Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations

Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry

As the flow becomes rarefied it has been seen that predictions of continuum formulations, such as the Navier–Stokes equations, become inaccurate. These inaccuracies stem from the linear approximations to the stress and heat flux in the viscous flux terms in the Navier–Stokes equations. Hence, it has long been conjectured that the inclusion of higher-order terms in the constitutive relations for the stress and heat flux may improve the predictive capabilities of such continuum formulations. Following this approach, second-order systems of hydrodynamic equations, such as the Burnett and Woods equations, were applied to the shock structure problem. While it was observed that these equations afforded a better description of the shock structure on coarse grids, they were prone to small wavelength instabilities when the grids were refined. The cause of this instability was subsequently traced to the fact that these equations can potentially violate the second law of thermodynamics when the local Knudsen number exceeds a critical limit. This leads to the fundamental question: is entropy consistency achievable in a system of second-order hydrodynamic equations? To answer this question, a novel set of equations, known as the BGK-Burnett equations, is constructed by taking moments of the Boltzmann equation for the second-order distribution function.The formulation of second-order hydrodynamic equations by moment methods is beset by three hurdles: (i) the highly nonlinear collision integral in the Boltzmann equation needs to be evaluated, (ii) the second-order distribution function does not satisfy the moment closure criterion and (iii) identification of the approximations to the material derivatives in the second-order distribution function that will correctly account for the difference in time scales between the first- and second-order fluxes. The first three terms of the Chapman–Enskog expansion, that defines the second-order distribution function, and the Bhatnagar–Gross–Krook model of the collision integral, form the basis of the BGK-Burnett equations. The entropy-consistent behaviour of the equations depend on the moment closure coefficients and the approximations to the material derivatives in the second-order fluxes. The requirement of moment closure alone, however, results in non-unique closure coefficients and a family of BGK-Burnett equations, from which an entropy-consistent set must be identified. From this family, two sets of BGK-Burnett equations have been considered, and this paper presents the details of the formulation of these two sets of equations, the identification of entropy-consistent approximations to the material derivatives by a novel entropy consistent relaxation technique, and shock structure computations in a monatomic hard sphere gas for a range of Mach numbers.

An approach to entropy consistency in second-order hydrodynamic equations

The paper discusses the shock structure in macroscopic multi-temperature model of gaseous mixtures, recently established within the framework of extended thermodynamics. The study is restricted to weak and moderate shocks in a binary mixture of ideal gases with negligible viscosity and heat conductivity. The model predicts the existence of temperature overshoot of heavier constituent, like more sophisticated approaches, but also puts in evidence its non-monotonic behavior not documented in other studies. This phenomenon is explained as a consequence of weak energy exchange between the constituents, either due to large mass difference, or large rarefaction of the mixture. In the range of small Mach number it is also shown that shock thickness (or equivalently, the inverse of Knudsen number) decreases with the increase of Mach number, as well as when the mixture tends to behave like a single-component gas (small mass difference and/or presence of one constituent in traces).

Shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures

A new model for the nonlinear Boltzmann equation for gas mixtures is constructed by the method employed in the derivation of the McCormack model in the linearized kinetic theory [F.J. McCormack, Phys. Fluids 16 (1973) 2095]. Then it is compared numerically with other existing models proposed in [P. Andries, K. Aoki, B. Perthame, J. Stat. Phys. 106 (2002) 993] and in [L.H. Holway Jr., Phys. Fluids 9 (1966) 1658] (the so-called ES-BGK model) as well as with the original Boltzmann equation. The new model is not restricted to the Maxwell molecule, can fit to general molecular models, and reproduces well solutions of the Boltzmann equation at least in the case of weak nonequilibrium. The numerical comparison is performed in the case of a binary gas mixture consisted of the hard-sphere or pseudo Maxwell molecules, after parameters concerning the molecular interaction are adjusted appropriately.

Model Boltzmann equation for gas mixtures : Construction and numerical comparison

A hierarchy kinetic model and its applications

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A hierarchy kinetic model and its applications