Meshless methods are a relatively new class of schemes for the numerical solution of partial differential equations. Their special characteristic is that they do not require a mesh but only need a distribution of points in the computational domain. The approximation at any point of spatial derivatives appearing in the partial differential equations is performed using a local cloud of points called the "connectivity" (or stencil). A point distribution can be more easily generated than a grid since we have less constraints to satisfy. The present work uses two meshless methods; an existing scheme called Least Squares Kinetic Upwind Method (LSKUM) and a new scheme called Kinetic Meshless Method (KMM). LSKUM is a "kinetic" scheme which uses a "least squares" approximation} for discretizing the derivatives occurring in the partial differential equations. The first part of the thesis is concerned with some theoretical properties and application of LSKUM to 3-D point distributions. Using previously established results we show that first order LSKUM in 1-D is positivity preserving under a CFL-like condition. The 3-D LSKUM is applied to point distributions obtained from FAME mesh. FAME, which stands for Feature Associated Mesh Embedding, is a composite overlapping grid system developed at QinetiQ (formerly DERA), UK, for store separation problems. The FAME mesh has a cell-based data structure and this is first converted to a node-based data structure which leads to a point distribution. For each point in this distribution we find a set of nearby nodes which forms the connectivity. The connectivity at each point (which is also the "full stencil" for that point) is split along each of the three coordinate directions so that we need six split (or half or one-sided) stencils at each point. The split stencils are used in LSKUM to calculate the split-flux derivatives arising in kinetic schemes which gives the upwind character to LSKUM. The "quality" of each of these stencils affects the accuracy and stability of the numerical scheme. In this work we focus on developing some numerical criteria to quantify the quality of a stencil for meshless methods like LSKUM. The first test is based on singular value decomposition of the over-determined problem and the singular values are used to measure the ill-conditioning (generally caused by a flat stencil). If any of the split stencils are found to be ill-conditioned then we use the full stencil for calculating the corresponding split flux derivative. A second test that is used is based on an accuracy measurement. The idea of this test is that a "good" stencil must give accurate estimates of derivatives and vice versa. If the error in the computed derivatives is above some specified tolerance the stencil is classified as unacceptable. In this case we either enhance the stencil (to remove disc-type degenerate structure) or switch to full stencil. It is found that the full stencil almost always behaves well in terms of both the tests. The use of these two tests and the associated modifications of defective stencils in an automatic manner allows the solver to converge without any blow up. The results obtained for a 3-D configuration compare favorably with wind tunnel measurements and the framework developed here provides a rational basis for approaching the connectivity selection problem. The second part of the thesis deals with a new scheme called Kinetic Meshless Method (KMM) which was developed as a consequence of the experience obtained with LSKUM and FAME mesh. As mentioned before the full stencil is generally better behaved than the split stencils. Hence the new scheme is constructed so that it does not require split stencils but operates on a full stencil (which is like a centered stencil). In order to obtain an upwind bias we introduce mid-point states (between a point and its neighbour) and the least squares fitting is performed using these mid-point states. The mid-point states are defined in an upwind-biased manner at the kinetic/Boltzmann level and moment-method strategy leads to an upwind scheme at the Euler level. On a standard 4-point Cartesian stencil this scheme reduces to finite volume method with KFVS fluxes. We can also show the rotational invariance of the scheme which is an important property of the governing equations themselves. The KMM is extended to higher order accuracy using a reconstruction procedure similar to finite volume schemes even though we do not have (or need) any cells in the present case. Numerical studies on a model 2-D problem show second order accuracy. Some theoretical and practical advantages of using a kinetic formulation for deriving the scheme are recognized. Several 2-D inviscid flows are solved which also demonstrate many important characteristics. The subsonic test cases show that the scheme produces less numerical entropy compared to LSKUM, and is also better in preserving the symmetry of the flow. The test cases involving discontinuous flows show that the new scheme is capable of resolving shocks very sharply especially with adaptation. The robustness of the scheme is also very good as shown in the supersonic test cases.

