Molecular Velocity Space Research Articles

The Half-Range Moment Method in Harmonically Oscillating Rarefied Gas Flows

The formulation of the half-range moment method (HRMM), well defined in steady rarefied gas flows, is extended to linear oscillatory rarefied gas flows, driven by oscillating boundaries. The oscillatory Stokes (also known as Stokes second problem) and the oscillatory Couette flows, as representative ones for harmonically oscillating half-space and finite-medium flow setups respectively, are solved. The moment equations are derived from the linearized time-dependent BGK kinetic equation, operating accordingly over the positive and negative halves of the molecular velocity space. Moreover, the boundary conditions of the “positive” and “negative” moment equations are accordingly constructed from the half-range moments of the boundary conditions of the outgoing distribution function, assuming purely diffuse reflection. The oscillatory Stokes flow is characterized by the oscillation parameter, while the oscillatory Couette flow by the oscillation and rarefaction parameters. HRMM results for the amplitude and phase of the velocity and shear stress in a wide range of the flow parameters are presented and compared with corresponding results, obtained by the discrete velocity method (DVM). In the oscillatory Stokes flow the so-called penetration depth is also computed. When the oscillation frequency is lower than the collision frequency excellent agreement is observed, while when it is about the same or larger some differences are present. Overall, it is demonstrated that the HRMM can be applied to linear oscillatory rarefied gas flows, providing accurate results in a very wide range of the involved flow parameters. Since the computational effort is negligible, it is worthwhile to consider the efficient implementation of the HRMM to stationary and transient multidimensional rarefied gas flows.

A topology optimization method in rarefied gas flow problems using the Boltzmann equation

This paper presents a topology optimization method in rarefied gas flow problems to obtain the optimal structure of a flow channel as a configuration of gas and solid domains. In this paper, the kinetic equation, the governing equation of rarefied gas flows, is extended over the entire design domain including solid domains assuming the solid as an imaginary gas for implicitly handling the gas-solid interfaces in the optimization process. Based on the extended equation, a 2D flow channel design problem is formulated, and the design sensitivity is obtained based on the Lagrange multiplier method and adjoint variable method. Both the rarefied gas flow and the adjoint flow are computed by a deterministic method based on a finite discretization of the molecular velocity space, rather than the DSMC method. The validity and effectiveness of our proposed method are confirmed through several numerical examples.

Nano-particle drag prediction at low Reynolds number using a direct Boltzmann–BGK solution approach

This paper outlines a novel approach for solution of the Boltzmann–BGK equation describing molecular gas dynamics applied to the challenging problem of drag prediction of a 2D circular nano-particle at transitional Knudsen number (0.0214) and low Reynolds number (0.25–2.0). The numerical scheme utilises a discontinuous-Galerkin finite element discretisation for the physical space representing the problem particle geometry and a high order discretisation for molecular velocity space describing the molecular distribution function. The paper shows that this method produces drag predictions that are aligned well with the range of drag predictions for this problem generated from the alternative numerical approaches of molecular dynamics codes and a modified continuum scheme. It also demonstrates the sensitivity of flow-field solutions and therefore drag predictions to the wall absorption parameter used to construct the solid wall boundary condition used in the solver algorithm. The results from this work has applications in fields ranging from diagnostics and therapeutics in medicine to the fields of semiconductors and xerographics.

Time-dependent conductive heat transfer in rarefied polyatomic gases confined between parallel plates

The transient conductive heat transfer through a rarefied gas confined between two infinite parallel plates due to a sudden jump in the temperature of one of the plates is investigated in the whole range of the Knudsen number via kinetic theory. More specifically, the time-dependent heat transfer flow is modelled by the Holway kinetic model subject to diffuse boundary conditions. The governing integro-differential equation is numerically solved using the discrete velocity method in the molecular velocity space and typical finite control volume schemes in time and physical spaces. The time evolution of the density and temperature distributions as well as of the translational and rotational heat fluxes in terms of the two parameters characterizing the heat flow, namely the Knudsen number and the imposed temperature ratio between the plates is provided. The investigation is focused on the effect of the rotational degrees of freedom and a comparison between monatomic and polyatomic gases is performed. It is found that the time needed to reach the steady-state conditions varies between monatomic and polyatomic gases. In all cases the total time to recover the stationary solution in terms of the rarefaction parameter exhibits a minimum close to the well-known Knudsen minimum.

