Numerical Comparison of Solutions of Kinetic Model Equations

• Main Author: A. A. Frolova
• Format: Article
• Language: Russian
• Published: MGTU im. N.È. Baumana 2016-06-01
• Series: Matematika i Matematičeskoe Modelirovanie
• Subjects: boltzmann equation; model equations; the equation of bgk; ellipsoidal statistical model (es); shakhov model (s-model)
• Online Access: https://www.mathmelpub.ru/jour/article/view/39

The collision integral approximation by different model equations has created a whole new trend in the theory of rarefied gas. One widely used model is the Shakhov model (S-model) obtained by expansion of inverse collisions integral in a series of Hermite polynomials up to the third order. Using the same expansion with another value of free parameters leads to a linearized ellipsoidal statistical model (ESL).Both model equations (S and ESL) have the same properties, as they give the correct relaxation of non-equilibrium stress tensor components and heat flux vector, the correct Prandtl number at the transition to the hydrodynamic regime and do not guarantee the positivity of the distribution function.The article presents numerical comparison of solutions of Shakhov equation, ESL- model and full Boltzmann equation in the four Riemann problems for molecules of hard spheres.We have considered the expansion of two gas flows, contact discontinuity, the problem of the gas counter-flows and the problem of the shock wave structure. For the numerical solution of the kinetic equations the method of discrete ordinates is used.The comparison shows that solution has a weak sensitivity to the form of collision operator in the problem of expansions of two gas flows and results obtained by the model and the kinetic Boltzmann equations coincide.In the problem of the contact discontinuity the solution of model equations differs from full kinetic solutions at the point of the initial discontinuity. The non-equilibrium stress tensor has the maximum errors, the error of the heat flux is much smaller, and the ESL - model gives the exact value of the extremum of heat flux.In the problems of gas counter-flows and shock wave structure the model equations give significant distortion profiles of heat flux and non-equilibrium stress tensor components in front of the shock waves. This behavior is due to fact that in the models under consideration there is no dependency of the collision frequency on the molecular velocity.As calculations show, the ESL-model describes more accurately the non-equilibrium flow regime, but gives a greater deviation from the Boltzmann equation, than the Shahov model in front of shock waves.DOI: 10.7463/mathm.0615.0823537