Differential equations, in particular partial differential equations, are used to mathematically describe many physical phenomenon. The importance of being able to solve these types of equations can therefore not be overstated. This thesis is going to elucidate one method, the method of characteristics, which can in some cases be used to solve partial differential equations. To further the reader’s understanding on the method this paper will provide some important insights on differential equations as well as show examples on how the method of characteristics can be used to solve partial differential equations of various complexity. We will also in this paper present some important geometric complications for linear partial differential equations which one might have to take into consideration when using the method.

The method of characteristics (MOC) is a deterministic transport method that has been applied to large-scale problems including those in reactor physics and radiative transfer. Long characteristics, (LC) methods, have been used extensively to discretize and solve transport problems in the spatial domain. There is a need for an equally adequate time-dependent discretization for these transport problems. The new contributions from this research include the development of a space-time long characteristic (STLC) method with various source approximations including several that employ a piece-wise linear (PWL) approximation spatially. In the prism-PWL (PPWL) method the coefficient of each PWL spatial function is linear in time in each space-time cell. Along with STLC, a PWL-LC method is developed for steady-state problems in (x, y) and (x, y, z). The methods developed in this work use least-squares projections to determine the coefficients of their source approximations. This work presents a detailed asymptotic analysis of the PWL-LC and STLC methods in the thick diffusion limit, which is of special interest in radiative transfer problems. This is the first such analysis reported for LC methods and these new methods are the first that are expected to perform well in this limit. Results from test problems executed with a modified version of the Parallel Deterministic Transport code, PDT, show the PWL-LC and STLC methods are more accurate than current methods for streaming problems. From asymptotic analysis and test problems, it is found that the steady-state PWL-LC method is accurate in the thick diffusion limit with solutions similar to those of analogous discontinuous finite element method, DFEM, solutions. Similarly, the PPWL-STLC method is found to be accurate in time-dependent thick diffusive problems. STLC is also a promising method for massively parallel applications because it permits the use of track-based sweeping, which appears to have significant advantages over cell-based sweeping. This is a key topic recommended for further research.

Discretization and solving of the transport equation has been an area of great research where many methods have been developed. Under the deterministic transport methods, the method of characteristics, MOC, is one such discretization and solution method that has been applied to large-scale problems. Although these MOC, specifically long characteristics, LC, have been thoroughly applied to discretize and solve transport problems in the spatial domain, there is a need for an equally adequate time-dependent discretization. A method has been developed that uses LC discretization of the time and space variables in solving the transport equation. This space-time long characteristic, STLC, method is a discrete ordinates method that applies LC discretization in space and time and employs a least-squares approximation of sources such as the scattering source in each cell. This method encounters the same problems that previous spatial LC methods have dealt with concerning achieving all of the following: particle conservation, exact solution along a ray, and smooth variation in reaction rate for specific problems. However, quantities that preserve conservation in each cell can also be produced with this method and compared to the non-conservative results from this method to determine the extent to which this STLC method addresses the previous problems. Results from several test problems show that this STLC method produces conservative and non-conservative solutions that are very similar for most cases and the difference between them vanishes as track spacing is refined. These quantities are also compared to the results produced from a traditional linear discontinuous spatial discretization with finite difference time discretization. It is found that this STLC method is more accurate for streaming-dominate and scattering-dominate test problems. Also, the solution from this STLC method approaches the steady-state diffusion limit solution from a traditional LD method. Through asymptotic analysis and test problems, this STLC method produces a time-dependent diffusion solution in the thick diffusive limit that is accurate to O(E) and is similar to a continuous linear FEM discretization method in space with time differencing. Application of this method in parallel looks promising, mostly due to the ray independence along which the solution is computed in this method.