In this thesis, chemically reacting flows are studied mainly for detonation problems under 1D, cylindrical and spherical symmetry conditions. The mathematical formulation of chemically reacting, inviscid, unsteady flows with species conservation equations and finite-rate chemistry is described. The Euler equations with finite-rate chemistry are discretized by Finite-Volume method and solved implicitly by using a time-spliting method. Inviscid fluxes are computed using Roe Flux Difference Splitting Model. The numerical solution is implemented in parallel using domain decomposition and PVM library routines for inter-process communication. The solution algorithm is validated first against the numerical and experimental data for a shock tube problem with and without chemical reactions and for a cylindrical and spherical propagation of a shock wave. 1D, cylindrically and spherically symmetric detonations of H2:O2:Ar mixture are studied next.

A Cartesian method for solution of the steady two-dimensional Euler equations is produced. Dynamic data structures are used and both geometric and solution-based adaptations are applied. Solution adaptation is achieved through solution-based gradient information. The finite volume method is used with cell-centered approach. The solution is converged to a steady state by means of an approximate Riemann solver. Local time step is used for convergence acceleration. A multistage time stepping scheme is used to advance the solution in time. A number of internal and external flow problems are solved in order to demonstrate the efficiency and accuracy of the method.

Le présent travail a consisté à évaluer, améliorer et valider plus avant une méthode de couplage faible CFD/CAA, notamment relativement à son application à des problèmes réalistes de bruit avion. Entre autres choses, il a été ici montré dans quelle mesure une telle méthode hybride peut effectivement (i) s’accommoder des contraintes inhérentes aux applications réalistes, (ii) sans être menacée par certains de ses inévitables effets de bord (tels que la dégradation du signal auxquelles sont soumises les données CFD, lorsqu’elles sont traitées ou exploitées acoustiquement). / The present work consisted in improving, assessing and validating further the CFD/CAA surface weak coupling methodology, with respect to its application to realistic problems of aircraft noise. In particular, it was here shown how far such hybrid methodology could (i) cope with all stringent constraints that are dictated by real-life applications, (ii) without being jeopardized by some of the unavoidable side-effects (such as the signal degradation to which CFD data are subjected, when processed or being then acoustically exploited).

