A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, for the degree of Master of Science. School of Computer Science and Applied Mathematics. November 7, 2016. / The equations for the flow of a thin lm of molten glass on a bath of molten tin are extended to the case in which the viscosity of the molten glass depends on the temperature. The continuity equation for an incompressible fluid, the Navier-Stokes equation and the energy balance equation are written in the lubrication (thin fluid lm) approximation. The kinematic boundary condition and the boundary conditions for the normal and tangential stress and the normal heat flux are derived on the upper and lower surfaces of the glass ribbon. It is found for the lubrication approximation that only one equation is obtained for four unknowns which are the two horizontal velocity components, the absolute temperature difference and the thickness of the molten glass rib- bon. The remaining three equations are obtained by taking the calculation to the next order in the square of the ratio of the thickness to length of the glass ribbon. The kinematic edge condition and the edge conditions for the normal and tangential stress and the normal heat flux are derived. The four edge conditions and the boundary conditions at the inlet and outlet give the boundary conditions for the four partial differential equations. It is not the aim of the dissertation to solve the boundary value problem which has been derived, either numerically or analytically. / LG2017

The exact governing equations of fluid dynamics are too computationally expensive to solve on a computer for practical applications. Hence, it is currently not possible to analytically describe the behavior of a turbulent flow -in particular its internal structures-, making turbulence one of the major remaining unsolved problems in Classical Physics. One solution to computationally predict the performance of engineering applications involving fluids is the formulation of alternative and computationally tractable equations. This work demonstrates the feasibility of modeling turbulence as a collection of interacting particles with intrinsic orientation. It also discusses current efforts regarding its accuracy and computational overhead in numerous turbulent flows. The goal of this thesis is to focus on numerical implementation as well as model evaluation and validation. The Oriented-Eddy Collision Model is tested for basic flow cases and incorporated inhomogeneity. The project is successful in demonstrating that with appropriate extensions, the model can be applied to a very wide variety of turbulent flows with high predictive accuracy.

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One of the key open problems in the field of theoretical fluid mechanics concerns the possibility of the singularity formation in solutions of the 3D Navier-Stokes system in finite time. This phenomenon is associated with the behaviour of the enstrophy, which is an L2 norm of the vorticity and must become unbounded if such a singularity occurs. Although there is no blow-up in the 2D Navier-Stokes equation, we would like to investigate how much enstrophy can a planar incompressible flow in a bounded domain produce given certain initial enstrophy. We address this issue by formulating an optimization problem in which the time derivative of the enstrophy serves as the objective functional and solve it using tools of the optimization theory and calculus of variations. We propose an efficient computational approach which is based on the iterative steepest-ascent procedure. In addition, we introduce an easy-to-implement method of computing the gradient of the objective functional. Finally, we present computational results addressing the key question of this project and provide numerical evidence that the maximum enstrophy growth exhibits the scaling dE/dt ~ C*E*E for C>0 and very small E. All computations are performed using the Chebyshev spectral method. / Thesis / Master of Science (MSc) / For many decades, scientists have been investigating fundamental aspects of the Navier-Stokes equation, a central mathematical model arising in fluid mechanics. Although the equation is widely used by engineers to describe numerous flow phenomena, it is still an open question whether the Navier-Stokes system always admits physically meaningful solutions. To address this issue, we want to explore its mathematical aspects deeper by analyzing the behaviour of the enstrophy, which is a quantity associated with the vorticity of the flow and a convenient measure of the regularity of the solution. In this study, we consider a planar and incompressible flow bounded by solid walls. Using basic tools of mathematical analysis and optimization theory, we propose a computational method enabling us to find out how much enstrophy can such a flow produce instantaneously. We present numerical evidence that this instantaneous growth of enstrophy has a well-defined asymptotic behavior, which is consistent with physical assumptions.

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Solving the linear quadratic regulator (LQR) problem for partial differential equations (PDEs) leads to many computational challenges. The primary challenge comes from the fact that discretization methods for PDEs typically lead to very large systems of differential or differential algebraic equations. These systems are used to form algebraic Riccati equations involving high rank matrices. Although we restrict our attention to control problems with small numbers of control inputs, we allow for potentially high order control outputs. Problems with this structure appear in a number of practical applications yet no suitable algorithm exists. We propose and analyze solution strategies based on applying model order reduction methods to Chandrasekhar equations, Lyapunov/Sylvester equations, or combinations of these equations. Our numerical examples illustrate improvements in computational time up to several orders of magnitude over standard tools (when these tools can be used). We also present examples that cannot be solved using existing methods. These cases are motivated by flow control problems that are solved by computing feedback controllers for the linearized system. / Ph. D.

