Browsing by Author "Rossi, Louis F."
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Item Achieving High-Order Convergence Rates with Deforming Basis Functions(Department of Mathematical Sciences, 2003) Rossi, Louis F.This article studies the use of moving, deforming elliptical Gaussian basis functions to compute the evolution of passive scalar quantities in a two-dimensional, incompressible flow field. We compute an evolution equation for the velocity, rotation, extension and deformation of the com- putational elements as a function of flow quantities. We find that if one uses the physical flow velocity data calculated from the basis function centroid, the method has only second order spatial accuracy. However, by computing the residual of the numerical method, we can determine adjustments to the centroid data so that the scheme will achieve fourth-order spatial accuracy. Simulations with nontrivial flow parameters demonstrate that the methods exhibit the properties predicted by theory.Item A Comparative Study of Lagrangian Methods Using Axisymmetric and Deforming Blobs(Department of Mathematics, 2003) Rossi, Louis F.This paper presents results from a head-to-head comparison of two Lagrangian methods for solutions to the two-dimensional, incompressible convection-diffusion equations. The first Lagrangian method is an axisymmetric core spreading method using Gaussian basis functions. The second method uses deforming elliptical Gaussian basis functions. Previous results show that the first method has second-order spatial accuracy and the second method has fourth-order spatial accuracy. However, the deforming basis functions require more computational effort per element, so this paper examines computational performance as well as overall accuracy. The test problem is the deformation and diffusion of ellipsoidal distribution of scalar with an underlying flow field that has closed circular streamlines. The test suite includes moderate, high and infinite Peclet number problems. The results indicate that the performance tradeoff for the sample flow calculation occur at modest problem sizes, and that the fourth-order method offers distinct advantages as a general approach for challenging problems.Item Evaluation of the biot-savart integral for deformable elliptical gaussian vortex elements(Mathematical Sciences Department, 2005) Rossi, Louis F.This paper introduces two techniques for approximating the Biot-Savart integral for deforming elliptical Gaussian functions. The primary motivation is to develop a high spatial accuracy vortex method. The first technique is a regular perturbation of the streamfunction in the small parameter ε = (a-1)/(a+1)where a^2 is the aspect ratio of the basis function. This perturbative technique is suitable for direct interactions. In the far field, the paper studies the applicability of the fast multipole method for deforming elliptical Gaussians since the multipole series are divergent. The noncompact basis functions introduce a new computational length scale that limits the efficiency of the multipole algorithm but by imposing a lower bound on the finest mesh size, one can approximate the far-field stream function to any specified tolerance.Item A High Order Langrangian Scheme for Flow Through Unsaturated Porous Media(2001) Rossi, Louis F.A new high order Lagrangian method uses moving basis functions to represent the moisture content in an unsaturated porous material. The basis functions are localized elliptical Gaussians that can move, spread, elongate and rotate under the action of the local velocity, velocity deviations, and diffusivity with corrections for local material properties. The velocity and its deviations are calculated from the local moisture potential and its derivatives which can be obtained from published experiments. This numerical technique is natu- rally adaptive in the sense that computational effort is expended only where there is moisture and nowhere else, and this method is capable of capturing infiltration instabilities in the wetting front observed by experimentalists and predicted by linear stability analysis. This method borrows many ideas from high Reynolds number vortex methods and other applications of Lagrangian schemes to nonlinear partial differential equations.Item High Order Vortex Methods With Deforming Elliptical Gaussian Blobs 1: Derivation and Validation(Department of Mathematical Sciences, 2001) Rossi, Louis F.This manuscript introduces a new vortex method based on elliptical Gaussian basis functions. Each basis function translates, nutates, elongates and spreads through the action of the local flow field and fluid viscosity. By allowing elements to deform, the method captures the effects of local flow deviations with a fourth order spatial accuracy. This method uses a fourth order asymptotic