Gas kinetic algorithm using Boltzmann model equation

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Based on the Boltzmann–BGK model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented by the aid of the basic characteristic on molecular movement and collision approaching to equilibrium. The optimum Golden Section principle is extended and applied to the discrete velocity ordinate method in order to discretize the corresponding velocity components, and then the molecular velocity distribution function equation will be cast into hyperbolic conservation laws form with non-linear source terms. In view of the unsteady characteristic of molecular convective movement and colliding relaxation, the time-splitting method is applied to decompose the velocity distribution function equations into the colliding relaxation equations with non-linear source terms and the convective motion equations. Based on the second-order Runge–Kutta method and the non-oscillatory, containing no free parameters, and dissipative (NND) finite difference method, the gas kinetic finite difference second-order scheme is constructed to directly solve the discrete velocity distribution functions. Four types of discrete velocity quadrature rules, such as the modified Gauss–Hermite formula and the Golden Section number-theoretic integral method, are developed and applied to evaluate the macroscopic flow moments of the distribution functions over the velocity space. As a result, a unified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime. To test the reliability of the present method, the one-dimensional shock wave structures, the flows past two-dimensional circular cylinder and the three-dimensional flows over sphere with various Knudsen numbers are simulated. The computational results are found in high resolution of the flow fields and good agreement with the theoretical, DSMC, and experimental results.