Dynamics of rarefied gas flows - Asymptotic and numerical analyses of the Boltzmann equation

2001 ◽  
Author(s):  
Kazuo Aoki
Keyword(s):  
Boltzmann Equation ◽  
Rarefied Gas ◽  
Numerical Analyses ◽  
Gas Flows ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation

Towards the development of a multiscale, multiphysics method for the simulation of rarefied gas flows

2010 ◽  
Vol 661 ◽  
pp. 262-293 ◽  
Author(s):  
DAVID A. KESSLER ◽  
ELAINE S. ORAN ◽  
CAROLYN R. KAPLAN
Keyword(s):  
Heat Flux ◽  
Boltzmann Equation ◽  
Conservation Laws ◽  
Rarefied Gas ◽  
Multiscale Methods ◽  
Time Interval ◽  
Gas Flows ◽  
Transition Regime ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation

We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continuum field, and the cycle is repeated. The target applications for this type of method are transition-regime gas flows for which standard continuum models (e.g. Navier–Stokes equations) cannot be used, but solution of Boltzmann's equation is prohibitively expensive. The use of molecular-level data to close the conservation laws significantly extends the range of applicability of the continuum conservation laws. In this study, the CM3 is used to perform two proof-of-principle calculations: a low-speed Rayleigh flow and a thermal Fourier flow. Velocity, temperature, shear-stress and heat-flux profiles compare well with direct-simulation Monte Carlo solutions for various Knudsen numbers ranging from the near-continuum regime to the transition regime. We discuss algorithmic problems and the solutions necessary to implement the CM3, building upon the conceptual framework of the heterogeneous multiscale methods.

scholarly journals Simulation of 2D rarefied gas flows based on the numerical solution of the Boltzmann equation

2017 ◽  
Author(s):  
Sergey O. Poleshkin ◽  
Ewgenij A. Malkov ◽  
Alexey N. Kudryavtsev ◽  
Anton A. Shershnev ◽  
Yevgeniy A. Bondar ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Numerical Solution ◽  
Rarefied Gas ◽  
Gas Flows ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation

scholarly journals Kinetic Methods for Solving Unsteady Problems with Jet Flows

2019 ◽  
pp. 34-51
Author(s):  
A. A. Frolova ◽  
V. A. Titarev
Keyword(s):  
Boltzmann Equation ◽  
Model Equation ◽  
Rarefied Gas ◽  
Strong Dependence ◽  
Gas Flows ◽  
Original Text ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation ◽  
Uniform Grids ◽  
Model Equations

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order  to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model,  Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space.  The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model)  and makes it possible to calculate on non-structured non uniform grids in velocity and  physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

scholarly journals Kinetic Methods for Solving Non-stationary Jet Flow Problems

2018 ◽  
pp. 27-44
Author(s):  
A. A. Frolova ◽  
V. A. Titarev
Keyword(s):  
Boltzmann Equation ◽  
Rarefied Gas ◽  
Velocity Space ◽  
Gas Flows ◽  
Study Objective ◽  
Software Complex ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation ◽  
Model Equations ◽  
3D Software

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

scholarly journals Assessment and development of the gas kinetic boundary condition for the Boltzmann equation

2017 ◽  
Vol 823 ◽  
pp. 511-537 ◽  
Author(s):  
Lei Wu ◽  
Henning Struchtrup
Keyword(s):  
Experimental Data ◽  
Boundary Condition ◽  
Boltzmann Equation ◽  
Rarefied Gas ◽  
Thermal Slip ◽  
Thermal Transpiration ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation ◽  
Kinetic Boundary ◽  
Slip Coefficients

Gas–surface interactions play important roles in internal rarefied gas flows, especially in micro-electro-mechanical systems with large surface area to volume ratios. Although great progress has been made to solve the Boltzmann equation, the gas kinetic boundary condition (BC) has not been well studied. Here we assess the accuracy of the Maxwell, Epstein and Cercignani–Lampis BCs, by comparing numerical results of the Boltzmann equation for the Lennard–Jones potential to experimental data on Poiseuille and thermal transpiration flows. The four experiments considered are: Ewart et al. (J. Fluid Mech., vol. 584, 2007, pp. 337–356), Rojas-Cárdenas et al. (Phys. Fluids, vol. 25, 2013, 072002) and Yamaguchi et al. (J. Fluid Mech., vol. 744, 2014, pp. 169–182; vol. 795, 2016, pp. 690–707), where the mass flow rates in Poiseuille and thermal transpiration flows are measured. This requires that the BC has the ability to tune the effective viscous and thermal slip coefficients to match the experimental data. Among the three BCs, the Epstein BC has more flexibility to adjust the two slip coefficients, and hence for most of the time it gives good agreement with the experimental measurements. However, like the Maxwell BC, the viscous slip coefficient in the Epstein BC cannot be smaller than unity but the Cercignani–Lampis BC can. Therefore, we propose to combine the Epstein and Cercignani–Lampis BCs to describe gas–surface interaction. Although the new BC contains six free parameters, our approximate analytical expressions for the viscous and thermal slip coefficients provide useful guidance to choose these parameters.

scholarly journals On the accuracy of macroscopic equations for linearized rarefied gas flows

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Lei Wu ◽  
Xiao-Jun Gu
Keyword(s):  
Boltzmann Equation ◽  
Volume Diffusion ◽  
Rarefied Gas ◽  
Hermite Polynomials ◽  
Velocity Distribution Function ◽  
Gas Flows ◽  
Burnett Equations ◽  
Moment Equations ◽  
Rarefied Gas Flows ◽  
Macroscopic Equations

AbstractMany macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and “thermo-mechanically consistent" Burnett equations based on the argument of “volume diffusion”. This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on “volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

Shock polar investigation in supersonic rarefied gas flows over a circular cylinder

2021 ◽  
Vol 33 (5) ◽  
pp. 052006
Author(s):  
Hassan Akhlaghi ◽  
Ehsan Roohi ◽  
Abbas Daliri ◽  
Mohammad-Reza Soltani
Keyword(s):  
Circular Cylinder ◽  
Rarefied Gas ◽  
Gas Flows ◽  
Rarefied Gas Flows
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