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Rarefied Gas Dynamics (RGD)

Rarefied gas dynamics is the study of gas flow at high Knudsen numbers where the phenomena being studied requires consideration of the molecular nature of the gas, a kinetic description, and use of the Boltzmann equation . In rarefied gas dynamics, the non-dimensional Knudsen number has a value approximately $K_{n}>0.01$ implying the gas flow is at low density and/or involves very small length scales. A simple example is flight in the upper atmosphere where there is a decrease in the ambient air density. Consequently, the gas cannot be described by continuum mechanics 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. Rarefied gas dynamics is a sub-branch of fluid dynamics.

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

A range of problems which fall under the regime of rarefied gas dynamics include high speed vehicles at high altitudes including spacecraft returning from orbit or hypersonic cruise vehicles. At very high in the atmosphere, the density is so low that there are very few collisions between the molecules and atoms around the vehicle. Hence, it is their operation in a low-density environment, or high in the atmosphere, that results in rarefied gas dynamics. 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 modeled appropriately in order to determine the drag and heating to the vehicle and its payload.

On the other hand, very small length scales can also result in rarefied gas phenomena. For example, 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 . An example of rarefied flow phenomena that combines both low density and small length scales involve thrusters on spacecraft used for maneuvering in outer space. These thrusters are typically supersonic involving low-pressure gas with length scales of a few centimeters giving rise to rarefied gas dynamics . Other situations where the flow is classified as rarefied gas dynamics includes satellites in low earth orbit , rocket plumes impinging on surfaces, gas flow around carbon-fiber materials at the microscale when designing heat shields for spacecraft, and nuclear fusion devices. Rarefied gas dynamics also includes understanding and controlling the formation, motion, reactions, and evolution of particles of varying composition and shapes ranging from a diameter of ${\ce {.001 \mu m}}$ to $50\mu m$, where understanding how they move in space and time under gradients of concentration, pressure, and temperature is important. This includes understanding the damage impact of atmospheric particles including rain droplets or ice particles on hypersonic vehicles.

Introduction

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. The mean free path is the average value of the distance between two subsequent collisions of a particle (atom or molecule). An example of the characteristic length scale in a flow being considered could be the radius of a pipe or the diameter of a reentry spacecraft, etc) .

$\mathrm {Kn} \ ={\frac {\lambda }{L}}$

Different flow regimes (from a continuum to free-molecular) exists based on the value of the Knudsen number. In most cases of engineering relevance, the $\mathrm {Kn} \ <<1$ and the underlying molecular nature of the gas can be ignored in favor of the continuum description. Flows with a nominal $\mathrm {Kn} \ <0.01$ are typically considered fully continuum. Hence, at small Knudsen numbers or the continuum regime, the flow is not rarefied and 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 .

The other limit is at large Knudsen numbers or the free-molecular regime where the flow is typically described by the Boltzmann Equation. Most problems in rarefied gas dynamics fall into the central region between these two limiting behaviors. Situations of high Knudsen numbers or in the range of $0.01<K_{n}<10$, defines the rarefied gas dynamics regime implying the gas is at low density and/or involves small length scales . Rarefied gas dynamics can occur high in the atmosphere around a hypersonic vehicle where the gas density is low (large mean free paths) resulting in large Knudsen numbers. In addition, rarefied gas dynamics can occur when the length scale is exceedingly small as in Micro-Electro-Mechanical systems (MEMS) and Nanoelectromechanical systems (NEMS), which involve the motion of gases at very small length scales.

Based on the Knudsen number and hence the flow regime, certain numerical modeling approaches are more suitable than others. For solving problems in the rarefied flow regime, particle-based methods, which can resolve a wide range of Knudsen numbers, are typically used. Hence, the application of statistical ideas or kinetic theory must be used such as probability density functions owing to the very large volumes of information involved in tracking the behavior of every single particle in a real gas flow. For simulating a dilute rarefied gas, the direct-Simulation-Monte-Carlo (DSMC) method is popular, which effectively simulates the Boltzmann equation the governing equation of kinetic theory . For rarefied gas dynamics problems, the importance of numerical simulations have grown immensely due to the steady increase in computational power. As the flow becomes continuum, the mean free path and mean collision time scales that must be resolved within a DSMC simulation, require prohibitively large computational resources.

Early History of Rarefied Gas Dynamics

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.

