Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

Abstract

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann-Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integration method can be developed and adopted to attack complex flows with different Mach numbers. HPF parallel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarily

Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation

“…In order to have a correct value for the Prandtl number, the M f in the BGK equation is replaced by the local equilibrium distribution function N f from the Shakhov model [6,9,16]. The function N f is taken as the asymptotic expansion in Hermite polynomials with local Maxwellian M f as the weighting function: 5 5 ].…”

“…All of the macroscopic flow variables of gas dynamics in consideration can be evaluated by the moments of the velocity distribution function over the velocity space [1,[3][4][5][6][7]. In the following computation, all of the variables will have been nondimensionalized, and the "~" sign in the equations will be dropped for the simplicity and concision without causing any confusion.…”

“…y z V V  In order to replace its continuous dependency on the velocity space, the discrete velocity ordinate method [3][4][5][6][7][14][15][16] can be employed in the x V velocity space, the velocity distribution function equation can be transformed into hyperbolic conservation form with nonlinear source terms at each of the discrete velocity ordinate points, as follows: …”

“…From the point of view of the mesoscopic theory [1,2] of Boltzmann equation describing molecular transport phenomena from various flow regimes, the gas-kinetic unified algorithm for flows from rarefied transition to continuum has been developed [3][4][5][6][7], in which the single velocity distribution function equation can be translated into hyperbolic conservation systems with nonlinear source terms in physical space and time by first developing the discrete velocity ordinate method in the gas kinetic theory. Then the gas-kinetic numerical schemes are constructed by using the time-splitting method for unsteady equation and the finite difference technique.…”

Gas-Kinetic Unified Algorithm for Re-Entering Complex Problems Covering Various Flow Regimes by Solving Boltzmann Model Equation

“…As a model form of the Boltzmann equation, the simplified velocity distribution function equation relied by the algorithm is used to describe the molecular transport phenomena from various flow regimes. It is not exact as the original Boltzmann equation, but reliable to qualitatively describe gas flows from various regimes, even as is described in Li (2001); Li & Zhang (2007,2008,2009a. The accuracy of the discrete velocity ordinate method mostly rests with the relevant discrete velocity numerical quadrature technique and the size of the discrete velocity domain.…”