Desde o importante trabalho de Barth e Frederickson (1990), um certo número de pesquisadores têm estudado o método de Volumes Finitos de alta-ordem k-exato, por exemplo o grupo do Prof. Ollivier-Gooch: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Outras discretizações espaciais de alta-ordem bastante populares são o método Galerkin Descontínuo e o método de Diferença Espectral; processos iterativos que involucram estes esquemas tem sido acelerados, nos últimos anos, por métodos p-multigrid. Porém, esta aceleração não tem sido aplicada no contexto do método de Volumes Finitos de alta-ordem, pelo menos para conhecimento do autor desta tese. Por isso, o objetivo desta pesquisa é adaptar o p-multigrid desenvolvido por Liang et al. (2009b) no contexto da Diferença Espectral, para o ambiente dos Volumes Finitos estudado pelo Prof. Ollivier-Gooch. A pesquisa começa implementando o solver VF-RK, de Volumes Finitos com avanço Runge-Kutta, para resolver as equações de advecção-difusão e de Euler aplicados a problemas estacionários, por exemplo, o escoamento transônico ao redor do NACA 0012. Depois, estuda-se o método p-multigrid no contexto da Diferença Espectral; o p-multigrid acelera o processo iterativo comutando níveis polinomiais de alta e de baixa-ordem. Após esse estudo, a adaptação ao âmbito dos Volumes Finitos é realizada resultando num p-multigrid relativamente mais simples porque, em contraposição com o p-multigrid para Diferença Espectral, não precisa de operadores de restrição e prolongação para a comunicação entre diferentes níveis polinomiais. A pesquisa conclui com uma comparação com o método de Volumes Finitos de 4a ordem sem p-multigrid (solver VF-RK). Nesse sentido, implementa-se o solver pMG, baseado no p-multigrid proposto, para resolver os problemas estacionários considerados na primeira parte do trabalho; o smoother do p-multigrid é o esquema Runge-Kutta do código VF-RK, e cada problema estacionário é resolvido utilizando diferentes Vciclos procurando sempre soluções de 4a ordem. Os resultados indicam que o método p-multigrid proposto é mais eficiente que o método de Volumes Finitos de 4a ordem sem p-multigrid, isto é, os dois métodos oferecem a mesma precisão mas o primeiro pode levar menos de 50% do tempo de CPU do segundo. / Since Barth and Frederickson\'s important work (Barth e Frederickson, 1990), a number of researchers have studied high-order k-exact Finite Volume method, for example Prof. Ollivier-Gooch\'s group: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Other quite popular high-order spatial discretizations are the Discontinuous Galerkin methods and the Spectral Difference methods; the iterative processes involving these schemes have been accelerated in recent years by p-multigrid methods. However, this acceleration has not been applied in the context of the high-order Finite Volume method, at least for the knowledge of the author of this thesis. Therefore, the objective of this research is to adapt the p-multigrid developed by Liang et al. (2009b) in the context of Spectral Difference methods, to the environment of Finite Volume studied by Prof. Ollivier-Gooch. This research begins by implementing the solver VF-RK, Finite Volume solver with Runge-Kutta advance, to compute the advection-diffusion equation and Euler equations applied to steady state problems, for example, the transonic flow around NACA 0012. Then, it is studied the p-multigrid method in the context of Spectral Difference schemes; p-multigrid accelerates the iterative process by switching polynomial levels of high- and low-order. After this study, the adaptation to the context of the Finite Volume scheme is performed resulting in a relatively simple p-multigrid because, in contrast to the p-multigrid for Spectral Difference schemes, it doesn\'t need restriction and prolongation operators for communication between different polynomial levels. The research concludes with a comparison with 4th order Finite Volume method without p-multigrid (solver VF-RK). Accordingly, the solver pMG, based on the proposed p-multigrid, is implemented to resolve the steady state problems considered in the first part of the work; the p-multigrid smoother is the Runge-Kutta scheme from VF-RK code, and each steady state problem is solved using different Vcycles, looking for 4th order solutions ever. The results indicate that the proposed p-multigrid method is more efficient than the 4th order Finite Volume method without p-multigrid: the two methods give the same accuracy but the first one can take less than 50% of second one\'s CPU time.

Este trabalho apresenta um estudo sobre a solução numérica da equação geral de advecção-difusão usando uma metodologia numérica conservativa. Para a discretização espacial, é usado o Método de Volumes Finitos devido à natureza conservativa da equação em questão. O método é configurado de modo a ter suas variáveis centradas em centro de célula e, para as variáveis, como a velocidade, centradas nas faces um método de interpolação de segunda ordem é utilizado para um ajuste numérico ao centro. Embora a implementação computacional tenha sido feita de forma paramétrica de maneira a acomodar outros esquemas numéricos, a discretização temporal dá ênfase ao Método de Crank-Nicolson. Tal método numérico, sendo ele implícito, dá origem a um sistema linear de equações que, aqui, é resolvido empregando-se o Método Multigrid-Multinível. A corretude do código implementado é verificada a partir de testes por soluções manufaturadas, de modo a checar se a ordem de convergência prevista em teoria é alcançada pelos métodos numéricos. Um jato laminar é simulado, com o acoplamento entre a equação de Navier-Stokes e a equação geral de advecção-difusão, em um domínio computacional tridimensional. O jato é uma forma de vericar se o algoritmo de geração de malhas adaptativas funciona corretamente. O módulo produzido neste trabalho é baseado no código computacional AMR3D-P desenvolvido pelos grupos de pesquisa do IME-USP e o MFLab/FEMEC-UFU (Laboratório de Dinâmica de Fluidos da Universidade Federal de Uberlândia). A linguagem FORTRAN é utilizada para o desenvolvimento da metodologia numérica e as simulações foram executadas nos computadores do LabMAP(Laboratório da Matemática Aplicada do IME-USP) e do MFLab/FEMEC-UFU. / This work presents a study about the numerical solution of variable coecients advectiondi usion equation, or simply, general advection-diusion equation using a conservative numerical methodology. The Finite Volume Method is choosen as discretisation of the spatial domain because the conservative nature of the focused equation. This method is set up to have the scalar variable in a cell centered scheme and the vector quantities, such velocity, are face centered and they need a second order interpolation to get adjusted to the cell center. The computational code is parametric, in which, any implicit temporal discretisation can be choosen, but the emphasis relies on Crank-Nicolson method, a well-known second order method. The implicit nature of aforementioned method gives a linear system of equations which is solved here by the Multilevel-Multigrid method. The correctness of the computational code is checked by manufactured solution method used to inspect if the theoretical order of convergence is attained by the numerical methods. A laminar jet is simulated, coupling the Navier-Stokes equation and the general advection-diusion equation in a 3D computational domain. The jet is a good way to check the corectness of adaptative mesh renement algorithm. The module designed here is based in a previous implemented code AMR3D-P designed by IME-USP and MFLab/FEMEC-UFU (Fluid Dynamics Laboratory, Federal University of Uberlândia). The programming language used is FORTRAN and the simulations were run in LabMAP(Applied Mathematics Laboratoy at IME-USP) and MFLab/FEMEC-UFU computers.

This work deals with the design of a hybrid time integrator that couples spatially explicit and implicit time integrators. In order to cope with the industrial solver of Ariane Group called FLUSEPA, the explicit scheme of Heun and the implicit scheme of Crank-Nicolson are hybridized using the transition parameter : the whole technique is called AION time integration. The latter is studied into details with special focus on spectral behaviour and on its ability to keep the accuracy. It is shown that the hybrid technique has interesting dissipation and dispersion properties while maintaining precision and avoiding spurious waves. Moreover, this hybrid approach is validated on several academic test cases for both convective and diffusive fluxes. And as expected the method is more interesting in term of computational time than standard time integrators. For the extension of this hybrid approach to the temporal adaptive method implemented in FLUSEPA, it was necessary to improve some treatments in order to maintain conservation and acceptable spectral properties. Finally the hybrid time integration was also applied to a RANS/LES turbulent test case with interesting computational time while capturing the flow physics.

Dans cette thèse, on présente une étude numérique de la convection forcée turbulente dans un dissipateur thermique muni d'une chicane transversale dans le by-pass. Le premier modèle est composé d’ailettes planes et le second consiste à ajouter des ailettes broches entre les ailettes planes. Les équations gouvernantes basées sur le modèle de turbulences k- SSt sont discrétisées et résolues par la méthode des volumes finis et l'algorithme SIMPLE. Les résultats dynamiques sont présentés en en termes de champs de vitesse, des profils de vitesse axiales dans des sections choisies ainsi que la perte de charge. L'étude thermique est présentée en terme de champs de température et de distribution du nombre de Nusselt. Un rapport entre les performances thermique et dynamique est présenté pour évaluer les différents dissipateurs thermiques. / In this thesis, we present an numerical study of turbulent forced convection in a heat sink provided with a transverse baffle in the bypass. The first model is composed of plates fins and the second consists of adding pin fins between the plates fins. The governing equations, based on the k- SSt turbulence model, are disscredized and solved by the finite volume method and the SIMPLE algorithm. Dynamic results are presented in terms of velocity fields, profiles of the axial velocities in selected sections and pressure drop. The thermal study is presented in terms of temperature fields and the distribution of Nusselt number. A ratio between the thermal and dynamic performances is presented to evaluate the different heat sinks.

The main objective of this dissertation is to present an adaptation of some finite volume methods used in the resolution of problems arising in sedimentation processes of flocculated suspensions (or sedimentation with compression).<br />This adaptation is based on the utilization of multiresolution techniques, originally designed to reduce the computational cost incurred in solving using high resolution schemes in the numerical solution of hyperbolic systems of conservation laws.

