Fluid motion has always been of great importance for humanity since much of our progress has been related to our understanding of fluid dynamics and to our control over the fluids surrounding us. In particular, the experimental techniques and the methods for numerical simulation developed during the last century allowed for great progresses both in creating new technologies and in improving old ones. Despite the great importance of experimental techniques, measuring all properties of a fluid throughout the whole domain, without intefering with the flow to be studied, is impossible. Also, building models even in scale is usually expansive. Both of these reasons have driven the development of numerical methods to the point they became an invaluable tool for fluid dynamic studies and the main tool for developing engineering solutions. If numerical methods are to be of any use, though, they have to correctly describe the problem geometry as well as capture the rich dynamics in a variety of flow situations, such as turbulence, boundary-layers and shock-waves. This thesis addresses two of these problems. In particular, I show modified versions of two immersed-interface methods to describe the geometry, simplifying their implementations with no impact to their applicability. I also introduce two methods for handling shock-waves: first aiming to minimize computational costs, then improving shock-wave resolution without increasing the number of grid points. / O movimento dos fluidos sempre foi de grande importância para a humanidade, dado que muito de nosso progresso esteve intimamente relacionado a um entendimento mais profundo de fluidodinâmica e de como controlar os flúidos ao nosso redor. Em particular, os métodos experimentais e de simulação computacional, desenvolvidos no último século, nos permitiram grandes avanços na criação de novas tecnologias e na otimização das já existentes. Apesar de sua grande importância, as dificuldades de se mensurar todas as propriedades de um flúido em todo o espaço, sem interferir com o comportamento do fluxo, além dos custos de se elaborar experimentos em tamanho real ou em escala, fez com que cada vez mais os métodos numéricos se tornassem uma importante ferramenta no estudo da fluido dinâmica e a principal ferramenta para o desenvolvimento de soluções de engenharia. Porém, para efetivamente substituir experimentos, os métodos numéricos tem que ser capazes de corretamente descrever a geometria do problema, além de capturarem todo tipo de comportamento apresentado pelos flúidos, como turbulência, camada limite e ondas de choque. Esta tese busca contribuir com dois destes desafios. Em particular, mostro versões modificadas de métodos de interface imersa para a descrição da geometria, simplificando as implementações originais sem prejudicar sua aplicabilidade. Também abordo métodos para tratar ondas de choque: primeiro buscando minimizar o esforço computacional e depois buscando aumentar a resolução do choque sem precisar refinar a malha computacional.

A weakly nonlinear stability analysis was conducted for the flow induced in an incompressible, Newtonian, viscous fluid lying between two infinite parallel plates which form a channel. The plates are oscillating synchronously in simple harmonic motion. The disturbed velocity of the flow was written in the form of a series in powers of a parameter which is a measure of the distance away from the linear theory neutral conditions. The individual terms of this series were decomposed using Floquet theory and Fourier series in time. The equations at second order and third order in were derived, and solutions for the Fourier coefficients were found using pseudospectral methods for the spatial variables. Various alternative methods of computation were applied to check the validity of the results obtained. The Landau equation for the amplitude of the disturbance was obtained, and the existence of equilibrium amplitude solutions inferred. The values of the coefficients in the Landau equation were calculated for the nondimensional channel half-widths h for the cases h = 5, 8, 10, 12, 14 and 16. It was found that equilibrium amplitude solutions exist for points in wavenumber Reynolds number space above the smooth portion of the previously determined linear stability neutral curve in all the cases examined. Similarly, Landau coefficients were calculated on a special feature of the neutral curve (called a ???finger???) for the case h = 12. Equilibrium amplitude solutions were found to exist at points inside the finger, and in a particular region outside near the top of the finger. Traces of the x-components of the disturbance velocities have been presented for a range of positions across the channel, together with the size of the equilibrium amplitude at these positions. As well, traces of the x-component of the velocity of the disturbed flow and traces of the velocity of the basic flow have been given for comparison at a particular position in the channel.

