On Numerical Algorithms for Fluid Flow Regularization Models

This thesis studies regularization models as a way to approximate a flow simulation at a lower computational cost. The Leray model is more easily computed than the Navier-Stokes equations (NSE), and it is more computationally attractive than the NS-α regularization because it admits a natural linearization which decouples the mass/momentum system and the filter system, allowing for efficient and stable computations. A major disadvantage of the Leray model lies in its inaccuracy. Thus, we study herein several methods to improve the accuracy of the model, while still retaining many of its attractive properties. This thesis is arranged as follows. Chapter 2 gives notation and preliminary results to be used in subsequent chapters. Chapter 3 investigates a nonlinear filtering scheme using the Vreman and Q-criteria based indicator functions. We define these indicator functions, prove stability and state convergence of the scheme to the NSE, and provide several numerical experiments which demonstrate its effectiveness over NSE and Leray calculations on coarse meshes. Chapter 4 investigates a deconvolution-based indicator function. We prove stability and convergence of the resulting scheme, verify the predicted convergence rates, and provide numerical experiments which demonstrate this scheme\u27s effectiveness. Chapter 5 then extends this scheme to the magnetohydrodynamic equations. We prove stability and convergence of our algorithm, and verify the predicted convergence rates. Chapter 6 provides a study of the Leray-α model. We prove stability and convergence for the fully nonlinear scheme, prove conditional stability for a linearized and decoupled scheme, and provide a numerical experiment which compares our scheme with the usual Leray-α model. Specifically, we show that choosing β \u3c α does indeed improve accuracy in computations. Chapter 7 investigates the Leray model with fine mesh filtering. We prove stability and convergence of the algorithm, then verify the increased convergence rate associated with the finer mesh, as predicted by the analysis. Finally, we present a benchmark problem which demonstrates the effectiveness of filtering on a finer mesh

Physicic-based algorithms and divergence free finite elements for coupled flow problems

This thesis studies novel physics-based methods for simulating incompressible fluid flow described by the Navier-Stokes equations (NSE) and magnetohydrodynamics equations (MHD). It is widely accepted in computational fluid dynamics (CFD) that numerical schemes which are more physically accurate lead to more precise flow simulations especially over long time intervals. A prevalent theme throughout will be the inclusion of as much physical fidelity in numerical solutions as efficiently possible. In algorithm design, model selection/development, and element choice, subtle changes can provide better physical accuracy, which in turn provides better overall accuracy (in any measure). To this end we develop and study more physically accurate methods for approximating the NSE, MHD, and related systems. Chapter 3 studies extensions of the energy and helicity preserving scheme for the 3D NSE developed in \cite{Reb07b}, to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme. In Chapter 4, we study a finite element scheme for the 3D NSE that globally conserves energy and helicity and, through the use of Scott-Vogelius elements, enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also show the method can be efficiently computed by exploiting a connection between this method, its associated penalty method, and the method arising from using grad-div stabilized Taylor-Hood elements. Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme. In Chapter 5, we extend the work done in \cite{CELR10} that proved, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. In \cite{CELR10} even though the analytical convergence rate was only shown to be $\gamma^{-\frac{1}{2}}$ (where $\gamma$ is the stabilization parameter), the computational results suggest the rate may be improvable $\gamma^{-1}$. We prove herein the analytical rate is indeed $\gamma^{-1}$, and extend the result to other incompressible flow problems including Leray-$\alpha$ and MHD. Numerical results are given that verify the theory. Chapter 6 studies an efficient finite element method for the NS-$\omega$ model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence. Several numerical experiments are given that demonstrate the performance of the scheme, and how the use of Scott-Vogelius elements can dramatically improve solutions. Chapter 7 extends Leray-$\alpha$-deconvolution modeling to the incompressible MHD. The resulting model is shown to be well-posed, and have attractive limiting behavior both in its filtering radius and order of deconvolution. Additionally, we present and study a numerical scheme for the model, based on an extrapolated Crank-Nicolson finite element method. We show the numerical scheme is unconditionally stable, preserves energy and cross-helicity, and optimally converges to the MHD solution. Numerical experiments are provided that verify convergence rates, and test the scheme on benchmark problems of channel flow over a step and the Orszag-Tang vortex problem

Analysis of a Reduced-Order Approximate Deconvolution Model and its interpretation as a Navier-Stokes-Voigt regularization

We study mathematical and physical properties of a family of recently introduced, reduced-order approximate deconvolution models. We first show a connection between these models and the NS-Voigt model, and that NS-Voigt can be re-derived in the approximate deconvolution framework. We then study the energy balance and spectra of the model, and provide results of some turbulent flow computations that backs up the theory. Analysis of global attractors for the model is also provided, as is a detailed analysis of the Voigt model's treatment of pulsatile flow

Advancements In Finite Element Methods For Newtonian And Non-Newtonian Flows

This dissertation studies two important problems in the mathematics of computational fluid dynamics. The first problem concerns the accurate and efficient simulation of incompressible, viscous Newtonian flows, described by the Navier-Stokes equations. A direct numerical simulation of these types of flows is, in most cases, not computationally feasible. Hence, the first half of this work studies two separate types of models designed to more accurately and efficient simulate these flows. The second half focuses on the defective boundary problem for non-Newtonian flows. Non-Newtonian flows are generally governed by more complex modeling equations, and the lack of standard Dirichlet or Neumann boundary conditions further complicates these problems. We present two different numerical methods to solve these defective boundary problems for non-Newtonian flows, with application to both generalized-Newtonian and viscoelastic flow models. Chapter 3 studies a finite element method for the 3D Navier-Stokes equations in velocity- vorticity-helicity formulation, which solves directly for velocity, vorticity, Bernoulli pressure and helical density. The algorithm presented strongly enforces solenoidal constraints on both the velocity (to enforce the physical law for conservation of mass) and vorticity (to enforce the mathematical law that div(curl)= 0). We prove unconditional stability of the velocity, and with the use of a (consistent) penalty term on the difference between the computed vorticity and curl of the computed velocity, we are also able to prove unconditional stability of the vorticity in a weaker norm. Numerical experiments are given that confirm expected convergence rates, and test the method on a benchmark problem. Chapter 4 focuses on one main issue from the method presented in Chapter 3, which is the question of appropriate (and practical) vorticity boundary conditions. A new, natural vorticity boundary condition is derived directly from the Navier-Stokes equations. We propose a numerical scheme implementing this new boundary condition to evaluate its effectiveness in a numerical experiment. Chapter 5 derives a new, reduced order, multiscale deconvolution model. Multiscale deconvolution models are a type of large eddy simulation models, which filter out small energy scales and model their effect on the large scales (which significantly reduces the amount of degrees of freedom necessary for simulations). We present both an efficient and stable numerical method to approximate our new reduced order model, and evaluate its effectiveness on two 3d benchmark flow problems. In Chapter 6 a numerical method for a generalized-Newtonian fluid with flow rate boundary conditions is considered. The defective boundary condition problem is formulated as a constrained optimal control problem, where a flow balance is forced on the inflow and outflow boundaries using a Neumann control. The control problem is analyzed for an existence result and the Lagrange multiplier rule. A decoupling solution algorithm is presented and numerical experiments are provided to validate robustness of the algorithm. Finally, this work concludes with Chapter 7, which studies two numerical algorithms for viscoelastic fluid flows with defective boundary conditions, where only flow rates or mean pressures are prescribed on parts of the boundary. As in Chapter 6, the defective boundary condition problem is formulated as a minimization problem, where we seek boundary conditions of the flow equations which yield an optimal functional value. Two different approaches are considered in developing computational algorithms for the constrained optimization problem, and results of numerical experiments are presented to compare performance of the algorithms

Turbulence: Numerical Analysis, Modelling and Simulation

The problem of accurate and reliable simulation of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. As the needs for accuracy increase and the applications expand beyond flows where extensive data is available for calibration, the importance of a sound mathematical foundation that addresses the needs of practical computing increases. This Special Issue is directed at this crossroads of rigorous numerical analysis, the physics of turbulence and the practical needs of turbulent flow simulations. It seeks papers providing a broad understanding of the status of the problem considered and open problems that comprise further steps

HIGHER ACCURACY METHODS FOR FLUID FLOWS IN VARIOUS APPLICATIONS: THEORY AND IMPLEMENTATION

This dissertation contains research on several topics related to Defect-deferred correction (DDC) method applying to CFD problems. First, we want to improve the error due to temporal discretization for the problem of two convection dominated convection-diffusion problems, coupled across a joint interface. This serves as a step towards investigating an atmosphere-ocean coupling problem with the interface condition that allows for the exchange of energies between the domains. The main diffuculty is to decouple the problem in an unconditionally stable way for using legacy code for subdomains. To overcome the issue, we apply the Deferred Correction (DC) method. The DC method computes two successive approximations and we will exploit this extra flexibility by also introducing the artificial viscosity to resolve the low viscosity issue. The low viscosity issue is to lose an accuracy and a way of finding a approximate solution as a diffusion coeffiscient gets low. Even though that reduces the accuracy of the first approximation, we recover the second order accuracy in the correction step. Overall, we construct a defect and deferred correction (DDC) method. So that not only the second order accuracy in time and space is obtained but the method is also applicable to flows with low viscosity. Upon successfully completing the project in Chapter 1, we move on to implementing similar ideas for a fluid-fluid interaction problem with nonlinear interface condition; the results of this endeavor are reported in Chapter 2. In the third chapter, we represent a way of using an algorithm of an existing penalty-projection for MagnetoHydroDynamics, which allows for the usage of the less sophisticated and more computationally attractive Taylor-Hood pair of finite element spaces. We numerically show that the new modification of the method allows to get first order accuracy in time on the Taylor-Hood finite elements while the existing method would fail on it. In the fourth chapter, we apply the DC method to the magnetohydrodynamic (MHD) system written in Elsásser variables to get second order accuracy in time. We propose and analyze an algorithm based on the penalty projection with graddiv stabilized Taylor Hood solutions of Elsásser formulations

