Computable model on the collision integral of Boltzmann equation and application to rarefied aerodynamics

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Due to its complexity in dealing with the collisional integral term of the Boltzmann equation and computational costs associated with multi-dimensional problems, deterministic methods are still restricted to simple flow such as one-dimensional linear flow. However, the recently emerged fast spectrum method has achieved breakthroughs in computational efficiency and accuracy, which can enable simulations for more realistic three-dimensional non-linear flows. In comparison with the dominant direct simulation Monte Carlo method, the deterministic method has advantages especially in simulating low-speed flows where statistical variations prevail. Here, we review the development of fast spectrum method and discuss its applications for practical flow simulations. In particular, extended Boltzmann model is required for polyatomic and dense gases where the Boltzmann equation may not be valid. We present the applications of extended Boltzmann model for polyatomic gases in predicting spectra of both spontaneous and coherent Rayleigh-Brillouin Scattering, and in simulating space vehicle reentries with a broad range of $Kn$. Finally, we discuss the gas-kinetic unified algorithm (GKUA) of computable model Boltzmann equation and applications to the hypersonic aerodynamics of space reentry covering various flow regimes.