approximation to the Biot-Savart integrals for elliptical Gaussian vorticity distributions to determine velocity and velocity derivatives. A robust adaptive refinement procedure reconfigures elements that spread beyond the specified resolution. The high order convergence rate is verified by comparing calculations with the vortex method to exact solutions in a variety of controlled experiments.Item A mathematical and experimental study of ant foraging line dynamics(Department of Mathematical Sciences, 2005-04-13) Rossi, Louis F.; Johnson, KatieIn this article, we present a mathematical model coupled to an experimental study of ant foraging lines. Our laboratory experiments do not support the common traffic modeling assumption that ant densities and velocities are directly correlated. Rather, we find that higher order effects play a major role in observed behavior, and our model reflects this by including inertial terms in the evolution equation. A linearization of the resulting system yields left- and right-moving waves, in agreement with laboratory measurements. The linearized system depends upon two Froude numbers reflecting a ratio of the energy stored in the foraging line to the kinetic energy of the ants. Furthermore, the model predicts and the measurements support the existence of two distinct phase velocities.Item A Model of Coupled Heat and Moisture Transport in an Annular Clay Barrier(Department of Mathematical Sciences, 2002) Rossi, Louis F.; Inyang, H. I.; Graham-Eagle, J.; Pennell, S.The design of repository seals for deeply buried high-level radioactive wastes incorporates densely compacted clayey barriers around metallic waste canisters. In this paper, a mathematical model that is based on conservation of thermal energy and mass is developed to describe the locations of moisture and temperature fronts within a barrier, around a cylindrical waste canister of 1-meter radius, containing radionuclides with half-lives that range from 100 – 10,000 years. The solution developed is axisymmetric: the moisture fraction, w, and temperature T, vary only with time t, and radial distance r from the axis of the cylindrical waste canister. The model produces parabolic partial differential equations (PDEs). The spatial domain is discretized such that ordinary differential equations (ODEs) that result are solved. Computations using a uniform mesh of 0.1 meters and a cooling coefficient of 6.7 x 10 -2 with assumed but typical data on material properties, indicate that coupling of transport processes would be negligible in the case of radionuclides with long half-lives. Also, a quasi-steady vaporization front can form and propagate outward over the course of several decades after waste emplacement. The evolution of the front is somewhat insensitive to the half-life used and the permeability of the clayey barrier material.Item On the Existence of Two-Dimensional, Localized, Rotating, Self-Similar Vortical Structures(Department of Mathematical Sciences, 2001) Rossi, Louis F.; Graham-Eagle, J.We prove that a Gaussian monopole, also known as the Lamb-Oseen vortex, is the only localized, rotating, self-similar solution to the two-dimensional, incompressible Navier-Stokes equations where level sets of vorticity and corotating streamfunction coincide. Our definition of self-similarity is restricted to the natural linear combination of space, time and viscous diffusion. We arrive at this conclusion by analytically determining the azimuthal Fourier modes for all possible solutions to this problem and then proving that the amplitude of all but the first (axisymmetric) is zero. Since coherent vortex multipoles are observed to be in a state where lines of vorticity and corotating streamfunction correspond, this casts doubt on the existence of any self-similar asymptotic structure other than the monopole.Item Slippage and Polymer Migration in a Model of Dilute Polymer Fluid(Department of Mathematical Sciences, 2003-04-23) Cook, L. Pamela; Rossi, Louis F.This paper introduces a new mathematical model for a dilute complex fluid based on a Hookean bead-spring mechanism. The new model couples constitutive equations with number density and includes bead slippage which manifests itself in higher- order corrections. In the case of simple shear flows, we compute steady solutions and determine the linear stability of this model along the flow curve. The linear stability indicates a selection mechanism for multi-valued regions of the flow curve in stress- controlled experiments. We find that the model provides a physically reasonable extension to existing models and exhibits desirable properties such as shear thinning and shear banding. Finally, it predicts hysteretic behavior in the effective viscosity qualitatively similar to that which has been observed in laboratory experiments.