Diffuse Reflection Boundary Condition Research Articles

Numerical analysis of a gas flow in a square cavity driven by spanwise lid motion on the basis of kinetic theory: Behavior of the gas near a sharp corner

A gas flow in a square cavity driven by a lid sliding in the direction of its line of contact with the cavity wall is considered. The steady behavior of the gas is numerically investigated based on the linearized Bhatnagar–Gross–Krook kinetic equation and the diffuse reflection boundary condition. When one applies the Stokes equation and the no-slip boundary condition to the system considered here, the flow velocity becomes multivalued at the corner between the lid and the cavity wall, and the shear stress diverges at the corner inversely proportionally to the distance from there, which is known as the so-called corner singularity. In the present work, the behavior of the gas near the corner is examined based on numerical results obtained from the kinetic theory. Although the range of the flow velocity value in the kinetic solution is limited due to the significant velocity slip near the corner, the flow velocity is, nevertheless, multivalued at the corner. The shear stress varies inversely proportionally to the distance from the corner up to the position that is a few tens of mean free paths away from there. The increase in the stress is suppressed at positions closer to the corner and its magnitude remains bounded. Thus, the total forces acting on the lid and the side cavity walls are bounded as well. Due to the distinctive behavior of the stress near the corner, the resulting nondimensional total forces behave with an unconventional rate Kn ln Kn for small Knudsen numbers Kn.

Hölder regularity of the Boltzmann equation past an obstacle

AbstractRegularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].

Poiseuille and thermal transpiration flows of a dense gas between two parallel plates

The Poiseuille and thermal transpiration flows of a dense gas between two parallel plates are investigated on the basis of the Enskog kinetic equation under the diffuse reflection boundary condition. In contrast to the case of an ideal gas, the density and the gradients of pressure and the normal stress component in the flow direction are not uniform in the direction normal to the plates for a dense gas. The non-uniform normal stress gradient contributes also to the acceleration or deceleration of the thermal transpiration flow for small Knudsen numbers. The profiles of mass and heat flows as well as the net mass flows are obtained for various Knudsen numbers and ratios of the molecular diameter to the distance of plates. In the analysis of the Poiseuille flow, most characteristics of a force-driven flow with a small force are recovered. However, for the case of a dense gas, differences between the force-driven and the present pressure-driven flows are observed even within the linearized regime for small force and pressure gradient, especially at the microscopic level. The behaviour of the velocity distribution functions, in particular, the way in which they approach the ones for the Boltzmann equation as the molecular diameter becomes smaller, is clarified.

Transition Gas Flow Between Two Parallel Plates with a Slit-Type Obstacle of Various Geometry by Event Driven Molecular Dynamics Simulation

In this study, pressure-driven flow through a slit-type obstacle with various length (L) and height (H) placed in between two parallel plates was investigated by Event Driven Molecular Dynamics (EDMD) simulation. Mach number, temperature and pressure distributions were obtained along the channel in the transition regime. The change in these macroscopic properties and flow rate were examined for different cases created by changing Knudsen number (Kn) of the gas, the geometry of the slit and the outlet/inlet pressure ratio of the flow. Collision of gas molecules with plates and the obstacle were modeled with diffuse reflection boundary condition. The flow rate showed a sudden change in the transition regime and significant differences in the molecular regime depending on the pressure ratio. Except for the Kn, H and L dimensions were found to be effective in Mach disc formation. Pressure drops at the exit of the slit were shaped differently in normalized pressure profiles depending on Kn, H and L dimensions. In addition, the structure of the vortices formed at the entrance and exit of the slit varies depending on Kn. Some of the results obtained were confirmed to be consistent with the similar studies in the literature.

Heat transfer in a dense gas between two parallel plates

Time-dependent heat transfer in a dense gas between two parallel plates, which is initiated by an abrupt change in temperature of one plate, is numerically investigated on the basis of the Enskog equation under the diffuse reflection boundary condition. Numerical computation is carried out by a finite-difference scheme combined with the Fourier spectral method for the efficient computation of the collision term of the Enskog equation. As a result, macroscopic quantities of the gas, such as heat flux and temperature, are obtained for various Knudsen numbers and ratios of the molecular diameter to the distance between plates. Compared to the case of an ideal gas, the heat flux in the stationary state is enhanced due to an effect of the finite size of molecules for not only small but also intermediate Knudsen numbers. The results imply that the finite-size effect also affects the propagation of disturbances in the initial stage, particularly for small Knudsen numbers.

Incompressible Euler Limit from Boltzmann Equation with Diffuse Boundary Condition for Analytic Data

A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance, we capture the Navier-Stokes equations of $$\begin{aligned} \textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}} \end{aligned}$$satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of the Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative \(L^p\)-\(L^\infty \) estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green’s function approach. The incompressible Euler limit follows as a byproduct of our framework.

