A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed by hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying low-speed slightly rarefied gas flows

Oxygen transport properties estimation by classical trajectory-direct simulation Monte Carlo

Coupling direct simulation Monte Carlo (DSMC) simulations with classical trajectory calculations is a powerful tool to improve predictive capabilities of computational dilute gas dynamics. The considerable increase in computational effort outlined in early applications of the method can be compensated by running simulations on massively parallel computers. In particular, Graphics Processing Unit acceleration has been found quite effective in reducing computing time of classical trajectory (CT)-DSMC simulations. The aim of the present work is to study dilute molecular oxygen flows by modeling binary collisions, in the rigid rotor approximation, through an accurate Potential Energy Surface (PES), obtained by molecular beams scattering. The PES accuracy is assessed by calculating molecular oxygen transport properties by different equilibrium and non-equilibrium CT-DSMC based simulations that provide close values of the transport properties. Comparisons with available experimental data are presented and discussed in the temperature range 300–900 K, where vibrational degrees of freedom are expected to play a limited (but not always negligible) role

Rayleigh-Brillouin scattering in molecular Oxygen by CT-DSMC simulations

Rayleigh-Brillouin scattering spectra (RBS) in molecular Oxygen have been simulated by DSMC. Different scattering models have been implemented based either on the Larsen-Borgnakke relaxation model and on the Classical Trajectories technique. Results are compared with recent experimentally measured spectra showing good agreement. It is suggested that DSMC-based models be used in the interpretation of light scattering experiments in place of the simplified kinetic models, widely used for the interpretation of RBS experiments. Actually, the former have a firmer physical ground and are readily extended to treat gas mixtures of arbitrary complexity

GPU Acceleration of Rarefied Gas Dynamic Simulations

Kinetic equations are used to mathematically model gas flows that are far from equilibrium due to their more general applicability than typical hydrodynamic equations. However, their complex mathematical structure requires time consuming algorithms to obtain accurate numerical solutions for realistic flow geometries. In this chapter, we show GPU-accelerated algorithms for direct solutions of kinetic equations. The efficiency of the GPU-accelerated codes is demonstrated on the two-dimensional driven cavity flow. Experimental results show that the GPU-accelerated codes run about two orders of magnitude faster than their sequential counterparts whose execution time is comparable to those reported in the literature. The algorithms described can be extended to three-dimensional flows and gas mixtures

DSMC simulation of Rayleigh-Brillouin scattering in binary mixtures

Rayleigh-Brillouin scattering spectra (RBS) in dilute gas mixtures have been simulated by the Direct Simulation Monte Carlo method (DSMC). Different noble gas binary mixtures have been considered and the spectra have been simulated adopting the hard sphere collision model. It is suggested that DSMC simulations can be used in the interpretation of light scattering experiments in place of approximate kinetic models. Actually, the former have a firmer physical ground and can be readily extended to treat gas mixtures of arbitrary complexity. The results obtained confirm the capability of DSMC to predict experimental spectra and clears the way towards the simulation of polyatomic gas mixtures of interest for actual application (notably, air) where tractable kinetic model equations are still lacking

Oxygen transport properties estimation by DSMC-CT simulations

Coupling DSMC simulations with classical trajectories calculations is emerging as a powerful tool to improve predictive capabilities of computational rarefied gas dynamics. The considerable increase of computational effort outlined in the early application of the method (Koura,1997) can be compensated by running simulations on massively parallel computers. In particular, GPU acceleration has been found quite effective in reducing computing time (Ferrigni,2012; Norman et al.,2013) of DSMC-CT simulations. The aim of the present work is to study rarefied Oxygen flows by modeling binary collisions through an accurate potential energy surface, obtained by molecular beams scattering (Aquilanti, et al.,1999). The accuracy of the method is assessed by calculating molecular Oxygen shear viscosity and heat conductivity following three different DSMC-CT simulation methods. In the first one, transport properties are obtained from DSMC-CT simulations of spontaneous fluctuation of an equilibrium state (Bruno et al, Phys. Fluids, 23, 093104, 2011). In the second method, the collision trajectory calculation is incorporated in a Monte Carlo integration procedure to evaluate the Taxman’s expressions for the transport properties of polyatomic gases (Taxman,1959). In the third, non-equilibrium zero and one-dimensional rarefied gas dynamic simulations are adopted and the transport properties are computed from the non-equilibrium fluxes of momentum and energy. The three methods provide close values of the transport properties, their estimated statistical error not exceeding 3%. The experimental values are slightly underestimated, the percentage deviation being, again, few percent

Direct solution of the Boltzmann equation for a binary mixture on GPUs

Abstract. We show how to accelerate the numerical solution of the Boltzmann equation for a binary gas mixture by using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we adopt a semi-regular method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by studying the propagation of plane harmonic waves of small amplitude in a binary gas mixture of hard spheres for a wide range of Knudsen numbers and wave frequencies. The GPU-based code is about two order of magnitudes faster than the CPU version thus proving that GPUs can substantially speedup the numerical solution of kinetic equations

Effect of vibrational degrees of freedom on the heat transfer in polyatomic gases confined between parallel plates

Conductive stationary heat transfer through rarefied nonpolar polyatomic gases, confined between parallel plates maintained at different temperatures, is investigated. It is assumed that gas molecules possess both rotational and vibrational degrees of freedom, described by the classical rigid rotator and quantum harmonic oscillator models, respectively. The flow structure is computed by the Holway kinetic model and the Direct Simulation Monte Carlo method. In both approaches the total collision frequency is computed according to the Inverse Power Law intermolecular potential. Inelastic collisions in DSMC simulations are based on the quantum version of the Borgnakke–Larsen collision model. Results are presented for N2, O2, CO2, CH4 and SF6 representing diatomic as well as linear and nonlinear polyatomic molecules with 1 up to 15 vibrational modes. The translational, rotational, vibrational and total temperatures and heat fluxes are computed in a wide range of the rarefaction parameter and for various ratios of the hot over the cold plate temperatures. Very good agreement, between the Holway and DSMC results is observed as well as with experiments. The effect of the vibrational degrees of freedom is demonstrated. In diatomic gases the vibrational heat flux varies from 5% up to 25% of the total one. Corresponding results in polyatomic gases with a higher number of vibrational modes show that even at low reference temperatures the contribution of the vibrational heat flux may be considerably higher. For example in the case of SF6 at 300 K and 500 K the vibrational heat flux is about 67% and 76% respectively of the total heat flux. Furthermore, it is numerically proved that the computed solutions are in agreement with the Chapman–Enskog approximation in a central strip of the computational domain even at moderately large values of the rarefaction parameter, as found in previous investigations. This property has been used to compute the gas thermal conductivity predicted by the adopted models

DSMC simulation of rarefied gas mixtures flows driven by arrays of absorbing plates

Gas flows induced by arrays of absorbing surfaces of various shapes are met in several vacuum technology devices, like NEG or cryogenic pumps. In order to obtain a simplified model of low pressure gas dynamics driven by surface absorption, flows of rarefied gas binary mixtures past arrays of absorbing plates are studied by numerical solution of a system of coupled and spatially two-dimensional Boltzmann equations. The overall absorption rate is obtained as a function of problem parameters which include the characteristic flow Knudsen number and wall absorption probabilities. Particular attention is devoted to mixtures flows in which one of the components is present in small amounts and it is weakly absorbed