Matthew W. Farthing

Isogeometric analysis of free-surface flow

Abstract This paper presents the first application of isogeometric analysis, a new computational technology built on higher-order and higher-continuity basis functions employed in Computer-Aided Design and computer graphics, to the computation of free-surface phenomena described using the level set approach. The method is based on the variational framework that is suitable for discretization by standard finite elements as well as the basis functions employed in isogeometric analysis. The underlying numerical formulation globally conserves mass and preserves a sharp air–water interface for the entire length of the simulation. The numerical tests indicate that the proposed methodology gives an accurate description of the free-surface behavior in both quasi-steady and dynamic regimes. Furthermore, very good per-degree-of-freedom accuracy is obtained when higher-order and higher-continuity isogeometric discretizations are employed in free-surface computations.

WITHDRAWN: Influence of Porous Media Heterogeneity on NAPL Dissolution Fingering and Upscaled Mass Transfer

The utility of existing models for describing upscaled mass transfer from non-aqueous phase liquid (NAPL) were examined when preferential dissolution pathways form in NAPL-contaminated zones. Such models are needed because the coarse discretizations often used in simulators cannot capture the details of local-scale preferential dissolution pathways. Laboratory experiments were conducted in two well-characterized, heterogeneous packings that differed only in the correlation lengths of the permeability field. Experimental results were used to validate a numerical simulator for capturing the growth of centimeter-scale preferential NAPL dissolution patterns. Using data from these experiments and simulations, three methods for upscaling the mass transfer rate coefficient for NAPL dissolution [Imhoff et al., 2003; Saenton and Illangasekare, 2007; Christ et al., 2006] and an equilibrium stream tube model for predicting contaminant flux [Basu et al., 2008] were evaluated. These models account for the influence of dissolution fingering [Imhoff et al., 2003] or NAPL architecture [Saenton and Illangasekare, 2007; Christ et al., 2006; Basu et al., 2008] on contaminant flux from NAPL source zones, and were applied here to local-scale preferential NAPL dissolution. When the correlation length of permeability perpendicular to the mean water flow direction was 6.0 cm, greater than the scale of the dissolution fingers, only 4.2% of the NAPL resided in pools. Dissolution fingers formed in this experiment and all models predicted effluent concentrations with similar accuracy, with root mean square errors (RMSEs) between 0.03 and 0.05. When the correlation scale was smaller (1.0 cm), 31.3% of the NAPL was in pools and preferential dissolution pathways were dominated by channeling, preferential dissolution caused by spatial variations in aqueous phase velocity and NAPL dissolution rates. For this experiment all models performed poorly, with RMSEs between 0.12 and 0.35. Dissolution fingering was important when the permeability correlation length was sufficiently large that dissolution finger formation was not disrupted and NAPL pools were not dominant.

Free-Surface Flow and Fluid-Object Interaction Modeling With Emphasis on Ship Hydrodynamics

Abstract : This paper presents our approach for the computation of free-surface/rigid-body interaction phenomena with emphasis on ship hydrodynamics. We adopt the level set approach to capture the free-surface. The rigid body is described using six-degree-of-freedom equations of motion. An interface-tracking method is used to handle the interface between the moving rigid body and the fluid domain. An Arbitrary Lagrangian Eulerian version of the residual-based variational multiscale formulation for the Navier Stokes and level set equations is employed in order to accommodate the fluid domain motion. The free-surface/rigid body problem is formulated and solved in a fully coupled fashion. The numerical results illustrate the accuracy and robustness of the proposed approach.

A conservative level set method suitable for variable-order approximations and unstructured meshes

This paper presents a formulation for free-surface computations capable of handling complex phenomena, such as wave breaking, without excessive mass loss or smearing of the interface. The formulation is suitable for discretizations using finite elements of any topology and order, or other approaches such as isogeometric and finite volume methods. Furthermore, the approach builds on standard level set tools and can therefore be used to augment existing implementations of level set methods with discrete conservation properties. Implementations of the method are tested on several difficult two- and three-dimensional problems, including two incompressible air/water flow problems with available experimental results. Linear and quadratic approximations on unstructured tetrahedral and trilinear approximations on hexahedral meshes were tested. Global conservation and agreement with experiments as well as computations by other researchers are obtained.

Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow

Abstract Richards’ equation (RE) is commonly used to model flow in variably saturated porous media. However, its solution continues to be difficult for many conditions of practical interest. Among the various time discretizations applied to RE, the method of lines (MOL) has been used successfully to introduce robust, accurate, and efficient temporal approximations. At the same time, a mixed-hybrid finite element method combined with an adaptive, higher order time discretization has shown benefits over traditional, lower order temporal approximations for modeling single-phase groundwater flow in heterogeneous porous media. Here, we extend earlier work for single-phase flow and consider two mixed finite element methods that have been used previously to solve RE using lower order time discretizations with either fixed time steps or empirically based adaption. We formulate the two spatial discretizations within a MOL context for the pressure head form of RE as well as a fully mass-conservative version. We conduct several numerical experiments for both spatial discretizations with each formulation, and we compare the higher order, adaptive time discretization to a first-order approximation with formal error control and adaptive time step selection. Based on the numerical results, we evaluate the performance of the methods for robustness and efficiency.

Efficient steady-state solution techniques for variably saturated groundwater flow

Abstract We consider the simulation of steady-state variably saturated groundwater flow using Richards’ equation (RE). The difficulties associated with solving RE numerically are well known. Most discretization approaches for RE lead to nonlinear systems that are large and difficult to solve. The solution of nonlinear systems for steady-state problems can be particularly challenging, since a good initial guess for the steady-state solution is often hard to obtain, and the resulting linear systems may be poorly scaled. Common approaches like Picard iteration or variations of Newton’s method have their advantages but perform poorly with standard globalization techniques under certain conditions. Pseudo-transient continuation has been used in computational fluid dynamics for some time to obtain steady-state solutions for problems in which Newton’s method with standard line-search strategies fails. Here, we examine the use of pseudo-transient continuation as well as Newton’s method combined with standard globalization techniques for steady-state problems in heterogeneous domains. We investigate the methods’ performance with direct and preconditioned Krylov iterative linear solvers. We then make recommendations for robust and efficient approaches to obtain steady-state solutions for RE under a range of conditions.

A comparison of high-resolution, finite-volume, adaptive-stencil schemes for simulating advective-dispersive transport

We investigate a set of adaptive‐stencil, finite-volume schemes used to capture sharp fronts and shocks in a wide range of fields. Our objective is to determine the most promising methods available from this set for solving sharp-front advective‐dispersive transport problems. Schemes are evaluated for a range of initial conditions, and for Peclet and Courant numbers. Based upon results from this work, we identify the most promising schemes based on eAciency and robustness. ” 2000 Elsevier Science Ltd. All rights reserved.

Modeling NAPL dissolution fingering with upscaled mass transfer rate coefficients

The dissolution of nonaqueous phase liquids (NAPLs) at residual saturation in porous media has sometimes resulted in the development of preferential dissolution pathways or NAPL dissolution fingers. While NAPL dissolution fingering may be modeled using numerical simulators with fine discretization, this approach is computational intensive. We derived an expression for an upscaled mass transfer rate coefficient that accounts for the growth of dissolution fingers within porous media contaminated uniformly with residual NAPL. This expression was closely related to the lengthening of the dissolution front. Data from physical experiments and numerical simulations in two dimensions were used to examine the growth of the dissolution front and the corresponding upscaled mass transfer rate coefficient. Using this upscaled mass transfer rate coefficient, the time when dissolution fingering results in a reduction in the overall mass transfer rate and thus controls the rate of NAPL dissolution was determined. This crossover time is a convenient parameter for assessing the influence of dissolution fingering on NAPL removal. For the physical experiments and numerical simulations analyzed in this study, the crossover time to dissolution fingering control always occurred before the dissolution front had moved 14 cm within NAPL-contaminated porous media, which is small compared to the scale of typical systems of concern. To verify the utility of this approach, data from a three-dimensional physical experiment were predicted reasonably well using an upscaled mass transfer rate coefficient that was determined independently from this experiment.

Mixed finite element methods and higher-order temporal approximations

The accurate numerical approximation of subsurface flow and transport processes in heterogeneous aquifers remains difficult. A necessary step in this task is the accurate representation of fluid velocity fields. In the recent past, mixed finite element methods have been investigated, since they provide velocity approximations that both conserve mass over individual mesh elements and are continuous across element interfaces. But, little work has been done to this point for fully three-dimensional problems. Furthermore, the existing temporal discretizations have been restricted to traditional first- and second-order approximations. In this work, we consider a fully three-dimensional mixed-hybrid finite element spatial discretization together with an adaptive higher-order time discretization applied to single-phase groundwater flow in heterogeneous porous media. We compare the adaptive higher-order temporal approximation, which is robust and provides formal error control, to traditional lower-order methods for accuracy and efficiency for a set of problems. The numerical experiments demonstrate that this approach provides several benefits with negligible computational overhead.

Finite element methods for variable density flow and solute transport

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.