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 H1 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.

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Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

A uniform supersonic flow of a rarefied gas past a flat plate at zero angle of attack is considered, and the steady behavior of the gas around the plate is investigated numerically on the basis of the Boltzmann–Krook–Welander equation (or the so-called BGK model) and the diffuse reflection boundary condition. An accurate finite-difference analysis, which gives the correct description of the discontinuity of the velocity distribution function of the gas molecules occurring in the gas, is carried out, and the features of the flow field (the velocity distribution function and the macroscopic variables such as the density, temperature, and flow velocity of the gas), in particular, those around the leading and trailing edges, are clarified for a wide range of the Knudsen number. The drag acting on the plate and the energy transferred to it are also obtained accurately. In addition, on the basis of the results for small Knudsen numbers, the behavior of the gas around the leading edge of a semi-infinite plate is...

Numerical analysis of a supersonic rarefied gas flow past a flat plate

The linearized binary model described by Hamel [Phys. Fluids 8, 418 (1964)] is used to obtain a set of kinetic equations and boundary conditions for the Couette flow problem. The derived set of two coupled integrodifferential equations is solved by iteration implementing standard discretization techniques. Highly accurate numerical results are presented for the mass velocity distribution and the total shear stress of the binary gas system.

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Couette flow of a binary gas mixture

Velocity Space Description of Certain Turbulent Free Shear Flow Characteristics

High-order accurate nonoscillatory schemes based on the discrete ordinate method and the characteristic flux difference splitting method are presented for solving the nonlinear Krook-type kinetic model equations for both monatomic gases and gases possessing internal degrees of freedom. The methods are tested for one-dimensional unsteady shock tube flow and for two-dimensional steady rarefied stream past a NACA 0012 airfoil at angles of attack. The computed results are found in good agreement both with theoretical solutions from Euler and Navier-Stokes calculations and with experimental results. Comparisons of results using kinetic models for both monatomic and diatomic gases are included.

Nonoscillatory schemes for kinetic model equations for gases with internal energy states

The sharp leading edge problem has been studied for both monatomic and diatomic gases using the Boltzmann equation with the Bhatnagar-Gross-Krook type models as the governing equation and the discrete ordinate method with a closed-boundary value approach as a tool. Plate length relative to the freestream mean free path is taken to be 52. The gas-surface interaction law is assumed to be diffuse reflection. The local distribution functions of molecular velocities and internal energies (for the diatomic gas) for the entire flowfield have been calculated for a freestream Mach number of 6.1. Comparisons are made between the calculated results and experimental data.

R. Srinivasan

Papers

High‐speed leading edge problem