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Mathematical Approaches to Rarefied Gas Dynamics

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On one hand, the field of Rarefied Gas Dynamics may be considered as the regime of fluid dynamics that does not satisfy the continuum assumption or a subset of compressible flow. In this view, one may consider cases where there may be some rarefaction effects, but the Knudsen number Kn is still quite low (<0.05), sometimes called “slip” regime (no-slip condition). Here one may use a Navier-Stokes solver for the flow and apply slip corrections near appropriate boundaries. More details may be found in references such as: Kogan pp. 367-400[1], Aoki et al.[2], Shakurova et al.[3]

On the other hand, the field of Rarefied Gas Dynamics may be developed directly from the Kinetic Theory of gases and the Boltzmann Equation (BE), which is valid for dilute gases.[4] The Kinetic Theory of gases is very complete for perfect gases at equilibrium conditions. When not in full thermodynamic equilibrium, then transport phenomena must be derived under various assumptions. For a simple gas in near-equilibrium conditions, a linearization of the BE may be used.[5] The Chapman-Enskog method is one of the most well-developed for obtaining transport properties.  Also, for far-from equilibrium flows one may reference Nagnibeda and Kustova [6].

Methods for complete dynamics of specific flows via the BE directly may be solved only in certain very simple cases.  See various text books on this page for gas in a slab, Poiseuille flow or Couette flow.

In some practical cases, a solution has been possible using numerical methods along with a simplified and approximate collision model or “Model Eqn.” Several text books in the references[1][7][8] provide details on linearization of the BE and Model Eqns.

Two commonly used Model Eqns are the Bhatnagar Gross Krook (BGK) model and the BGK-ES model blockquote here:

“A different kind of correction to the BGK model is obtained when a complete agreement with the compressible Navier-Stokes equations is required for large values of the collision frequency. In fact the BGK model has only one parameter (at a fixed space point and time instant): the collision frequency nu; the latter can be adjusted to give a correct value for either the viscosity mu or the heat conductivity kappa, but not for both. This is shown by the fact that the Prandtl number Pr turns out to be unity for the BGK model, whereas it is about 2/3 for a monatomic gas (according to both experimental data and the Boltzmann Equation). In order to have a correct value for the Prandtl number, one is led to replacing the local Maxwellian in [the BGK equation by a different expression]. Only recently has this model (called Ellipsoidal Statistical ES model) been shown [to satisfy the H-theorem].”[7] Some examples of numerical and statistical BGK solvers are here.[8][9][10][11][12]

Recent work is leveraging advanced computing tools toward solving the full BE.[13]

In the limit of very rarefied flows, where Kn>100, the free-molecular or collisionless limit is approached, and various methods have been developed for these cases. For free-molecular cases, Kogan[1] explains: “If we are not interested in the flow field, and the problem consists only of determining the forces acting on the body and the energy transmitted to it, then in free-molecule flow there is no need to know the distribution function of the reflected particles… the momentum and energy transmitted to the surface are completely determined, if the accommodation coefficients are” specified. An example is the work of Moe.[14]

Much of RGD research has focused on the intermediate regime (sometimes called “transition regime”) where 0.01<Kn<100. In these cases, neither continuum assumptions nor collisionless flow assumptions are valid. The most common numerical tools used in these cases include numerical solutions of Model Equations, and two other methods described below.

Direct Simulation Monte Carlo (DSMC) is a statistical method proposed by Graeme A. Bird, who also led the early development of it.[15][16] DSMC is a probabilistic method for simulating a dilute gas; it simulates the Boltzmann equation and generates collisions stochastically with scattering rates and post-collision velocity distributions and energy states determined from the kinetic theory of dilute gases[17][8][18]. It can also simulate chemical reactions and real gas effects via probabalistic methods. The DSMC method has similarities to Molecular Dynamics (MD) but is much more efficient, since in MD the trajectory of every particle in the flow is computed from Newton's equations given an empirically determined interparticle potential. There is an entire field of research that has emerged on the DSMC method and its application to scientific and engineering problems.

Recent work has also developed a more computationally intesive version, Direct Molecular Simulation (DMS) that combines DSMC's collision-pair selection with MD's use of a realistic interparticle potential during each collision (ref Schwartzentruber).

Discrete velocity method solvers: For transient flow problems, direct numerical simulation of the Boltzmann equation via a discrete velocity model have shown to be more efficient than DSMC methods[19][20].

Other research areas within RGD

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In addition to study of these flows and appropriate solutions, the field of RGD encompasses understand the physics and behavior of boundary conditions not in the simplest limits: where the gas-surface interactions are complex, in boundary layers near surface, in boundary layer near a solid or liquid particle carried in a flow, inside shock layers. Chemical reactions, real gas effects, realistic PES, condensation.

Applications of RGD

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Need for Hybrid Methods

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Some of the many Application areas for RGD methods and research are given here with example references. Note that some of these flows have vastly different density/Kn in some spatial or temporal regions where they transition between continuum and free-molecular conditions (plumes; hypersonic flows around complex shapes) and thus may require hybrid solution methods.[21][22][23]

High altitude flight and hypersonic flows

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hypersonic flight, atmospheric entry: These application areas often need RGD methods[24][25][6]

Plumes and jets

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text [26] [27][28] [29][30]

Flow inside material microstructure

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text:  [31][32][33]

Micro-ElectroMechanical Systems MEMS

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text: [34][35][36][37]

Gas flows near particulates and droplets

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text: [38][39]

Crookes radiometer

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text: [40][41]

Knudsen pump

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text: [42][43][44].

Low-density plasmas and plasma processing

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text: [45][46] PIC hybrid Tumuklu, O., & Levin, D. A. (2018). Particle simulations of the effects of atomic oxygen on ion thruster plumes. Journal of Spacecraft and Rockets, 55(5), 1154-1165.

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  3. ^ Shakurova, L.; Armenise, I.; Kustova, E. (2023-08-01). "State-specific slip boundary conditions in non-equilibrium gas flows: Theoretical models and their assessment". Physics of Fluids. 35 (8). doi:10.1063/5.0161025. ISSN 1070-6631.
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  8. ^ a b c Sharipov, Felix (2016-02-23). Rarefied Gas Dynamics: Fundamentals for Research and Practice. John Wiley & Sons. ISBN 978-3-527-41326-3.
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  10. ^ Tumuklu, Ozgur; Li, Zheng; Levin, Deborah A. (2016-12). "Particle Ellipsoidal Statistical Bhatnagar–Gross–Krook Approach for Simulation of Hypersonic Shocks". AIAA Journal. 54 (12): 3701–3716. doi:10.2514/1.j054837. ISSN 0001-1452. {{cite journal}}: Check date values in: |date= (help)
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