The incompressible Navier-Stokes equations are among the most important partial differential systems arising from classical physics. They are utilized to model a wide range of fluids, from water moving around a naval vessel to blood flowing through the arteries of the cardiovascular system. Furthermore, the secrets of turbulence are widely believed to be locked within the Navier-Stokes equations. Despite the enormous applicability of the Navier-Stokes equations, the underlying behavior of solutions to the partial differential system remains little understood. Indeed, one of the Clay Mathematics Institute's famed Millenium Prize Problems involves the establishment of existence and smoothness results for Navier-Stokes solutions, and turbulence is considered, in the words of famous physicist Richard Feynman, to be "the last great unsolved problem of classical physics." Numerical simulation has proven to be a very useful tool in the analysis of the Navier-Stokes equations. Simulation of incompressible flows now plays a major role in the industrial design of automobiles and naval ships, and simulation has even been utilized to study the Navier-Stokes existence and smoothness problem. In spite of these successes, state-of-the-art incompressible flow solvers are not without their drawbacks. For example, standard turbulence models which rely on the existence of an energy spectrum often fail in non-trivial settings such as rotating flows. More concerning is the fact that most numerical methods do not respect the fundamental geometric properties of the Navier-Stokes equations. These methods only satisfy the incompressibility constraint in an approximate sense. While this may seem practically harmless, conservative semi-discretizations are typically guaranteed to balance energy if and only if incompressibility is satisfied pointwise. This is especially alarming as both momentum conservation and energy balance play a critical role in flow structure development. Moreover, energy balance is inherently linked to the numerical stability of a method. In this dissertation, novel B-spline discretizations for the generalized Stokes and Navier-Stokes equations are developed. The cornerstone of this development is the construction of smooth generalizations of Raviart-Thomas-Nedelec elements based on the new theory of isogeometric discrete differential forms. The discretizations are (at least) patch-wise continuous and hence can be directly utilized in the Galerkin solution of viscous flows for single-patch configurations. When applied to incompressible flows, the discretizations produce pointwise divergence-free velocity fields. This results in methods which properly balance both momentum and energy at the semi-discrete level. In the presence of multi-patch geometries or no-slip walls, the discontinuous Galerkin framework can be invoked to enforce tangential continuity without upsetting the conservation and stability properties of the method across patch boundaries. This also allows our method to default to a compatible discretization of Darcy or Euler flow in the limit of vanishing viscosity. These attributes in conjunction with the local stability properties and resolution power of B-splines make these discretizations an attractive candidate for reliable numerical simulation of viscous incompressible flows. / text

Submitted by Marcele Costal de Castro (costalcastro@gmail.com) on 2017-10-04T17:13:38Z No. of bitstreams: 1 PAULO AUGUSTO BERQUÓ DE SAMPAIO D.pdf: 6576641 bytes, checksum: 71355f6eedcf668b2236d4c10f1a2551 (MD5) / Made available in DSpace on 2017-10-04T17:13:38Z (GMT). No. of bitstreams: 1 PAULO AUGUSTO BERQUÓ DE SAMPAIO D.pdf: 6576641 bytes, checksum: 71355f6eedcf668b2236d4c10f1a2551 (MD5) Previous issue date: 1991-09 / The basic difficulties associated with the numerical solution of the incompressible Navier-Stokes equations in primitive variables are identified and analysed. These difficulties, namely the lack of self-adjointness of the flow equations and the requirement of choosing compatible interpolations for velocity and pressure, are addressed with the development of consistent Petrov-Galerkin formulations. In particular, the solution of incompressible viscous flow problems using simple equal order interpolation for all variables becomes possible .

