Abstract. The convergence rate of the discrete velocity method (DVM), which has been applied extensively in the area of rarefied gas dynamics, is studied via a Fourier stability analysis. The spectral radius of the continuum form of the iteration map is found to be equal to one, which justifies the slow convergence rate of the method. Next the efficiency of the DVM is improved by introducing various acceleration schemes. The new synthetic-type schemes speed up significantly the iterative convergence rate. The spectral radius of the acceleration schemes is also studied and the so-called H 1 acceleration method is found to be the optimum one. Finally, the two-dimensional flow problem of a gas through a rectangular microchannel is solved using the new fast iterative DVM. The number of required iterations and the overall computational time are significantly reduced, providing experimental evidence of the analytic formulation. The whole approach is demonstrated using the BGK and S kinetic models.Key words. iterative methods, acceleration schemes, rarefied gas flows AMS subject classifications. 65B99, 76P05PII. S1064827502406506 Introduction. After the early work of Broadwell [4], Huang et al. [10], and Cabannes [6], the discrete velocity method (DVM) has been developed into one of the most common techniques for solving the Boltzmann equation [7,12] and simplified model equations [14,23,15] in the area of rarefied gas dynamics. The method has also been applied to solve mixture problems [20,8]. An extensive review article on internal rarefied gas flows including DVM applications has been given lately by Sharipov [16]. Very recently, new models of discrete velocity gases [5] and mixtures [9] have been introduced indicating that the method can be extended into more general models including polyatomic gases with chemical reactions.The method is based on a discretization of the velocity and space variables by choosing a suitable set of discrete velocities and by applying a consistent finite difference scheme, respectively. Then the collision integral term is approximated by an appropriate quadrature, and the resulting discrete system of equations is solved in an iterative manner. Researchers implementing the DVM are well aware, however, of its slow convergence, particularly when domains with thick subregions are considered [17]. In these cases a large number of iterations are required, and calculations are amenable to accumulated round-off error. Special attention is needed to sustain acceptable accuracy. Even more when multidimensional physical systems are examined, computational effort and time are drastically increased. These types of calculations are now needed to solve in an efficient and accurate manner fluidics applications in micro-electrical-mechanical systems and nanotechnology problems. In these applications, due to the fact that the Navier-Stokes equations are restricted by the hydrodynamic regime, kinetic-type equations and corresponding numerical approaches must

A new experimental setup for flow rate measurement of gases through microsystems is presented. Its principle is based on two complementary techniques, called droplet tracking method and constant-volume method. Experimental data on helium and argon isothermal flows through rectangular microchannels are presented and compared with computational results based on a continuum model with second-order boundary conditions and on the linearized kinetic BGK equation. A very good agreement is found between theory and experiment for both gases, assuming purely diffuse accommodation at the walls. Also, some experimental data for a binary mixture of monatomic gases are presented and compared with kinetic theory based on the McCormack model.

The flow of a rarefied gas in a rectangular enclosure due to the motion of the upper wall is solved over the whole range of the Knudsen number. The formulation is based on the two-dimensional linearized Bhatnagar-Gross-Krook ͑BGK͒ kinetic equation with Maxwell diffuse-specular boundary conditions. The integro-differential equations are solved numerically implementing the discrete velocity method. The discontinuity at the boundaries between stationary and moving walls is treated accordingly. A detailed investigation of the rarefaction effects on the flow pattern and quantities is presented over the whole range of the Knudsen number and various aspect ͑height/width͒ ratios. Numerical results of flow characteristics, including the streamlines, the velocity profiles, the pressure and temperature contours, and the drag force of the moving wall, are presented for different aspect ratios and various degrees of gas rarefaction from the free molecular through the transition up to the continuum limit. On several occasions, depending upon the flow parameters, in addition to the main vortex, corner eddies are created. As the depth of the cavity is increased, these eddies grow and merge into additional vortices under the top one. The mesoscale kinetic-type approach proves to be efficient and suitable for problems that incorporate multiscale physics, such as the present nonequilibrium flow.

Articles you may be interested inImpact of the gas-surface scattering and gas molecule-molecule interaction on the mass flow rate of the rarefied gas through a short channel into a vacuum J. Vac. Sci. Technol. A 28, 1393 (2010); 10.1116/1.3504596Rarefied gas flow through a thin slit into vacuum simulated by the Monte Carlo method over the whole range of the Knudsen number Experimental and numerical modeling of rarefied gas flows through orifices and short tubes AIP Conf.A rarefied gas flow into vacuum through a tube of finite length is investigated over the whole range of gas rarefaction by the direct simulation Monte Carlo method. The nonequilibrium effects at the inlet and outlet of the tube have been considered by including in the computational domain large volumes of the upstream and downstream reservoirs. Results for the dimensionless flow rate and for the flow field are presented for a wide range of the gas rarefaction and for various values of the length to radius ratio in the range from 0 to 10. The influence of the gas-surface interaction model, as well as the effect of the intermolecular potential model on the gas flow, is examined. A good agreement has been obtained between the present numerical results and the corresponding experimental ones available in the literature.

The flow of binary gaseous mixtures through rectangular microchannels due to small pressure, temperature, and molar concentration gradients over the whole range of the Knudsen number is studied. The solution is based on a mesoscale approach, formally described by two coupled kinetic equations, subject to diffuse scattering boundary conditions. The model proposed by McCormack substitutes the complicated collision term and the resulting kinetic equations are solved by an accelerated version of the discrete velocity method. Typical results are presented for the flow rates and the heat fluxes of two different binary mixtures (Ne–Ar and He–Xe) with various molar concentrations, in two-dimensional microchannels of different aspect (height to width) ratios. The formulation is very efficient and can be used instead of the classical method of solving the Navier–Stokes equations with slip boundary conditions, which is restricted by the hydrodynamic regime. Moreover, the present formulation is a good alternative to the direct simulation Monte Carlo method, which often becomes computationally inefficient.

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A computational and experimental study has been performed for the investigation of fully developed rarefied gas flows through channels of circular, orthogonal, triangular, and trapezoidal cross sections. The theoretical-computational approach is based on the solution of the Bhatnagar-Gross-Krook kinetic equation subject to Maxwell diffuse-specular boundary conditions by the discrete velocity method. The experimental work has been performed at the vacuum facility "TRANSFLOW," at Forschungszentrum Karlsruhe and it is based on measuring, for assigned flow rates, the corresponding pressure differences. The computed and measured mass flow rates and conductance are in all cases in very good agreement. In addition, in order to obtain some insight in the flow characteristics, the reference Knudsen, Reynolds, and Mach numbers characterizing the flow at each experimental run have been estimated. Also, the pressure distribution along the channel for several typical cases is presented. Both computational and experimental results cover the whole range of the Knudsen number.