Draft:Rarefied gas dynamics

Particle simulation of Argon gas flowing over a flat plate at angle of attack

Rarefied gas dynamics is a branch of fluid mechanics where the continuum assumption is no longer accurate as a characteristic length scale in the gas (e.g. radius of a body in a flow)  becomes comparable to the mean free path of the gas. Consequently, the gas cannot be described as a continuum and the Boltzmann transport equation must be used to understand the physics of the system; hence rarefied gas dynamics is studied using the application of statistical ideas as described by the kinetic theory of gases. In rarefied gas dynamics, the Boltzmann transport equation is the appropriate mathematical model in describing the gas.

Definition

To determine if a gas can be classified as rarefied, the non-dimensional Knudsen number is typically calculated. The Knudsen number ${\ce {\mathrm {Kn} }}$ is defined as the ratio of the mean free path $\lambda$, to a characteristic length scale $L$ in the flow. Different flow regimes (ranging from a continuum to a free-molecular regime) exists based on the value of the Knudsen number. When describing a rarefied gas, a microscopic description is used where the molecular nature of the gas must be accounted for. The gas is described as consisting of many discrete particles each with its own velocity and position. Considering the behavior of particles at the microscopic scale, two main processes occur: translational motion in space due to their velocity, and intermolecular collisions with other particles in the gas. In rarefied gas dynamics statistical ideas or kinetic theory must be used since it would be an impossible computational task in storing the very large volumes of information involved in tracking the behavior of every single particle in a real gas flow. A statistical description is made possible because in practice changes in the macroscopic state of the gas described by quantities such as density, bulk velocity, temperature, stresses, and heat flux are related to suitable averages of the gas at the microstate.

Knudsen number

Based on the value of Knudsen number, the gas can be characterized as a continuum, slip, transitional, or free-molecular flow regime based on the value of the Knudsen number. The value of the Knudsen number tells you whether a molecular modeling approach must be used or a macroscopic description is sufficient. High Knudsen numbers means that the gas is in the rarefied regime while small Knudsen numbers imply that the gas is in the continuum regime. The various flow regimes based on the Knusden number are described below.

• $K_{n}<0.01$ Continuum Flow (Not Rarefied)
• $0.01<K_{n}<0.10$ Slip Flow (Rarefied)
• $0.1<K_{n}<10$ Transitional Flow (Rarefied)
• $K_{n}>10$ Free-molecular flow (Rarefied)

Based on the value of Knudsen number, the gas can be characterized as a continuum, slip, transitional, or free-molecular flow regime. When the Knudsen number is close to a value of 0 or $K_{n}\ <0.01$ a continuum modeling approach is appropriate. The gas is not rarefied and hence can be described by the Euler or Navier-Stokes equations using traditional computational fluid dynamics. However, the lack of accounting for particle collisions in the Naiver-Stokes equations makes the physics of the Naiver-Stokes equations invalid in the rarefied gas regime.

However, for large Knudsen numbers or the free-molecular regime $K_{n}\ >10$ very few collisions occur within the volume of interest. Most problems in rarefied gas dynamics fall into the central region between these two limiting behaviors or when $0.01<K_{n}<10$ implying the gas is at low density and/or involves small length scales. For modeling problems in the rarefied flow regime, particle-based methods, which can resolve a wide range of Knudsen numbers, are typically used.

Modeling

A microscopic description is different from a macroscopic description. In a macroscopic description, the molecular nature of the gas is neglected and the spatial and temporal variations of the flow are described in terms of its fluid properties such as velocity, density, pressure, and temperature. The macroscopic description is used for gases where the continuum assumption is accurate; here the Navier-Stokes equations provides the appropriate mathematical model in describing the fluid.

However, in rarefied gas dynamics the Boltzmann transport equation is the appropriate mathematical model in describing the gas. In rarefied gas dynamics, the molecular nature of the gas must be considered where the gas is described by a velocity distribution function $f(r,v)$ where $r$ is a vector of spatial coordinates and $v$ is a velocity of molecules. The velocity distribution function provides a statistical description of the gas on the molecular level. Taking moments of the velocity distribution function results in macroscopic properties and provides the connection back to fluid dynamics. The evolution of the velocity distribution function is described by the Boltzmann equation which provides a valid description of a dilute gas or as long as the density of the gas is sufficiently low. In a dilute gas, the mean molecular spacing $\delta$ is large compared to the molecular size of the particles $d$ or $\delta \gg d$ meaning collisions are predominantly binary in nature. Another important assumption in the derivation of the Boltzmann transport equation is molecular chaos, which assumes velocities of colliding particles are uncorrelated, and independent of position.

In studying rarefied gas dynamics, in addition to the velocity distribution function, an additional piece of information required is the interaction of the gas molecules with the solid surfaces. It is this interaction that produces the drag and lift produced on an aerospace vehicle like an airplane and this interaction dictates the heat transfer between the gas and the solid surface. The study of gas-surface interaction is regarded as the bridge between the kinetic theory of gases and solid state physics. This implies that the Boltzmann equation must be accompanied by boundary conditions which describe the interaction of the gas molecules with the solid wall. For modeling gas flows in the rarefied flow regime, numerical simulations are typically performed using particle-based methods, which can resolve a wide range of Knudsen numbers.

