Computers and Fluids 168 (2018) 1–13 Contents lists available at ScienceDirect Computers and Fluids journal homepage: www.elsevier.com/locate/compfuid Assessment of the cubic Fokker–Planck–DSMC hybrid method for hypersonic rarefied flows past a cylinder Eunji Jun a,∗ , M. Hossein Gorji b , Martin Grabe a , Klaus Hannemann a a German Aerospace Center (DLR), Göttingen 37073, Germany b Center for Computational Engineering Science, RWTH Aachen, Aachen 52062, Germany a r t i c l e i n f o Article history: Received 31 January 2018 Revised 10 March 2018 Accepted 20 March 2018 Available online 20 March 2018 Keywords: Hypersonic rarefied flow Multiscale flow DSMC Fokker–Planck(FP) FP–DSMC Hybrid SPARTA a b s t r a c t Hypersonic vehicles experience a wide range of Knudsen number regimes due to changes in atmospheric density. The Direct Simulation Monte Carlo (DSMC) method is physically accurate for all flow regimes, however it is relatively computationally expensive in high density, and low Knudsen number regions. Recent advances in the Fokker–Planck (FP) kinetic models have addressed this issue by approximating the particle collisions involved in the Boltzmann collision integral with continuous stochastic processes. Furthermore, a coupled FP–DSMC solution method has been devised aiming at a universally efficient yet accurate solution algorithm for rarefied gas flows. Well known Lofthouse case of a generic hypersonic flow about a cylinder (Mach 10, Kn 0.01, Argon) is selected to investigate the performance of a hybrid FP–DSMC implementation. The effect of molecular potential on the accuracy of the scheme is mainly an- alyzed. Furthermore, spatial resolution of cubic FP scheme is studied. Finally, detailed study of accuracy and efficiency of FP–DSMC hybrid scheme is discussed. It is found that the presented adaptive grid to- gether with the FP–DSMC method results in a factor of six speed up for considered hypersonic flow about a cylinder. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Rarefied gas dynamics is concerned with flows at low density where the molecular mean free path is not negligible. Under these conditions, the continuum assumption breaks down and the gas no longer behaves according to the conventional hydrodynamics. Im- portant modifications in the aerodynamic and heat transfer char- acteristics occur which are associated with the basic molecular structure of the gas. The degree of rarefaction is generally char- acterized through the Knudsen number Kn = λ/L, where λ is the mean free path of the gas and L is a characteristic length scale. The flow regime is classified as free molecular, transitional, and continuum, depending on the Knudsen number [1]. It is generally accepted that free molecular flow is an accurate assumption for Kn ≥ 10, whereas Kn ≪ 1 chracterizes the hydrodynamic limit. In the latter case, one can disregard microscopic phenomena in the gas and consider only macroscopic fields such as density, velocity and temperature as the relevant physical quantities. Due to the suf- ficiently large number of collisions the distribution F (V) of particle velocity remains close to an equilibrium distribution, and the con- ventional Navier–Stokes or Euler equations are appropriate models. ∗ Corresponding author. E-mail address: eunji.jun@dlr.de (E. Jun). Yet as the Knudsen number increases, the local velocity distribu- tion may depart significantly from equilibrium, and the flow may not be accurately described by the Navier–Stokes equations. There- fore, the Boltzmann equation should be regarded as the governing model [1]. The Boltzmann equation provides the evolution of the velocity distribution according to DF Dt = 1 m  R 3  4π 0 ( F ( V ∗ ) F ( V ∗ 1 ) − F ( V ) F ( V 1 )) gσ ( θ , g ) dθ d 3 V 1 (1) with D(. . .)/Dt = ∂ (...)/∂ t+V i ∂ (...)/∂ x i + G i ∂ (...)/∂ V i . Here the m is the mass of a single gas molecule, the velocity pair (V ∗ 1 , V ∗ ) is the post collision state of the pair (V 1 , V), σ is the differential cross section of the collision, θ the solid angle which provides the orien- tation of the post collision relative velocity vector and g = |V − V 1 |. Furthermore, G represents the external force normalized by the molecular mass. For engineering applications, the direct numerical solution of the Boltzmann equation may become computationally demanding, due to non-linearity of the collision operator and the high dimensionality of the solution domain. Alternative to the direct discretization of the distribution func- tion, particle Monte-Carlo approximations can deal with high di- mensionality very effectively. In particular, solution algorithms based on DSMC, employ computational particles and Monte-Carlo https://doi.org/10.1016/j.compfluid.2018.03.059 0045-7930/© 2018 Elsevier Ltd. All rights reserved.