Efficient Numerical Methods for Magnetohydrodynamic Flow

This dissertation studies efficient numerical methods for approximating solu-tions to viscous, incompressible, time-dependent magnetohydrodynamic (MHD) flows and computing MHD flows ensembles. Chapter 3 presents and analyzes a fully discrete, decoupled efficient algorithm for MHD flow that is based on the Els¨asser variable formulation, proves its uncondi-tional stability with respect to the timestep size, and proves its unconditional con-vergence. Numerical experiments are given which verify all predicted convergence rates of our analysis, show the results of the scheme on a set of channel flow problems match well the results found when the computation is done with MHD in primitive variables, and finally illustrate that the scheme performs well for channel flow over a step. In chapter 4, we propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep, yet maintains un-conditional stability with respect to timestep size. The scheme is optimally accu-rate in space, and behaves like second order in time in practice. The proposed method chooses θ ∈ [0, 1], dependent on the viscosity ν and magnetic diffusiv-ity νm, so that unconditionally stability is achieved, and gives temporal accuracy O(∆t2 + (1 − θ)|ν − νm|∆t). In practice, ν and νm are small, and so the method be-haves like second order. We show the θ-method provides excellent accuracy in cases where usual BDF2 is unstable. Chapter 5 proposes an efficient algorithm and studies for computing flow en-sembles of incompressible MHD flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through an efficient algo-rithm that, at each time step, uses the same coefficient matrix for each of the J system solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Els¨asser variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments. This work concludes with chapter 6, which proposes, analyzes and tests high order algebraic splitting methods for MHD flows. The key idea is to applying Yosida-type algebraic splitting to the incremental part of the unknowns at each time step. This reduces the block Schur complement by decoupling it into two Navier-Stokes-type Schur complements, each of which is symmetric positive definite and the same at each time step. We prove the splitting is third order in ∆t, and if used together with (block-)pressure correction, is fourth order. A full analysis of the solver is given, both as a linear algebraic approximation, and as a finite element discretization of an approximation to the un-split discrete system. Numerical tests are given to illustrate the theory and show the effectiveness of the method. Finally, conclusions and future works are discussed in the final chapter

A Penalty-projection based Efficient and Accurate Stochastic Collocation Method for Magnetohydrodynamic Flows

We propose, analyze, and test a penalty projection-based efficient and accurate algorithm for the Uncertainty Quantification (UQ) of the time-dependent Magnetohydrodynamic (MHD) flow problems in convection-dominated regimes. The algorithm uses the Els\"asser variables formulation and discrete Hodge decomposition to decouple the stochastic MHD system into four sub-problems (at each time-step for each realization) which are much easier to solve than solving the coupled saddle point problems. Each of the sub-problems is designed in a sophisticated way so that at each time-step the system matrix remains the same for all the realizations but with different right-hand-side vectors which allows saving a huge amount of computer memory and computational time. Moreover, the scheme is equipped with ensemble eddy-viscosity and grad-div stabilization terms. The stability of the algorithm is proven rigorously. We prove that the proposed scheme converges to an equivalent non-projection-based coupled MHD scheme for large grad-div stabilization parameter values. We examine how Stochastic Collocation Methods (SCMs) can be combined with the proposed penalty projection UQ algorithm. Finally, a series of numerical experiments are given which verify the predicted convergence rates, show the algorithm's performance on benchmark channel flow over a rectangular step, and a regularized lid-driven cavity problem with high random Reynolds number and magnetic Reynolds number.Comment: 28 pages, 13 figure

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

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

Isogeometric finite element methods for liquid metal magnetohydrodynamics

A fusion blanket is a key component in a fusion reactor which extracts heat energy, protects the surrounding structure and possibly produces tritium, one of the fuels required for the deuterium-tritium fusion reaction. Interest in magneto-hydrodynamic (MHD) effects in the fusion blanket has been growing due to the promising prospect of a liquid breeder blanket, due to its high power density and the possibility of sustainable production of tritium. However, MHD effects can significantly influence the operating performance of the fusion blanket and an accurate and reliable analysis of the MHD effects are critical in its design. Significant progress in the numerical study of MHD has been made recently, due in large part to the advancement in computing power. However, its maturity has not yet reached a point comparable with standard CFD solvers. In particular, complex domains and complex externally applied magnetic fields present additional challenges for numerical schemes in MHD. For that reason, the application of isogeometric analysis is considered in this thesis. Isogeometric Analysis (IGA) is a new class of numerical method which integrates Computer Aided Design (CAD) into Finite Element Analysis (FEA). In IGA, B-splines and NURBS, which are the building blocks used to construct a geometry in CAD, are also used to build the finite element spaces. This allows to represent geometries more accurately, and in some cases exactly. This may help advance the progress of numerical studies of MHD effects, not only in fusion blanket scenarios, but more widely. In this thesis, we develop and study a number of types of IGA based MHD solver; a fully-developed MHD flow solver, a steady-state MHD solver and a time-dependent MHD solver. These solvers are validated using analytical methods and methods of manufactured solution and are compared with other numerical schemes on a number of benchmark problems.Open Acces

Low Complexity, Time Accurate, Model Accurate Algorithms in Computational Fluid Dynamics

Computational fluid dynamics is an essential research area that is of crucial importance in comprehending of fluid flows in mechanical and hydrodynamic processes. Accurate, efficient and reliable simulation of flows occupies a central place in the development of computational science. In this work, we explore various numerical methods and utilize them to improve flow predication. Four research projects are conducted and show evidence in enhancement of accuracy, efficiency and reliability of prediction of fluid motion. We first propose a low computationally complex, stable and adaptive method for time accurate approximation of the evolutionary stokes Darcy system and Navier-Stokes equations. The improved method post-processes the solutions of the Backward Euler scheme by adding no more than three lines to an existing program. Time accuracy is increased from first to second order and the overdamping of the Backward Euler method is removed. The second project is to develop an efficient method to describe magnetohydrodynamic flows at low magnetic Reynolds numbers. The decoupled method is based on the artificial compression and partitioned schemes. Computational efficiency is greatly improved because we only need to solve linear problems at each time step with systems decouple by physical processes. Last but not least, we introduce a way to correct the Baldwin-Lomax model for non-equilibrium turbulence, which is often considered impossible to simulate due to backscatter. The corrected Baldwin-Lomax model not only shows that effects of fluctuations on means are dissipative on time average but also can have bursts for which energy flow reverses. For each project, we present comprehensive error and stability analysis and provide different numerical experiments to further support theoretical theories

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