Student Projects Reports Kinetics of Gases Graduate Course Fluid Dynamics 5C1505

1 Introduction to the gas simulation methods The Navier-Stokes equations commonly encountered in fluid mechanics applications builds on the continuum approximation. When the ratio of the mean free path λ to the macroscopic length scale L, formalised by the non-dimensional Knudsen number Kn = λ/L, becomes large this approximation breaks down. The macroscopic length scale should be based on the local field (i.e. velocity, pressure, density) to gradient ratio, thereby removing the question on how to defining this length scale for complex flows. For small Knudsen numbers Kn < 0.1 the continuum approximation is valid. However for larger Knudsen numbers one has to resort to some sort of methodology which considers interaction between particles. The Boltzmann equation is the accepted mathematical model of a gas flow at the molecular level. Numerically this equation can be solved using conventional CFD techniques, with the velocity distribution function being the obtained solution on a computational grid. Since this is a six-dimensional partial differential equation it quickly becomes computationally heavy. Further the collision terms represented by the right hand side integral is a challenging task to evaluate. An alternative to solving the Boltzmann equation is to employ direct simulation methods, where a large number of simulated molecules are traced in space and time, and their velocities and internal states are modified through collisions and boundary interactions. The most fundamental approach within this class is that of the deterministic Molecular Dynamics (MD) in which actual collisions between particles are calculated based on Newton's law or quantum mechanics. Note however that this quickly becomes computationally intractable since as soon as a geometry and the gas characteristics are set there are no tunable parameters, making MD methods applicable only to dense gases. In the next section we will describe briefly the probabilistic approach to direct simulation. The Direct Simulation Monte Carlo (DSMC) (Bird, 1995) method is suitable for simulating dilute gases by means of a probabilistic approach. The term dilute gas means that a typical molecular spacing δ is much larger than a typical molecule diameter d, 1 formally stated as δ/d ≫ 1. In terms of Knutsen number this would approximately mean that O(0.1) ≤ Kn ≤ O(10). The fundamental idea of DSMC is to track a large number of statistically representative particles where each of the particles contains a cluster of molecules. The particles motion and interactions are used to modify their positions and velocities. …