Fast, High Order Vortex Methods Based on Deforming Basis Functions

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9971800This project expands and enhances existing knowledge of a new category of vortex method based on deforming basis functions. Vortex methods approximate the vorticity field of a flow as a linear combination of localized basis functions. These basis functions move with the flow velocity at the basis function centroid, and the velocity field is calculated from the vorticity field through a Biot-Savart integral. For flows dominated by isolated regions of vorticity, vortex methods offer considerable advantages because they are naturally adaptive in the sense that computational elements are dedicated exclusively to regions that have vorticity. Most vortex methods use rigid, axisymmetric basis functions though there are some exceptions. This project studies vortex methods that use basis functions which deform with local flow deviations, and builds upon the recent development of a high spatial order vortex method based on deforming elliptical Gaussians. Elliptical Gaussians are special in the sense that they represent self-similar, exact solutions to the relevant advection-diffusion equation. To implement such a scheme, one must develop methods for evaluating the Biot-Savart integral for an elliptical Gaussian and identify the relevant convergence parameters for the method as a whole. Though a two-dimensional scheme has already been developed, the Principle Investigator will enhance this method by developing a fast summation algorithm. The Principle Investigator will extend this method to three dimensions, providing a crucial link between vortex stretching and the geometry of vortex filaments. The new method will be used to perform careful calculations of a variety of problems including vortex dipole collisions and jet transients. Finally, the Principle Investigator will extend this approach to moisture transport through unsaturated porous media.This work provides a means of quickly and accurately calculating fluid flow properties on a wide range of problems including but not limited to aerospace applications, industrial processes, ocean currents and atmospheric flows of all sorts. The Principal Investigator will also extend these concepts along new lines to develop methods that can accurately calculate the motion and diffusion of moisture and contaminants in unsaturated soils. This type of method, called a "vortex method", is unusual in that these schemes are naturally adaptive. Naturally adaptive methods dedicate computational resources exclusively to the dominant regions of the flow. These methods simulate flows by calculating the evolution of the local angular momentum in the fluid. Often, the angular momentum is restricted to a small fraction of the total volume of the fluid. For instance, weather systems are driven by a collection of storm systems that represent concentrated regions of angular momentum. Using such a method, one can reconstruct the entire flow field based solely on the evolution of the angular momentum which means that the computer only need perform calculations over a small area to capture the entire flow field. Vortex methods have the added advantage of being easily parallelized on multiple processor computers so that one can take full advantage of supercomputing facilities and networks of connected computers. Also, the Principal Investigator will apply these techniques to an entirely new type of problem involving flow through unsaturated porous media such as dry soils, clays or concrete. Similar to vortex methods, this new technique takes advantage of the fact that the movement of moisture is dominated by narrow "preferred paths" occupying a relatively small fraction of the total domain of interest. These paths are created by moisture-media interactions, and the computer need only dedicate its resources to those regions containing moisture. Finally, many aspects of these activities make excellent undergraduate research projects. These research activities will enhance students' interests and knowledge in mathematics, environmental science and high performance computing. Thus, in addition to its scientific merits, this project will give students at the University of Massachusetts Lowell meaningful research experiences and enhance collaboration across several disciplines.