Improved Accuracy for Fluid Flow Problems Via Enhanced Physics

This thesis is an investigation of numerical methods for approximating solutions to fluid flow problems, specifically the Navier-Stokes equations (NSE) and magnetohydrodynamic equations (MHD), with an overriding theme of enforcing more physical behavior in discrete solutions. It is well documented that numerical methods with more physical accuracy exhibit better long-time behavior than comparable methods that enforce less physics in their solutions. This work develops, analyzes and tests finite element methods that better enforce mass conservation in discrete velocity solutions to the NSE and MHD, helicity conservation for NSE, cross-helicity conservation in MHD, and magnetic field incompressibility in MHD

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Large scale finite element solvers for the large eddy simulation of incompressible turbulent flows

Premi extraordinari doctorat UPC curs 2015-2016, àmbit d’Enginyeria CivilIn this thesis we have developed a path towards large scale Finite Element simulations of turbulent incompressible flows. We have assessed the performance of residual-based variational multiscale (VMS) methods for the large eddy simulation (LES) of turbulent incompressible flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. We show that VMS thought as an implicit LES model can be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, since it provides pressure stabilization. In this work, we also consider a different approach, based on inf-sup stable elements and convection-only stabilization. In order to do so, we define a symmetric projection stabilization of the convective term using an orthogonal subscale decomposition. The accuracy and efficiency of this method compared with residual-based algebraic subgrid scales and orthogonal subscales methods for equal-order interpolation is also assessed in this thesis. Furthermore, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. Precisely, the symmetric projection stabilization approach is suitable for segregated Runge-Kutta time integration schemes. This combination, together with the use of block-preconditioning techniques, lead to elasticity-type and Laplacian-type problems that can be optimally preconditioned using the balancing domain decomposition by constraints preconditioners. The weak scalability of this formulation have been demonstrated in this document. Additionally, we also contemplate the weak imposition of the Dirichlet boundary conditions for wall-bounded turbulent flows. Four well known problems have been mainly considered for the numerical experiments: the decay of homogeneous isotropic turbulence, the Taylor-Green vortex problem, the turbulent flow in a channel and the turbulent flow around an airfoil.En aquesta tesi s'han desenvolupat diferents algoritmes per la simulació a gran escala de fluxos turbulents incompressibles mitjançant el mètode dels Elements Finits. En primer lloc s'ha avaluat el comportament dels mètodes de multiescala variacional (VMS) basats en el residu, per la simulació de grans vòrtexs (LES) de fluxos turbulents. S'han considerat diferents models VMS tenint en compte diferents aproximacions de les subescales, que inclouen tant subescales estàtiques o dinàmiques, una definicó lineal o nolineal, i diferents seleccions de l'espai de les subescales. S'ha demostrat que els mètodes VMS pensats com a models LES poden ser una alternativa als models basats en la física del problema. Aquest tipus de mètode normalment es combina amb l'ús de parelles de velocitat i pressió amb igual ordre d'interpolació. En aquest treball, també s'ha considerat un enfocament diferent, basat en l'ús d'elements inf-sup estables conjuntament amb estabilització del terme convectiu. Amb aquest objectiu, s'ha definit un mètode d'estabilització amb projecció simètrica del terme convectiu mitjançant una descomposició ortogonal de les subescales. En aquesta tesi també s'ha valorat la precisió i eficiència d'aquest mètode comparat amb mètodes basats en el residu fent servir interpolacions amb igual ordre per velocitats i pressions. A més, s'ha proposat un esquema d'integració en temps basat en els mètodes de Runge-Kutta que té dues propietats destacables. En primer lloc, el càlcul de la velocitat i la pressió es segrega al nivell de la integració temporal, sense la necessitat d'introduir tècniques de fraccionament del pas de temps. En segon lloc, els esquemes segregats de Runge-Kutta proposats, mantenen el mateix ordre de precisió tant per les velocitats com per les pressions. Precisament, els mètodes d'estabilització amb projecció simètrica són adequats per ser integrats en temps mitjançant esquemes segregats de Runge-Kutta. Aquesta combinació, juntament amb l'ús de tècniques de precondicionament en blocs, dóna lloc a problemes tipus elasticitat i Laplacià que poden ser òptimament precondicionats fent servir els anomenats \textit{balancing domain decomposition by constraints preconditioners}. La escalabilitat dèbil d'aquesta formulació s'ha demostrat en aquest document. Adicionalment, també s'ha contemplat la imposició de forma dèbil de les condicions de contorn de Dirichlet en problemes de fluxos turbulents delimitats per parets. En aquesta tesi principalment s'han considerat quatre problemes ben coneguts per fer els experiments numèrics: el decaïment de turbulència isotròpica i homogènia, el problema del vòrtex de Taylor-Green, el flux turbulent en un canal i el flux turbulent al voltant d'una ala.Award-winningPostprint (published version