In this study the phenomenon of compressible flow and heat transfer in a porous media is modeled based on fundamental principles. The conservation equations for the two phases are transformed by the method of volume averaging which is an analytic method used to unite the microscale and macroscale effects characteristic to porous media flows. Unique to this analysis is that the model is valid for oscillatory, cryogenic flows such as that occurring in a regenerative cryogenic refrigerator such as a Pulse Tube Cryocooler (PTC.) In a PTC the forced flow drive oscillations in the regenerator create Reynolds numbers high enough such that microscale inertial effects dominate the momentum equation. This phenomenon, known as the Forchheimer Effect, can be predicted and modeled based solely on fundamental principles and the method of volume averaging. The coefficients that characterize the Forchheimer momentum equation are determined experimentally. In addition to pressure gradients, heat transfer within a porous media occurs due to temperature gradients. Conduction within the solid and fluid phases is made evident by volume averaging, but the determination of the conductivity coefficients requires numerical experiments and is unique to the geometry and conductivities of the two phases. Convection between the two phases is the dominant mode of heat transfer within the porous media. Determination of the convective heat transfer coefficient for a porous media requires physical experiments. Heat transfer and pressure gradients in the porous media are always competing effects leading to a model which requires coupling of the momentum and energy equations. These competing effects are united with the concept of entropy generation which relies on the second law of thermodynamics. All real processes generate entropy, and the most efficient processes which balance pressure gradients and heat transfer generate minimum entropy. This concept of minimum entropy generation is unique. As a result, minimum entropy generation should always be used as the criteria for thermodynamic optimization of thermohydraulic systems.

Orientador: Messias Meneguette Júnior / Banca: Vanessa Avansini Botta Pirani / Banca: Heloisa Helena Marino Silva / Resumo: O estudo de equações diferenciais parciais tem merecido muito destaque nos últimos anos. O fato é que se trata de uma área muito utilizada em vários ramos da Ciência como Matemática, Física e Engenharia. Além disso, permite a modelagem de muitos problemas encontrados em nosso cotidiano e na natureza em geral. Porém, a sua utilização se torna complicada uma vez que, tais equações nem sempre apresentam o que chamamos de solução analítica. Isto só acontece com uma "pequena" classe de equações (ver [19]). Fazse então necessário, buscar outras alternativas para a resolução de tais equações e daí os métodos numéricos de resolução desempenham um papel muito importante. O método das linhas, conhecido como um método de semi-discretização, representa uma alternativa para encontrar tais soluções e tem recebido atenção na atualidade. O presente trabalho, abrange, um estudo do método das linhas em sua forma original, bem como o estudo da estabilidade desse método utilizando a dinâmica de um cabo flexível. O método não foi satisfatório tanto para o cabo inextensível quanto para o cabo extensível, logo após poucos passos no tempo, a solução se deteriorou, representando, ao nosso ver, a instabilidade do método / Abstract: The study of partial differential equations has received much attention in the recent years. The fact is that this is an area widely used in various branches of science such as Mathematics, Physics and Engineering. Furthermore, it allows the modeling of many problems encountered in our activities and nature in general. However, their use becomes complicated since these equations do not always have what we call analytical solution. This only happens with a " small" class of equations (see [19]). So, it is necessary to seek other alternatives for solving these equations, hence the numerical resolution methods play an important role. The method of lines, known as a semi-discretization method, represents an alternative to find such solutions and has received attention in the literature. This work includes a study of the method of lines in its original form, as well as to study the stability of this method using the dynamics of a flexible cable. This method was not satisfactory for both the inextensible cable and to extensible cable, after a few steps in time, the solution has deteriorated, representing, in our view, the instability of the method / Mestre

