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Direct Numerical Solution of the Three-Dimensional Generalized Boltzmann Equation for Hypersonic Non-Equilibrium Flows

Date of Award

Summer 8-15-2010

Author's School

School of Engineering & Applied Science

Author's Department

Mechanical Engineering & Materials Science

Degree Name

Doctor of Philosophy (PhD)

Degree Type



The development and applications of a computer code for solving the three-dimensional generalized Boltzmann equation (GBE) using a direct numerical method are presented. The Boltzmann solver of Professor Felix G. Tcheremissine of the Russian Academy of Science serves as the foundation for the development effort. This jet code includes only the translational and rotational energy states of a diatomic gas and has been applied to simulate the flow field of a jet issuing into a vacuum. In this dissertation, this code is employed to accomplish three distinct developmental steps. First, the code is extended for calculating hypersonic shock waves in an inert mixture of gases. For this purpose, the GBE is formulated in an impulse space (instead of the conventional velocity space). The computational methodology is then applied to a binary mixture of gases, which requires the simultaneous solution of four GBE’s. Simulations are performed using a gas mixture including both diatomic and monatomic gases in proportions similar to that in air. The solutions are validated against existing hypersonic shock wave experimental data for a single specie gas (nitrogen) in rotational-translational non-equilibrium and available computational data for a binary mixture of monatomic gases. Simulations are then performed for an inert binary mixture of monatomic and diatomic gases in translational non-equilibrium for various concentrations. The effect of mass ratio and molecular diameter ratio of the gases on the structure of the shock is also investigated. Second, boundary conditions necessary for accurately simulating the flows around immersed bodies are developed and evaluated. This research on boundary conditions constitutes a significant advancement beyond the adsorptive boundary condition used in the original Boltzmann solver of Tcheremissine. Five types of boundary conditions at the solid boundary are investigated: (a) the standard adsorptive boundary condition, (b) the specular reflection boundary condition, (c) the diffuse reflection boundary condition, (d) the Maxwellian boundary condition, and (e) the adsorptive Maxwellian boundary condition with different values for the accommodation coefficient. These boundary conditions are tested for hypersonic flow past a flat plate to evaluate their accuracy. Third, the original Boltzmann code, hard-coded for solving the flow field of a jet issuing into a vacuum, is modified to enable simulations of rarefied flows around immersed bodies. The computations are performed for three benchmark geometries, extensively used in the literature for Navier-Stokes simulations, at various hypersonic inflow conditions for flow of a diatomic gas (N2) in rotational-translational non-equilibrium. The three geometries used in the simulations are an axisymmetric blunt body, an axisymmetric bicone, and an axisymmetric hollow-flared-cylinder. Initially, a relatively coarse Cartesian grid was employed in the three-dimensional simulations because of the limitations of physical memory on the available computers. As a result, a shared memory parallel computing platform was developed and built for the sole purpose of being able to perform the fine grid solutions. Consequently, refined grid solutions were generated on the parallel computing platform. For this purpose, the code was parallelized and the parallelization issues for a Boltzmann type solver were addressed. A comparison between the coarse and refined grid solutions is presented to show the influence of grid density on solution accuracy. In light of these results, the issues of accuracy and efficiency of the three-dimensional Boltzmann solver are addressed.


English (en)


Ramesh Agarwal

Committee Members

Ramio Hakkinen, Kenneth Jerina, Mori Mani


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