Transient behaviour of a rarefied gas around a sphere caused by impulsive rotation

Abstract

On the motion of slightly rarefied gas induced by a discontinuous surface temperature

The motion of a slightly rarefied gas in a long straight two-dimensional channel caused by a discontinuous surface temperature is investigated on the basis of kinetic theory with a special interest in the fluid-dynamic description. More precisely, the channel is longitudinally divided into two parts and each part is kept at a uniform temperature different from each other, so that the surface temperature of the whole channel has a jump discontinuity at the junction. Under the assumption that the amount of jump in the surface temperature is small, the steady behaviour of the gas induced in the channel is studied on the basis of the linearized Boltzmann equation and the diffuse reflection boundary condition in the case where the Knudsen number, defined by the ratio of the molecular mean free path and the width of the channel, is small. Using a matched asymptotic expansion method combined with Sone’s asymptotics, a Stokes system describing the overall macroscopic behaviour of the gas inside the channel is derived, with a new feature of the ‘slip boundary condition’ for the flow velocity due to the jump discontinuity in the surface temperature of the channel. This condition takes the form of a diverging singularity with source and sink located at the point of discontinuity, with a multiplicative factor determined through the analysis of a spatially two-dimensional Knudsen-layer (or a Knudsen-zone) problem. Some numerical demonstrations based on the Bhatnagar–Gross–Krook equation are also presented.

Interaction of rigid body motion and rarefied gas dynamics based on the BGK model

In this paper we present simulations of moving rigid bodies immersed in a rarefied gas. The rarefied gas is simulated by solving the Bhatnager-Gross-Krook (BGK) model for the Boltzmann equation. The Newton-Euler equations are solved to simulate the rigid body motion. The force and the torque on the rigid body is computed from the surrounded gas. An explicit Euler scheme is used for the time integration of the Newton-Euler equations. The BGK model is solved by the semi-Lagrangian method suggested by Russo &amp; Filbet ^{[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]}. Due to the motion of the rigid body, the computational domain for the rarefied gas (and the interface between the rigid body and the gas domain) changes with respect to time. To allow a simpler handling of the interface motion we have used a meshfree method for the interpolation procedure in the semi-Lagrangian scheme. We have considered a one way, as well as a two-way coupling of rigid body and gas flow. We use diffuse reflection boundary conditions on the rigid body and also on the boundary of the computational domain. In one space dimension the numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. In the two-dimensional case results are compared with DSMC simulations for the Boltzmann equation and with results obtained by other researchers. Several test problems and applications illustrate the versatility of the approach.

A microchannel flow application of a linearized kinetic Bhatnagar-Gross-Krook-type model for inert gas mixtures with general intermolecular forces

The flow of binary gaseous mixtures in microchannels, driven by a gradient of pressure, is investigated using the linearized Boltzmann equation based on a Bhatnagar-Gross-Krook-type model, able to describe general collision kernels, and diffuse reflection boundary conditions. Semi-analytical solutions have been obtained through a transformation in integral equations and the results compared with those derived by the McCormack model, which have revealed a good consistency with the experimental data.

Regularity of Boltzmann Equation with External Fields in Convex Domains of Diffuse Reflection

We consider the Boltzmann equation with external fields in strictly convex domains with diffuse reflection boundary condition. As long as the normal derivative of external fields satisfy some sign condition on the boundary (1.8) we construct classical $C^1$ solutions away from the grazing set. As a consequence we construct solutions of Vlasov-Poisson-Boltzmann system having bounded derivatives away from the grazing set (weighted $W^{1,\infty}$ estimate). In particular this improves the recent regularity estimate of such system in weighted $W^{1,p}$ space for $p<6$ in [1].

The Boltzmann equation with large-amplitude initial data in bounded domains

The paper is devoted to constructing the global solutions around global Maxwellians to the initial-boundary value problem on the Boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries. We allow a class of non-negative initial data which have arbitrary large amplitude and even contain vacuum. The result shows that the oscillation of solutions away from global Maxwellians becomes small after some positive time provided that they are initially close to each other in L2. This yields the disappearance of any initial vacuum and the exponential convergence of large-amplitude solutions to equilibrium in large time. The isothermal diffuse reflection boundary condition plays a vital role in the analysis. The most key ingredients in our strategy of the proof include: (i) Lx,v2–L∞xLv1–Lx,v∞ estimates along a bootstrap argument; (ii) Pointwise estimates on the upper bound of the gain term by the product of L∞ norm and L2 norm; (iii) An iterative procedure on the nonlinear term.