Orientadores: Helena Judith Nussenzveig Lopes, Milton da Costa Lopes Filho / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T19:55:10Z (GMT). No. of bitstreams: 1 Bronzi_AnneCaroline_D.pdf: 2733933 bytes, checksum: 87251756cbb4b7f97bdfe363274bea04 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho estudamos o conceito de solução fraca de equações que modelam fluidos ideais incompressíveis. Mais precisamente, estudamos exemplos que evidenciam deficiências na definição de solução fraca das equações de Euler. Um exemplo é o fluxo de Shnirelman, que é uma solução fraca das equações de Euler, no toro bidimensional, com suporte compacto no tempo. Isso implica que as soluções fracas das equações de Euler não são únicas. Nesse trabalho construímos uma aproximação numérica do fluxo de Shnirelman, com o objetivo de obter uma visualização da estrutura do fluxo. Em um trabalho conjunto com Shnirelman, modificamos a construção original a fim de obter um fluxo com uma estrutura física mais interessante e através da qual a visualização da cascata inversa de energia se torna mais clara. Recentemente, De Lellis e Székelyhidi também construíram soluções fracas das equações de Euler, no espaço todo, com suporte compacto no tempo e espaço. A técnica utilizada por eles é inovadora e se mostrou eficiente na construção de contra-exemplos variados. Utilizamos a técnica desenvolvida por De Lellis e Székelyhidi para construir soluções fracas das equações de Euler 2D com traçador passivo que tenham suporte compacto no tempo e espaço. Por fim, em nosso trabalho também estudamos as equações de Euler com simetria helicoidal, tendo demonstrado existência global, no tempo, de soluções fracas, na ausência de rodopio helicoidal, desde que a vorticidade inicial esteja em Lp, com p > 4=3, e seja de suporte compacto no plano, periódico na direção axial. Este resultado representa uma melhoria em relação ao estado da arte, devido a Ettinger e Titi, que é a boa-colocação no caso de domínio limitado e com vorticidade inicial limitada / Abstract: In this work we study the concept of weak solution of the incompressible ideal flow equations. More precisely, we study examples that highlight the shortcomings of the definition of weak solution for the Euler equations. An example is Shnirelman's flow, which is a weak solution of the Euler equations, on the bidimensional torus, compactly supported in time. This implies that weak solutions of the Euler equations are not unique. In this work we construct a numerical approximation of Shnirelman's flow, in order to visualize the structure of the flow. In joint work with Shnirelman, we modified the original construction in order to obtain a flow with more interesting physical structure whereby the visualization of the inverse energy cascade is clearer. Recently, De Lellis and Székelyhidi also constructed weak solutions of the Euler equations, in the whole space, with compact support in time and space. The technique used by them is innovative and has proved to be very effective in the construction of several counter-examples. We used the technique developed by De Lellis and Székelyhidi in order to construct weak solutions of the 2D Euler equations, coupled with a passive tracer, which are compactly supported in time and space. Finally, in our work we also studied the Euler equations with helical symmetry; we proved global existence, in time, of weak solutions, in the absence of helical swirl, provided that the initial vorticity lies in Lp, with p > 4=3, and has compact support in the plane, periodic in the axial direction. This result represents an improvement with respect to the state of art, due to Ettinger and Titi, who established the well-posedness, for bounded helical domains, assuming that the initial vorticity is bounded / Doutorado / Matematica / Doutor em Matemática

Orientadores: Helena Judith Nussenzveig Lopes, Milton da Costa Lopes Filho / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T23:48:14Z (GMT). No. of bitstreams: 1 Benvenutti_MaiconJose_D.pdf: 1286407 bytes, checksum: 1303364ce9343f87bf8bf6fddeb273be (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, abordamos questões de existência de soluções fracas e de estabilidade não-linear para as equações de Euler incompressíveis. Mais precisamente, analisamos dois tópicos distintos dentro destes assuntos. No primeiro, consideramos as equações de Euler sob a simetria helicoidal e com a restrição geométrica de ser livre de rodopio. Assim, utilizando as reduções provenientes da simetria, estendemos as técnicas de estabilidade desenvolvidas por Burton e por Wan e Pulvirenti para o caso helicoidal. Consequentemente, para um domínio helicoidal, simplesmente conexo, suave e limitado nos planos horizontais, demonstramos que o ponto de máximo estrito da energia cinética restrito à classe de rearranjos de uma função helicoidal qualquer é uma vorticidade helicoidal estacionária e estável num sentido não-linear. Além disto, em um domínio cilíndrico, mostramos também que há uma vorticidade helicoidal estacionária e estável que pode ser vista como uma extensão do vortex patch circular. No segundo tópico, consideramos as equações de Euler bidimensionais e com dados iniciais que não decaem no infinito. Demonstramos que os vortex patches iniciais abrangidos pelo Teorema de Existência de Soluções de Serfati (isto é, soluções com velocidades e vorticidades limitadas) não podem conter bolas arbitrariamente grandes. Além disto, construímos um contra exemplo de um vortex patch com velocidade associada limitada e tal que existe um subconjunto cujo vortex patch não possui uma velocidade associada limitada / Abstract: In this work, we approach issues regarding weak solutions existences and nonlinear stability for the incompressible Euler equations. More precisely, we analyze two distinct issues within these topics. At first, we consider the Euler equations with helical symmetry and with no swirl. Then, we use the reduction through symmetry to extend the stability techniques developed by Burton and by Wan and Pulvirenti to the helical case. Consequently, for a simply connected, bounded in horizontal planes and smooth helical domain, we prove that the strict maximiser of kinetic energy relative to all rearrangement of an arbitrary helical function is a steady and stable helical vorticity. Furthermore, in a cylindrical domain, we also prove that there exists a steady and stable helical vorticity which can be seen as an extension of the circular vortex patch. On the second issue, we consider the two-dimensional Euler equations and with initial data that do not decay at infinity. We show that initial vortex patches covered by Serfati Existence of Solutions Theorem (that is, solutions with bounded velocities and vorticities) cannot contain arbitrarily large balls. In addition, we construct a counterexample of a vortex patch for which there exists an associated bounded velocity and such that there exists a subset in which the vortex patch does not have any associated bounded velocity / Doutorado / Matematica / Doutor em Matemática

