For turbulent flow calculations, some of the well-known turbulence models in the literature are applied on a previously developed Navier-Stokes solver designed to handle laminar flows. A finite volume formulation, which is cell-based for inviscid terms and cell-vertex for viscous terms, is used for numerical discretization of the Navier-Stokes equations in conservative form. This formulation is combined with one-step, explicit time marching Lax-Wendroff numerical scheme that is second order accurate in space. To minimize non-physical oscillations resulting from the numerical scheme, second and fourth order artificial smoothing terms are added. To increase the convergence rate of the solver, local time stepping technique is applied. Before applying turbulence models, Navier-Stokes solver is tested for a case of subsonic, laminar flow over a flat plate. The results are in close agreement with Blasius similarity solutions. To calculate turbulent flows, Boussinesq eddy-viscosity approach is utilized. The eddy viscosity (also called turbulent viscosity), which arises as a consequence of this approach, is calculated using Cebeci-Smith, Michel et. al., Baldwin-Lomax, Chien&rsquo / s k-epsilon and Wilcox&rsquo / s k-omega turbulence models. To evaluate the performances of these turbulence models and to compare them with each other, the solver has been tested for a case of subsonic, laminar - transition fixed - turbulent flow over a flat plate. The results are verified by analytical solutions and empirical correlations.

The aim of this study is to extend a parallel Fortran90 code to compute three&ndash / dimensional laminar and turbulent flowfields over rotary wing configurations. The code employs finite volume discretization and the compact, four step Runge-Kutta type time integration technique to solve unsteady, thin&ndash / layer Navier&ndash / Stokes equations. Zero&ndash / order Baldwin&ndash / Lomax turbulence model is utilized to model the turbulence for the computation of turbulent flowfields. A fine, viscous, H type structured grid is employed in the computations. To reduce the computational time and memory requirements parallel processing with distributed memory is used. The data communication among the processors is executed by using the MPI ( Message Passing Interface ) communication libraries. Laminar and turbulent solutions around a two bladed UH &ndash / 1 helicopter rotor and turbulent solution around a flat plate is obtained. For the rotary wing configurations, nonlifting and lifting rotor cases are handled seperately for subsonic and transonic blade tip speeds. The results are, generally, in good agreement with the experimental data.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The present work is dedicated to perform the mathematical modeling and DNS and LES simulations of a three-dimensional, temporally evolving incompressible plane wake are performed, seeking to evidence differences in stability, transition and onset of both coherent and small scale structures, when the flow is subjected to random perturbations of different amplitudes. The perturbations are generated using the Random-Flow-Generator (RFG) technique, being imposed to the flow as initial conditions. The Navier-Stokes equations are solved in a prismatic domain, with periodic boundary conditions in all directions, using Fourier pseudospectral method. The invariants of the velocity gradient tensor, Q and R, are analyzed for random perturbations with magnitudes 10−3, 10−4 and 10−5, showing the onset of their characteristic teardrop correlation map. Moreover, maps of the second and third invariants of the rate-of-strain tensor, QS and RS, are shown, in order to evidence the differences in local flow strain and topological characteristics of the dissipation of kinetic energy. Isosurface plots of Q and QW, as well as vorticity contours are shown, allowing visual identification of the coherent structures and confirming patterns predicted by the invariant maps. / O presente trabalho se dedica a modelagem matemática e a simulações numéricas DNS e LES de uma esteira tridimensional, incompressível, em desenvolvimento temporal, buscando evidenciar diferenças na estabilidade, transição e no desenvolvimento de estruturas coerentes e de pequena escala, quando o escoamento é submetido a perturbações randômicas de diferentes amplitudes. As perturbações são geradas utilizando-se a técnica Random Flow Generator (RFG), sendo sobrepostas à condição inicial do escoamento. As equações de Navier-Stokes são resolvidas em um domínio prismático, com condições de contorno periódicas em todas as direções, utilizando-se o método pseudoespectral de Fourier. Os invariantes do tensor gradiente de velocidade, Q e R, são analisados para perturbações de magnitude 10−3, 10−4 and 10−5, mostrando a formação de uma correlação no formato de gota, característica da resolução das equações de Navier-Stokes. Além disso, são apresentados mapas do segundo e terceiro invariante do tensor taxa de deformação, QS e RS, a fim de evidenciar as diferenças locais no escoamento e as características topológicas na taxa de dissipação de energia cinética. Isosuperfícies de Q e QW, bem como contornos de vorticidade são apresentados, possibilitando a identificação visual das estruturas coerentes, e confirmando os padrões de estruturas previstos pelos mapas de invariância. / Mestre em Engenharia Mecânica

