Taku Ohwada

Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard‐sphere molecules

The Poiseuille and thermal transpiration flows of a rarefied gas between two parallel plates are investigated on the basis of the linearized Boltzmann equation for hard‐sphere molecules and diffuse reflection boundary condition. The velocity distribution functions of the gas molecules as well as the gas velocity and heat flow profiles and mass fluxes are obtained for the whole range of the Knudsen number with good accuracy by the numerical method recently developed by the authors.

Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules

Shear flow and thermal creep flow (flow induced by the temperature gradient along the boundary wall) of a rarefied gas over a plane wall are considered on the basis of the linearized Boltzmann equation for hard‐sphere molecules and diffuse reflection boundary condition. These fundamental rarefied gas dynamic problems, typical half‐space boundary‐value problems of the linearized Boltzmann equation, are analyzed numerically by the finite‐difference method developed recently by the authors, and the velocity distribution functions, as well as the macroscopic variables, are obtained with good accuracy. From the results, the shear and thermal creep slip coefficients and their associated Knudsen layers of a slightly rarefied gas flow past a body are derived. The results for the slip coefficients and Knudsen layers are compared with experimental data and various results by the Boltzmann–Krook–Welander (BKW) equation, the modified BKW equation, and a direct simulation method.

Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard‐sphere molecules

A semi‐infinite expanse of a rarefied gas over a plane wall where there is a constant heat flow normal to the wall from infinity is considered. The behavior of the gas is analyzed numerically by a finite difference method on the basis of the standard linearized Boltzmann equation for hard‐sphere molecules with diffuse reflection at the wall. From the result the temperature jump coefficient and its associated Knudsen layer of a slightly rarefied gas flow around a body are derived.

Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite‐difference analysis of the nonlinear Boltzmann equation for hard‐sphere molecules

Heat flow and temperature and density distributions in a rarefied gas between two parallel plates at rest with different uniform temperatures are analyzed numerically on the basis of the full nonlinear Boltzmann equation for hard‐sphere molecules and the Maxwell‐type boundary condition by a finite difference method where the collision term is computed direct numerically. The accurate results are presented for the case in the density measurement by Teagan and Springer [Phys. Fluids 11, 497 (1968)], where the temperature ratio is 1.326, the value of the accommodation coefficient is 0.826, and the ratio of mean free path to plate spacing (Knudsen number Kn) is 0.0658≤Kn≤0.7582. It is found that there is a considerable difference between the present density distribution and the experimental data. The reason for this discrepancy is also discussed. The accurate numerical results of the linearized problem are also presented for comparison.Heat flow and temperature and density distributions in a rarefied gas between two parallel plates at rest with different uniform temperatures are analyzed numerically on the basis of the full nonlinear Boltzmann equation for hard‐sphere molecules and the Maxwell‐type boundary condition by a finite difference method where the collision term is computed direct numerically. The accurate results are presented for the case in the density measurement by Teagan and Springer [Phys. Fluids 11, 497 (1968)], where the temperature ratio is 1.326, the value of the accommodation coefficient is 0.826, and the ratio of mean free path to plate spacing (Knudsen number Kn) is 0.0658≤Kn≤0.7582. It is found that there is a considerable difference between the present density distribution and the experimental data. The reason for this discrepancy is also discussed. The accurate numerical results of the linearized problem are also presented for comparison.

Evaporation and condensation on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation for hard‐sphere molecules

The behavior of a semi‐infinite expanse of a gas bounded by its plane condensed phase, where evaporation or condensation is taking place, is considered on the basis of the linearized Boltzmann equation for hard‐sphere molecules. The half‐space boundary‐value problem of the linearized Boltzmann equation for hard‐sphere molecules is solved numerically by the finite difference method introduced in the temperature jump problem by the authors. The local behavior of the gas (the velocity distribution function of gas molecules, the density and temperature of the gas, etc.) as well as the relations among the macroscopic variables on the condensed phase and at infinity is obtained.