<p>The MUSCL-Hancock finite volume method with different slope limiters has been analyzed in the context of a fast transient flow problem. A derivation and analysis of the axial forces inside a pipe system due to a flow transient is also performed. </p><p>The following slope limiters were implemented and compared: MC, van Leer, van Albada, Minmod and Superbee. The comparison was based on the method's ability to calculate the forces due to a flow transient inside a pipe system.</p><p>The tests and comparisons in this thesis show that the MC, van Leer, van Albada and Minmod limiters behave very much the same for the flow transient problem. If one would rank these four limiters with respect to the numerical error, the order would be the one presented above, the MC limiter being the most accurate. The error the four limiters produce is mainly of diffusive nature and it is just the magnitude of the diffusion that seems to differ between the methods. One should also note that the workload rank of the four limiters is the same as the order presented above. The MC limiter being the least efficient of the four and the Minmod limiter the most efficient.</p><p>In most of the tests performed the Superbee limiter display a rather negative unpredictable behavior. For some relatively simple cases this particular approach shows big difficulties maintaining the dynamical properties of the force. However, the upside of the Superbee limiter is its remarkable ability to maintain the maximum value of the forces present in the pipe system, preventing underestimation of the maximum magnitude of the force.</p>

In this thesis study, Detached Eddy Simulation turbulence model is studied in two dimension mainly for flow over single element airfoils in high Reynolds numbers to gain experience with model before applying it to a three dimensional simulations. For this aim, Spalart-Allmaras and standard DES ,DES97, turbulence models are implemented to parallel, viscous, hybrid grid flow solver. The flow solver ,Set2d, is written in FORTRAN language. The Navier-Stokes equations are discretized by first order accurately cell centered finite volume method and solved explicitly by using Runge-Kutta dual time integration technique. Inviscid fluxes are computed using Roe flux difference splitting method. The numerical simulations are performed in parallel environment using domain decomposition and PVM library routines for inter-process communications. To take into account the effect of unsteadyness after the convergence is ensured by local time stepping technique for four order magnitude drop in density residual, global time stepping is applied for 20000 iterations. The solution algorithm is validated aganist the numerical and experimental studies for single element airfoils in subsonic and transonic flows. It is seen that Spalart-Allmaras and DES97 turbulence models give the same results in the non-seperated flows. Grey area is investigated by changing $C_{DES}$ coefficient. Modeled Stress Depletion which cause reduction of eddy viscosity is observed.

Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system. The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state.  Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions. We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle  discontinuities in a robust way is used. Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.

Control of the transition of laminar flow to turbulence would result in lower drag and reduced energy consumption in many engineering applications. A spectral state-space model of linearised plane Poiseuille flow with wall transpiration ac¬tuation and wall shear measurements is developed from the Navier-Stokes and continuity equations, and optimal controllers are synthesized and assessed in sim¬ulations of the flow. The polynomial-form collocation model with control by rate of change of wall-normal velocity is shown to be consistent with previous interpo¬lating models with control by wall-normal velocity. Previous methods of applying the Dirichlet and Neumann boundary conditions to Chebyshev series are shown to be not strictly valid. A partly novel method provides the best numerical behaviour after preconditioning. Two test cases representing the earliest stages of the transition are consid¬ered, and linear quadratic regulators (LQR) and estimators (LQE) are synthesized. Finer discretisation is required for convergence of estimators. A novel estimator covariance weighting improves estimator transient convergence. Initial conditions which generate the highest subsequent transient energy are calculated. Non-linear open- and closed-loop simulations, using an independently derived finite-volume Navier-Stokes solver modified to work in terms of perturbations, agree with linear simulations for small perturbations. Although the transpiration considered is zero net mass flow, large amounts of fluid are required locally. At larger perturbations the flow saturates. State feedback controllers continue to stabilise the flow, but estimators may overshoot and occasionally output feedback destabilises the flow. Actuation by simultaneous wall-normal and tangential transpiration is derived. There are indications that control via tangential actuation produces lower highest transient energy, although requiring larger control effort. State feedback controllers are also synthesized which minimise upper bounds on the highest transient energy and control effort. The performance of these controllers is similar to that of the optimal controllers.

The MUSCL-Hancock finite volume method with different slope limiters has been analyzed in the context of a fast transient flow problem. A derivation and analysis of the axial forces inside a pipe system due to a flow transient is also performed.  The following slope limiters were implemented and compared: MC, van Leer, van Albada, Minmod and Superbee. The comparison was based on the method's ability to calculate the forces due to a flow transient inside a pipe system. The tests and comparisons in this thesis show that the MC, van Leer, van Albada and Minmod limiters behave very much the same for the flow transient problem. If one would rank these four limiters with respect to the numerical error, the order would be the one presented above, the MC limiter being the most accurate. The error the four limiters produce is mainly of diffusive nature and it is just the magnitude of the diffusion that seems to differ between the methods. One should also note that the workload rank of the four limiters is the same as the order presented above. The MC limiter being the least efficient of the four and the Minmod limiter the most efficient. In most of the tests performed the Superbee limiter display a rather negative unpredictable behavior. For some relatively simple cases this particular approach shows big difficulties maintaining the dynamical properties of the force. However, the upside of the Superbee limiter is its remarkable ability to maintain the maximum value of the forces present in the pipe system, preventing underestimation of the maximum magnitude of the force.