# Browsing by Subject "Fluid dynamics--Mathematical models"

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Item Analysis of a Darcy-Stokes system modeling flow through vuggy porous media(2004) Lehr, Heather Lyn; Arbogast, Todd James, 1957-Show more Our goal is to accurately model flow through subsurface systems composed of vuggy porous media. A vug is a small cavity in a porous medium which is large relative to the intergranular pore size. A vuggy porous medium is a porous medium with vugs scattered throughout it. While the vugs are often small, they can have a tremendous effect on the flow of fluid through the medium. We first introduce our microscale mathematical model for flow of an incompressible, viscous fluid in vuggy porous media. Our next step is to obtain a homogenized macroscale model. In order to do so, we assume periodicity of the medium. We obtain necessary existence and uniqueness results, error estimates, and slight generalizations of two-scale convergence results for bi-modal media. First using formal homogenization and then the rigorous two-scale convergence method, we show that our microscale model homogenizes to give a much simpler modified Darcy’s law macroscale model. In this homogenized model, the permeability tensor is modified to capture the effects of the vugs on the flow through the medium. In order to compute the homogenized permeability tensor, we essentially compute our microscale system on a (much smaller) representative cell. Toward this end, we introduce two numerical methods for the microscale model. We combine a discontinuous Galerkin method with a low order RaviartThomas element and obtain suboptimal convergence rates for the first method. The second method differs only slightly from the first, but yields optimal convergence rates. Unfortunately, it is less efficient in practical implementations.Show more Item A computational fluid dynamics simulation model for flare analysis and control(2006) Castiñeira Areas, David; Edgar, Thomas F.Show more Industrial flares are units designed to safely dispose of waste hydrocarbon gases from chemical and petrochemical plants by burning gases to carbon dioxide and steam, which are then released to the atmosphere. There is still great uncertainty about flare efficiency and the resultant gas emissions under different operating conditions. For this reason, environmental agencies have encouraged the development of predictive models for flare gas combustion systems, so effective control and mitigation strategies can be implemented. The principal focus of this dissertation is to develop mathematical models of industrial flares that predict the efficiency of these industrial combustion systems. For this purpose, a computational fluid dynamics (CFD) simulation model is implemented to analyze the effects of variables such as ambient wind velocity, gas heating value, and steam injection on flare combustion efficiency. Some advanced chemistry and turbulence submodels are also implemented to describe the complex flare flow phenomena. Simulation results show that flares may represent an important source of gas emissions due to inefficient operation at high crosswinds and large steam/fuel ratios. The predictive models presented in this work will allow for better estimation of the resulting gas emissions from industrial plants. Use of these simulation models will also yield economic savings for environmental studies compared to setting up expensive flare experiments. In addition, these predictive models allow for a detailed analysis of species concentration profiles and turbulent flow patterns within the flames, data which is not available experimentally. Furthermore, several instrumentation and control strategies for industrial flares are analyzed in this dissertation. A new approach for flare monitoring based on multivariate image analysis is proposed so that flare combustion efficiency can be measured in real-time.Show more Item A discontinuous Galerkin method for two- and three-dimensional shallow-water equations(2004) Aizinger, Vadym; Dawson, Clinton N.Show more The shallow-water equations (SWE), derived from the incompressible Navier-Stokes equations using the hydrostatic assumption and the Boussinesq approximation, are a standard mathematical representation valid for most types of flow encountered in coastal sea, river, and ocean modeling. They can be utilized to predict storm surges, tsunamis, floods, and, augmented by additional equations (e.g., transport, reaction), to model oil slicks, contaminant plume propagation, temperature and salinity transport, among other problems. An analytical solution of these equations is possible for only a handful of particular cases. However, their numerical solution is made challenging by a number of factors. The SWE are a system of coupled nonlinear partial differential equations defined on complex physical domains arising, for example, from irregular land boundaries. The bottom sea bed (bathymetry) is also often very irregular. Shallow-water systems are subjected to a wide range of external forces, such as the Coriolis force, surface wind stress, atmospheric pressure gradient, and tidal potential forces. As a result, flow regimes can vary greatly throughout the domain, from very smooth to high gradients and shock waves. The solution of the system is further complicated by the difficulties connected with the mathematical nature of the SWE. Most important is the coupling between the gravity forcing and the