Deterministic approach for unsteady rarefied flow simulations in complex geometries and its application to gas flows in microsystems.

摘要

Micro-electro-mechanical systems (MEMS) are widely used in automotive, communications and consumer electronics applications with microactuators, micro gyroscopes and microaccelerometers being just a few examples. However, in areas where high reliability is critical, such as in aerospace and defense applications, very few MEMS technologies have been adopted so far. Further development of high frequency microsystems such as resonators, RF MEMS, microturbines and pulsed-detonation microengines require improved understanding of unsteady gas dynamics at the micro scale. Accurate computational simulation of such flows demands new approaches beyond the conventional formulations based on the macroscopic constitutive laws. This is due to the breakdown of the continuum hypothesis in the presence of significant non-equilibrium and rarefaction because of large gradients and small scales, respectively. More generally, the motion of molecules in a gas is described by the kinetic Boltzmann equation which is valid for arbitrary Knudsen numbers. However, due to the multidimensionality of the phase space and the complex non-linearity of the collision term, numerical solution of the Boltzmann equation is challenging for practical problems.;In this thesis a fully deterministic, as opposed to a statistical, finite volume based three-dimensional solution of Boltzmann ES-BGK model kinetic equation is formulated to enable simulations of unsteady rarefied flows. The main goal of this research is to develop an unsteady rarefied solver integrated with finite volume method (FVM) solver in MEMOSA (MEMS Overall Simulation Administrator) developed by PRISM: NNSA center for Prediction of Reliability, Integrity and Survivability of Microsystems (PRISM) at Purdue and apply it to study micro-scale gas damping. Formulation and verification of finite volume method for unsteady rarefied flow solver based on Boltzmann-ESBGK equations in arbitrary three-dimensional geometries are presented. The solver is based on the finite volume method in the physical space and the discrete ordinate method in the velocity space with an implicit time discretization. A conservative discretization of the collision term has been incorporated. Verification was carried out for an unsteady approach to equilibrium, steady one-dimensional Couette and Fourier flows and a two-dimensional quasi-steady gas damping for a moving microbeam.;The solver was directly compared with a 2D steady ESBGK solver using reduced distribution functions (rdf) for the squeeze film damping problem and was compared to theory for a 2D conduction in a thin rectangular plate. The solver was also validated with experiments for a free cantilever damping problem. An approach for coupling with other deterministic solvers such as the Navier-Stokes solver in MEMOSA has been presented. A new equilibrium breakdown parameter based on entropy generation rate is introduced. The proposed continuum-rarefied coupling scheme was verified with analytical solution for Couette flow. An immersed boundary method was formulated for the ES-BGK equations and the implementation in 1D Couette flow was carried out. Finally, the application of the full 3D parallel solver is considered to simulate unsteady microscale gas damping in a micro-electro-mechanical system switch. Simulation results with half a billion unknowns on 128 processors are presented and suggest that, with the advent of petascale computing platforms, it has become practical to solve full 3D unsteady rarefied flow problems for complex geometries.