Made available in DSpace on 2014-06-11T19:30:22Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-04-18Bitstream added on 2014-06-13T18:40:24Z : No. of bitstreams: 1 colnago_m_me_prud.pdf: 550851 bytes, checksum: 6c5305e58ea0da83bf0f1477b3ee0756 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Universidade Estadual Paulista (UNESP) / O estudo de equações diferenciais parciais tem merecido muito destaque nos últimos anos. O fato é que se trata de uma área muito utilizada em vários ramos da Ciência como Matemática, Física e Engenharia. Além disso, permite a modelagem de muitos problemas encontrados em nosso cotidiano e na natureza em geral. Porém, a sua utilização se torna complicada uma vez que, tais equações nem sempre apresentam o que chamamos de solução analítica. Isto só acontece com uma “pequena” classe de equações (ver [19]). Fazse então necessário, buscar outras alternativas para a resolução de tais equações e daí os métodos numéricos de resolução desempenham um papel muito importante. O método das linhas, conhecido como um método de semi-discretização, representa uma alternativa para encontrar tais soluções e tem recebido atenção na atualidade. O presente trabalho, abrange, um estudo do método das linhas em sua forma original, bem como o estudo da estabilidade desse método utilizando a dinâmica de um cabo flexível. O método não foi satisfatório tanto para o cabo inextensível quanto para o cabo extensível, logo após poucos passos no tempo, a solução se deteriorou, representando, ao nosso ver, a instabilidade do método / The study of partial differential equations has received much attention in the recent years. The fact is that this is an area widely used in various branches of science such as Mathematics, Physics and Engineering. Furthermore, it allows the modeling of many problems encountered in our activities and nature in general. However, their use becomes complicated since these equations do not always have what we call analytical solution. This only happens with a “ small” class of equations (see [19]). So, it is necessary to seek other alternatives for solving these equations, hence the numerical resolution methods play an important role. The method of lines, known as a semi-discretization method, represents an alternative to find such solutions and has received attention in the literature. This work includes a study of the method of lines in its original form, as well as to study the stability of this method using the dynamics of a flexible cable. This method was not satisfactory for both the inextensible cable and to extensible cable, after a few steps in time, the solution has deteriorated, representing, in our view, the instability of the method

The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.

In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the spatial variable over an N-point mesh, thereby converting each partial differential equation into N ordinary differential equations. These equations can then be solved using numerical routines defined for ordinary differential equations.

The research work presented in this thesis deals with modelling, design and characterisation of passive and active optical waveguide devices. The rst part of the thesis is related to algorithm development and numerical modelling of planar optical waveguides and gratings using the Method of Lines (MoL). The basic three-point central-di erence approximation of the δ2=δx2 operator used in the Helmholtz equation is extended to a new ve-point and seven-point approximation with appropriate interface conditions for the TE and TM elds. Di erent structures such as a high-contrast waveguide and a TM surface plasmon mode waveguide are simulated, and improved numerical accuracy for calculating the optical mode and propagation constant is demonstrated. A new fast and stable non-paraxial bi-directional beam propagation method, called Cascading and Doubling algorithm, is derived to model deep gratings with many periods. This algorithm is applied to model a quasi-guided multi-layer anti-resonant reecting optical waveguide (ARROW) grating polarizing structure. In the second part of the thesis, our focus is on active optical devices such as vertical-cavity and edge-emitting lasers. With a view to improve the bandwidth of directly modulated laser, an InGaAsP quantum well with InGaAlAs barrier is studied due to its favorable band o set for hole injection as well as for electron con nement. Quantum wells with di erent barrier bandgap are grown and direct carrier transport measurements are done using time and wavelength resolved photoluminescence upconversion. Semi-insulating regrown Fabry-Perot lasers are manufactured and experimentally evaluated for light-current, optical gain, chirp and small-signal performance. It is shown that the lasers having MQW with shallow bandgap InGaAlAs barrier have improved carrier transport properties, better T0, higher di erential gain and lower chirp. For lateral current injection laser scheme, it is shown that a narrow mesa is important for gain uniformity across the active region. High speed directly modulated DBR lasers are evaluated for analog performance and a record high spurious free dynamic range of 103 dB Hz2=3 for frequencies in the range of 1-19 GHz is demonstrated. Large signal transmission experiment is performed at 40 Gb/s and error free transmission for back-to-back and through 1 km standard single mode ber is achieved. / QC 20100827

The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.

