The primary objective of my research is to develop numerical techniques to accurately model practical micron-scale gaseous flows across a wide range of flow regimes. The regime in which a flow exists can be characterized by the Knudsen number (Kn = lambda/l) where "lambda" is the mean free path that a gas particle will travel before it undergoes a collision and "l" is the characteristic length of the flow. In flows with very low Knudsen numbers, typically Kn < 0.01, collisional processes will dominate and the flow will remain in local thermodynamic equilibrium; this is known as the continuum regime. In this regime, traditional fluid dynamic equation sets, such as the Euler and Navier-Stokes equations, are valid. On the opposite side of the spectrum are high Knudsen number flows, typically Kn > 10. This is the free-molecular regime. In this regime, particle collisions occur infrequently and the fluid can deviate drastically from local thermodynamic equilibrium. Typically in this regime, the non-equilibrium flows are modeled using an approach such as the direct-simulation Monte Carlo method (DSMC). This classification leaves a range of Knudsen numbers (0.01 < Kn < 10) where the flow characteristics are in transition between continuum flow and free-molecular flow. In this transition regime, traditional fluid dynamic equation sets are not valid because collisional processes occur infrequently and there can exist significant deviations from local thermodynamic equilibrium (such as anisotropic pressures), however the computational expense required for direct simulation is often prohibitively expensive. This deficiency of current numerical techniques is unfortunate as there are many problems of practical interest that lie within the transition regime, such as flows in micro-electromechanical systems (MEMS), fabrication techniques used in micro-processor production such as chemical vapour deposition (CVD) or high atmospheric flight. It is obvious that the development of algorithms that can span this gap would be advantageous.

The approach that will be taken in the development of these algorithms will be to use techniques from gaskinetic theory, specifically moment closures of the Boltzmann equation. In classical gaskinetic theory, fluids are represented by a probability density function that exists in six-dimensional phase space. Moment closures offer a means by which the evolution of macroscopically observable characteristics of a gas can be obtained. The proposed moment closure considered for this research is the 10-moment Gaussian closure. This closure yields a robust hyperbolic set of equations that can be easily used in upwind-based solution schemes. Some very promising initial work has already been done in this area. In addition to having the ability to model non-equilibrium effects, this closure requires the evaluation of only first derivatives. This makes this model relatively insensitive to grid irregularities that will inevitably occur when unstructured or adaptive gridding is used for the treatment of complex geometries. This is unlike the Navier-Stokes or Burnett equations, both of which have an elliptic nature and require the evaluation of higher-order derivatives.

During the course of my research, a three-dimensional upwind finite volume flow solver based on the 10-moment model will be constructed. This solver will use block-based adaptive meshing to deal with complex geometries and will designed to operate efficiently in parallel on large distributed memory computers. For low-Mach-number flows, the use of a preconditioner will be investigated to improve convergence speed. Currently, the 10-moment model is deficient in that it cannot account for heat transfer; it can therefore currently only be applied to problems in which heat transfer is not significant. In order to increase the applicability of this closure, an extension that allows for heat transfer will be sought. The development of this solver will be of both theoretical and practical interest.