Numerical Error Analysis for Deterministic Kinetic Solutions of Low-Speed Flows

The computational cost of the direct simulation Monte Carlo solutions of rarefied flows increases with decreasing average flow velocity due to larger noise-to-signal ratio and a longer time to reach the steady state. For such low-speed flows, the discrete-ordinate solution of kinetic model equations can provide accurate and computationally efficient numerical modeling. In this work, analysis of the numerical errors of discrete-ordinate solution of the kinetic model equation is carried out using the Richardson extrapolation on non-uniform meshes. The procedure is illustrated for a second-order finitedifference solution of ellipsoidal statistical kinetic model equation for a two-dimensional low-speed rarefied flow generated by a non-uniformly heated plate. INTRODUCTION Starting from the pioneering work by Bird in early 1960s[1] on the development of stochastic numerical methods for kinetic description of gas flows, the direct simulation Monte Carlo (DSMC) method has evolved into a powerful numerical tool that has provided accurate and efficient solutions to many important problems of rarefied gas dynamics.[2, 3] However, the computational cost of the DSMC method increases with decreasing Mach number due to: (a) explicit time integration in the DSMC algorithm and, therefore, long time to reach a steady-state; (b) larger sample sizes to attain a required signal-to-noise ratio when the average gas velocity is small in comparison to thermal speed[4]. Additionally, coordinate transformation in physical space domain that could have greatly increased computational efficiency for a high-aspect ratio geometries can not be implemented in an atomistic simulation. Last but not least, there are inherent difficulties in coupling a stochastic method for gas flow modeling with conventional deterministic simulations of solid material response for the integrated system analysis including coupled thermal-fluid-structural interactions. The aforementioned computational issues stimulate applications of deterministic kinetic methods for numerical modeling of low-speed microscale flows. APPLICATION OF KINETIC MODELS FOR LOW-SPEED FLOWS For modeling of low-speed microscale flows we consider the solution of the steady-state Boltzmann equation with a simplified collision operator. The two-dimensional single-collision time approximation of the Boltzmann equation has the form: U ∂ f ∂x +V ∂ f ∂y = ν( f0 − f ) (1) where f = f (x,y,U,V,W ) is the distribution function, x and y are Cartesian coordinates, U,V, and W are velocity components, and ν is the collision frequency. For the Bhatnagar-Gross-Krook (BGK) model f0 = fM = n(2πRT)−3/2exp(− ~ C 2 2RT ) is the local equilibrium Maxwell distribution. For the ellipsoidal statistical (ES) model[5] f0 = fG is the local isotropic three-dimensional Gaussian fG = n