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In the present study, using the Generalized Integral Transform Technique (GITT), we derive a semi-analytical solution of the Navier-Stokes equation to obtain the mean wind profile in the atmospheric boundary layer. The technique combines series expansion and an integration employing an inverse-transform pair. The PBL is discretized into N sub-intervals in such manner that inside each sub-region the eddy diffusivity is the average value, this allows the use of realistic eddy diffusivity profiles, which depend on the physical characteristics of the energy-containing eddies. The nonlinear terms are written in terms of kinematical properties of the flow, such as divergence and vorticity, allowing the solutions to be interpreted in terms of large-scale synoptic conditions. The model results are compared to observed wind profiles obtained from the classical Wangara experiment. In addition, the results obtained by the proposed model are compared with the unidimensional model solved by the method of discretization, the model of two layers, with the logarithmic law and the Ekman model. The method used was efficient for the problem studied, since it has presented results consistent with those available in literature. / No presente estudo, usando a Técnica da Transformada Integral Generalizada (GITT), deriva-se uma solução semianalítica para as Equações de Navier-Stokes aplicada à Camada Limite Planetária (CLP). A técnica combina uma expansão em série com uma integração por meio de um par de transformada-inversa. A CLP é discretizada em N subintervalos de maneira que, dentro de cada sub-região, os coeficientes de difusão assumam valores médios, o que nos permite utilizar perfis mais realísticos para o coeficiente de difusão e que dependem das características dos turbilhões mais energéticos. Os termos não-lineares são escritos em função das propriedades cinemáticas do escoamento, como divergência e vorticidade, permitindo que a solução seja interpretada em termos das condições sinóticas de grande escala. O desempenho do modelo estudado foi comparado com dados experimentais de vento medidos durante os experimentos de Wangara. Adicionalmente, os resultados obtidos através do modelo proposto são comparados com o modelo unidimensional resolvido pelo método de discretização, com o modelo de duas camadas, com a Lei Logarítmica e com o modelo de Ekman. O método empregado mostrou-se eficiente para o problema estudado, uma vez que apresentou resultados coerentes com os disponíveis na literatura.

Title: Numerical Solution of the Three-dimensional Compressible Flow Author: Martin Kyncl Department: Department of Numerical Mathematics Supervisor: Doc. RNDr. Jiří Felcman, CSc. Abstract: This thesis deals with a fluid flow in 3D in general. The system of the equations, describing the compressible gas flow, is solved numerically, with the aid of the finite volume method. The main purpose is to describe particular boundary conditions, based on the analysis of the incomplete Riemann problem. The analysis of the original initial-value problem shows, that the right hand-side initial condition, forming the Riemann problem, can be partially replaced by the suitable complementary condition. Several modifications of the Riemann problem are introduced and analyzed, as an original result of this work. Algorithms to solve such problems were implemented and used in code for the solution of the compressible gas flow. Numerical experiments documenting the suggested methods are performed. Keywords: compressible fluid flow, the Navier-Stokes equations, the Euler equations, boundary conditions, finite volume method, the Riemann problem, numerical flux, tur- bulent flow