horizontal velocity field, which could lead to spurious spatial oscillations if the numerical algorithms are not chosen with care. One has to note, though, that most existing numerical methods for the SWE have serious drawbacks with regard to stability, local conservation, and ability to accommodate parallel implementation and hp-adaptivity. These problems become even more evident if we try to simulate problems involving discontinuities, shock waves, etc. The discontinuous Galerkin (DG) methods are an attempt to marry the most favorable features of the continuous finite element and finite volume schemes. On the one hand, they can employ approximating spaces of any order (not necessarily polynomial), and, on the other, the numerical fluxes on the inter-element boundaries are evaluated exactly as in the finite volume method – by solving a Riemann problem. As a result, these numerical schemes enjoy the same stability properties as the finite volume method. In addition, most DG methods guarantee local conservation of mass and momentum, which is, in many cases, a highly desirable quality reflecting the physical nature of the processes we are trying to model. In this thesis, we formulate the local discontinuous Galerkin method for the 2- and 3D shallow-water equations and derive stability and a priori error estimates for a simplified form of the 2D shallow-water equations and conduct stability analysis of our 3D scheme. In a series of numerical studies, we test both formulations using problems with discontinuous solutions as well as typical tidal flow problems. In addition, we demonstrate adaptive capabilities of the method using a shock-detection algorithm as an error indicator.Show more Item Numerical modeling of Stokesian emulsions(2002) Overfelt, James Robert; Rodin, G. J.; Van de Geijn, Robert A.Show more The principle objective of this dissertation was to develop a fast solution method for the accurate large scale modeling of fluid dynamics problems involving emulsions. The objective has been achieved by combining boundary element methods with fast iterative solvers based on the fast multipole method. Our solution method required extending the fast multipole method to generalized electrostatics problems to periodic domains. In addition, we developed an efficient hierarchical time stepping scheme and adopted accurate integration schemes, particularly useful in simulation of dense emulsions. Our solution method was implemented on parallel computers and used to model a creaming experiment of an emulsion.Show more Item A priori prediction of macroscopic properties of sedimentary rocks containing two immiscible fluids(2005) Gladkikh, Mikhail Nikolaevich; Bryant, Steven L.Show more The processes in porous media governed by capillary forces, such as drainage and imbibition cycles, infiltration from surface water, flow, transport, adsorption and dissolution of contaminants, are of great importance in soil science, subsurface geochemistry, petroleum engineering, and hydrology. The methodology proposed in this work is devoted to the pore-scale modeling of such processes. The goal is to be able to make a priori predictions of the macroscopic properties of model sedimentary rocks, such as capillary pressure curves, interfacial area, relative permeabilities and electrical properties. The idea is to conduct a theoretical investigation in a simple but physically representative model porous medium. The model is a random packing of equal spheres for which the coordinates of the centers have been measured. Knowledge of the coordinates determines the grain space and the void space in the packing, thereby overcoming a long-standing difficulty for theoretical approaches to pore-level modeling. Geological processes, such as isopachous and pore-filling cementation, are simulated in the sphere pack, thus creating simple models of sedimentary rocks with predetermined pore space geometry. Despite the simplicity of the model porous medium, it is a powerful tool for investigation of the flow and transport in soils and sedimentary rocks. In particular it allows a priori predictions of macroscopic behavior. This capability is the most important aspect of the approach. Because there are no arbitrarily prescribed parameters, the predictions can be compared directly to experiments, providing a much stronger test than is possible with many other modeling approaches. For example, knowledge of the pore space geometry and wettability conditions allows computing the exact configuration of liquid phases in porous media. This capability enables the simulation of imbibition of a wetting phase into the model porous medium using a physically consistent dynamic criterion for the imbibition of individual pores. This approach allows a priori, quantitative prediction of the configuration of fluid phases during imbibition. This in turn allows a quantitative understanding of how different macroscopic processes and parameters (e.g. relative permeabilities or resistivity index) depend on the geologic (e.g. type and amount of cement) and physical (e.g. wettability) features of porous media.Show more Item Simulating fluid flow in vuggy porous media(2005) Brunson, Dana Sue; Arbogast, Todd James, 1957-Show more We develop and analyze a mixed finite element method for the solution of an elliptic system modeling a porous medium with large cavities, called vugs. It consists of a second order elliptic (i.e., Darcy) equation on part of the domain coupled to a Stokes equation on the rest of the domain, and a slip boundary condition (due to Beavers, Joseph, and Saffman) on the interface between them. The tangential velocity is not continuous on the interface. We consider a vuggy porous medium with many small cavities throughout its extent, so the interface is not isolated. We use a certain conforming Stokes element on rectangles, slightly modified near the interface to account for the tangential discontinuity. This gives a mixed finite element method for the entire Darcy-Stokes system with a regular sparsity pattern that is easy to implement, no matter how complex the vug geometry may be. We prove optimal global first order L 2 convergence of the velocity and pressure, as well as of the velocity gradient, in the Stokes domain. Numerical results verify these rates of convergence, and even suggest somewhat better convergence in v certain situations. Finally, we present a lower dimensional space that uses Raviart-Thomas elements in the Darcy domain and uses our new modified elements near the interface in transition to the Stokes elements. We present two computational studies to illustrate and verify an homogenized macro-model of flow in a vuggy medium. And finally, we compare the effect of the Beavers-Joseph slip condition to using a no slip condition on the interface in a few simple examples.Show more Item A three dimensional finite element method and multigrid solver for a Darcy-Stokes system and applications to vuggy porous media(2007-05) San Martin Gomez, Mario, 1968-; Arbogast, Todd James, 1957-Show more A vuggy porous medium is one with many small cavities called vugs, which are interconnected in complex ways forming channels that can support high flow rates. Flow in such a medium can be modeled by combining Darcy flow in the rock matrix with Stokes flow in the vugs. We develop a finite element for the numerical solution of this problem in three dimensions, which converges at the optimal rate. We design a multigrid method to solve a saddle point linear system that comes from this discretization. The intertwining of the Darcy and Stokes subdomains in a natural vuggy medium makes the resulting matrix highly oscillating, or ill-conditioned. The velocity field we are trying to compute is also very irregular and its tangential component might be discontinuous at the Darcy-Stokes interface. This imposes a difficulty in defining intergrid transfer v operators. Our definition is based on mass conservation and the analysis of the orders of magnitude of the solution. A new smoother is developed that works well for this ill-conditioned problem. We prove that coarse grid equations at all levels are well posed saddle point systems. Our algorithm has a measured convergence factor independent of the size of the system. We then use our solver to study transport and flow properties of vuggy media by simulations. We analyze the results of our transport simulations and compare them to experimental results. We study the influence of vug geometry on the macroscopic flow properties of a three dimensional vuggy porous medium.Show more Item The transport of suspensions in geological, industrial and biomedical applications(2008-12) Oguntade, Babatunde Olufemi; Bonnecaze, R. T. (Roger T.)Show more Suspension flows in varied settings and at different concentrations of particles are studied theoretically using various modeling techniques. Particulate suspension flows are dispersion of particles in a continuous medium and their properties are a consequence of the interplay among hydrodynamic, buoyancy, interparticle and Brownian forces. The applicability of continuum modeling techniques to suspension flows at different particle concentration was assessed by studying systems at different time and length scales. The first two studies involve the use of modeling techniques that are valid in systems where the forces between particles are negligible, which is the case in dilute suspension flows. In the first study, the growth and progradation of deltaic geologic bodies from the sedimentation of particles from dilute turbidity currents is modeled using the shallow water equations or vertically averaged equations of motions coupled with a particle conservation equation. The shallow water model provides a basis for extracting grain size and depositional history information from seismic data. Next, the Navier-Stokes equations of motion and the convection-diffusion equation are used to model suspension flow in a biomedical application involving the flow and reaction of drug laden nanovectors in arteries. Results from this study are then used prescribe the best design parameters for optimal nanovector uptake at the desired sites within an artery. The third study involves the use of macroscopic two phase models to describe concentrated suspension flows where interparticle hydrodynamic forces cannot be neglected. The isotropic form of both the diffusion-flux and the suspension balance models are solved for a buoyant bidisperse pressure-driven flow system. The model predictions are found to compare fairly well with experimental results obtained previously in our laboratory. Finally, the power of discrete type models in connecting macroscopic observations to structural details is demonstrated by studying a system of aggregating colloidal particles via Brownian dynamics. The results from the simulations match experimental shear rheology and also provide a structural explanation for the observed macroscopic behavior of aging.Show more