scholarly journals Discretization of the Velocity Space in the Solution of the Boltzmann Equation

1998 ◽  
Vol 80 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Xiaowen Shan ◽  
Xiaoyi He
Keyword(s):  
Boltzmann Equation ◽  
Velocity Space ◽  
The Boltzmann Equation

Discretization of the Boltzmann equation in velocity space using a Galerkin approach

2000 ◽  
Vol 129 (1-3) ◽  
pp. 91-99 ◽  
Author(s):  
Jonas Tölke ◽  
Manfred Krafczyk ◽  
Manuel Schulz ◽  
Ernst Rank
Keyword(s):  
Boltzmann Equation ◽  
Velocity Space ◽  
Galerkin Approach ◽  
The Boltzmann Equation

INCLUSION OF FLOW TERMS IN THE CARTESIAN TENSOR EXPANSION OF THE BOLTZMANN EQUATION

1963 ◽  
Vol 41 (11) ◽  
pp. 1776-1786 ◽  
Author(s):  
I. P. Shkarofsky
Keyword(s):  
Distribution Function ◽  
Boltzmann Equation ◽  
Flow Velocity ◽  
Velocity Space ◽  
Pressure Tensor ◽  
Moment Equations ◽  
Ionized Gas ◽  
The Boltzmann Equation ◽  
Integrated Distribution Function ◽  
Boltzmann's Equation

The Cartesian tensor expansion of Boltzmann's equation as given by Johnston (1960) is extended to include terms denoting gradients in flow velocity. The expansion is performed in intrinsic velocity space. The gradient velocity terms yield a linear contribution to the tensor (f2) part of the angle-integrated distribution function from which the zero-trace pressure tensor is calculable. It is shown that the standard moment equations are obtained by further integration over the magnitude of velocity. For the case of a completely ionized gas, collisional terms are inserted appropriately.

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

1957 ◽  
Vol 3 (3) ◽  
pp. 255-260 ◽  
Author(s):  
Akira Sakurai
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Mach Number ◽  
Interpolation Formula ◽  
Velocity Space ◽  
Unique Determination ◽  
Finite Region ◽  
Wave Problem ◽  
The Boltzmann Equation

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.

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.

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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