published_or_final_version / Mathematics / Master / Master of Philosophy

Explosions are a common phenomena in the Universe. Beginning with the Big Bang, one could say the history of the Universe is narrated by a series of explosions. Yet no matter how large, small, or complex, all explosions occur through a series of similar physical processes beginning with their initiation to their dynamical interaction with the environment. Of particular interest to this study is how these processes are modified in a magnetized medium. The role of the magnetic field is investigated in two scenarios. The first scenario addresses how a magnetic field alters the propagation of a gaseous detonation where the application of interest is the modification of a condensed-phase explosion. The second scenario is focused on the aftermath of the explosion event and addresses how fluid mixing changes in a magnetized medium. A primary focus of this thesis is the development of a numerical tool capable of simulating explosive phenomenon in a magnetized medium. While the magnetohydrodynamic (MHD) equations share many of the mathematical characteristics of the hydrodynamic equations, numerical methods developed for the conservation equations of a magnetized plasma are complicated by the requirement that the magnetic field must be divergent free. The advantages and disadvantages of the proposed method are discussed in relation to explosion applications.

Numerical simulations were used to investigate natural convection and radiation interactions in small enclosures of both two and three-dimensional geometries. The objectives of the research were to (1) determine the relative importance of natural convection and radiation, and to (2) estimate the natural convection heat transfer coefficients. Models are generated using Gambit, while numerical computations were conducted using the CFD code FLUENT. Dimensions for the two-dimensional enclosure were a height of 2.54 cm (1 inch), and a width that varied between 5.08 cm and 10.16 cm (2 inches and 4 inches). The three-dimensional model had a depth of 5.08 cm (2 inches) with the same height and widths as the two-dimensional model. The obstruction is located at the centroid of the enclosure and is represented as a circle in the two-dimensional geometry and a cylinder in the three-dimensional geometry. Obstruction diameters varied between .51 cm and 1.52 cm (0.2 inches and 0.6 inches). Model parameters used in the investigation were average surface temperatures, net total heat flux, and net radiation heat flux. These parameters were used to define percent temperature differences, percent heat flux contributions, convective heat transfer coefficients, Nusselt numbers, and Rayleigh numbers. The Rayleigh numbers varied between 0.005 and 300, and the convective heat transfer coefficients ranged between 2 and 25 W/m2K depending on the point in the simulation. The simulations were conducted with temperatures ranging between 310 K and 1275 K on the right boundary. For right boundary temperatures above 800 K, the estimated error on the obstruction temperature is less than 6.1% for neglecting natural convection and conduction from the heat transfer analysis. Lower right boundary temperatures such as 310 K had significant contributions, over 50%, from heat transfer modes other than radiation. For lower right boundary temperatures, a means of including natural convection should be included. When a bulk fluid temperature and average surface temperature values are available, a time average heat transfer coefficient of 6.73 W/m2K is proposed for simplifying the numerical calculations. In the transient right boundary temperature analysis, all modes of heat transfer other than radiation can be neglected to have an error below 8.1%.

Certain numerical methods have been well developed for solving one-dimensional two-phase flow (e.g. gas and liquid) problems in the literatures during the last two decades. Based on the existing methods, the present work compares the computational efficiency, accuracy, and robustness of various numerical schemes by predicting the numerical solutions of fluid properties for a specific case to find the proper numerical method. One of the numerical schemes introduced in this work is a practical, semi-implicit upwind method used for fluid flow simulations in different flow patterns,stratified flow and slug flow. This method implements the iterative and non-iterative schemes using a two-fluid model that consists of sets of non-hyperbolic equations. A numerical error term is applied in the pressure equation to maintain the volume balance of the two-phase flow model. If the temperature varies, the discretised energy equations use similar error terms as in the pressure equation. In some cases, the small values of the numerical errors are negligible and do not influence the numerical results. These errors are, however, important factors to consider when maintaining the stability and robustness of the above numerical schemes for strong non-linear cases. The computational efficiency ofthe non-iterative scheme, where the inner iterations are deactivated, is better than the iterative scheme. Different grid arrangements are compared with respect to computational accuracy and efficiency. A staggered structured grid implements the same semi-implicit upwind method as in the non-iterative scheme; the non-staggered grid arrangement uses an existing flux-splitting scheme (Evje and Flåtten, 2003) as a reference. All the above schemes produce numerical solutions with a single precision that normally satisfy the requirements of computational accuracy of industrial two-phase pipe flows. However, if one pursues a higher-order accuracy scheme, e.g. a Roe-averaged algorithm, the governing equations should be strictly a hyperbolic system of partial differential equations, which is achieved by introducing the nonviscous force terms in the two-fluid model (LeVeque, 2002).By properly incorporating the non-conservative terms in the formulation of the numerical fluxes, the capability of the Roe-averaged algorithm is demonstrated by capturing shock waves. Results from the present research include the following. A one-dimensional scheme that solves a system of discretised equations with the staggered semi-implicit upwind method is presented and validated for its computational efficiencyand robustness. This scheme can be widely used in the industry with sufficient accuracy. The other first-order semi-implicit numerical schemes producestable numerical results, especially in the dynamic cases of two-phase flow, except when the gas phase nearly disappears or appears in pipes. The Roe-averaged algorithm is recommended due to the high-resolution numerical results obtained, but at the costs of computational time and effort.

