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

Stability of the IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations

Abstract Stability is proven for two second order, two step methods for uncoupling a system of two evolution equations with exactly skew symmetric coupling: the Crank-Nicolson Leap Frog (CNLF) combination and the BDF2-AB2 combination. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. For CNLF we prove stability for the coupled system under the time step condition suggested by linear stability theory for the Leap-Frog scheme. This seems to be a first proof of a widely believed result. For BDF2-AB2 we prove stability under a condition that is better than the one suggested by linear stability theory for the individual methods. This report is an expended version of the one submitted for publication

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

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

Computational fluid dynamics of coupled free/porous regimes: a specialised case of pleated cartridge filter

The multidisciplinary project AEROFIL has been defined and coordinated with the idea of developing novel filter designs to be employed in aeronautic hydraulic systems. The cartridge filters would be constructed using eco-friendly filtration media supported by unconventional disposable or reusable solid components. My main contribution to this project is the development of a robust and cost-effective design and analysis tool for simulating the hydrodynamics in these pleated cartridge filters. The coupled free and porous flow regimes are generally observed in filtration processes. These processes have been the subject of intense investigation for researchers over the decades who are striving hard to resolve some of the critical issues related to the free/porous interfacial constraints and their mathematical representations concerning its industrial applications. [Continues.

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

Time-dependent Stokes-Darcy Flow with Deposition

This thesis investigates two nonlinear systems of time-dependent partial differential equations that model a filtration process. Existence and uniqueness results for the governing equations is established. For each system, a finite element scheme capable of approximating the solutions is investigated. Accompanying numerical experiments corroborate the analytical findings. Finally, an optimization application concerning the design of a filter is discussed and supported with a numerical study

Three dimensional hydrodynamic modelling of combined free/porous flow regimes

In the present scenario, as advances in research, technology and engineering application have been on a rise , thus persuading researchers and engineers to employ new computer modelling techniques for the design and analysis, mainly due to time, environmental and economic constraints. Moreover it also forms a basis for any observed anomalies, when comparing with the simulated and experimental results and taking steps to develop optimum design strategies. The present research work deals with the development of novel ftlter designs when employed in aeronautical hydraulic systems. These pleated cartridge ftlters would be fabricated using eco-friendly fIltering media supported by unconventional disposable or reusable solid components. The primary focus of the present research work to develop a robust cost-effective simulating tools for simulating the results in the hydrodynamic behaviour of the fluid in pleated cartridge ftlters. As observed in any ftltration process, it comprises of two flow regimes namely free flow and porous flow regimes. For over five decades, it had been a subject of intense research and investigation for researchers, scientist and engineers to resolve some of the critical and vital issues related to filtration process. The main problems, when compared to others, that are associated with such processes are the free/porous interfacial constraints along with boundary conditions and their mathematical representation with respect to the industrial applications. A three dimensional model has been developed to represent the momentum and mass conservation for creeping incompressible flow in coupled free/porous flow regimes. In order to take into consideration the rheological behaviour of the fluid, power law model has been included, which forms the constitutive equation, and the viscosity of the fluid has been updated for the highly viscous specially formulated hydraulic fluid. For any numerical technique of analysis, on vital aspect is the boundary conditions that are imposed on the surface/volume/edge of the domain under consideration. The free (Stokes) and porous flow (Darey) regimes have been linked and solved in conjunction with continuity equations on a perturbed continuity scheme based on the standard Galerkin weighted residual finite element method. The perturb continuity UVWP finite element scheme is based On the equal order interpolation approximations and the discretized working equations are then transformed into the local coordinate system using iso-parametric mapping. The elements used are linear (8 nodded) hexahedral elements. The integrals in the elemental stiffness equations were calculated using Gauss-Legendre quadrature. After evaluation of the members of the elemental stiffness matrix, they are assembled over the common nodes in the computational grid to obtain a system of algebraic equations. After substituting the boundary conditions, the system becomes determinate and the algebraic equations can be solved using a frontal solution method. The described simulations are carried out using an in-house developed lnte! Visual FORTRAN code. The time stepping technique used here is second order Taylor-Galerkin method. The concept of compression permeability model developed by Nassehi et aL Nassehi et aL, 2005J ( developed for two dimensional case and now extended to three dimensional case) has been used to into the flow model to take into account the effects arising due to the mtration area loss in pleated cartridge filters and degree or extent of compression of the fUter medium. Significant over-use of media material or the need for changes to the geometric or mechanical design can be identified using the procedures described.EThOS - Electronic Theses Online ServiceGBUnited Kingdo