L'objectif de cette thèse est de développer une méthode de décomposition de domaine pour la résolution du système de Stokes discrétisé avec les éléments finis mixtes stables où la pression est continue comme Hood-Taylor et Mini. La nouvelle méthode résulte dela combinaison de FETI qui est appliquée à la vitesse et de BDD qui est appliquée à la pression sans découpler les inconnues. Elle hérite et découple les projecteurs grossiers associés à FETI et à BDD. La méthodologie débouche sur un système linéaire symétrique,semi-défini positif que nous avons résolu par la méthode du gradient conjugué projeté préconditionné. La méthode contient deux préconditionneurs grossiers creux et des préconditionneurs locaux exacts qui assurent son extensibilié, sa robustesse et son efficacité. L'introduction de projecteurs locaux construits à partir des modes de pression des sousdomaines étend la méthode aux éléments finis mixtes discontinues en pression et rend le problème grossier de BDD facultatif même en présence de la pression aux interfaces.Nous avons aisément appliqué la méthode à l'élasticité incompressible et quasi-incompressible et elle peut s'étendre de la même façon au cadre plus général des systèmes de point-selle issus des problèmes de minimisation sous contraintes grâce à sa nature algébrique. / The purpose of this thesis is to develop a domain decomposition method suitable tosolve the Stokes system discretized with stable mixte finite elements where pressure is continuous such as Hood-Taylor and Mini. The new method arises from the combinaison of FETI applied to the velocity and BDD applied to the pressure without decoupling the unknowns. It inherits and decouples the coarse projectors included in FETI and BDD. The methodology leads to a symmetric, positive semi-definite linear system that we solveby projected preconditioned conjugate gradient. The method contains two sparse coarse preconditionners and exact local preconditionners that ensure its scalability, its robustness and its efficiency. We use local projectors constructed from the constant pressure modes of the subdomains that enable an extension to mixte finite elements with discontinuous pressure and that make the coarse problem of BDD optional even in the presence of pressure on the interfaces. We have easily applied the method to incompressibleand almost incompressible elasticity and it can be extended the same way to other saddle-point systems arising from minimization problems under constraints due to its algebraic property.

This work presents a novel method to model immiscible incompressible fluids in a stable manner. Here, the immiscible behavior of the flow is described by the incompressible Navier-Stokes-Cahn-Hilliard model, which is based on a diffuse interface method. We introduce buoyancy effects in the model through the Boussinesq approximation in a consistent manner. A structure-preserving discretization is used to guarantee the linear stability of the discrete problem and to satisfy the incompressibility of the discrete solution at every point in space by construction. For the solution of the model, we developed the Portable Extensible Toolkit for Isogeometric Analysis with Multi-Field discretizations (PetIGA-MF), a high-performance framework that supports structure-preserving spaces. PetIGA-MF is built on top of PetIGA and the Portable Extensible Toolkit for Scientific Computation (PETSc), sharing all their user-friendly, performance, and flexibility features. Herein, we describe the implementation of our model in PetIGA-MF and the details of the numerical solution. With several numerical tests, we verify the convergence, scalability, and validity of our approach. We use highly-resolved numerical simulations to analyze the merging and rising of droplets. From these simulations, we detailed the energy exchanges in the system to evaluate quantitatively the quality of our simulations. The good agreement of our results when compared against theoretical descriptions of the merging, and the small errors found in the energy analysis, allow us to validate our approach. Additionally, we present the development of an unconditionally energy-stable generalized-alpha method for the Swift-Hohenberg model that offers control over the numerical dissipation. A pattern formation example demonstrates the energy-stability and convergence of our method.

The specific turbulent kinetic energy, root-mean-square fluctuating vorticity, and mean-vortexwavelength distributions are presented for fully rough pipe flow. The distributions of these turbulence variables are obtained from a proposed turbulence model. Many of the turbulence models commonly used for computational fluid dynamics are based on an analogy between molecular and turbulent transport. However, traditional k-ε and k-ω models fail to exhibit proper dependence on the molecular viscosity. Based on a rigorous application of the Boussinesq’s hypothesis, Phillips proposed a vorticity-based transport equation for the turbulent kinetic energy. The foundation for this vorticity-based transport equation is presented. In future development of this model, a transport equation for the fluctuating vorticity is needed. In order to assess the model and evaluate closure coefficients, the resulting turbulent vorticity distribution must be compared to reference distributions. This dissertation presents reference distributions for the mean fluctuating vorticity and mean turbulent wavelength obtained for fully rough pipe flow. These distributions are obtained from a turbulence model, which involves the proposed transport equation for the turbulent kinetic energy and an empirical relation for the mean vortex wavelength. The empirical relation for the mean vortex wavelength requires numerous closure coefficients. These closure coefficients are determined through gradient-based optimization techniques. The current model gives excellent agreement with well established relations obtained for both the friction factor and velocity distribution.

