A discrete velocity direction model for the Boltzmann equation and applications to micro gas flows

A discrete velocity direction model for the Boltzmann equation is proposed in this paper, which provides an alternative technique to the rarefied gas flows. In this model, the directions of molecular velocities are discrete, which are restricted in eight fixed directions, while the molecular speed rate is still continuous. By this approximation, the Boltzmann equation in the six-dimensional phase space is replaced by eight differential-integral equations in three-dimensional space. Thus, the computational cost is reduced greatly by reduction of three dimensions. The number of discrete velocities is not fixed in the present model because the speed rate can be truncated arbitrarily. This is distinguished from the conventional discrete velocity models (DVM). To test this technique, it was applied to the Couette flow and Poiseuille flow. The computed results agree well with those by the linearized Boltzmann equation and the DSMC method.