Early history

The history of rarefied gas dynamics starts with the kinetic theory of gases. Maxwell published results on the law governing the distribution of molecular velocities for a uniform gas in equilibrium or the Maxwell Boltzmann Distribution and on the law of equipartition of the mean molecular energy in a mixture of gases. As a result, the assumption that all molecules move with the same speed was abandoned and the random nature of molecular motion was recognized. Boltzmann in 1872 then derived the H-theorem, which demonstrated how molecular collisions tend to increase entropy and that any initial distribution of molecular positions and velocities will almost certainly evolve into an equilibrium state. In the same paper, Boltzmann then derived an integro-differential equation (Boltzmann equation) to describe the evolution of the velocity distribution function in space and time.

Hilbert then proved the existence and uniqueness of a solution from the Boltzmann Equation and established a firm logical structure for kinetic theory. The connection between kinetic theory and fluid dynamics was done by Chapman and Enskog whom deduced expressions for the coefficients of viscosity and thermal conductivity for a non-uniform gas which was based upon a series solution of the Boltzmann’s equation. Furthermore, in the general theory of non-uniform gasses. Burnett came up with a method to calculate the velocity distribution function to any order of approximation for a simple gas. Grad then proved the equivalence of the equations of fluid dynamics to an asymptotic form of the Boltzmann equation. An important realization was the existence of different time scales e.g. the fluid dynamic description of a gas is much coarser than the time scale of kinetic theory. Rarefied gas dynamics has existed since the nineteenth century and came into the forefront with space exploration and the first international symposium on rarefied gas dynamics was held in Nice, France in July 1958.

Mathematical approaches

Mathematical development

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 Aoki et al., Shakurova et al.

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. 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. 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.

Mathematical and numerical solving

Methods for the 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 provide details on linearization of the BE and Model Eqns.

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

"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]."

Some examples of numerical and statistical BGK solvers are here.

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

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 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.

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 (see above), 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. 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. 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. (ref thr DSMC Workshop here?)

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

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

Applications

High altitude flight and hypersonic flows

hypersonic flight, atmospheric entry: Satellites in orbit, spacecraft or rockets re-entering Earth atmosphere, or entering atmosphere of another planet, or hypersonic cruise vehicles. These application areas often need RGD methods In addition, even within a mostly continuum flow around a high-speed vehicle, there may be local regions of rarefied flow as in the wake of the vehicle or near sharp-leading edges. These rarefied flow regions must be modelled appropriately in order to determine, e.g., the drag and heating to the vehicle and the radiation emitted and plasma blackout. It also includes understanding the damage impact of micrometer aerosol atmospheric particles on the material erosion and heating at the surface.

Plumes and jets

An example of rarefied gas dynamics that combines both low density and small length scales involve thrusters on spacecraft used for maneuvering in outer space as well as rocket plumes impinging on surfaces.

Flow inside material microstructure

Modeling gas flow around porous carbon-fiber materials at the microscale when designing heat shields for spacecraft with the objective of characterizing the aerothermal loading (e.g. heat flux, shear stress) and concentrations of atomic species inside the material microstructure. Here the mean free path approaches the length scale of the carbon-fibers of $10\mu m$.

MEMS and NEMS

Very small length scales can also result in rarefied gas phenomena. Micro-Electro-Mechanical systems (MEMS) and Nanoelectromechanical systems (NEMS), which involve the fabrication and operation of microscopic devices, involve the motion of gases at very small length scales including $1\mu m$ or $10^{-6}$ m resulting in rarefied gas dynamics

Gas flows near particulates and droplets

text:

Crookes radiometer

text:

Knudsen pump

text:

Low-density plasmas and materials processing and plasma processing

text:

Need for hybrid methods

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.

Conferences and symposiums

Rarefied Gas Symposium

Prompted by the problems raised by the space exploration of the early 1960s, when the study of the Boltzmann equation was in its infancy, the International Symposium on Rarefied Gas Dynamics (RGD) was established and its first symposium was held in Nice, France, in 1958. The Rarefied Gas Dynamics Symposium has been the place to discuss problems related to the study of the mathematical properties of the Boltzmann equation, the development of model kinetic equations that could be more easily solved while providing accurate approximations to the physics of interest, algorithms for the numerical solution of the Boltzmann equation (DSMC, spectral solvers, moment methods) and all related applications.

DSMC Conference

Held every two years in Santa Fe, New Mexico, the DSMC conference brings together outstanding DSMC researchers from around the world. The goal of this meeting is to bring together developers and practitioners of the Direct Simulation Monte Carlo (DSMC) method. Talks cover all types of DSMC-related work: theoretical foundations, verification and validation, convergence, computational efficiency, hydrodynamic fluctuations, flow instabilities, algorithm development, aerospace, hypersonics, microscale flows, nanoscale flows, plasmas, transport properties, collisional energy exchange, gas-phase chemical reactions and ionization, gas-surface interactions, planetary atmospheres, dense gases, liquids, granular flow, and experiments relevant to DSMC.

External links

• https://www.rarefiedgasdynamics.org/
• https://www.sandia.gov/dsmc/
• https://www.linkedin.com/company/rgd-next-gen/