Orientadores: Helena Judith Nussenzveig Lopes, Milton da Costa Lopes Filho / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T02:48:33Z (GMT). No. of bitstreams: 1 Silva_LuizAlbertoViana_D.pdf: 1535000 bytes, checksum: dc971611cbda75b2a6dd72a0bd80d05f (MD5) Previous issue date: 2012 / Resumo: Um problema clássico em aberto é determinar se, em domínios com fronteira, soluções das equações de Navier-Stokes convergem, em um sentido apropriado, a uma solução das equações de Euler quando a viscosidade do fluido tende a zero. Baseados nesta importante questão, Kelliher, Lopes Filho e Nussenzveig Lopes examinaram, em [21], o comportamento de escoamentos com viscosidade pequena em domínios limitados com fronteira afastada, e descreveram condições precisas para que o escoamento limite fosse regido pelas equações de Euler no espaço todo. O presente trabalho é uma continuação natural do artigo mencionado, pois analisamos a dinâmica de escoamentos tridimensionais incompressíveis com viscosidade pequena em torno de obstáculos distantes. Mais precisamente, apresentamos uma estimativa fina que indica um comportamento assintótico para famílias de soluções das equações de Navier-Stokes em termos da viscosidade do escoamento e da localização do obstáculo, e contrastamos a referida estimativa com aquela demonstrada no contexto dos escoamentos em domínios limitados / Abstract: It is a classical open problem to determine if the vanishing viscosity limit can be established in the presence of boundaries. Based on this important issue, Kelliher, Lopes Filho and Nussenzveig Lopes studied in [21] the behavior of viscous incompresible flow in an expanding bounded domain when the viscosity is very small. To be more precise, these three authors described conditions under which the limiting flow satisfies the full space Euler equations. The present work is natural continuation of the aforementioned research since we consider 3D incompressible viscous flows around a distant obstacle along with the vanishing viscosity limit. Specificly, we obtain such a polynomial decay which shows an asymptotic behavior of families of 3D incompressible viscous flows, in the exterior of a single smooth obstacle, in terms of both the obstacle position and the small viscosity. Our approach allows us to compare our rate of convergence to that ones proved in [21] / Doutorado / Matematica / Doutor em Matemática