O presente trabalho traz uma comparaÃÃo entre as aproximaÃÃes de primeira e segunda ordem para as equaÃÃes caracterÃsticas como soluÃÃo para o fenÃmeno transiente em uma tubulaÃÃo. Foram testadas oito situaÃÃes que resultaram em dezesseis cenÃrios de cÃlculo. Procurou-se demonstrar a eficiÃncia do mÃtodo das caracterÃsticas (MOC) com a aproximaÃÃo de 1 ordem para pequenas variaÃÃes do fator de atrito em relaÃÃo à aproximaÃÃo de 2 ordem comprovando ser suficiente a primeira, para os exemplos estudados levando em consideraÃÃo o termo de atrito como referÃncia de estabilidade. Demonstrou-se tambÃm um estudo hipotÃtico simulando a instabilidade para altos termos de atrito. / This paper provides a comparison between first and second order approaches equations for the characteristics method as a solution to the transient phenomenon in a pipeline. Eight cases were tested, which resulted in sixteen simulations calculations. Sought to demonstrate the efficiency method of characteristics (MOC) to the nearest 1st. order for small variations of the friction factor in relation to the nearest 2nd order confirming the first to be enough, for the examples studied taking into account the friction term stability as a reference. It was also demonstrated a study simulation the hypothetical instability in terms of high friction.

An Axisymmetric Rocket Ejector Simulation (ARES) was developed to numerically analyze various configurations of an air augmented rocket. Primary and secondary flow field visualizations are presented and performance predictions are tabulated. A parametric study on ejector geometry is obtained following a validation of the flow fields and performance values. The primary flow is calculated using a quasi-2D, irrotational Method of Characteristics and the secondary flow is found using isentropic relations. Primary calculations begin at the throat and extend through the nozzle to the location of the first Mach Disk. Combustion properties are tabulated before analysis to allow for propellant property selection. Secondary flow calculations employ the previously calculated plume boundary and ejector geometry to form an isentropic solution. Primary and secondary flow computations are iterated along the new pressure distributions established by the 1D analysis until a convergence tolerance is met. Thrust augmentation and Specific Impulse values are predicted using a control volume approach. For the validation test cases, the nozzle characteristic net is very similar to that of previous research. Plume characteristics are in good agreement but fluctuate in accuracy due to flow structure formulation. The individual unit processes utilized by the Method of Characteristics are found to vary their outputs by up to 0.025% when compared to existing sources. Rocket thrust and specific impulse are increased by up to 22% for a static system and 15% for an ejector flow at Mach 0.5. Evidence of Fabri conditions were observed in the flow visualization and graphically through the performance predictions. It was determined that the optimum ejector divergence angle for an air augmented rocket greatly depends on the stagnation pressure ratio between the primary and secondary flows.

Two-dimensional supersonic flows out of rocket nozzles are one of three flow types: over-, perfectly-, or under-expanded. In under-expanded flows, an expansion fan is centered at the top and bottom tip of the rocket nozzle. When the waves from the expansion wave cross through the centerline and intersect the free boundary, the waves are reflected as compression waves. For higher exit-to-ambient pressure ratios, the compression waves coalesce and eventually form a barrel shock. The purpose of this study was to use the Method of Characteristics (MOC), a mathematical procedure for solving hyperbolic partial differential equations, to simulate the formation of the barrel shock. A MOC code was developed in the Python programming language to accomplish task. Results of the MOC code compared favorably with CFD results.

The Air Inlet Method of Characteristics Analysis Tool (AIMCAT), a tool based in Python 3, is developed to model supersonic air inlet geometries during the early phases of design. The method of characteristics (MOC) is used to solve the governing equations for an inviscid, irrotational, isentropic, steady, supersonic flowfield. A comparison is made between modeling shock waves implicitly using Mach wave coalescence and modeling them explicitly using oblique shock relations. Multiple test cases are used to assess the accuracy of the tool by comparing to experimental wind tunnel data. Good general agreement was achieved over the majority of the supersonic portion of the flowfield for all test cases. The implicit shock mesh achieved better accuracy for shock wave positions compared to the explicit shock mesh. However, the explicit shock mesh captured total pressure losses across the shocks which is of value when assessing the efficiency of the inlet. Both approaches show their respective values and their suitability depends on the conditions being studied. AIMCAT has shown initial promise, however further development is need to improve its utility and robustness.