Qualitative properties of solutions to equations of fluid mechanics Mgr. Jakub Tichý Supervisor: doc. Mgr. Petr Kaplický, Ph.D. Department: Department of Mathematical Analysis Abstract This thesis is devoted to the boundary regularity of weak solutions to the system of nonlinear partial differential equations describing incompressible flows of a certain class of generalized Newtonian fluids in bounded domains. Equations of motion and continuity equation are complemented with perfect slip boundary conditions. For stationary generalized Stokes system in Rn with growth condi- tion described by N-function Φ the existence of the second derivatives of velocity and their regularity up to the boundary are shown. For the same system of equa- tions integrability of velocity gradients is proven. Lq estimates are obtained also for classical evolutionary Stokes system via interpolation-extrapolation scales. Hölder continuity of velocity gradients and pressure is shown for evolutionary generalized Navier-Stokes equations in R2 . Keywords Generalized Stokes and Navier - Stokes equations, incompressible fluids, perfect slip boundary conditions, regularity up to the boundary

In this thesis we are concerned with the numerical simulation of the in- teraction of compressible viscous flow and an elastic structure in 2D. For the elastic deformation we use a 2D linear model and nonlinear St. Venant- Kirchhoff and neo-Hookean models. The flow is described by the compressible Navier-Stokes equations written in the arbitrary Lagrangian-Eulerian (ALE) form in order to take into account the time-dependence of the flow domain. The discretization of both the flow problem and the elasticity problem is re- alized by the discontinuous Galerkin finite element method (DGM). We focus on testing the DGM applied to the solution of the flow and elasticity prob- lems. Furthermore, we discuss the coupling algorithm and the technique, how to deal with the deformation of the computational domain for the fluid flow problem. Our work is motivated by the biomedical applications. Numerical experiments include numerical simulation of vibrations of human vocal folds induced by the compressible viscous flow.

Chorin’s method, Incremental Pressure Correction Scheme (IPCS) and Crank-Nicolson’s method (CN) are three numerical methods that were investigated in this study. These methods were here used for solving the incompressible Navier-Stokes equations, which describe the motion of an incompressible fluid, in three different benchmark problems. The methods were stabilized using residual based artificial viscosity, which was introduced to avoid instability. The methods were compared in terms of accuracy and computational time. Furthermore, a theoretical study of adaptivity was made, based on an a posteriori error estimate and an adjoint problem. The implementation of the adaptivity is left for future studies. In this study we consider the following three well-known benchmark problems: laminar 2D flow around a cylinder, Taylor-Green vortex and lid-driven cavity problem. The difference of the computational time for the three methods were in general relatively small and differed depending on which problem that was investigated. Furthermore the accuracy of the methods also differed in the benchmark problems, but in general Crank-Nicolson’s method gave less accurate results. Moreover the stabilization technique worked well when the kinematic viscosity of the fluid was relatively low, since it managed to stabilize the numerical methods. In general the solution was affected in a negative way when the problem could be solved without stabilization for higher viscosities.

L'objet de cette thèse est l'étude de la stabilisation de modèles d'interaction fluide-structure par des contrôles de dimension finie agissant sur la frontière du domaine fluide. L'écoulement du fluide est décrit par les équations de Navier-Stokes incompressibles tandis que l'évolution de la structure, située à la frontière du domaine fluide, satisfait une équation d'Euler-Bernoulli avec amortissement. Dans le chapitre 1, nous étudions le cas où le contrôle est une condition aux limites de Dirichlet sur les équations du fluide (contrôle par soufflage/aspiration). Nous obtenons des résultats de stabilisation locale du système non-linéaire autour d'une solution stationnaire instable de ce système. Dans les chapitres 2 et 3, nous nous intéressons au cas où le contrôle est une force appliquée sur la structure (contrôle par déformation de paroi). Dans le chapitre 2, nous considérons un modèle simplifié, où l'équation d'Euler-Bernoulli pour la structure est remplacée par un système de dimension finie. Nous construisons des lois de contrôle pour les systèmes de dimension infinie, ou pour leurs approximations semi-discrètes, capables de stabiliser les systèmes linéarisés avec un taux de décroissance exponentielle prescrit, et localement les systèmes non-linéaires. Nous présenterons des résultats numériques permettant de vérifier l'efficacité de ces lois de contrôles. / The aim of this thesis is to study the stabilization of fluid-structure interaction models by finite dimensional controls acting at the boundary of the fluid domain. The fluid flow is described by the incompressible Navier-Stokes equations while the displacement of the structure, localized at the boundary of the fluid domain, satisfies a damped Euler-Bernoulli beam equation. First, we study the case where the control is a Dirichlet boundary condition in the fluid equations (control by suction/blowing). We obtain local feedback stabilization results around an unstable stationary solution of this system. Chapters 2 and 3 are devoted to the case where control is a force applied to the structure (control by boundary deformation). We consider, in chapter 2, a simplified model where the Euler-Bernoulli equation for the structure is replaced by a system of finite dimension. We construct feedback control laws for the infinite dimensional systems, or for their semi-discrete approximations, able to stabilize the linearized systems with a prescribed exponential decay rate, and locally the nonlinear systems. We present some numerical results showing the efficiency of the feedback laws.