This thesis is on the existence and uniqueness of weak solutions to the Navier-Stokes equations in R3 which govern the velocity of incompressible fluid with viscosity ν. The solution is obtained in the space of tempered distributions on R3 given an initial condition and forcing data which are dominated by majorizing kernels. The solution takes the form of an expectation of functionals on a Markov process indexed by a binary branching tree. / Graduation date: 2012

We present an immersed interface method for the incompressible Navier Stokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the no-slip condition on the boundary is satisfied, singular forces are applied on the fluid at the immersed boundary. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. / Singapore-MIT Alliance (SMA)

In this thesis, the two-dimensional, unsteady, laminar and incompressible fluid flow problems governed by partial differential equations are solved by using dual reciprocity boundary element method (DRBEM). First, the governing equations are transformed to the inhomogeneous modified Helmholtz equations, and then the fundamental solution of modified Helmholtz equation is used for obtaining boundary element method (BEM) formulation. Thus, all the terms in the equation except the modified Helmholtz operator are considered as inhomogeneity. All the inhomogeneity terms are approximated by using suitable radial basis functions, and corresponding particular solutions are derived by using the annihilator method. Transforming time dependent partial differential equations to the form of inhomogeneous modified Helmholtz equations in DRBEM application enables us to use more information from the original governing equation. These are the main original parts of the thesis. In order to obtain modified Helmholtz equation for the time dependent partial differential equations, the time derivatives are approximated at two time levels by using forward finite difference method. This also eliminates the need of another time integration scheme, and diminishes stability problems. Stream function-vorticity formulations are adopted in physical fluid dynamics problems in DRBEM by using constant elements. First, the procedure is applied to the lid-driven cavity flow and results are obtained for Reynolds number values up to $2000.$ The natural convection flow is solved for Rayleigh numbers between $10^3$ to $10^6$ when the energy equation is added to the Navier-Stokes equations. Then, double diffusive mixed convection flow problem defined in three different physical domains is solved by using the same procedure. Results are obtained for various values of Richardson and Reynolds numbers, and buoyancy ratios. Behind these, DRBEM is used for the solution of natural convection flow under a magnetic field by using two different radial basis functions for both vorticity transport and energy equations. The same problem is also solved with differential quadrature method using the form of Poisson type stream function and modified Helmholtz type vorticity and energy equations. DRBEM and DQM results are obtained for the values of Rayleigh and Hartmann numbers up to $10^6$ and $300,$ respectively, and are compared in terms of accuracy and computational cost. Finally, DRBEM is used for the solution of inverse natural convection flow under a magnetic field using the results of direct problem for the missing boundary conditions.

In this thesis, the two-dimensional, unsteady, laminar and incompressible fluid flow problems governed by partial differential equations are solved by using dual reciprocity boundary element method (DRBEM). First, the governing equations are transformed to the inhomogeneous modified Helmholtz equations, and then the fundamental solution of modified Helmholtz equation is used for obtaining boundary element method (BEM) formulation. Thus, all the terms in the equation except the modified Helmholtz operator are considered as inhomogeneity. All the inhomogeneity terms are approximated by using suitable radial basis functions, and corresponding particular solutions are derived by using the annihilator method. Transforming time dependent partial differential equations to the form of inhomogeneous modified Helmholtz equations in DRBEM application enables us to use more information from the original governing equation. These are the main original parts of the thesis. In order to obtain modified Helmholtz equation for the time dependent partial differential equations, the time derivatives are approximated at two time levels by using forward finite difference method. This also eliminates the need of another time integration scheme, and diminishes stability problems. Stream function-vorticity formulations are adopted in physical fluid dynamics problems in DRBEM by using constant elements. First, the procedure is applied to the lid-driven cavity flow and results are obtained for Reynolds number values up to 2000. The natural convection flow is solved for Rayleigh numbers between 10^3 to 10^6 when the energy equation is added to the Navier-Stokes equations. Then, double diffusive mixed convection flow problem defined in three different physical domains is solved by using the same procedure. Results are obtained for various values of Richardson and Reynolds numbers, and buoyancy ratios. Behind these, DRBEM is used for the solution of natural convection flow under a magnetic field by using two different radial basis functions for both vorticity transport and energy equations. The same problem is also solved with differential quadrature method using the form of Poisson type stream function and modified Helmholtz type vorticity and energy equations. DRBEM and DQM results are obtained for the values of Rayleigh and Hartmann numbers up to 10^6 and 300, respectively, and are compared in terms of accuracy and computational cost. Finally, DRBEM is used for the solution of inverse natural convection flow under a magnetic field using the results of direct problem for the missing boundary conditions.

