Domain Decomposition for Stokes-Darcy Flows with Curved Interfaces

AbstractA non-overlapping domain decomposition method is developed for coupled Stokes-Darcy flows in irregular domains. The Stokes region is discretized by standard Stokes finite elements while the Darcy region is discretized by the multipoint flux mixed finite element method. The subdomain grids may not match on the interfaces and mortar finite elements are employed to impose weakly interface continuity conditions. The interfaces can be curved and matching conditions are imposed via appropriate mappings from physical grids to reference grids with flat interfaces. The global problem is reduced to a mortar interface problem, which is solved by the conjugate gradient method. Each iteration involves solving subdomain problems of either Stokes or Darcy type, which is done in parallel. Computational experiments are presented to illustrate the convergence of the discretization and the condition number of the interface operator

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.

Development of a predictive mathematical model for coupled Stokes–Darcy flows in cross-flow membrane filtration

Free flow regimes accompanied by porous walls feature commonly in a variety of natural processes and industrial applications such as groundwater flows, packed beds, arterial blood flows and cross-flow and dead-end filtrations. Cross-flow microfiltration or ultrafiltration processes are generally employed in a range of industrial situations ranging from oil to medical applications. The coupled free/porous fluid transport phenomenon plays an equally important role along with the particle transport mechanisms concerning the separation efficiency of cross-flow membrane filtration. To provide a theoretical background for the experimental outcomes of cross-flow filtration, a mathematically sound model is desired which can reliably represent the interfacial boundary whilst maintaining the continuity of flow field variables across the interface between the free and porous flow regimes. Notwithstanding the numerous attempts reported in the literature, the development of a generic mathematical model for coupled flows has been prohibited by the complexities of interactions between the free and the porous flow systems. Henceforth, the aim of present work is to gain a better mathematical understanding of the interfacial phenomena encountered in coupled free and porous flow regimes applicable to cross-flow filtration systems. The free flow dynamics can be justifiably represented by the Stokes equation whereas the non-isothermal, non-inertial and incompressible flow in a low permeability porous medium can be handled by the Darcy equation. Solutions to the system of partial differential equations (PDE’s) are obtained using the finite element method employing mixed interpolations for the primary field variables which are velocity and pressure. A nodal replacement scheme previously developed by the same authors has been effectively enforced as the boundary constraint at the free-porous interface for coupling the two physically different flow regimes in a single mathematical model. A series of computational experiments for permeability values of the porous medium ranging between 10-6 -10-12 m2 have been performed to examine the susceptibility of the developed model towards complex and irregular shaped geometries. Our results indicate that at high permeability values, the discrepancy in mass balance calculations is observed to be significant for a curved porous surface, which may be attributed to the inability of the Darcy equation to represent the flow dynamics in a highly permeable medium. At a low permeability, a very small amount of fluid permeated through the free/porous interface as most of the fluid leaves the domain through the free flow exit. The geometry and permeability of the free/porous interface are found to affect the amount of fluid passing through the porous medium significantly. All the numerical solutions that are presented have been theoretically validated for their accuracy by computing the overall mass continuity across the computational domains

Coupling Stokes-Darcy flow with transport on irregular geometries

This thesis studies a mathematical model, in which Stokes-Darcy flow system is coupled with a transport equation. The objective is to develop stable and convergent numerical schemes that could be used in environmental applications. Special attention is given to discretization methods which can handle irregular geometry. First, we will use a multiscale mortar finite element method to discretize coupled Stokes-Darcy flows on irregular domains. Especially, we will utilize a special discretization method called multi-point flux mixed finite element method to handle Darcy flow. This method is accurate for rough grids and rough full tensor coefficients, and reduces to a cell-centered pressure scheme. On quadrilaterals and hexahedra the method can be formulated either on the physical space or on the reference space, leading to a non-symmetric or symmetric scheme, respectively. While Stokes region is discretized by standard inf-sup stable elements. The mortar space can be coarser and it is used to approximate the normal stress on the interface and to impose weakly continuity of normal flux. The interfaces can be curved and matching conditions are imposed via appropriate mappings from physical grids to reference grids with flat interfaces. Another approach that we use to deal with the flow equations is based on non-overlapping domain decomposition. Domain decomposition enables us to solve the coupled Stokes-Darcy flow problem in parallel by partitioning the computational domain into subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling of the subdomain problems is removed through an iterative procedure. We investigate the properties of this method and derive estimates for the condition number of the associated algebraic system. To discretize the transport equation we develop a local discontinuous Galerkin mortar method. In the method, the subdomain grids need not match and the mortar grid may be much coarser, giving a two-scale method. We weakly impose the boundary condition on the inflow part of the interface and the Dirichlet boundary condition on the elliptic part of the interface via Lagrange multipliers. We develop stability for the concentration and the diffusive flux in the transport equation

