Partitioned methods for coupled fluid flow problems

Many flow problems in engineering and technology are coupled in their nature. Plenty of turbulent flows are solved by legacy codes or by ones written by a team of programmers with great complexity. As knowledge of turbulent flows expands and new models are introduced, implementation of modern approaches in legacy codes and increasing their accuracy are of great concern. On the other hand, industrial flow models normally involve multi-physical process or multi-domains. Given the different nature of the physical processes of each subproblem, they may require different meshes, time steps and methods. There is a natural desire to uncouple and solve such systems by solving each subphysics problem, to reduce the technical complexity and allow the use of optimized legacy sub-problems' codes. The objective of this work is the development, analysis and validation of new modular, uncoupling algorithms for some coupled flow models, addressing both of the above problems. Particularly, this thesis studies: i) explicitly uncoupling algorithm for implementation of variational multiscale approach in legacy turbulence codes, ii) partitioned time stepping methods for magnetohydrodynamics flows, and iii) partitioned time stepping methods for groundwater-surface water flows. For each direction, we give comprehensive analysis of stability and derive optimal error estimates of our proposed methods. We discuss the advantages and limitations of uncoupling methods compared with monolithic methods, where the globally coupled problems are assembled and solved in one step. Numerical experiments are performed to verify the theoretical results

PARTITIONED METHODS FOR COUPLED FLUID FLOW AND TURBULENCE FLOW PROBLEMS

Numerical simulation of different physical processes in different regions is one of the wide variety of real world applications. Many important applications such as coupled surface water groundwater flows require the accurate solution of multi-domain, multi-physics coupling of unobstructed flows with filtration or porous media flows. There are large advantages in efficiency, storage, accuracy and programmer effort in using partitioned methods build from components optimized for the individual sub-processes. On the other hand, the multi-domain or multi-physical process describes different natures of the physical processes of each subproblem. They may require different meshes, time steps and methods. There is a natural desire to uncouple and solve such systems by solving each sub physics problem, to reduce the technical complexity and allow the use of optimized, legacy sub-problems' codes in fluid flow. Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear. Including a particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to many of the current models which use linear filters to select the eddies for damping. The objective of the first part of this work is the development, analysis and validation of new partitioned algorithms for some coupled flow models, addressing some of the above problems. Particularly, this thesis studies: i) long time stability of four methods for splitting the evolutionary Stokes-Darcy problem into Stokes and Darcy sub problems, ii) analysis of a multi-rate splitting method for uncoupling evolutionary groundwater-surface water flows, and iii) a connection between filter stabilization and eddy viscosity models. For each problem, we give comprehensive analysis of stability and derive optimal error estimates of our proposed methods. Numerical experiments are performed to verify the theoretical results

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-Order, Strongly Stable Methods for Uncoupling Groundwater-Surface Water Flow

Many environmental problems today involve the prediction of the migration of contaminants in groundwater-surface water flow. Sources of contaminated groundwater-surface water flow include: landfill leachate, radioactive waste from underground storage containers, and chemical run-off from pesticide usage in agriculture, to name a few. Before we can track the transport of pollutants in environmental flow, we must first model the flow itself, which takes place in a variety of physical settings. This necessitates the development of accurate numerical models describing coupled fluid (surface water) and porous media (groundwater) flow, which we assume to be described by the fully evolutionary Stokes-Darcy equations. Difficulties include finding methods that converge within a reasonable amount of time, are stable when the physical parameters of the flow are small, and maintain stability and accuracy along the interface. Ideally, because there exist a wide variety of physical scenarios for this coupled flow, we desire numerical methods that are versatile in terms of stability and practical in terms of computational cost and time. The approach to model this flow studied herein seeks to take advantage of existing efficient solvers for the separate sub-flows by uncoupling the flow so that at each time level we may solve a separate surface and groundwater problem. This approach requires only one (SPD) Stokes and one (SPD) Darcy sub-physics and sub-domain solve per time level for the time-dependent Stokes-Darcy problem. In this dissertation, we investigate several different methods that uncouple groundwater-surface water flow, and provide thorough analysis of the stability and convergence of each method along with numerical experiments

Efficient and Long-Time Accurate Second-Order Methods for the Stokes-Darcy System