Plunger pumps are needed for heavy duty sludge pumping at wastewater treatment facilities. America's leading pump manufacturer Wastecorp Inc. brought their plunger pump problem to us in late 2009. It was found that when the ow rate reaches a critical value, the plunger pump starts to generate a clicking noise. A one-dimensional model was built for studying the ow of a typical plunger pump operation. The velocities and pressures are calculated at certain interesting locations. Pressure jumps have been found while opening or closing the valves. The valve motion is then modeled with considerations to its geometry. The results show that as the plunger speed reaches a critical value, the valve moves more rapidly and more likely to hit the wall and generates a noise. We also provide a methodology to study the ow across the valve in higher resolution. A nite-di erence approach to the Navier-Stokes equations are presented with the immersed boundary method. / UOIT

The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. In this work, a high-order Central Essentially Non-Oscillatory (CENO) finite-volume scheme is developed for the Euler and Navier-Stokes equations in three dimensions. The proposed CENO scheme is based on a hybrid solution reconstruction procedure using a fixed central stencil. A solution smoothness indicator facilitates the hybrid switching between a high-order k-exact reconstruction technique, and a monotonicity preserving limited piecewise linear reconstruction algorithm. The resulting scheme is applied to the compressible forms of the Euler and Navier-Stokes equations in three dimensions. The latter of which includes the application of this high-order work to the Large Eddy Simulation (LES) of turbulent non-reacting flows.

The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. In this work, a high-order Central Essentially Non-Oscillatory (CENO) finite-volume scheme is developed for the Euler and Navier-Stokes equations in three dimensions. The proposed CENO scheme is based on a hybrid solution reconstruction procedure using a fixed central stencil. A solution smoothness indicator facilitates the hybrid switching between a high-order k-exact reconstruction technique, and a monotonicity preserving limited piecewise linear reconstruction algorithm. The resulting scheme is applied to the compressible forms of the Euler and Navier-Stokes equations in three dimensions. The latter of which includes the application of this high-order work to the Large Eddy Simulation (LES) of turbulent non-reacting flows.

A 3d parallel CFD code is written to investigate the characteristics of and differences between Large Eddy Simulation (LES) models in the context of simulating a thermal buoyant plume. An efficient multigrid scheme is incorporated to solve the Poisson equation, resulting from the fractional step, projection method used to solve the Low Mach Number (LMN) Navier-Stokes equations. A wide range of LES models are implemented, including a variety of eddy models, structure models, mixed models and dynamic models, for both the momentum stresses and the temperature fluxes. Generalised gradient flux models are adapted from their RANS counterparts, and also tested. A number of characteristics are observed in the LES models relating to the thermal plume simulation in particular and turbulence in general. Effects on transition, dissipation, backscatter, equation balances, intermittency and energy spectra are all considered, as are the impact of the governing equations, the discretisation scheme, and the effect of grid coarsening. Also characteristics to particular models are considered, including the subgrid kinetic energy for the one-equation models, and constant histories for dynamic models. The argument that choice of LES model is unimportant is shown to be incorrect as a general statement, and a recommendation for when the models are best used is given.