In this thesis the solutions of the two-dimensional (2D) and three-dimensional (3D) lid-driven cavity problem are obtained by solving the steady Navier-Stokes equations at high Reynolds numbers. In 2D, we use the streamfunction-vorticity formulation to solve the problem in a square domain. A numerical method is employed to discretize the problem in the x and y directions with a spectral collocation method. The problem is coded in the MATLAB programming environment. Solutions at high Reynolds numbers are obtained up to $Re=25000$ on a fine grid of 131 * 131. The same method is also used to obtain the numerical solutions for the steady separated corner flow at high Reynolds numbers are generated using a for various domain sizes, at various Reynolds number which are much higher than those obtained by other researchers.Finally, the numerical solutions for the three-dimensional lid-driven cavity problem are obtained by solving the velocity-vorticity formulation of the Navier-Stokes equations for various Reynolds numbers. A spectral collocation method is employed to discretize the y and z directions and finite difference method is used to discretize the x direction. Numerical solutions are obtained for Reynolds number up to 200.

The present work deals with the development of a direct simulation strategy for solving the motion of spherical particles in non-Newtonian liquids. The purely viscous (non-elastic) non-Newtonian liquids are described by Bingham and thixotropy models. Validation of the strategy is performed for single phase (lid driven cavity flow) and two phase flows (sphere sedimentation). Lid driven cavity flow results illustrate the flow evolution of thixotropic liquid and subtle differences between thixotropic rheology and pseudo Bingham rheology. Single sphere sedimentation in Bingham liquid is shown to be influenced by the yield stress of the liquid. Time-dependent properties such as aging prominently affect the settling of a sphere in thixotropic liquid. The hydrodynamic interactions between two spheres are also studied at low and moderate Reynolds numbers. In thixotropic liquid, an intriguing phenomenon is observed where the separation distance between the spheres first increases and then rapidly decreases. / Chemical Engineering

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Fluid mixing is a challenging problem in laminar flow systems. Even in microfluidic systems, diffusion is often negligible compared to advection in the flow. The idea of chaotic advection can be applied in these systems to enhance mixing efficiency. Topological chaos can also lead to efficient and rapid mixing. In this dissertation, an approach to enhance fluid mixing in laminar flows without internal rods is demonstrated by using the idea of topological chaos. Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. This approach is in contrast to standard techniques. Appropriate stirrer motions are determined using the Thurston-Nielsen classification theorem, which also predicts a lower bound on the complexity of the dynamics in the flow. Work in this area has focused largely on using physical rods as stirrers, but the theorem also applies when the ``stirrers'' are passive fluid particles. In this thesis, topological chaos is theoretically and numerically investigated in lid-driven cavities and rectangular channels without internal rods. When a two-dimensional incompressible Newtonian flow is in the Stokes flow regime, the stream function satisfies the two-dimensional biharmonic equation. When the flow occurs in a lid-driven cavity with solid side walls, this equation can be solved using a method that is similar to the traditional Fourier expansion but uses an asymptotic approximation for the sum of high order terms. When the flow occurs between two infinite plates with spatially periodic boundary conditions, an exact solution in a rectangle with finite width, which represents the flow in this infinitely-wide cavity, can be obtained by using the principle of superposition. A fully developed, three-dimensional flow in a rectangular channel can be decomposed into an unperturbed Poiseuille flow in the axial direction and a lid-driven cavity secondary flow in the cross section. This model can be applied to numerically simulate either pressure-driven flow in a rectangular channel with surface grooves or electro-osmotic flow in a rectangular channel with variations in surface potential. In this dissertation, the occurrence of topological chaos in unsteady two-dimensional flows as well as steady three-dimensional flows without internal rods is demonstrated. For appropriate choices of boundary velocity on the top and/or bottom walls, there exist three periodic points in the flows that produce a chaos-generating motion. In steady flow through a three-dimensional rectangular channel, the axial direction plays the role of time and the periodic points lie on streamtubes that "braid" the surrounding fluid as it moves through the duct. When appropriate motion is applied on the boundary of the wide cavity or channel, topological chaos can also be generated in the flow. The stretching rate of non-trivial material lines in all these flows agrees with the prediction of the lower bound of topological entropy provided by the Thurston-Nielsen theorem. Ghost rod structures are found and analyzed in the lid-driven cavity and rectangular channel flows with solid side walls. The results suggest that the no-slip boundary condition on the stationary internal surfaces is one of the reasons for poor mixing in steady laminar three-dimensional flows considered previously with solid braided internal rods. / Ph. D.

