A molecular collision operator of adjustable direction for the discrete velocity direction model

The discrete velocity direction model is an approximate method to the Boltzmann equation. A developed molecular collision operator for the model is presented in this paper. Under the new operator, the discrete directions of molecules are adjustable, namely, both the number and the angles of discrete directions can be changed as needed in the discrete velocity direction model. At the same time, the governing equations will keep unchanged when the number of discrete directions changes. In fact, with the continuous molecular speed, the discrete velocity direction model has been able to employ any discrete velocities in numerical calculations. The discrete velocity direction model under the new collision operator was applied into some benchmark flows in micro scales in this paper, and the influence of the number of discrete velocities on the computational accuracy was analyzed. The numerical results show that the accuracy of the discrete velocity direction model can be improved significantly by employing more discrete directions, especially for the gas flows at large Knudsen number. With appropriate discrete velocities, this model has been able to give accurate numerical results in all flow regimes. In addition, it is proved that the discrete velocity direction model under the new collision operator satisfies a global H theorem unconditionally, which means that the new operator further improves the intrinsic stability of the discrete velocity direction model.