Discretizations and Solvers for Coupling Stokes-Darcy Flows With Transport

This thesis studies a mathematical model, in which Stokes-Darcy flow system is coupled with a transport equation. The objective is to develop stable and convergent numerical schemes that could be used in environmental applications. Special attention is given to discretization methods that conserve mass locally. First, we present a global saddle point problem approach, which employs the discontinuous Galerkin method to discretize the Stokes equations and the mimetic finite difference method to discretize the Darcy equation. We show how the numerical scheme can be formulated on general polygonal (polyhedral in three dimensions) meshes if suitable operators mapping from degrees of freedom to functional spaces are constructed. The scheme is analyzed and error estimates are derived. A hybridization technique is used to solve the system effectively. We ran several numerical experiments to verify the theoretical convergence rates and depending on the mesh type we observed superconvergence of the computed solution in the Darcy region.Another approach that we use to deal with the flow equations is based on non-overlapping domain decomposition. Domain decomposition enables us to solve the coupled Stokes-Darcy flow problem in parallel by partitioning the computational domain into subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling of the subdomain problems is removed through an iterative procedure. We investigate the properties of this method and derive estimates for the condition number of the associated algebraic system. Results from computer tests supporting the convergence analysis of the method are provided. To discretize the transport equation we use the local discontinuous Galerkin (LDG) method, which can be thought as a discontinuous mixed finite element method, since it approximates both the concentration and the diffusive flux. We develop stability and convergence analysis for the concentration and the diffusive flux in the transport equation. The numerical error is a combination of the LDG discretization error and the error from the discretization of the Stokes-Darcy velocity. Several examples verifying the theory and illustrating the capabilities of the method are presented

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

A discretization method with non-matching grids is proposed for the coupled Stokes-Darcy problem that uses a mortar variable at the interface to couple the marker and cell (MAC) method in the Stokes domain with the Raviart-Thomas mixed finite element pair in the Darcy domain. Due to this choice, the method conserves linear momentum and mass locally in the Stokes domain and exhibits local mass conservation in the Darcy domain. The MAC scheme is reformulated as a mixed finite element method on a staggered grid, which allows for the proposed scheme to be analyzed as a mortar mixed finite element method. We show that the discrete system is well-posed and derive a priori error estimates that indicate first order convergence in all variables. The system can be reduced to an interface problem concerning only the mortar variables, leading to a non-overlapping domain decomposition method. Numerical examples are presented to illustrate the theoretical results and the applicability of the method

Pore-scale modeling of phase change in porous media

The combination of high-resolution visualization techniques and pore-scale flow modeling is a powerful tool used to understand multiphase flow mechanisms in porous media and their impact on reservoir-scale processes. One of the main open challenges in pore-scale modeling is the direct simulation of flows involving multicomponent mixtures with complex phase behavior. Reservoir fluid mixtures are often described through cubic equations of state, which makes diffuse-interface, or phase-field, theories particularly appealing as a modeling framework. What is still unclear is whether equation-of-state-driven diffuse-interface models can adequately describe processes where surface tension and wetting phenomena play important roles. Here we present a diffuse-interface model of single-component two-phase flow (a van der Waals fluid) in a porous medium under different wetting conditions. We propose a simplified Darcy-Korteweg model that is appropriate to describe flow in a Hele-Shaw cell or a micromodel, with a gap-averaged velocity. We study the ability of the diffuse-interface model to capture capillary pressure and the dynamics of vaporization-condensation fronts and show that the model reproduces pressure fluctuations that emerge from abrupt interface displacements (Haines jumps) and from the breakup of wetting films

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems