scholarly journals Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

2012 ◽  
Vol 50 (3) ◽  
pp. 1111-1135 ◽  
Author(s):  
Giovanni Russo ◽  
Pietro Santagati ◽  
Seok-Bae Yun
Keyword(s):  
Boltzmann Equation ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Lagrangian Scheme

THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE

2013 ◽  
Vol 25 (01) ◽  
pp. 1340025
Author(s):  
RODDAM NARASIMHA
Keyword(s):  
Boltzmann Equation ◽  
Gas Dynamics ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Early Years ◽  
Plane Shock ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Nonequilibrium Flows ◽  
Space Age

The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.

The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model

2018 ◽  
Author(s):  
Chang-jiang Liu ◽  
Song Pang ◽  
Qiang Xu ◽  
Ling He ◽  
Shao-peng Yang ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Two Phase Flow ◽  
Three Dimensional ◽  
Phase Flow ◽  
Bgk Model ◽  
Two Phase ◽  
The Boltzmann Equation

scholarly journals Steepest Entropy Ascent Models of the Boltzmann Equation: Comparisons With Hard-Sphere Dynamics and Relaxation-Time Models for Homogeneous Relaxation From Highly Non-Equilibrium States

2013 ◽  
Author(s):  
Gian Paolo Beretta ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Power Law ◽  
Collision Frequency ◽  
Equilibrium States ◽  
Space Distribution ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Far From Equilibrium ◽  
Non Equilibrium

We present a family of steepest entropy ascent (SEA) models of the Boltzmann equation. The models preserve the usual collision invariants (mass, momentum, energy), as well as the non-negativity of the phase-space distribution, and have a strong built-in thermodynamic consistency, i.e., they entail a general H-theorem valid even very far from equilibrium. This family of models features a molecular-speed-dependent collision frequency; each variant can be shown to approach a corresponding BGK model with the same variable collision frequency in the limit of small deviation from equilibrium. This includes power-law dependence on the molecular speed for which the BGK model is known to have a Prandtl number that can be adjusted via the power-law exponent. We compare numerical solutions of the constant and velocity-dependent collision frequency variants of the SEA model with the standard relaxation-time model and a Monte Carlo simulation of the original Boltzmann collision operator for hard spheres for homogeneous relaxation from near-equilibrium and highly non-equilibrium states. Good agreement is found between all models in the near-equilibrium regime. However, for initial states that are far from equilibrium, large differences are found; this suggests that the maximum entropy production statistical ansatz is not equivalent to Boltzmann collisional dynamics and needs to be modified or augmented via additional constraints or structure.

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