A generalized Conjugate Gradient like method is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. Preconditioning techniques are employed with the Conjugate Gradient like method to enhance the stability and convergence rate of the overall iterative method. The superiority of the new solver is established by comparisons with a conventional Line GaussSeidel Relaxation (LGSR) solver. Comparisons are based on 'number of iterations required to converge to a steady-state solution' and 'total CPU time required for convergence'. Three test cases representing widely varying flow physics are chosen to investigate the performance of the solvers. Computational test results for very low speed (incompressible flow over a backward facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade) and hypersonic flow (shockon- shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the 1vfach 0.1 case, speed-up factors of 30 (in terms of iterations) and 20 (in terms of CPU time) are found in favor of the new solver when compared with the LGSR solver. The corresponding speed-up factors for the transonic flow case are 20 and 18, respectively. The hypersonic case shows relatively lower speed-up factors of 5 and 4, respectively. This study reveals that preconditioning can greatly enhance the range of applicability and improve the performance of Conjugate Gradient like methods. / Ph. D.

An optimal shape control problem for the steady-state Navier-Stokes equations is considered from an analytical point of view. We examine a rather specific model problem dealing with 2-dimensional channel flow of incompressible viscous fluid: we wish to determine the shape of a bump on a part of the boundary in order to minimize the energy dissipation. To formulate the problem in a comprehensive manner, we study some properties of the Navier-Stokes equations. The penalty method is applied to relax the difficulty of dealing with incompressibility in conjunction with domain perturbations and regularity requirements for the solutions. The existence of optimal solutions for the penalized problem is presented. The computation of the shape gradient and its treatment plays central role in the shape sensitivity analysis. To describe the domain perturbation and to derive the shape gradient, we study the material derivative method and related shape calculus. The shape sensitivity analysis using the material derivative method and Lagrange multiplier technique is presented. The use of Lagrange multiplier techniques,from which an optimality system is derived, is justified by applying a method from functional analysis. Finite element discretizations for the domain and discretized description of the problem are given. We study finite element approximations for the weak penalized optimality system. To deal with inhomogeneous essential boundary condition, the framework of a Lagrange multiplier technique is applied. The split formulation decoupling the traction force from the velocity is proposed in conjunction with the penalized optimality system and optimal error estimates are derived. / Ph. D.

Accurate and efficient numerical algorithms for solving the three dimensional Navier Stokes equations with a generalized thermodynamic and chemistry model and a one equation turbulence model on structured and unstructured mesh topologies are presented. In the thermo-chemical modeling, particular attention is paid to the modeling of the chemical source terms, modeling of equilibrium thermodynamics, and the modeling of the non-equilibrium vibrational energy source terms. In this work, nonequilibrium thermo-chemical models are applied in the unstructured environment for the first time. A three-dimensional, second-order accurate k-exact reconstruction algorithm for the inviscid and viscous fluxes is presented. Several new methods for determining the stencil required for the inviscid and viscous k-exact reconstruction are discussed. A new simplified method for the computation of the viscous fluxes is also presented. Implementation of the one equation Spalart and Allmaras turbulence model is discussed. In particular, an new integral formulation is developed for this model which allows flux splitting to be applied to the resulting convective flux. Solutions for several test cases are presented to verify the solution algorithms discussed. For the thermo-chemical modeling, inviscid solutions to the three dimensional Aeroassist Flight Experiment vehicle and viscous solutions for the axi-symmetric Ram-II C are presented and compared to experimental data and/or published results. For the hypersonic AFE and Ram-II C solutions, focus is placed on the effects of the chemistry model in flows where ionization and dissociation are dominant characteristics of the flow field. Laminar and turbulent solutions over a flat plate are presented and compared to exact solutions and experimental data. Three dimensional higher order solutions using the k-exact reconstruction technique are presented for an analytic forebody. / Ph. D.