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 the present study, a fifth-order weighted essentially non-oscillatory (WENO) scheme was built for solving the surface-capturing level-set equation. Combined with the level-set equation, the three-dimensional Reynolds averaged Navier-Stokes (RANS) equations were employed for the prediction of nonlinear wave-interaction and wave-breaking phenomena over sloping beaches. In the level-set finite-analytic Navier-Stokes (FANS) method, the free surface is represented by the zero level-set function, and the flows are modeled as immiscible air-water two phase flows. The Navier-Stokes equations for air-water two phase flows are formulated in a moving curvilinear coordinate system and discretized by a 12-point finite-analytical scheme using the finite-analytic method on a multi-block over-set grid system. The Pressure Implicit with Splitting of Operators / Semi-Implicit Method for Pressure-Linked Equation Revised (PISO/SIMPLER) algorithm was used to determine the coupled velocity and pressure fields. The evolution of the level-set method was solved using the third-order total variation diminishing (TVD) Runge-Kutta method and fifth-order WENO scheme. The accuracy was confirmed by solving the Zalesak's problem. Two major subjects are discussed in the present study. First, to identify the WENO scheme as a more accurate scheme than the essentially non-oscillatory scheme (ENO), the characteristics of a nonlinear monochromatic wave were studied systematically and comparisons of wave profiles using the two schemes were conducted. To eliminate other factors that might produce wave profile fluctuation, different damping functions and grid densities were studied. To damp the reflection waves efficiently, we compared five damping functions. The free-surface elevation data collected from gauges distributed evenly in a numerical wave tank are analyzed to demonstrate the damping effect of the beach. Second, as a surface-tracking numerical method built on curvilinear coordinates, the level-set RANS model was tested for nonlinear bichromatic wave trains and breaking waves on a sloping beach with a complex free surface. As the wave breaks, the velocity of the fluid flow surface became more complex. Numerical modeling was performed to simulate the two-phase flow velocity and its corresponding surface and evolution when the wave passed over different sloping beaches. The breaking wave test showed that it is an efficient technique for accurately capturing the breaking wave free surface. To predict the breaking points, different wave heights and beach slopes are simulated. The results show that the dependency of wave shape and breaking characteristics to wave height and beach slope match the results provided by experiments.

La thèse de Jean-François Simon est consacrée au développement d'un outil de simulation numérique de l'écoulement dans les turbomachines axiales. La démarche proposée a pour but de réduire l'empirisme présent dans les outils de calcul méridien en turbomachines. Cet objectif est atteint en deux étapes, l'une consistant à traiter de manière cohérente les effets dus à la viscosité du fluide dans le plan méridien, l'autre à injecter dans le calcul méridien les tensions déterministes et circonférentielles ainsi que les forces d'aubes. Les équations de Navier-Stokes sont moyennées azimutalement et sont résolues par une approche volumes-finis. La capture des effets dus à la viscosité du fluide le long des parois de carter et de moyeu permet d'éviter l'introduction d'un coefficient de blocage, ou le recours à un calcul couche-limite séparé. Jean-François Simon prolonge en outre l'approche d'Adamczyk par un opérateur de moyenne circonférentielle. Différents termes additionnels apparaissent alors dans les équations du modèle et traduisent entre autres linfluence des phénomènes non axisymétriques. L'importance relative de ces différentes contributions est analysée. Les méthodologies développées sont appliquées à plusieurs cas-tests (rotor simple, étages de compresseur ou de turbine, compresseur multi- étagé), qui permettent d'illustrer l'intérêt de la démarche proposée.

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.

A two-dimensional thin-layer Navier-Stokes cascade flow solver for turbomachinery is developed. A second-order finite-difference scheme and a second and fourth-difference dissipation scheme are used. Periodic and non-reflecting inlet and outlet boundary conditions are implemented into the approximate-factorization numerical method. Turbulence is modeled through the one-equation Spalart-Allmaras model. A two-dimensional turbomachinery cascade structured grid generator is developed to produce six-block H-type grids. The validity of this work is tested in various ways. A grid convergence study is performed showing the effect of grid density. The non-reflecting inlet and outlet boundary conditions are tested for boundary placement influence. Comparisons of the flow solver numerical results are performed against experimental results. A Mach number sweep and angle of attack sweep are performed on two similar transonic turbine cascades.

A two-dimensional thin-layer Navier-Stokes cascade flow solver for turbomachinery is developed. A second-order finite-difference scheme and a second and fourth-difference dissipation scheme are used. Periodic and non-reflecting inlet and outlet boundary conditions are implemented into the approximate-factorization numerical method. Turbulence is modeled through the one-equation Spalart-Allmaras model. A two-dimensional turbomachinery cascade structured grid generator is developed to produce six-block H-type grids. The validity of this work is tested in various ways. A grid convergence study is performed showing the effect of grid density. The non-reflecting inlet and outlet boundary conditions are tested for boundary placement influence. Comparisons of the flow solver numerical results are performed against experimental results. A Mach number sweep and angle of attack sweep are performed on two similar transonic turbine cascades.