A comprehensive survey of available numerical methods and models was performed on the open source computational fluid dynamics solver OpenFOAM version 2.0 for incompressible turbulent bluff body flows. Numerical methods are illuminated using source code for side-by-side comparison. For validation, the accuracy of flow predictions over a sphere in the subcritical regime and delta wing with sharp leading edge is assessed. Solutions show mostly good agreement with experimental data and data obtained from commercial software. A demonstration of the numerical implementation of a dynamic hybrid RANS/LES framework is also presented, including results from test studies.

The utilization of steady state flow of air over tandem circular cylinders has several applications in engineering systems. Incompressible flow over circular cylinders in tandem at different spacing with and without plasma actuation on the leading cylinder will be investigated in this paper to understand the effects of plasma actuation on flow properties and wake region of the two cylinders in cross flow. The principal focus of the research is on the use of experimental and computational methods to study and provide valid results, the research will analyze the wake region, the effect of Reynolds number and the longitudinal spacing between cylinder on vortex shedding, aerodynamic parameters i.e. lift, drag, pressure differential, etc. The research will be conducted for steady flow at Reynold number, Re = U∞ L/v between 5000 and 8000 for air. The turbulence of the wake and dynamic instability of the experimental is characterized by the Strouhal number, St = fL/U∞ frequency of the vortex shedding in the wake which is directly proportional to the spacing, λ from center to center of cylinders between 3 to 5 inches. The dependencies on critical values of Re and St in symmetric flow over cylinders to show the instability of the flow regime in previous research. At Re = 5000 the vortex co-shedding on the second cylinder would occur at critical spacing, λccharacterized by the Re - St relationship at 3 ≤ λ ≤ 5 in the flow regime. The use of plasma actuation in fluid dynamics to control flow velocity by generating momentum to force atmospheric pressure and velocity in external flow with Single- Dielectric Barrier Discharge(SDBD) for both two and three-dimensional, 2D and 3D actuator (straight and segmented actuator). The SDBD actuators are mounted spanwise on the leading cylinder for both 2D and 3D to impact momentum, therefore, forcing the wake regime. Computational Analysis is utilized for result and data pre-processing. The steady three-dimensional flow of tandem cylinders can be studied through Large Eddy Simulation (LES) using a subgrid-scale model to compare numerical and experimental results for the same setup and physical conditions. Particle Image Velocimetry (PIV) is used to resolve time series images from flow visualization of the experiment, the images are processed to visualize velocity vectors of the flow regimes. The velocity profile of the flow can be averaged and plotted for all instantaneous time-series images processed in PIV by Dynamic Mode Decomposition (DMD) or Proper Orthogonal Decomposition (POD) to generate common eigenvalues and eigenvector of the large dimension PIV data which shows the average properties of the flow properties.

Of the surface capturing schemes, the levelset and multi-phase models are implemented and extensively examined. First, the levelset method is shown, and its weaknesses are identified; a mis- appropriation of changes in momentum, a strong dependence on the density by the eigenvalues of the inviscid flux Jacobian, and a prescribed density transition. These weaknesses are specifically addressed and overcome by the formulation of the multi-phase model. Consequently, the multi-phase model is chosen for this work. Previous surface fitting techniques simply absorb the gravitational source term into the pressure. It must be noted that this absorbtion is valid only for single density flows; since the surface fitting approach is solving only one side of the interface, there is no significant change in the density througout the domain. Consequently, absorbing the gravitational source into the pressure term is not possible in a surface capturing scheme in which both sides of the interface are solved. Thus, a new treatment of the gravitational source term is required and is presented in this work. A multi-phase model is implemented into a parallel, three-dimensional, unsteady, incompressible Navier-Stokes flow solver for the purpose of examining free surface flows on unstructured meshes. The reasons for choosing this model above others are presented, and the multi-phase model is discussed. The base algorithm is briefly examined with emphasis given to the areas which require additional care. The construction of the gravity source term which drives the formation of the waves is explained in detail, and its effects on the rest of the algorithm are identified. Finally, the method is carefully compared with available data on a submerged NACA 0012 airfoil, the Wigley Hull, the Series 60 Cb=0.6 ship, and the DTMB 5415 ship.

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