Novel sequential and parallel computational fluid dynamic (CFD) codes based on method of lines (MOL) approach were developed for the numerical simulation of multi-component reacting flows using detailed transport and thermodynamic models. Both codes were applied to the prediction of a confined axisymmetric laminar co-flowing methane-air diffusion flame for which experimental data were available in the literature. Flame-sheet model for infinite-rate chemistry and one-, two-, and five- and ten-step reduced finite-rate reaction mechanisms were employed for methane-air combustion sub-model. A second-order high-resolution total variation diminishing (TVD) scheme based on Lagrange interpolation polynomial was proposed in order to alleviate spurious oscillations encountered in time evolution of flame propagation. Steady-state velocity, temperature and species profiles obtained by using infinite- and finite-rate chemistry models were validated against experimental data and other numerical solutions. They were found to be in reasonably good agreement with measurements and numerical results. The proposed difference scheme produced accurate results without spurious oscillations and numerical diffusion encountered in the classical schemes and hence was found to be a successful scheme applicable to strongly convective flow problems with non-uniform grid resolution. The code was also found to be an efficient tool for the prediction and understanding of transient combustion systems. This study constitutes the initial steps in the development of an efficient numerical scheme for direct numerical simulation (DNS) of unsteady, turbulent, multi-dimensional combustion with complex chemistry.

Predictive accuracy of the previously developed coupled code for the solution of the time-dependent Navier-Stokes equations in conjunction with the radiative transfer equation was first assessed by applying it to the prediction of thermally radiating, hydrodynamically developed laminar pipe flow for which the numerical solution had been reported in the literature. The effect of radiation on flow and temperature fields was demonstrated for different values of conduction to radiation ratio. It was found that the steady-state temperature predictions of the code agree well with the benchmark solution. In an attempt to test the predictive accuracy of the coupled code for turbulent radiating flows, it was applied to fully developed turbulent flow of a hot gas through a relatively cold pipe and the results were compared with the numerical solution available in the literature. The code was found to mimic the reported steady-state temperature profiles well. Having validated the predictive accuracy of the coupled code for steady, laminar/turbulent, radiating pipe flows, the performance of the code for transient radiating flows was tested by applying it to a test problem involving laminar/turbulent flow of carbon dioxide through a circular pipe for the simulation of simultaneous hydrodynamic and thermal development. The transient solutions for temperature, velocity and radiative energy source term fields were found to demonstrate the physically expected trends. In order to improve the performance of the code, a parallel algorithm of the code was developed and tested against sequential code for speed up and efficiency. It was found that the same results are obtained with a reasonably high speed-up and efficiency.

Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative method for the numerical solution of PDEs. Effective methods for the numerical inversion are based on the approximation of the Bromwich integral. In this thesis, a numerical study is undertaken to compare the efficiency of the Laplace inversion method with more conventional time integrator methods. Particularly, we consider the method-of-lines based on MATLAB’s ODE15s and the Crank-Nicolson method. Our studies include an introductory chapter on the Laplace inversion method. Then we proceed with spectral methods for the space discretization where we introduce the interpolation polynomial and the concept of a differentiation matrix to approximate derivatives of a function. Next, formulas of the numerical differentiation formulas (NDFs) implemented in ODE15s, as well as the well-known second order Crank-Nicolson method, are derived. In the Laplace method, to compute the Bromwich integral, we use the trapezoidal rule over a hyperbolic contour. Enhancement to the computational efficiency of these methods include the LU as well as the Hessenberg decompositions. In order to compare the three methods, we consider two criteria: The number of linear system solves per unit of accuracy and the CPU time per unit of accuracy. The numerical results demonstrate that the new method, i.e., the Laplace inversion method, is accurate to an exponential order of convergence compared to the linear convergence rate of the ODE15s and the Crank-Nicolson methods. This exponential convergence leads to high accuracy with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the Crank-Nicolson method as the results show. Finally, we apply with satisfactory results the inversion method to the axial dispersion model and the heat equation in two dimensions. / AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die benadering van die Bromwich integraal. In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer op MATLAB se ODE15s en die Crank-Nicolson metode. Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode. Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie, waar ons die interpolasie polinoom invoer sowel as die konsep van ’n differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word. Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste van al hierdie metodes word verbeter met die LU sowel as die Hessenberg ontbindings. Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode, akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson metode. Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.