},\tau$) plane at $\mathrm{Ma}=134$. DSMC results are from Ref. [26]. (b) Zoomed-in view near the upstream critical point.

Figure 2

Nondimensional entropy generation $[{\dot{\sigma }}^{*}=\dot{\sigma }l/\left({\rho }_0{u}_{0}R\right)]$ inside the shock at $\mathrm{Ma}=134$.

Figure 3

Variation of hydrodynamic fields (a) velocity, (b) density, (c) temperature, and (d) heat flux across the shock for $\mathrm{Ma}=134$. Molecular dynamics (MD) results are from Ref. [46].

Figure 4

Variation of hydrodynamics fields' (a) velocity, (b) density, (c) temperature, and (d) heat flux across the shock for $\mathrm{Ma}=3$.

Figure 5

Comparison of OBurnett results for density and heat flux profiles with those of R13 moment equations. Nondimensionalization and normalization of variables were performed as done in Torrilhon and Struchtrup [28]. (a) $\mathit{Ma}$ = 3, (b) $\mathit{Ma}$ = 4