Testing continuum descriptions of low-Mach-number shock structures

Abstract

Numerical experiments have been performed on normal shock waves with Monte Carlo Direct Simulations (MCDS's) to investigate the validity of continuum theories at very low Mach numbers. Results from the Navier—Stokes and the Burnett equations are compared to MCDS's for both hard-sphere and Maxwell gases. It is found that the maximum-slope shock thicknesses are described equally well (within the MCDS computational scatter) by either of the continuum formulations for Mach numbers smaller than about 1.2. For Mach numbers greater than 1.2, the Burnett predictions are more accurate than the Navier—Stokes results. Temperature—density profile separations are best described by the Burnett equations for Mach numbers greater than about 1.3. At lower Mach numbers the MCDS scatter is too great to differentiate between the two continuum theories. For all Mach numbers above one, the shock shapes are more accurately described by the Burnett equations.

Regularized 13-moment equations: shock structure calculations and comparison to Burnett models

“…For the actual setup, such as interval length, upstream temperature, etc. we adopted the values of Pham- Van-Diep et al (1991). Note that the calculation of a single low-Mach-number shock structure by a standard DSMC program takes several hours which is several orders of magnitude slower than the calculation by a continuum model.…”

“…The shock thickness values obtained with these calculations are linearly interpolated in order to obtain continuous curves. The results for the shock thickness are compared with DSMC calculations from Pham- Van-Diep et al (1991) which also have been reproduced by our DSMC simulations. Figure 10 shows the theoretical curves for the inverse shock thickness, see (6.1), of the various continuum models together with the DSMC results as symbols.…”

“…Even worse, the NSF theory cannot even reproduce trends for the shock asymmetry (see also § 6), as is shown e.g. in Alsmeyer (1976), Pham- Van-Diep, Erwin & Muntz (1991) and Au (2001). These results reflect the fact that the NSF equations are restricted to near-equilibrium flows.…”

Internal Energy Mode Relaxation in High Speed Continuum and Rarefied Flows

“…The advantage of the shock structure problems is that it does not involve the nonequihbrium effects of gas-surface interactions, an area of research not considered in the proposed paper. Numerical experiments performed by Pham-Van-Diep, et al [7] in testing continuum descriptions showed that the continuum solutions of the shock structure in a monatomic gas were able to match the rarefied solution at a Mach number of 1.2. At higher Mach numbers the solutions obtained with continuum approach do not match those by the DSMC method.…”

Model Reduction in Kinetic Theory

“…The relevant parameter for the shock is the inflow Mach number M0 = v0 ºÕ 5 3 θ0 . We compare the shock structures obtained from macroscopic equations for rarefied flows to DSMC results obtained with Bird's code [5], and plot results in dimensionless form [51,15,3]. Figure 2 shows the density and heat flux profiles of a M0 = 2 shock calculated with the NSF and Grad13 equations as well as with the Burnett and super-Burnett equations for Maxwell molecules.…”