Important Concepts in Rarefied Gas Dynamics

Given the very large volumes of information involved in tracking the behavior of every single particle in a real gas flow, it would be an impossible computational task in describing the state of the gas by specifying the microscopic state (i.e. or the position and velocity of every gas particle) . Doing so would require a huge number of real variables (of the order of $10^{20}$). Hence, in rarefied gas dynamics we must use statistical ideas or kinetic theory. A statistical description is a description based on the average behavior of large numbers of molecules and is much simpler than an approach which follows the detailed behavior of the large number of particles. 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.

Information is required about the spatial distribution of gas particles and their velocity distribution as well. The velocity distribution allows one to extract the momentum and energy of the flow which play an important role in gas dynamics. Hence, a rarefied gas is described in terms of a distribution function which contains information on both the distribution of molecules in the system under consideration and the distribution of molecular velocities . Macroscopic properties of the gas are found by taking moments of the distribution function. Ludwig Boltzmann derived an evolution equation (Boltzmann Equation) for the distribution function. The Boltzmann equation 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 is large compared to the molecular size of the particles meaning collisions are predominantly binary in nature. Such binary collisions occur over time scales that are much shorter than the mean collision time of the gas and hence occur instantaneously.

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. 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.

• $K_{n}<0.01$ Continuum
• $0.01<K_{n}<0.10$ Slip Flow (Rarefied)
• $0.1<K_{n}<10$ Transitional Flow (Rarefied)
• $K_{n}>10$ Free-molecular flow (Rarefied)

In studying rarefied gas dynamics, in addition to the description based on the Boltzmann equation, 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 the body and this interaction dictates the heat transfer between the gas and the solid surface. For example, in designing heat shields for spacecraft reentering on planets, ablative heat shields or thermal protection systems are used. As the spacecraft reenters the atmosphere, air gas particles like oxygen and nitrogen atoms chemically react with the surface resulting in ablation or recession. 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 .

Real World Applications

Applications involving rarefied gas dynamics include satellites in low earth orbit, rocket plumes impinging on surface, study of thermal loading at the microscale for protection of spacecraft, microfluidic devices, nuclear fusion devices. For example, rarefied gas occurs when examining the flow through the small crevices on the surface of a spacecraft where instrumentation is usually mounted (e.g. pressure sensors) as the mean free path of the surrounding gas approaches the length scale of interest (the diameter of the pressure sensor).

In addition, applications involves 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.

Research in rarefied gas dynamics also involve examining the the physics of molecular interactions including gas surface interactions, kinetic theory, astronomical observations, gas transport, multi-phase flows, combustion, non-equilibrium hypersonic gas dynamics, and plasma processing. A summary of some real world applications in rarefied gas dynamics are listed below:

• Boltzmann and related equations: mathematical properties of the Boltzmann equation, proofs of existence of solutions in particular cases, model kinetic equations
• Monte Carlo methods and numerical solutions
• Moment methods
• Experimental methods
• Gas-surface interactions
• Aerospace. Examples include high speed flows, shock waves, nozzle expansions, high altitude aerodynamics.
• Jets and plumes
• Internal flows and vacuum systems
• Reactive gas dynamics
• Rarefied plasmas
• MEMs and NEMs
• Granular flows
• Porous media

Methods of Solving Problems

Typically in solving rarefied gas dynamics problems where the continuum approach is no longer accurate, particle-based approaches are used . The Direct Simulation Monte Carlo (DSMC) technique is a probabilistic method for simulating a dilute gas. The method directly simulates the Boltzmann equation and generates collisions stochastically with scattering rates and post-collisions velocity distributions determined from the kinetic theory of dilute gases. Its removes the continuum constraint and models the gas on a microscopic level through interactions and convection of individual particles. The DSMC method is an efficient alternative than molecular dynamics (MD), 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. The DSMC method was originally developed by aerospace engineers for the simulation of rarefied gas flows. 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 .

For situations where the flowfield is comprised of both continuum and rarefied flows regimes, hybrid methods are a popular approach where the DSMC method is coupled with a Navier-Stokes solver. In regions where the continuum assumption is valid, a Euler or Navier-Stokes solver is used. In the remaining portion a DSMC technique is employed. In the remaining portion a DSMC technique is employed. Applications include blunt aerodynamic vehicles with complex geometries and control surfaces at high-speeds in the atmosphere. For example, Modular-particle continuum (MPC) methods have been developed where the DSMC method is used in rarefied flow regimes and the Navier-Stokes equations are solved elsewhere .

Conferences/Symposiums

1. 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 .
2. 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/