THESIS
2014

xv leaves, 163 pages : illustrations ; 30 cm

**Abstract**
Boltzmann equation based on kinetic theory is well accepted as an alternative approach to describe
the motion of gas flows. Comparing with Navier-Stokes equation, Boltzmann equation offers a
wider regime of gas flows. However, the high computational cost always renders the numerical
solutions of Boltzmann equation unaffordable for practical engineering applications, even with
simplified collision models. As macroscopic quantities appeared in the collision model are obtained
by taking hydrodynamic moments, quadrature method to evaluate the moment integrals plays an
important role in solving the Boltzmann equation. The main obstacle for the general
implementation of quadrature method occurs on flows at high Mach number and with large thermal
inhomogeneity, which requires huge quad...[

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Boltzmann equation based on kinetic theory is well accepted as an alternative approach to describe
the motion of gas flows. Comparing with Navier-Stokes equation, Boltzmann equation offers a
wider regime of gas flows. However, the high computational cost always renders the numerical
solutions of Boltzmann equation unaffordable for practical engineering applications, even with
simplified collision models. As macroscopic quantities appeared in the collision model are obtained
by taking hydrodynamic moments, quadrature method to evaluate the moment integrals plays an
important role in solving the Boltzmann equation. The main obstacle for the general
implementation of quadrature method occurs on flows at high Mach number and with large thermal
inhomogeneity, which requires huge quadrature points to cover the effective range of velocity
space. This leads to low convergence rate. Recently, a novel quadrature method, called Dynamic
Quadrature Scheme (DQS), was proposed for relieving the low convergence rate issue. The
advancement of DQS is that the hydrodynamic moment integrals with respect to the velocity-space
coordinates are transformed into non-dimensional Gaussian-like forms before taking a quadrature procedure. As a result, the quadrature error is independent of macroscopic quantities and occurs at
the second order of Knudsen number for small Knudsen number flows, which are commonly
encountered in engineering problems.

The aim of this thesis is to offer a comprehensive detail on the implementation of dynamic
quadrature scheme with a discrete ordinate method. This method is referred as ”Dynamics Discrete
Ordinate Method” (DDOM). The main feature of the numerical implementation for DDOM is to
estimate the macroscopic variables at next time-step for proper velocity-space coordinate
transformation of DQS. From the parameterized behavior of macroscopic variables in the
equilibrium distribution function, the velocity-space coordinate transformation can be approximated
with the free-advection. This method has greatly reduces significantly the computational cost, and is
termed as “Non-Stationary Transformation” (NST). To show the computational efficiency of DQS,
1-D and 2-D DDOM codes were developed based on NST of DQS, with the BGK collision model
for Boltzmann equation. The validity, as well as the efficiency and accuracy, of the 1-D and 2-D
DDOM codes are tested with the benchmark flows of 1-D Riemann problems, 2-D Riemann
problems, backward-step problem, shock reflection problem, as well as cavity flows. It was found
that only 3 quadrature points are needed for each velocity dimension by DDOM. An efficiency of
20-times faster in computational-time is achieved by 2-D DDOM than by traditional DOM, for 2-D
problems, while 5 times faster for 1-D problems. Adaptive quadrature scheme and simplified solver
for low Mach number flows are introduced to enhance further the numerical efficiency of 2-D
DDOM. The enhanced 2-D DDOM code is 14% faster than Godunov solver based on Roe
approximation.

For 3-D simulations, a 3-D DDOM code has been developed. The 3-D DDOM code is parallelized
along with the 2-D DDOM to reduce the computational time. A hybrid programming (OpenMP,
MPI and CUDA) approach is used to enhance the performance of a parallelized Dynamic Discrete
Ordinate Method (DDOM) solver. This hybrid parallelism model is extended to a Central
Processing Unit (CPU) and Graphics Processing Unit (GPU) clusters. Parallelized 2-D DDOM and
3-D DDOM codes are preliminarily tested by Beowulf clusters. The clusters with 64 CPU cores are
connected by gigabit switch. The results of tests show an achievement of over 90% parallel
efficiency. By using massive multicore GPUs, the CUDA-accelerated code achieves a speed 250
times faster with a single GPU and over 780 times faster with a Quad-GPU cluster versus the
identical process running on a single thread of CPU. The present results demonstrate that 3-D
DDOM solver provides good scalability on CPU and GPU clusters.

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