Approximate Riemann solvers for moment models of dilute gases.

Abstract

Two sets of transport equations based on the 10- and 35-moment models of kinetic theory are presented as possible models for the prediction of transitional flows. The 10-moment model is based on the Gaussian distribution function and is deficient in heat transfer while the 35-moment distribution function is an expansion of the Gaussian and is capable of modelling heat transfer. Detailed eigensystem and linearized dispersion analyses on these systems revealed hyperbolicity of the transport equations with elegant wave structures. This hyperbolicity allows for the use of powerful numerical schemes which take advantage of the wave-like nature of the physics. Roe-type approximate Riemann solvers are developed for both of the moment models with three Riemann solvers given for the 35-moment model. Two of the 35-moment solvers use eigenstructures obtained at near-equilibrium conditions along with a correction so that the correct flux jumps are obtained. The third solver uses the non-equilibrium eigenstructure with the eigenvectors expressed as functions of the eigenvalues. A relationship exists among terms which make up the non-equilibrium characteristic polynomial of the 35-moment model which is exploited to provide efficient numerical solution of the non-equilibrium eigenvalues. These solvers were significantly less expensive in terms of computational costs than a 35-moment solver which obtained the eigenvectors numerically. An additional condition is presented for Property U which leads to unique Roe-averages for moment models. This condition must be satisfied if the left eigenvectors obtained from the primitive form of the transport equations are used in the solver. The first known solutions of these models are obtained for the problem of one-dimensional shock structure for a monatomic gas with comparison made to the Navier-Stokes and direct simulation Monte Carlo methods. The solutions are extended to second-order accuracy using Hancock's Predictor/Corrector technique. The results reveal that for the 10-moment model there is no upper limit on the inflow Mach number while solutions are unobtainable above an inflow Mach number of approximately two for the 35-moment model. At low inflow Mach numbers the 35-moment model provides solutions in excellent agreement with the DSMC solutions at a computational cost several orders of magnitude less expensive. At higher inflow Mach numbers the agreement, while not excellent, remains good. An efficiency study reveals that even though the cell cost of the moment models is greater, with respect to the Navier-Stokes cell cost, there is significant saving in iteration count at the higher Mach numbers.