We propose and study two second order in time implicit-explicit methods for the coupled Stokes-Darcy system that governs flows in karst aquifers and other subsurface flow systems. the first method is a combination of a second-order backward differentiation formula and the second order Gear\u27s extrapolation approach. the second is a combination of the second-order Adams-Moulton and second-order Adams-Bashforth methods. Both algorithms only require the solution of decoupled Stokes and Darcy problems at each time-step. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. the uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long-time scales. Error estimates for fully discretized schemes using finite element spatial discretization\u27s are derived. Numerical examples are provided that illustrate the accuracy, efficiency, and long-time stability of the two schemes. © 2013 Society for Industrial and Applied Mathematics

An Efficient and Long-Time Accurate Third-Order Algorithm for the Stokes–Darcy System

A third order in time numerical IMEX-type algorithm for the Stokes–Darcy system for flows in fluid saturated karst aquifers is proposed and analyzed. a novel third-order Adams–Moulton scheme is used for the discretization of the dissipative term whereas a third-order explicit Adams–Bashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. the scheme is efficient in the sense that one needs to solve, at each time step, decoupled Stokes and Darcy problems. Therefore, legacy Stokes and Darcy solvers can be applied in parallel. the scheme is also unconditionally stable and, with a mild time-step restriction, long-time accurate in the sense that the error is bounded uniformly in time. Numerical experiments are used to illustrate the theoretical results. to the authors\u27 knowledge, the novel algorithm is the first third-order accurate numerical scheme for the Stokes–Darcy system possessing its favorable efficiency, stability, and accuracy properties

A sensitivity study of artificial viscosity in a defect-deferred correction method for the coupled Stokes/Darcy model

This paper analyzes the sensitivity of the artificial viscosity in the defect deferred correction method for the non-stationary coupled Stokes/Darcy model. For the defect step and the deferred correction step of the defect deferred correction method, we respectively give the corresponding sensitivity systems related to the change of artificial viscosity. Finite element schemes are devised for computing solutions to the sensitivity systems. Finally, we will verify the theoretical analysis results through numerical experiments. Our results reveal that the solution is sensitive for small values of the artificial viscosity, and when the viscosity/hydraulic conductivity coefficients are small

COUPLED SURFACE AND GROUNDWATER FLOWS: QUASISTATIC LIMIT AND A SECOND-ORDER, UNCONDITIONALLY STABLE, PARTITIONED METHOD

In this thesis we study the fully evolutionary Stokes-Darcy and Navier-Stokes/Darcy models for the coupling of surface and groundwater flows versus the quasistatic models, in which the groundwater flow is assumed to instantaneously adjust to equilibrium. Further, we develop and analyze an efficient numerical method for the Stokes-Darcy problem that decouples the sub-physics flows, and is 2nd-order convergent, uniformly in the model parameters. We first investigate the linear, fully evolutionary Stokes-Darcy problem and its qua- sistatic approximation, and prove that the solution of the former converges to the solution of the latter as the specific storage parameter converges to zero. The proof reveals that the quasistatic problem predicts the solution accurately only under certain parameter regimes. Next, we develop and analyze a partitioned numerical method for the evolutionary Stokes- Darcy problem. We prove that the new method is asymptotically stable, and second-order, uniformly convergent with respect to the model parameters. As a result, it can be used to solve the quasistatic Stokes-Darcy problem. Several numerical tests are performed to support the theoretical efficiency, stability, and convergence properties of the proposed method. Finally, we consider the nonlinear Navier-Stokes/Darcy problem and its quasistatic ap- proximation under a modified balance of forces interface condition. We show that the solution of the fully evolutionary problem converges to the quasistatic solution as the specific stor- age converges to zero. To prove convergence in three spatial dimensions, we assume more regularity on the solution, or small data

Decoupling Techniques for Coupled PDE Models in Fluid Dynamics

We review decoupling techniques for coupled PDE models in fluid dynamics. In particular, we are interested in the coupled models for fluid flow interacting with porous media flow and the fluid structure interaction (FSI) models. For coupled models for fluid flow interacting with porous media flow, we present decoupled preconditioning techniques, two-level and multilevel methods, Newton-type linearization-based two-level and multilevel algorithms, and partitioned time-stepping methods. The main theory and some numerical experiments are given to illustrate the effectiveness and efficiency of these methods. For the FSI models, partitioned time-stepping algorithms and a multirate time-stepping algorithm are carefully studied and analyzed. Numerical experiments are presented to highlight the advantages of these methods