Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM

The method presented in Aubry et al. (Comput Struc 83:1459–1475, 2005) for the solution of an incompressible viscous fluid flow with heat transfer using a fully Lagrangian description of motion is extended to three dimensions (3D) with particular emphasis on mass conservation. A modified fractional step (FS) based on the pressure Schur complement (Turek 1999), and related to the class of algebraic splittings Quarteroni et al. (Comput Methods Appl Mech Eng 188:505–526, 2000), is used and a new advantage of the splittings of the equations compared with the classical FS is highlighted for free surface problems. The temperature is semi-coupled with the displacement, which is the main variable in a Lagrangian description. Comparisons for various mesh Reynolds numbers are performed with the classical FS, an algebraic splitting and a monolithic solution, in order to illustrate the behaviour of the Uzawa operator and the mass conservation. As the classical fractional step is equivalent to one iteration of the Uzawa algorithm performed with a standard Laplacian as a preconditioner, it will behave well only in a Reynold mesh number domain where the preconditioner is efficient. Numerical results are provided to assess the superiority of the modified algebraic splitting to the classical FS

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

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

Complex flow and transport phenomena in porous media

This thesis analyzes partial differential equations related to the coupled surface and subsurface flows and develops efficient high order discontinuous Galerkin (DG) methods to solve them numerically. Specifically, the coupling of the Navier-Stokes and the Darcy's equations, which is encountered in the environmental problem of groundwater contamination through lakes and rivers, is considered. Predicting accurately the transport of contaminants by this coupled flow is of great importance for the remediation strategies. The first part of this thesis analyzes a weak formulation of the time-dependent Navier-Stokes equation coupled with the Darcy's equation through the Beavers-Joseph-Saffman condition. The analysis changes depending on whether the inertial forces are included in the interface conditions or not. The inclusion of the inertial forces (Model I) remedies the difficulty in the analysis caused by the nonlinear convection term; however, it does not reflect the physical interactions on the interface correctly. Hence, I also analyze the weak problem by omitting these forces (Model II) which complicates the analysis and necessitates an extra small data condition. For Model I, a fully discrete scheme based on the DG method and the Crank-Nicolson method is introduced. The convergence of the scheme is proven with optimal error estimates. The second part couples the surface flow and a convection-diffusion type transport with miscible displacement in the subsurface. Initially, I consider the coupled stationary Stokes and Darcy's equations for the flow and establish the existence of a weak solution. Next, imposing additional assumptions on the data, I extend the result to the nonlinear case where the surface flow is given by the Navier-Stokes equation. The analysis also applies to the particular case where the flow is loosely coupled to the transport, that is, the velocity field obtained from the flow is an input for the transport equation. The flow is discretized by combinations of the continuous finite element method and the DG method whereas the discretization of the transport is done by a combined DG and backward Euler methods. The scheme yields optimal error estimates and its robustness for fractured porous media is shown by a numerical example

Modeling studies and numerical analyses of coupled PDEs system in electrohydrodynamics

Electrohydrodynamics (EHD) is the term used for the hydrodynamics coupled with electrostatics, whose governing equations consist of the electrostatic potential (Poisson) equation, the ionic concentration (Nernst-Planck) equations, and Navier-Stokes equations for an incompressible, viscous dielectric liquid. In this dissertation, we focus on a specic application of EHD - fuel cell dynamics - in the eld of renewable and clean energy, study its traditional model and attempt to develop a new fuel cell model based on the traditional EHD model. Meanwhile, we develop a series of ecient and robust numerical methods for these models, and carry out their numerical analyses on the approximation accuracy. In particular, we analyze the error estimates of nite element method for a simplied 2D isothermal steady state two-phase transport model of Proton Exchange Membrane Fuel Cell (PEMFC) as well as its transient version. On the aspect of hydrodynamics arising in the fuel cell system, the fluid flow through the open channels and porous media at the same time, both Navier-Stokes equations and Darcy\u27s law are involved in the fluid domains, leading to a Navier-Stokes-Darcy coupling problem. In this dissertation, we study a one-continuum model approach, so-called Brinkman model, to overcome this problem in a more ecient way. To develop a new fuel cell model based on EHD theory, in addition to the two-phase transport model of fuel cells, we carry out numerical analyses for Poisson-Nernst-Planck (PNP) equations using both standard FEM and mixed FEM, which are the essential governing equations involved by EHD model. Finally, we are able to further extend the traditional fuel cell model to more general cases in view of EHD characteristics, and develop a new fuel cell model by appropriately combining PNP equations with the traditional fuel cell model. We conduct the error analysis for PNP-Brinkman system in this dissertation

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

Approximating fast, viscous fluid flow in complicated domains

Typical industrial and biological flows often occur in complicated domains that are either infeasible or impossible to resolve. Alternatives to solving the Navier-Stokes equations (NSE) for the fluid velocity in the pores of these problems must be considered. We propose and analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity tends to infinity and permeability K tends to 0 in solid obstacles, and K tends to infinity in fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider inhomogeneous Dirichlet boundary conditions and non-solenoidal velocity (to model sources/sinks). Coupling between these two conditions makes even existence of solutions subtle. We establish conditions for the well-posedness of the continuous and discrete problem. We also establish convergence as Brinkman viscosity tends to infinity and K tends to 0 in solid obstacles, as K tends to infinity in fluid region, and as the mesh width vanishes. We prove similar results for time-dependent Brinkman equations for backward-Euler (BE) time-stepping. We provide numerical examples confirming theory including convergence of velocity, pressure, and drag/lift.We also investigate the stability and convergence of the fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We show herein that no such restriction is required. Moreover, we propose a new linear extrapolation of the convecting velocity for CNLE so that the approximating velocities converge without without time-step restriction in l^{infty}(H^1) along with the discrete time derivative of the velocity in l^2(L^2). The new extrapolation ensures energetic stability of CNLE in the case of inhomogeneous boundary data. Such a result is unknown for conventional CNLE (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE

Analysis of an implicitly extended Crank-Nicolson scheme for the heat equation on a time-dependent domain

We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version