Numerous efforts have contributed to the study of phase-change problems for over a century|both analytical and numerical. Among those numerical approximations applied to solve phase-transition problems, phase-field models attract more and more attention because they not only capture two important effects, surface tension and supercooling, but also enable explicitly labeling the solid and liquid phases and the position of the interface. In the research of this dissertation, a phase-field model has been employed to simulate 2-D dendrite growth of pure nickel without a flow, and 2-D ice crystal growth in a high-Reynolds-number lid-driven-cavity flow. In order to obtain the details of ice crystal structures as well as the flow field behavior during freezing for the latter simulation, it is necessary to solve the phase-field model without convection and the equations of motion on two different scales. To accomplish this, a heterogeneous multiscale method is implemented for the phase-field model with convection such that the phase-field model is simulated on a microscopic scale and the equations of motion are solved on a macroscopic scale. Simulations of 2-D dendrite growth of pure nickel provide the validation of the phase-field model and the study of dendrite growth under different conditions, e.g., degree of supercooling, interface thickness, kinetic coefficient, and shape of the initial seed. In addition, simulations of freezing in a lid-driven-cavity flow indicate that the flow field has great effect on the small-scale dendrite structure and the flow eld behavior on the large scale is altered by freezing inside it.

This thesis focuses on the phenomenon of aliasing and its mitigation with two explicit filters, i.e., Shuman and Padé filters. The Shuman filter is applied to velocity components of the Navier--Stokes equations. A derivation of this filter is presented as an approximation of a 1-D “pure math” mollifier and extend this to 2D and 3D. Analysis of the truncation error and wavenumber response is conducted with a range of grid spacings, Reynolds numbers and the filter parameter, β. Plots of the relationship between optimal filter parameter β and grid spacing, L2-norm error and Reynolds number to suggest ways to predict β are also presented. In order to guarantee that the optimal β is obtained under various stationary flow conditions, the power spectral density analysis of velocity components to unequivocally identify steady, periodic and quasi-periodic behaviours in a range of Reynolds numbers between 100 and 2000 are constructed. Parameters in Pade filters need not be changed. The two filters are applied to velocities in this paper on perturbed sine waves and a lid-driven cavity. Comparison is based on execution time, error and experimental results.

High-level parallel programming approaches, such as OpenACC, have recently become popular in complex fluid dynamics research since they are cross-platform and easy to implement. OpenACC is a directive-based programming model that, unlike low-level programming models, abstracts the details of implementation on the GPU. Although OpenACC generally limits the performance of the GPU, this model significantly reduces the work required to port an existing code to any accelerator platform, including GPUs. The purpose of this research is twofold: to investigate the effectiveness of OpenACC in developing a portable and maintainable GPU-accelerated code, and to determine the capability of OpenACC to accelerate large, complex programs on the GPU. In both of these studies, the OpenACC implementation is optimized and extended to a multi-GPU implementation while maintaining a unified code base. OpenACC is shown as a viable option for GPU computing with CFD problems. In the first study, a CFD code that solves incompressible cavity flows is accelerated using OpenACC. Overlapping communication with computation improves performance for the multi-GPU implementation by up to 21%, achieving up to 400 times faster performance than a single CPU and 99% weak scalability efficiency with 32 GPUs. The second study ports the execution of a more complex CFD research code to the GPU using OpenACC. Challenges using OpenACC with modern Fortran are discussed. Three test cases are used to evaluate performance and scalability. The multi-GPU performance using 27 GPUs is up to 100 times faster than a single CPU and maintains a weak scalability efficiency of 95%. / Master of Science / The research and analysis performed in scientific computing today produces an ever-increasing demand for faster and more energy efficient performance. Parallel computing with supercomputers that use many central processing units (CPUs) is the current standard for satisfying these demands. The use of graphics processing units (GPUs) for scientific computing applications is an emerging technology that has gained a lot of popularity in the past decade. A single GPU can distribute the computations required by a program over thousands of processing units. This research investigates the effectiveness of a relatively new standard, called OpenACC, for offloading execution of a program to the GPU. The most widely used standards today are highly complex and require low-level, detailed knowledge of the GPU’s architecture. These issues significantly reduce the maintainability and portability of a program. OpenACC does not require rewriting a program for the GPU. Instead, the developer annotates regions of code to run on the GPU and only has to denote high-level information about how to parallelize the code. The results of this research found that even for a complex program that models air flows, using OpenACC to run the program on 27 GPUs increases performance by a factor of 100 over a single CPU and by a factor of 4 over 27 CPUs. Although higher performance is expected with other GPU programming standards, these results were accomplished with minimal change to the original program. Therefore, these results demonstrate the ability of OpenACC to improve performance while keeping the program maintainable and portable.

A two-dimensional (2-D), mathematical model is adopted to investigate the development of circulation patterns for compressible, laminar, and shear driven flow inside a rectangular cavity. The bottom of the cavity is free to move at a specified speed and the aspect ratio of the cavity is changed from 1.0 to 1.5. The vertical sides and the bottom of the cavity are assumed insulated. The cavity is filled with a compressible fluid with Prandtl number, Pr =1. The governing equations are solved numerically using the commercial Computational Fluid Dynamics (CFD) package ANSYS FLUENT 2015 and compared with the results for the primitive variables of the problem obtained using in house CFD code based on Coupled Modified Strongly Implicit Procedure (CMSIP). The simulations are carried out for the unsteady, lid driven cavity flow problem with moving boundary (bottom) for different Reynolds number, Mach numbers, bottom velocities and high initial pressure and temperature.

