稀薄流到连续流的气体运动论模型方程算法研究

通过引入碰撞松弛参数和当地平衡态分布函数对BGK模型方程进行修正，确定含流态控制参数可描述不同流域气体流动特性的气体分子速度分布函数的简化控制方程。发展和应用离散速度坐标法于气体分子速度空间，利用一套在物理空间和时间上连续而速度空间离散的分布函数来代替原分布函数对速度空间的连续依赖性。基于非定常时间分裂数值计算方法和无波动、无自由参数的NND耗散差分格式，建立直接求解气体分子速度分布函数的气体运动论有限差分数值方法。推广应用改进的Gauss-Hermite无穷积分法和华罗庚－王元提出的以单和逼近重积分的黄金分割数论积分方法等，对离散速度空间进行宏观取矩获取物理空间各点的气体流动参数，由此发展一套从稀薄流到连续流各流域统一的气体运动论数值算法。通过对不同Knudsen数下一维激波管问题、二维圆柱绕流和三维球体绕流的初步数值实验表明文中发展的数值算法是可行的。

STUDY ON GAS KINETIC ALGORITHM FOR FLOWS FROM RAREFIED TRANSITION TO CONTINUUM USING BOLTZMANN MODEL EQUATION

Based on the BGK-Boltzmann model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be obtained by introducing the colliding relaxation parameters and the local equilibrium distribution function to revise the BGK equation. Based on the principle of probability statistics, the discrete velocity ordinate method is applied to the distribution function equation in order to replace its continuous dependency on the velocity space, the optimum Golden Section method is used to discretize velocity components, and then the equation will be cast into hyperbolic conservation law form with nonlinear source term. The unsteady time-splitting method is used to split the distribution function equations into the colliding relaxation equation and the convection movement equations. The non-oscillatory, containing no free parameters, and dissipative (NND) scheme is employed to solve the convection terms and the colliding relaxation equation is numerically simulated by the aid of the second order Itungc-Kutta method. The gas kinetic finite difference method is constructed for the computation of the discrete velocity distribution function. Four types of quadrature rules, such as the modified Gauss-Hcrmitc formula and the Golden Section number-theoretic integral method based on the thoughts of the Hua-Wang method, are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm based on the nonlinear Boltzmann model equation is established for flows from rarefied transition to continuum. To test the reliability of the present numerical method to solve the gas dynamical problems from rarefied transition to continuum, the one-dimensional shock-tube problems and the flows past two-dimensional circular cylinder and the flows past three-dimensional sphere with various Knudsen numbers are simulated. The computations indicate that both high resolution of the now fields and good qualitative agreement with the theoretical, DSMC, and experimental results can be obtained.