Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

Nonlinear subgrid finite element models for low Mach number flows coupled with radiative heat transfer

The general description of a fluid flow involves the solution of the compressible Navier-Stokes equations, a very complex problem whose mathematical structure is not well understood. It is widely accepted that these equations provide an accurate description of any problem in fluid mechanics which may present many different nonlinear physical mechanisms. Depending on the physics of the problem under consideration, different simplified models neglecting some physical mechanisms can be derived from asymptotic analysis. On the other hand, radiative heat transfer can strongly interact with convection in high temperature flows, and neglecting its effects may have significant consequences in the overall predictions. Problems as fire scenarios emphasized the need for an evaluation of the effect of radiative heat transfer. This work is directed to strongly thermally coupled low Mach number flows with radiative heat transfer. The complexity of these mathematical problem makes their numerical solution very difficult. Despite the important difference in the treatment of the incompressibility, the low Mach number equations present the same mathematical structure as the incompressible Navier-Stokes equations, in the sense that the mechanical pressure is determined from the mass conservation constraint. Consequently the same type of numerical instabilities can be found, namely, the problem of compatibility conditions between the velocity and pressure finite element spaces, and the instabilities due to convection dominated flows. These instabilities can be avoided by the use of stabilization techniques. Many stabilization techniques used nowadays are based on the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. The modeling of the subgrid scale and its influence leads to a modified coarse scale problem providing stability. The quality of the final approximation (accuracy, efficiency) depends on the particular model. The extension of these techniques to nonlinear and coupled problems is presented. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms appearing in the finite element equations and in the subgrid scale model. The first ingredient permits to obtain an improved time discretization scheme(higher accuracy, better stability). The second ingredient permits to prove global conservation properties, being also responsible of the higher accuracy of the method. This ingredient is intimately related to the problem of thermal turbulence modeling from a strictly numerical point of view. The capability for the simulation of turbulent flows is a measure of the ability of modeling the effect of the subgrid flow structures over the coarser ones. The performance of the model in predicting the behavior of turbulent flows is demonstrated. The radiation transport equation has been also approximated within the variational multiscale framework, the design and analysis of stabilized finite element methods is presented.La descripción general del movimiento de un flujo implica la solución de las ecuaciones de Navier-Stokes compresibles, un problema de muy compleja estructura matemática. Estas ecuaciones proporcinan una descripción detallada de cualquier problema en mecánica de fluidos, que puede presentar distintos mecanismos no lineales que interactúan entre si. En función de la física del problema que se esté considerando, pueden derivarse modelos simplificados de las ecuaciones de Navier-Stokes mediante analisis dimensional, que ignoran algunos fenómenos físicos. Por otro lado, la transferencia de calor por radiación puede interactuar con el movimiento de un fluido, e ignorar sus efectos puede tener consecuencias importantes en las predicciones del flujo. Problemas donde hay fuego implican la evaluacion del efecto del calor por radiación. El presente trabajo está dirigido a flujos a bajo número de Mach térmicamente acoplados, donde el calor por radiación afecta al flujo. Debido a la complejidad del problema matemático, la solución numérica es muy complicada. A pesar de las diferencia en el tratamiento de la incompresibilidad, las ecuaciones de flujo a bajo número de Mach poseen una estructura matemática similar a la de flujo incompresible, en el sentido que la presión mecánica se determina a partir de la ecuación de conservación de la masa. En consecuencia poseen el mismo tipo de inestabilidades numéricas, que son el problema de condiciones de compatibilidad entre los espacios de elementos finitos de velocidad y presión, y las inestabilidades debidas a flujos con convección dominante. Estas inestabilidades pueden evitarse mediante técnicas de estabilización numérica. Muchos métodos de estabilización utilizados hoy día se basan en el método de multiscalas variacionales, donde el espacio funcional de la solucion se divide en un espacio discreto y resolubre y un espacio infinito de subscalas. El modelado de las subescalas y su influencia modifican el problema discreto proporcionando estabilidad. La calidad de la aproximación numérica final (precisión, eficiencia) depende del modelo particular de subescalas. En este trabajo se extienden estas técnicas de estabilización a problemas no lineales y acoplados. Las características que distinguen a nuestra aproximación son considerar las subsecalas como transitorias y mantener la división de escalas en todos los términos no lineales que aparecen en las ecuaciones de elementros finitos y en las del modelo de subescalas. La primera característica permite obtener mayor precisión y mejor estabilidad en la solución, la segunda característica permite obtener esquemas donde las propiedades se conservan globalmente, y mayor precisión del método. El hecho de mantener la división de escalas en todos los términos no lineales está intimamemte relacionado con el modelado de turbulencia en flujos térmicamente acoplados desde un punto de vista estrictamente numérico. La capacidad de simulación de flujo turbulento es una medida de la habilidad de modelar el efecto de las estructuras de escala fina sobre las estructuras de escala gruesa. Se muestra en esta tesis el desempeño del método para de predecir flujo turbulento. La ecuación de transporte de radiación también se aproxima numéricamente en el marco de multiscala variacional. El diseño y análisis de este método se presenta en detalle en esta tesi

A Meshfree Lagrangian Method for Flow on Manifolds

In this paper, we present a novel meshfree framework for fluid flow simulations on arbitrarily curved surfaces. First, we introduce a new meshfree Lagrangian framework to model flow on surfaces. Meshfree points or particles, which are used to discretize the domain, move in a Lagrangian sense along the given surface. This is done without discretizing the bulk around the surface, without parametrizing the surface, and without a background mesh. A key novelty that is introduced is the handling of flow with evolving free boundaries on a curved surface. The use of this framework to model flow on moving and deforming surfaces is also introduced. Then, we present the application of this framework to solve fluid flow problems defined on surfaces numerically. In combination with a meshfree Generalized Finite Difference Method (GFDM), we introduce a strong form meshfree collocation scheme to solve the Navier-Stokes equations posed on manifolds. Benchmark examples are proposed to validate the Lagrangian framework and the surface Navier-Stokes equations with the presence of free boundaries

Micro-macro modeling and computation of ferrofluids

We present an innovative, effective micro-macro numerical approach for modeling ferrofluids under the presence of a magnetic field and a driven-cavity flow. Our multi-model approach combines the localized use of a microscopic Smoluchowski equation solver and a continuous constitutive law coupled with the macroscopic flow and an externally applied magnetic field. The model is confirmed with a direct simulation and results are compared with the closure approximation proposed by Shen and Doi. We systematically study the change in viscosity by incrementally changing the numerical parameter settings. Indeed, "negative change" in viscosity is observed

INITIAL ASSESSMENT OF THE COMPRESSIBLE POOR MAN\u27S NAVIER{STOKES (CPMNS) EQUATION FOR SUBGRID-SCALE MODELS IN LARGE-EDDY SIMULATION

Large-eddy simulation is rapidly becoming the preferred method for calculations involving turbulent phenomena. However, filtering equations as performed in traditional LES procedures leads to significant problems. In this work we present some key components in the construction of a novel LES solver for compressible turbulent flow, designed to overcome most of the problems faced by traditional LES procedures. We describe the construction of the large-scale algorithm, which employs fairly standard numerical techniques to solve the Navier{Stokes equations. We validate the algorithm for both transonic and supersonic ow scenarios. We further explicitly show that the solver is capable of capturing boundary layer effects. We present a detailed derivation of the chaotic map termed the \compressible poor man\u27s Navier{Stokes (CPMNS) equation starting from the Navier{Stokes equations themselves via a Galerkin procedure, which we propose to use as the fluctuating component in the SGS model. We provide computational results to show that the chaotic map can produce a wide range of temporal behaviors when the bifurcation parameters are varied over their ranges of stable behaviors. Investigations of the overall dynamics of the CPMNS equation demonstrates that its use increases the potential realism of the corresponding SGS model

High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality

In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability of pseudospectra in the field of hydrodynamic stability to obtain more information than a traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context

Analytical and Numerical Studies of Several Fluid Mechanical Problems

In this thesis, three parts, each with several chapters, are respectively devoted to hydrostatic, viscous and inertial fluids theories and applications. In the hydrostatics part, the classical Maclaurin spheroids theory is generalized, for the first time, to a more realistic multi-layer model, which enables the studies of some gravity problems and direct numerical simulations of flows in fast rotating spheroidal cavities. As an application of the figure theory, the zonal flow in the deep atmosphere of Jupiter is investigated for a better understanding of the Jovian gravity field. High viscosity flows, for example Stokes flows, occur in a lot of processes involving low-speed motions in fluids. Microorganism swimming is such typical a case. A fully three dimensional analytic solution of incompressible Stokes equation is derived in the exterior domain of an arbitrarily translating and rotating prolate spheroid, which models a large family of microorganisms such as cocci bacteria. The solution is then applied to the magnetotactic bacteria swimming problem and good consistency has been found between theoretical predictions and laboratory observations of the moving patterns of such bacteria under magnetic fields. In the analysis of dynamics of planetary fluid systems, which are featured by fast rotation and very small viscosity effects, three dimensional fully nonlinear numerical simulations of Navier-Stokes equations play important roles. A precession driven flow in a rotating channel is studied by the combination of asymptotic analyses and fully numerical simulations. Various results of laminar and turbulent flows are thereby presented. Computational fluid dynamics requires massive computing capability. To make full use of the power of modern high performance computing facilities, a C++ finite-element analysis code is under development based on PETSc platform. The code and data structures will be elaborated, along with the presentations of some preliminary results

The evolution of a magnetic field subject to Taylor′s constraint using a projection operator

In the rapidly rotating, low-viscosity limit of the magnetohydrodynamic equations as relevant to the conditions in planetary cores, any generated magnetic field likely evolves while simultaneously satisfying a particular continuous family of invariants, termed Taylor′s constraint. It is known that, analytically, any magnetic field will evolve subject to these constraints through the action of a time-dependent coaxially cylindrical geostrophic flow. However, severe numerical problems limit the accuracy of this procedure, leading to rapid violation of the constraints. By judicious choice of a certain truncated Galerkin representation of the magnetic field, Taylor′s constraint reduces to a finite set of conditions of size O(N), significantly less than the O(N3) degrees of freedom, where N denotes the spectral truncation in both solid angle and radius. Each constraint is homogeneous and quadratic in the magnetic field and, taken together, the constraints define the finite-dimensional Taylor manifolδ whose tangent plane can be evaluated. The key result of this paper is a description of a stable numerical method in which the evolution of a magnetic field in a spherical geometry is constrained to the manifold by projecting its rate of change onto the local tangent hyperplane. The tangent plane is evaluated by contracting the vector of spectral coefficients with the Taylor tensor, a large but very sparse 3-D array that we define. We demonstrate by example the numerical difficulties in finding the geostrophic flow numerically and how the projection method can correct for inaccuracies. Further, we show that, in a simplified system using projection, the normalized measure of Taylorization, t, may be maintained smaller than O(10-10) (where t= 0 is an exact Taylor state) over 1/10 of a dipole decay time, eight orders of magnitude smaller than analogous measures applied to recent low Ekman-number geodynamo model