The objective of this dissertation is to develop robust and accurate numerical methods for solving the compressible, non-linear Euler equations of gas dynamics in one and two space dimensions. In theory, solutions of the Euler equations can display various characteristics including shock waves, rarefaction waves and contact discontinuities. To capture these features correctly, highly accurate numerical schemes are designed. In this thesis, two different projects have been studied to show the accuracy and utility of these numerical schemes. Firstly, the compressible, non-linear Euler equations of gas dynamics in one space dimension are considered. Since the non-linear partial differential equations (PDEs) can develop discontinuities (shock waves), the numerical code is designed to obtain stable numerical solutions of the Euler equations in the presence of shocks. Discontinuous solutions are defined in a weak sense, which means that there are many different solutions of the initial value problems of PDEs. To choose the physically relevant solution among the others, the entropy condition was applied to the problem. This condition is then used to derive a bound on the solution in order to satisfy L2-stability. Also, it provides information on how to add an adequate amount of diffusion to smooth the numerical shock waves. Furthermore, numerical solutions are obtained using far-field and no penetration (wall) boundary conditions. Grid interfaces were also included in these numerical computations. Secondly, the two dimensional compressible, non-linear Euler equations are considered. These equations are used to obtain numerical solutions for compressible ow in a shock tube with a 90° circular bend for two channels of different curvatures. The cell centered finite volume numerical scheme is employed to achieve these numerical solutions. The accuracy of this numerical scheme is tested using two different methods. In the first method, manufactured solutions are used to the test the convergence rate of the code. Then, Sod's shock tube test case is implemented into the numerical code to show the correctness of the code in both ow directions. The numerical method is then used to obtain numerical solutions which are compared with experimental data available in the literature. It is found that the numerical solutions are in a good agreement with these experimental results.

Cette thèse contribue à un ensemble de travaux consacrés à l’étude d’un schéma ENO centré-sommet (CENO) d’ordre élevé ainsi qu’à l’adaptation de maillage anisotrope pour des calculs de Mécaniques des Fluides précis à l’ordre 3. La première partie des travaux de cette thèse est consacré à une analyse approfondie de la précision du schéma CENO et à la création de termes correcteurs pour améliorer ses propriétés dispersives et dissipatives en une et deux dimensions. On propose un schéma CENO quadratique précis à l’ordre 3, puis cubique précis à l’ordre 4, pour les équations d’Euler des gaz compressibles, ainsi qu’ une première version du schéma avec capture de choc monotone. La deuxième partie des travaux est consacrée à la mise au point d’une plateforme numérique d’adaptation de maillage anisotrope multi-échelle et basée fonctionnelle intégrant le schéma CENO. Nous proposons un nouvel estimateur d’ordre 3 du schéma quadratique basé sur une reconstruction de hessien équivalent et son application à des simulations d’acoustiques instationnaire et de Scramjet stationnaire utilisant nos limiteurs. / This thesis presents to an assembly of work dedicated to the study of high order vertex-centred ENO scheme (CENO) and to anisotropic mesh adaptation for third-order accurate Fluid Mecanics problems. The thesis is structured in two parts. The first part is devoted to a thorough analysis of the CENO scheme accuracy and to the constuction of some corrector terms meant for improving the dissipative and dispersive properties for 1D and 2D numerical problems. We proposed a quadratique third-order accurate CENO scheme, then a cubic fourth-order accurate one, applied to Euler equations for compressible flows. A first monotone, shock capturing version of these scheme is also introduced in the first part. The second part of the thesis focuses on the implementation of a numerical platform for anisotropic multi-scale and goal-oriented mesh adaptivity involving the CENO scheme. A new third-order error estimator for the quadratic scheme is proposed, here based on a reconstuction of the Hessian. Numerical exemples for unsteady acoustic problems and a steady Scramjet problem computed with monotony preserving limiters are presented for validation of the theoretical results.

A posteriori error estimation is an inseparable component of any reliable numerical method for solving partial differential equations. The aim of the goal-oriented a posteriori error estimates is to control the computational error directly with respect to some quantity of interest, which makes the method very convenient for many engineering applications. The resulting error estimates may be employed for mesh adaptation which enables to find a numerical approximation of the quantity of interest under some given tolerance in a very efficient manner. In this thesis, the goal-oriented error estimates are derived for discontinuous Galerkin discretizations of the linear scalar model problems, as well as of the Euler equations describing inviscid compressible flows. It focuses on several aspects of the goal-oriented error estimation method, in particular, higher order reconstructions, adjoint consistency of the discretizations, control of the algebraic errors arising from iterative solutions of both algebraic systems, and linking the estimates with the hp-anisotropic mesh adaptation. The computational performance is demonstrated by numerical experiments.