<p>In perifusion cell cultures, the culture medium flows continuously through a chamber containing immobilized cells and the effluent is collected at the end. In our main applications, gonadotropin releasing hormone (GnRH) or oxytocin is introduced into the chamber as the input. They stimulate the cells to secrete luteinizing hormone (LH), which is collected in the effluent. To relate the effluent LH concentration to the cellular processes producing it, we develop and analyze a mathematical model consisting of coupled partial differential equations describing the intracellular signaling and the movement of substances in the cell chamber. We analyze three different data sets and give cellular mechanisms that explain the data. Our model indicates that two negative feedback loops, one fast and one slow, are needed to explain the data and we give their biological bases. We demonstrate that different LH outcomes in oxytocin and GnRH stimulations might originate from different receptor dynamics. We analyze the model to understand the influence of parameters, like the rate of the medium flow or the fraction collection time, on the experimental outcomes. We investigate how the rate of binding and dissociation of the input hormone to and from its receptor influence its movement down the chamber. Finally, we formulate and analyze simpler models that allow us to predict the distortion of a square pulse due to hormone-receptor interactions and to estimate parameters using perifusion data. We show that in the limit of high binding and dissociation the square pulse moves as a diffusing Gaussian and in this limit the biological parameters can be estimated.</p> / Dissertation

ABSTRACT SIMULATION OF FLOW TRANSIENTS IN LIQUID PIPELINE SYSTEMS Ko&ccedil / , Gen&ccedil / er M.S., Department of Mechanical Engineering Supervisor: Prof. Dr. O. Cahit Eralp November 2007, 142 pages In liquid pipeline systems, transient flow is the major cause of pipeline damages. Transient flow is a situation where the pressure and flow rate in the pipeline rapidly changes with time. Flow transients are also known as surge and Waterhammer which originates from the hammering sound of the water in the taps or valves. In liquid pipelines, preliminary design parameters are chosen for steady state operations, but a transient check is always necessary. There are various types of transient flow situations such as valve closures, pump trips and flow oscillations. During a transient flow, pressure inside the pipe may increase or decrease in an unexpected way that cannot be foreseen by a steady state analysis. Flow transients should be considered by a complete procedure that simulates possible transient flow scenarios and by the obtained results, precautions should be taken. There are different computational methods that can be used to solve and simulate flow transients in computer environment. All computational methods utilize basic v flow equations which are continuity and momentum equations. These equations are nonlinear differential equations and some mathematical tools are necessary to make these equations linear. In this thesis a computer program is coded that utilizes &ldquo / Method of Characteristics&rdquo / which is a numerical method in solving partial differential equations. In pipeline hydraulics, two partial differential equations, continuity and momentum equations are solved together, in order to obtain the pressure and flow rate values in the pipeline, during transient flow. In this thesis, MATLAB 7.1 is used as the programming language and obtained code is converted to a C# language to be able to integrate the core of the program with a user friendly Graphical User Interface (GUI). The Computer program is verified for different scenarios with the available real pipeline data and results of various reputable agencies. The output of the computer program is the tabulated pressure and flow rate values according to time indexes and graphical representations of these values. There are also prompts for users warning about possible dangerous operation modes of the pipeline components.

A pressure transient is generated when a sudden change in injection rate occurs due to a valve closure or injector shutdown. This pressure transient, referred to as a water hammer, travels down the wellbore, is reflected back and induces a series of pressure pulses on the sand face. This study presents a semi-analytical model to simulate the magnitude, frequency and duration of water hammer in wellbores. An impedance model has been suggested that can describe the interface, between the wellbore and the formation. Pressure transients measured in five wells in an offshore field are history matched to validate the model. It is shown that the amplitude of the pressure waves may be up to an order of magnitude smaller at the sand face when compared with surface measurements. Finally, a model has been proposed to estimate fracture dimensions from water hammer data. / text