Link-wise artificial compressibility method

The artificial compressibility method (ACM) for the incompressible Navier-Stokes equations is (link-wise) reformulated (referred to as LW-ACM) by a finite set of discrete directions (links) on a regular Cartesian mesh, in analogy with the lattice Boltzmann method (LBM). The main advantage is the possibility of exploiting well established technologies originally developed for LBM and classical computational fluid dynamics, with special emphasis on finite differences (at least in the present paper), at the cost of minor changes. For instance, wall boundaries not aligned with the background Cartesian mesh can be taken into account by tracing the intersections of each link with the wall (analogously to LBM technology). LW-ACM requires no high-order moments beyond hydrodynamics (often referred to as ghost moments) and no kinetic expansion. Like finite difference schemes, only standard Taylor expansion is needed for analyzing consistency. Preliminary efforts towards optimal implementations have shown that LW-ACM is capable of similar computational speed as optimized (BGK-) LBM. In addition, the memory demand is significantly smaller than (BGK-) LBM. Importantly, with an efficient implementation, this algorithm may be among the few which are compute-bound and not memory-bound. Two- and three-dimensional benchmarks are investigated, and an extensive comparative study between the present approach and state of the art methods from the literature is carried out. Numerical evidences suggest that LW-ACM represents an excellent alternative in terms of simplicity, stability and accuracy.

Analytical study of transonic flows of a gas condensing onto its plane condensed phase on the basis of kinetic theory

Abstract A uniform flow of a gas condensing onto its plane condensed phase (commonly known as the half-space problem of condensation) is considered. The problem is studied analytically on the basis of the Boltzmann equation when the flow is in a transonic region. The paper clarifies the analytical structure of the solution, especially the mechanism by which the range of the parameters (the flow speed, pressure, and temperature of the uniform flow blowing from infinity) where a steady solution exists changes abruptly (from a surface to a domain in the parameter space) when the flow speed passes the sonic speed, the correspondence of a family of supersonic solutions to a subsonic solution, etc. The solutions constructed analytically are compared with new numerical solutions near the sonic point.

Artificial compressibility method revisited: Asymptotic numerical method for incompressible Navier-Stokes equations

The artificial compressibility method for the incompressible Navier-Stokes equations is revived as a high order accurate numerical method (fourth order in space and second order in time). Similar to the lattice Boltzmann method, the mesh spacing is linked to the Mach number. An accuracy higher than that of the lattice Boltzmann method is achieved by exploiting the asymptotic behavior of the solution of the artificial compressibility equations for small Mach numbers and the simple lattice structure. An easy method for accelerating the decay of acoustic waves, which deteriorate the quality of the numerical solution, and a simple cure for the checkerboard instability are proposed. The high performance of the scheme is demonstrated not only for the periodic boundary condition but also for the Dirichlet-type boundary condition.

Connection between kinetic methods for fluid-dynamic equations and macroscopic finite-difference schemes

The lattice Boltzmann method (LBM) for the incompressible Navier-Stokes (NS) equations and the gas kinetic scheme for the compressible NS equations are based on the kinetic theory of gases. In the latter case, however, it is shown that the kinetic formulation is necessary only in the discontinuous reconstruction of fluid-dynamic variables for shock capturing. Analogously we will discuss the reduction of a kinetic method for the incompressible case, where the LBM scheme will be shown to shrink to an artificial compressibility type finite-difference scheme. We will prove first that a simple and compact LBM scheme cannot catch rarefied effects beyond Navier-Stokes and hence that it is worth the effort to develop kinetic-based FD alternatives. Finally we will propose two improvements to existing kinetic-based FD schemes: first of all, (a) the proposed scheme is formulated purely in terms of macroscopic quantities on a compact stencil; secondly (b) the semi-implicit formulation is proposed in order to increase the stability. We think that this work may be useful to others in realizing the actual possibilities of simple LBM schemes beyond Navier-Stokes and in adopting the suggested improvements in their actual FD codes.

The error of the splitting scheme for solving evolutionary equations

The accuracy of splitting method is investigated in an abstract Cauchy problem and is shown to be first order in time for general evolutionary equations except for a special case. A general formula for the leading term is obtained. It is also shown as an immediate consequence of the formula that the accuracy is improved from first order to second order by a simple modification. Such a modification was first proposed by Strang [1] for PDEs. Thus, the Strang result is generalized in the present paper to the case of arbitrary evolutionary equations. In particular, it is valid for practically important cases of integro-differential nonlinear kinetic equations, and therefore, there is no need to make additional error estimations in each particular case.