A spectral-Lagrangian Boltzmann solver for a multi-energy level gas

Abstract

In this paper a spectral-Lagrangian method is proposed for the full, non-linear Boltzmann equation for a multi-energy level gas typical of a hypersonic re-entry flow. Internal energy levels are treated as separate species and inelastic collisions (leading to internal energy excitation and relaxation) are accounted for. The formulation developed can also be used for the case of a gas mixture made of monatomic gases without internal energy (where only elastic collisions occur). The advantage of the spectral-Lagrangian method lies in the generality of the algorithm in use for the evaluation of the elastic and inelastic collision operators, as well as the conservation of mass, momentum and energy during collisions. The latter is realized through the solution of constrained optimization problems. The computational procedure is based on the Fourier transform of the partial elastic and inelastic collision operators and exploits the fact that these can be written as weighted convolutions in Fourier space with no restriction on the cross-section model. The feasibility of the proposed approach is demonstrated through numerical examples for both space homogeneous and in-homogeneous problems. Computational results are compared with those obtained by means of the DSMC method in order to assess the accuracy of the proposed spectral-Lagrangian method.

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

“…Tcheremissine & Agarwal (2008) solved the WCU equation for normal shock waves in nitrogen, and found that the computational cost is larger by about two orders of magnitude than for a monatomic gas. Recently, a spectral-Lagrangian method with computational memory and cost of the order of N 4 e N 6 v has been proposed (Munafò et al 2014), where N e is the number of discrete internal energy levels; for N e = 5 and N v = 16, the memory needed is approximately 88 GB, while the time needed for calculating the collision operators once is approximately 3 s using 12 compute threads, which restricts its suitability when applied to real problems. The direct simulation Monte Carlo (DSMC) method proposed by Bird (1994), using the Larsen-Borgnakke collision rule (Borgnakke & Larsen 1975) for the translationalinternal energy exchange, is a good alternative because of its linear computational cost with the number of simulated particles and far smaller memory requirements.…”

A fast spectral method for the Boltzmann equation for monatomic gas mixtures

“…The method employs Fourier-Galerkin discretization in velocity space, and handles binary collisions in the corresponding frequency space. The method has spectral accuracy, and can deal with mixtures with large molecular mass ratios at a computational cost of O( p m r M 2 N 4 log N ), which is much smaller than the conventional spectral method's cost of O(m 3 r N 6 ) [57]. Numerical results have been presented for the spatially-homogeneous relaxation of pseudo-Maxwell molecules, and for spatially-inhomogeneous problems (such as normal shock waves, planar Fourier/Couette flows, and a two-dimensional temperature-driven flow) involving hardsphere gases with molecular mass ratios up to 36.…”

“…Note that the procedure in deriving this FSM for monatomic gas mixtures is similar to that for a single-species Boltzmann collision operator [35], so it can be proved that the FSM presented here conserves mass, while the errors in achieving momentum and energy conservation are spectrally small. These errors, however, can be removed using the method of Lagrangian multipliers [37,57].…”

“…For general cases, a multipoint conservative projectioninterpolation method has been developed for an arbitrary ratio of molecular masses [56], but its accuracy is not known, especially at large Knudsen numbers. Recently, a spectral-Lagrangian method with a computational cost of O(m 3 r N 6 ) has been proposed (where m r is the mass ratio of the heavier to the lighter species), and a normal shock wave in a binary mixture with a mass ratio of about 2 has been simulated [57]. As this method uses the same velocity discretization for each component, it cannot be applied to mixtures with large mass ratios, since the computational cost and storage are m 3 r and m 3/2 r times larger than that for the single-species BE, respectively.…”

Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations

“…We assume that we can obtain the symmetry relations (35), possibly after a scaling (34). Actually assuming a convenient reciprocity relation [26], this will be the case.…”