Milne problem for non-absorbing medium with extremely anisotropic scattering kernel in the case of specular and diffuse reflecting boundaries

The Milne problem is studied in one speed neutron transport theory using the linearly anisotropic scattering kernel which combines forward and backward scatterings (extremely anisotropic scattering) for a non-absorbing medium with specular and diffuse reflection boundary conditions. In order to calculate the extrapolated endpoint for the Milne problem, Legendre polynomial approximation ( $$P_{N}$$ method) is applied and numerical results are tabulated for selected cases as a function of different degrees of anisotropic scattering. Finally, some results are discussed and compared with the existing results in literature.

The Milne problem using the P$_{N}$ method for a nonabsorbing medium with linearly anisotropic scattering

The Milne problem, known as one of the classical problems of radiative transfer and neutron transport theory, is solved using the P$_{N}$ method for a nonabsorbing medium (c = 1) with a linearly anisotropic scattering kernel. Specular and diffuse reflection boundary conditions are taken into account. The numerical results are listed for different selected parameters. Some results are also compared with the literature.

Simulation of microscale gas flow in heterogeneous porous media based on the lattice Boltzmann method

A microscale multi-relaxation-time lattice Boltzmann model with the regularization procedure is adopted to simulate gas flow in different porous media. The diffuse reflection boundary condition is used to deal with the random solid boundaries. Because of the complex geometry of the pores, the characteristic length is no longer a constant but a function of the pore locations for the porous media. A rational method is proposed to obtain the local characteristic lengths of the porous media for the microscale gas flow simulations. The simulation results show that gas flow characteristics in different flow regions are notably different. In the continuum flow region and slip flow region, the gas flow abilities in different pores are quite different. The effect of heterogeneity of the porous media on gas velocity distribution is very obvious. As the Knudsen number increases, the differences of gas flow abilities in different pores decrease. For gas flow in the strong transition flow region and free molecular flow region, the gas flow abilities in small pores are similar to those in large pores and the effect of heterogeneity becomes small. Such phenomena are mainly caused by the different gas flow mechanisms in different flow regions. In addition, two commonly used apparent permeability calculation models are evaluated by the simulation results and a better coefficient for the Klinkenberg model is proposed according to the simulation results.

The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

Boundary effects are central to the dynamics of the dilute particles governed by Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for Boltzmann equation with soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an $L^2$ argument and its interplay with intricate $L^\infty$ analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in $L^\infty$ space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the {\it a priori} $L^\infty$ estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the $L^2-L^\infty$ theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted $L^\infty$ theory so that we could deal with the greater difficulties stemmed from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in $L^\infty$ space with the aid of the $L^2$ theory and a bootstrap argument. These methods in the latter case can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

High-frequency sound wave propagation in binary gas mixtures flowing through microchannels

The propagation of high-frequency sound waves in binary gas mixtures flowing through microchannels is investigated by using the linearized Boltzmann equation based on a Bhatnagar-Gross-Krook (BGK)-type approach and diffuse reflection boundary conditions. The results presented refer to mixtures whose constituents have comparable molecular mass (like Ne-Ar) as well as to disparate-mass gas mixtures (composed of very heavy plus very light molecules, like He-Xe). The sound wave propagation model considered in the present paper allows to analyze the precise nature of the forced-sound modes excited in different gas mixtures.

Application of mixed quadrature lattice Boltzmann models for the simulation of Poiseuille flow at non-negligible values of the Knudsen number

We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffuse reflection boundary conditions apply. We present a systematic procedure for the construction of the force term in lattice Boltzmann models based on mixed Cartesian quadratures, where the quadrature on each axis is selected independently. We find that, at non-negligible value of the Knudsen number, half-range quadratures outperform the full-range Gauss–Hermite quadratures for the direction perpendicular to the diffuse-reflecting plates, while the quadrature on the periodic direction along the flow is the full-range Gauss–Hermite quadrature. Our results are validated against numerical results available in the literature.

Singularity of the Velocity Distribution Function in Molecular Velocity Space

We study the boundary singularity of the solutions to the Boltzmann equation in the kinetic theory. The solution has a jump discontinuity in the microscopic velocity \({\zeta}\) on the boundary and a secondary singularity of logarithmic type around the velocity tangential to the boundary, \({\zeta_{n} \sim 0_{-}}\), where \({\zeta_{n}}\) is the component of molecular velocity normal to the boundary, pointing to the gas. We demonstrate this secondary singularity by obtaining an asymptotic formula for the derivative of the solution on the boundary with respect to \({\zeta_{n}}\) that diverges logarithmically when \({\zeta_{n} \sim 0_{-}}\). Our study is for the thermal transpiration problem between two plates for the hard sphere gases with sufficiently large Knudsen number and with the diffuse reflection boundary condition. The solution is constructed and its singularity is studied by an iteration procedure.

Second-Order Knudsen-Layer Analysis for the Generalized Slip-Flow Theory II: Curvature Effects

Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists.