The development of a program of calculation to determine the heat distribution in multilayered plates

Разработана методика и программа расчета для численного моделирования распределения тепла в многослойных тонких пластинах на основе решения одномерного не стационарного уравнения теплопроводности. Предполагается, что межд

Des avancées dans la réduction de modèle de type PGD pour les EDPs d’ordre élevé, le traitement des géométries complexes et la résolution des équations de Navier-Stokes instationnaires

The main purpose of this work is to describe a simulation method for the use of aPGD-based Model reduction Method (MOR) for solving high order partial differentialequations. First, the PGD method is used for solving fourth order PDEs and thealgorithm is illustrated on a lid-driven cavity problem. Transformations of coordinatesfor changing the complex physical domain into the simple computational domain arealso studied, which lead to extend the spatial PGD method to complex geometrydomains. Some numerical examples for different kinds of domain are treated toillustrate the potentialities of this methodology.Finally, a PGD-based space-time separation is introduced to solve the unsteadyStokes or Navier-Stokes equations. This decomposition makes use of common tem-poral modes for both velocity and pressure, which lead to velocity spatial modessatisfying individually the incompressibility condition. The adaptation and imple-mentation of a PGD approach into a general purpose finite volume framework isdescribed and illustrated on several analytic and academic flow examples. A largereduction of the computational cost is observed on most of the treated examples.L’objectif principal de ce travail est de proposer une nouvelle approche de simulationbasée sur une Méthode de réduction du modèle (MOR) utilisant une décompositionPGD. Dans ce travail, cette approche est d’abord utilisée pour résoudre des équationsaux dérivées partielles d’ordre élevé avec un exemple numérique pour les équations auxdérivées partielles du quatrième ordre sur le problème de la cavité entraînée. Ensuiteun changement de coordonnées pour transformer le domaine physique complexe enun domaine de calcul simple est étudié, ce qui conduit à étendre la méthode PGDau traitement de certaines géométries complexes. Divers exemples numériques pourdifférents types de domaines géométriques sont ainsi traités avec l’approche PGD.Enfin, une séparation espace-temps est proposée pour résoudre les équations deNavier-Stokes instationnaires à l’aide d’une approche PGD. Cette décompositionest basée sur le choix de modes temporels communs pour la vitesse et la pression,ce qui conduit à une décomposition basée sur des modes spatiaux satisfaisant in-dividuellement la condition d’incompressibilité. L’adaptation d’une formulationvolumes finis à cette décomposition PGD est présentée et validée sur de premiersexemples analytiques ou académiques pour les équations de Stokes ou Navier-Stokesinstationnaires. Une importante réduction des temps calculs est observée sur lespremiers exemples traités

Fast Algorithms for Biharmonic Problems and Applications to Fluid Dynamics

Many areas of physics, engineering and applied mathematics require solutions of inhomogeneous biharmonic problems. For example, various problems on Stokes flow and elasticity can be cast into biharmonic boundary value problems. Hence the slow viscous flow problems are generally modeled using biharmonic boundary value problems which have widespread applications in many areas of industrial problems such as flow of molten metals, flow of particulate suspensions in bio-fluid dynamics, just to mention a few. In this dissertation, we derive, implement, validate, and apply fast and high order accurate algorithms to solve Poisson problems and inhomogeneous biharmonic problems in the interior of a unit disc in the complex plane. In particular, we use two methods to solve inhomogeneous biharmonic problems: (i) the double-Poisson method which is based on transforming biharmonic problems into solving a sequence of Poisson problems (sometime also one homogeneous biharmonic problem) and then making use of the fast Poisson solver developed in this dissertation.; (ii) the direct method which uses the fast biharmoninc solver also developed in this dissertation. Both of these methods are analyzed for accuracy, complexity and efficiency. These biharmonic solvers have been compared with each other and have been applied to solve several Stokes flow problems and elasticity problems. The fast Poisson algorithm is derived here from exact analysis of the Green’s function formulation in the complex plane. This algorithm is essentially a recast of the fast Poisson algorithm of Borges and Daripa from the real plane to the complex plane. The fast biharmonic algorithms for several boundary conditions for use in the direct method mentioned above have been derived in this dissertation from exact analysis of the representation of their solutions in terms of problem specific Green’s function in the complex plane. The resulting algorithms primarily use fast Fourier transforms and recursive relations in Fourier space. The algorithms have been analyzed for their accuracy, complexity, efficiency, and subsequently tested for validity against several benchmark test problems. These algorithms have an asymptotic complexity of O(log N ) per degree of freedom with very low constant which is hidden behind the order estimate. The direct and double-Poisson methods have been applied to solving the steady, incompressible slow viscous flow problem in a cir- cular cylinder and some problems from elasticity. The numerical results from these computations agree well with existing results on these problems

Application of Reynolds Stress Model Using Direct Modeling and Actuator Disk Approaches for a Small-Scale Wind Turbine

The Reynolds Stress Model (RSM) has been avoided for turbulence closure in CFD simulations of wind turbines, largely due to the computational expense and the high potential for numerical instability. The advantage of using RSM is having access to shear stresses that are not available from two-equation RANS-based closure models like k-e and k-w. Access to the shear stresses will aide in the understanding of how the blade design will affect the wake, particularly in the near-wake region. In this research, the RSM turbulence model has been successfully applied in simulating a three-bladed small-scale wind turbine through a direct-model approach and an actuator disk approach. In the direct-model method, the turbine blades were discretized within a rotating subdomain and in the actuator disk method, the turbine blades were modeled as a rotating disk using the Virtual Disk model available in Star CCM+. The transient Rigid Body Motion (RBM) simulation was able to accurately predict velocity deficit and tip vortices that compared well with hot-wire measurements and high speed images. The actuator disk method is more practical in simulating wind farms due to the simplified mesh and requires accurate information for lift and drag coefficients. Experimental results showed interaction between the tower and rotating blades can create significant turbulence in the wake. Experiments with multiple turbines showed how each turbine contributed to the velocity deficit and total turbulence intensity. For the experimental blade design, the velocity deficit recovered and turbulence intensity had dissipated within three rotor diameters

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques

Computational Fluid Dynamics

This collection of papers was presented at the Computational Fluid Dynamics (CFD) Conference held at Ames Research Center in California on March 12 through 14, 1991. It is an overview of CFD activities at NASA Lewis Research Center. The main thrust of computational work at Lewis is aimed at propulsion systems. Specific issues related to propulsion CFD and associated modeling will also be presented. Examples of results obtained with the most recent algorithm development will also be presented

Computation of viscous incompressible flows

Incompressible Navier-Stokes solution methods and their applications to three-dimensional flows are discussed. A brief review of existing methods is given followed by a detailed description of recent progress on development of three-dimensional generalized flow solvers. Emphasis is placed on primitive variable formulations which are most promising and flexible for general three-dimensional computations of viscous incompressible flows. Both steady- and unsteady-solution algorithms and their salient features are discussed. Finally, examples of real world applications of these flow solvers are given