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Controlled cooling is an important part of steel production industry that affects the properties of the outcome steel. Many of the researches done in controlled cooling are experimental. Due to progress in the numerical techniques and high cost of experimental works in this field the numerical work seems more feasible. Heat transfer analysis is the necessary element of successful controlled cooling and ultimately achievement of novel properties in steel. Heat transfer on the surface of the plate normally contains different regimes such as film boiling, nucleate boiling, transition boiling and radiation heat transfer. This makes the analysis more complicated. In order to perform the heat transfer analysis often empirical correlations are being used. In these correlations the velocity and pressure within the fluid domain is involved. Therefore in order to obtain a better understanding of heat transfer process, study of hydrodynamics of the fluid becomes necessary. Circular jet due to its high efficiency has been used vastly in the industry. Although some experimental studies of round jet arrays have been done, yet the characteristics of a single jet with industrial geometric and flow parameters on the surface of a flat plate is not fully understood. Study of hydrodynamics of the jet impingement is the first step to achieve better understanding of heat transfer process. Finite element method as a popular numerical method has been used vastly to simulate different domains. Traditional approaches of finite element method, Lagrangian and Eulerian, each has its own benefits and drawbacks. Lagrangian approach has been used widely in solid domains and Eulerian approach has been widely used in fluid fields. Jet impingement problem, due to its unknown free surface and the change in the boundary, falls in the category of special problems and none of the traditional approaches is suitable for this application. The Arbitrary Lagrangian Eulerian (ALE) formulation has emerged as a technique that can alleviate many of the shortcomings of the traditional Lagrangian and Eulerian formulations in handling these types of problems. Using the ALE formulation the computational grid need not adhere to the material (Lagrangian) nor be fixed in space (Eulerian) but can be moved arbitrarily. Two distinct techniques are being used to implement the ALE formulation, namely the operator split approach and the fully coupled approach. This thesis presents a fully coupled ALE formulation for the simulation of flow field. ALE form of Navier-Stokes equations are derived from the basic principles of continuum mechanics and conservation laws in the fluid. These formulations are then converted in to ALE finite element equations for the fluid flow. The axi-symmetric form of these equations are then derived in order to be used for jet impingement application. In the ALE Formulation as the mesh or the computational grid can move independent of the material and space, an additional set of unknowns representing mesh movement appears in the equations. Prescribing a mesh motion scheme in order to define these unknowns is problem-dependent and has not been yet generalized for all applications. After investigating different methods, the Winslow method is chosen for jet impingement application. This method is based on adding a specific set of partial differential Equations(Laplace equations) to the existing equations in order to obtain enough equations for the unknowns. Then these set of PDEs are converted to finite element equations and derived in axi-symmetric form to be used in jet impingement application. These equations together with the field equations are then applied to jet impingement problem. Due to the number of equations and nonlinearity of the field equations the solution of the problem faces some challenges in terms of convergence characteristics and modeling strategies. Some suggestions are made to deal with these challenges and convergence problems. Finally the numerical treatment and results of analyzing hydrodynamics of the Jet Impingement is presented. The work in this thesis is confined to the numerical simulation of the jet impingement and the specifications of an industrial test setup only have been used in order to obtain the parameters of the numerical model.

The analysis of reinforced concrete structures under blast and impact loads is an area of research that has become increasingly relevant in recent years. Complex hydrocodes are typically used for impact analyses, although single-degree-of-freedom methods have also been developed. There are a number of disadvantages associated with both methods, and the Canadian Nuclear Safety Commission (CNSC) is looking for a tool that can be used in conjunction with hydrocodes to analyze hard and soft missile impacts, with target damage ranging from flexural cracking to perforation. The VecTor programs, a suite of nonlinear finite element programs developed at the University of Toronto for the analysis of reinforced concrete structures, can potentially be developed into such tools. The analytical work done in this study serves to investigate the current impact and impulse loading analysis capabilities in VecTor2 and VecTor3, and to identify areas where work should be focused in the future.