Os mecanismos de instabilidade hidrodinâmica têm um papel importante no processo da transição do escoamento de laminar para turbulento. A análise da instabilidade hidrodinâmica em uma cavidade com tampa deslizante foi realizada através da decomposição em modos globais (biglobal) para avaliar o efeito da compressibilidade neste fenômeno. O escoamento base foi obtido através de simulação numérica direta (DNS). Para tal, foi desenvolvido um código DNS compressível com discretização espacial por diferenças finitas compactas de alta resolução espectral e capacidade de processamento paralelo, com um método de decomposição de domínio que mantém a precisão das diferenças finitas compactas. O escoamento base é usado para montar o problema de autovalor oriundo das equações de Navier-Stokes linearizadas para a perturbação, discretizadas por diferenças finitas explícitas. O uso de diferenças finitas em conjunto com a implementação em matrizes esparsas reduz sensivelmente o uso de memória. Através do algoritmo de Arnoldi, a ordem do problema de autovalor é reduzida e os autovalores de interesse são recuperados. Os resultados indicam o efeito estabilizante da compressibilidade nos modos dominantes da cavidade e revelam modos inerentes ao escoamento compressível, para os quais a compressibilidade tem efeito desestabilizante. Dentre estes modos compressíveis, estão presentes modos de propagação sonora em dutos e modos relacionados à geração de som na cavidade. / Hydrodynamic instability mechanisms play an important role in laminar to turbulent transition. Hydrodynamic instability analysis of a lid-driven cavity flow was performed by global mode decomposition (biglobal) to evaluate compressibility effects on this phenomenon. The basic flow was calculated by direct numerical simulation (DNS). A compressible DNS code was developed with spectral-like compact finite difference spatial discretization. The code allows parallel processing with a domain decomposition method that preserves the compact finite difference accuracy. The basic flow is used to form the eigenvalue problem associated to the linear Navier- Stokes equations for the perturbation, which were discretized by an explicit finite difference scheme. The combination of sparse matrix techniques and finite difference discretization leads to a significant memory reduction. The order of the eigenvalue problem was reduced using the Arnoldi algorithm and the eigenvalues of interest were calculated. Results show the stabilizing effect of compressibility on the leading modes and reveal some modes intrinsic to compressible flow, for which compressibility has a destabilizing effect. Among these compressible modes, there are some related to sound propagation in ducts and to sound generation inside the cavity.

Os mecanismos de instabilidade hidrodinâmica têm um papel importante no processo da transição do escoamento de laminar para turbulento. A análise da instabilidade hidrodinâmica em uma cavidade com tampa deslizante foi realizada através da decomposição em modos globais (biglobal) para avaliar o efeito da compressibilidade neste fenômeno. O escoamento base foi obtido através de simulação numérica direta (DNS). Para tal, foi desenvolvido um código DNS compressível com discretização espacial por diferenças finitas compactas de alta resolução espectral e capacidade de processamento paralelo, com um método de decomposição de domínio que mantém a precisão das diferenças finitas compactas. O escoamento base é usado para montar o problema de autovalor oriundo das equações de Navier-Stokes linearizadas para a perturbação, discretizadas por diferenças finitas explícitas. O uso de diferenças finitas em conjunto com a implementação em matrizes esparsas reduz sensivelmente o uso de memória. Através do algoritmo de Arnoldi, a ordem do problema de autovalor é reduzida e os autovalores de interesse são recuperados. Os resultados indicam o efeito estabilizante da compressibilidade nos modos dominantes da cavidade e revelam modos inerentes ao escoamento compressível, para os quais a compressibilidade tem efeito desestabilizante. Dentre estes modos compressíveis, estão presentes modos de propagação sonora em dutos e modos relacionados à geração de som na cavidade. / Hydrodynamic instability mechanisms play an important role in laminar to turbulent transition. Hydrodynamic instability analysis of a lid-driven cavity flow was performed by global mode decomposition (biglobal) to evaluate compressibility effects on this phenomenon. The basic flow was calculated by direct numerical simulation (DNS). A compressible DNS code was developed with spectral-like compact finite difference spatial discretization. The code allows parallel processing with a domain decomposition method that preserves the compact finite difference accuracy. The basic flow is used to form the eigenvalue problem associated to the linear Navier- Stokes equations for the perturbation, which were discretized by an explicit finite difference scheme. The combination of sparse matrix techniques and finite difference discretization leads to a significant memory reduction. The order of the eigenvalue problem was reduced using the Arnoldi algorithm and the eigenvalues of interest were calculated. Results show the stabilizing effect of compressibility on the leading modes and reveal some modes intrinsic to compressible flow, for which compressibility has a destabilizing effect. Among these compressible modes, there are some related to sound propagation in ducts and to sound generation inside the cavity.