Poiseuille Flow and Thermal Transpiration of a Rarefied Gas Between Parallel Plates: Effect of Nonuniform Surface Properties of the Plates in the Transverse Direction

Poiseuille flow and thermal transpiration of a rarefied gas between parallel plates with nonuniform surface properties in the transverse direction are studied based on kinetic theory. We considered a simplified model in which one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary on which the accommodation coefficient varies periodically and smoothly in the transverse direction. The spatially two-dimensional (2D) problem in the cross section is studied numerically based on the linearized Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The flow behavior, i.e., the macroscopic flow velocity and the mass flow rate of the gas as well as the velocity distribution function, is studied over a wide range of the mean free path of the gas and the parameters of the distribution of the accommodation coefficient. The mass flow rate of the gas is approximated by a simple formula consisting of the data of the spatially one-dimensional (1D) problems. When the mean free path is large, the distribution function assumes a wavy variation in the molecular velocity space due to the effect of a nonuniform surface property of the plate.

Oscillatory heating in a microchannel at arbitrary oscillation frequency in the whole range of the Knudsen number

The oscillatory heating in a microchannel is investigated on the basis of linear kinetic theory. In particular, the linearized unsteady Shakhov kinetic model, subject to Maxwell boundary conditions is numerically solved in a fully deterministic manner based on finite differencing scheme in the physical space and in the discrete velocity method in the molecular velocity space. The solution of the problem is determined by two parameters: the Knudsen number and the ratio of the intermolecular collision frequency over the oscillation frequency. The numerical calculations are carried out for a wide range of both parameters and results are presented for the amplitude and the phase of all macroscopic quantities of physical interest.

Pressure driven rarefied gas flow through a slit and an orifice

The flow of a monatomic gas through a slit and an orifice due to an arbitrarily large pressure difference is examined on the basis of the nonlinear Bhatnagar–Gross–Krook (BGK) model equation, subject to Maxwell diffuse boundary conditions. The governing kinetic equation is discretized by a second-order control volume scheme in the physical space and the discrete velocity method in the molecular velocity space. The nonlinear fully deterministic algorithm is optimized to reduce the computational effort by introducing memory usage optimization, grid refinement and parallelization in the molecular velocity space. Results for the flow rates and the macroscopic distributions of the flow field are presented in a wide range of the Knudsen number for several pressure ratios. The effect of the various geometric and physical parameters on the flow field is examined. Comparison with previously reported corresponding Direct Simulation Monte Carlo (DSMC) results indicates a very good agreement, which clearly demonstrates the accuracy of the kinetic algorithm and furthermore the reliability of the BGK model for simulating pressure driven flows.

Pressure- and Temperature-Driven Flow Through Triangular and Trapezoidal Microchannels

A detailed study of pressure- and temperature-driven flows through long channels of triangular and trapezoidal cross sections is carried out. Due to the imposed pressure and temperature gradients there is a combined gas flow consisting of a thermal creep flow from the cold toward the hot reservoir and a Poiseuille flow from the high- toward the low-pressure reservoir. The formulation is based on the linearized Shakhov model subject to Maxwell boundary conditions, and it is solved numerically using a finite-difference scheme in the physical space and the discrete velocity method in the molecular velocity space. The results are valid in the whole range of the Knudsen number. In addition to the dimensionless flow rates, a methodology is presented to estimate for a certain set of input data the mass flow rates and the pressure distribution along the channel. Finally, special attention is given to the case of zero net mass flow and to the computation of the coefficient of the thermomolecular pressure difference.

Numerical simulation of rarefied gas flow through a slit

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas from one reservoir to another through a two-dimensional slit. The cases considered are for hard vacuum downstream pressure, finite pressure ratios, and isobaric pressure with thermal diffusion, which are not well established in spite of the simplicity of the flow field. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, three kinds of collision sampling techniques, the time counter (TC) method, the null collision (NC) method, and the no time counter (NTC) method, are used.

A numerical solution of inviscid transonic flow using the Boltzmann equation.

An efficient numerical method of the Boltzmann equation has been developed for inviscid transonic flow. In this method, the convective terms in the Boltzmann equation are approximated by the second order upwind finite difference with limiter: The rational Runge-Kutta scheme is used for the corresponding time integration. The residual averaging technique is incorporated into the method so that the rate of convergence is improved to a steady-state solution. The solution of the collision integral is obtained as thermodynamic equilibrium states. The number of grid points in molecular velocity space is reduced to 4 for each direction to shorten the CPU time in the present approach. As an example, the one-dimensional unsteady shock-tube problem is solved and the results are compared with the exact solution. The feasibility of the present method is demonstrated for the two-dimensional transonic flow in a channel, and over the NACA0012 airfoil and the RAE2822 airfoil. Those results are compared with the results of the Euler equations.

A note on Mott-Smith's solution of the Boltzmann equation for a shock wave

After a Modification, the interpolation formula of Mott-Smith (1951) for the shock wave problem is found to be a solution of the Boltzmann equation at large Mach number in a finite region of molecular velocity space. This modification gives a unique determination of the shock wave thickness, removing the ambiguity for this in Mott-Smith's formula.