A Study Of Discrete Velocity Model For Micro Gas Flows

With the rapid development of MEMS technology and micro-power systems, microscale flow problems were paid great attention. The relative scientific researches also rapidly developed. However, the insufficient understanding of the physical mechanisms beneath these problems and the limited methodologies had seriously stunted further progress of these newly developed industrial fields. At present, research methodology for micrscale gas flows and heat transfer mainly relies on numerical simulation. However, computational expense of directly simulating the Boltzmann equation, the governing equation for micro gas flows, is extremely high. In this paper, a new collision operator for Discrete Velocity Direction (DVD) model was proposed, which significantly improved its numerical stability and accuracy. This paper mainly contains the following contents.The molecular collision operator was rebuilt and its governing equations were derived. In this operator, the molecular velocity directions are discretized while the speed distribution function is still continuous. The governing equations for all velocity directions are of uniform mathematical expressions:when the selection of velocity directions changes, the mathematical expressions would remain the same. Thus, the new operator could arbitrarily change the selections of velocity directions. It effectively increases the number of velocity directions selected, which significantly improves the numerical accuracy of DVD model.In this paper, the governing equations were theoretically and numerically analyzed. Their characteristics and physical essences were also discussed. An H theorem of governing equations was mathematically proved, indicating intrinsic numerical stability of this model. A BKW distribution was simulated numerically, confirmed the trend of this model to achieve uniformity and homogeneity, which accords with the physical reality. A dimensionless type of governing equations was derived, which not only further simplified the equations but also indicated that the Knudsen number was the only similarity criterion number of the governing equations. The differential scheme of the governing equations was also constructed, the second-order upwind scheme was chosen for the discretization of convection term, with an absolute implicit expression for the time term.Computational programs were encoded using FORTRAN language. With those programs, low-speed benchmark flows in transition regime were simulated. These flows contain Couette flows, Rayleigh flows and cavity flows. Through grid analysis, appropriate time and spatial steps were selected. The spatial spacing of the mesh should be no more than a mean free path (MFP) of the simulated gases. The time steps for unsteady flow should be less than the time interval, in which molecules with average speed progress a mesh spacing. The numerical results of new DVD model were compared with the corresponding results of Linearized Boltzmann Equation (LBE), Direct Simulation Monte Carlo (DSMC) method, Information Preservation (IP) method, which showed that the new model could obtain qualitatively correct results in the whole transition regime. The comparisons between the same results of new and previous DVD model showed new model had significant improvement in accuracy, which promoted the adaptability of DVD model for broader flow regimes.The influences of discrete speeds and discrete velocity directions were systematically studied through numerical simulations. For flows at large Knudsen number, increasing the number of discrete speeds could improve the accuracy of the numerical results with a decreasing trend of improvement. That means, after achieved a certain number, further increase of the number of discrete speeds has only slight positive effects. Increasing the number of discrete velocity direction could significantly improve the accuracy of the results for larger Knudsen numbers, while it only has little improvement in small Knudsen numbers. Increasing the number of discrete speed and discrete velocity directions would both significantly increase computational expenses. Besides the number of discrete velocities, the directions themselves could also affect the results. For shearing flows like Couette flows and Rayleigh flows, the more accurate the velocity and shearing stress provided by the discrete velocity direction are, the more accurate the numerical results would be.