The numerical simulation of fluid flow involving complex geometries is greatly limited by the required spatial grid resolution. These flows often contain small regions with complex motions, while the remaining flow is relatively smooth. Adaptive mesh refinement (AMR) enables the spatial grid to be refined in local regions that require finer grids to resolve the flow. This work describes an approach to parallelization of a structured adaptive mesh refinement (SAMR) algorithm. This type of methodology is based on locally refined grids superimposed on coarser grids to achieve the desired resolution in numerical simulations. Parallel implementations of SAMR methods offer the potential for accurate simulations of high complexity fluid flows. However, they present interesting challenges in dynamic resource allocation, data-distribution and load-balancing. The overall efficiency of parallel SAMR applications is limited by the ability to partition the underlying grid hierarchies at run-time to expose all inherent parallelism, minimize communication and synchronization overheads, and balance load. The methodology is based on a message passing interface model (MPI) using the recursive coordinate bisection (RCB) for domain partition. For this work, a semi-implicit projection method has been implemented to solve the incompressible Navier Stokes equations. All numerical implementations are an extension of a sequential Fortran 90 code, called "AMR3D", developed in the work of Nós (2007) .The efficiency and robustness of the applied methodology are verified via convergence analysis using the method of manufactured solutions. Validations were performed by simulating an incompressible jet flow and a lid driven cavity flow. / A simulação numérica de escoamentos envolvendo geometrias complexas é fortemente limitada pela resolução da malha espacial. Na grande maioria dos escoamentos, há pequenas regiões do domínio onde o fluido se movimenta de forma complexa gerando gradientes elevados, enquanto que no restante do domínio o escoamento é relativamente calmo". O Refinamento Adaptativo de Malhas (Adaptive Mesh Refinement - AMR), possibilita que o refinamento da malha espacial seja mais apurado em regiões especificas, enquanto que nas demais regiões o refinamento pode ser mais grosseiro. O presente trabalho consiste no desenvolvimento de uma metodologia de paralelização para a solução das equações de Navier-Stokes em malhas adaptativas bloco-estruturadas (Structured Adaptive Mesh Refinement - SAMR) utilizando a interface MPI (Message Passing Interface) e o método de bisseção por coordenadas RCB (Recursive Coordinate Bisection) para o balanço de carga. Implementações de métodos SAMR em processamento paralelo oferecem a possibilidade de simulações precisas de escoamentos de elevada complexidade. No entanto, apresentam desafios interessantes quanto à dinamicidade na alocação e distribuição dos dados e no balanceamento de carga. Cabe ressaltar que a é ciência total das aplicações envolvendo métodos SAMR em processamento paralelo é fortemente dependente da qualidade do particionamento dinâmico de domínio, efetuado em tempo de execução, para que se garanta os menores custos de comunicação e sincronização possíveis, além de uma boa distribuição da carga computacional. Neste trabalho, utilizou-se o esquema semi-implícito proposto por Ceniceros et al. (2010) para avanço temporal. Todas as implementações foram efetuadas como uma extensão do código AMR3D", proposto por Nós (2007). A é ciência e a robustez do método proposto são verificadas por meio do método das soluções manufaturadas. As validações foram feitas por meio da simulação do escoamento em uma cavidade com tampa deslizante e de um jato incompressível. / Doutor em Engenharia Mecânica