A Generalized Birkhoff-Rott Equation for Two-Dimensional Active Scalar Problems

In this paper we derive evolution equations for the two-dimensional active scalar problem when the solution is supported on one-dimensional curves. These equations are a generalization of the Birkhoff–Rott equation when vorticity is the active scalar. The formulation is Lagrangian and it is valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, incompressible flow) and the collapse of delta ring solutions (in the case of pure aggregation), finding excellent agreement. We then study two examples with kernels of mixed type, i.e., kernels that contain both incompressible and gradient flows. The first example is a vortex density model which arises in superfluids. We analyze the effect of the added gradient component on the Kelvin–Helmholtz instability. In the second example, we examine a nonlocal biological swarming model and study the dynamics of density rings which exhibit complicated milling behavior

Nonlinear and nonlocal models in fluid Mechanics

Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas . Fecha de lectura: 16-07-201

A surface vorticity method for wake–body interactions

The objective of this dissertation research is to develop a surface vorticity method for simulating high Reynolds number incompressible aerodynamic flows with strong unsteady interactions between wakes and lifting bodies. Examples of these types of flows include rotors in hover, propeller/wing installations, and impingement of vortex cores shed from wing strakes or flaps on downstream surfaces. Although higher-order panel codes provide good representation of potential flow around lifting bodies, their treatment of wakes is inadequate for our purpose. In the absence of significant boundary layer separation, the vorticity in these flows concentrates into thin shear layers. Therefore, vortex sheets are a natural mathematical representation of these flows. We leverage and extend rigorous methods from the vortex methods literature to model a wake as a free vortex sheet discretized as a triangulation of panels with linearly varying surface vorticity. The vorticity evolution equation is solved approximately by maintaining constant circulation along each half-edge in the triangulation, an approach that generalizes current methods for constant-strength elements. The vortex sheet is regularized with a smoothing parameter which provides an apparent thickness that mimics the limited viscous mixing in high Reynolds number flow. An adaptive paneling algorithm is implemented to maintain the desired level of detail as the wake triangulation stretches and deforms. The induced velocities from the wake vortex sheet are computed with a treecode implemented on a graphics processing unit (GPU) to allow computations with millions of panels. Lifting bodies are modeled with bound vortex sheets that are also triangulated with linear strength panels. These higher-order vorticity elements provide accurate velocity predictions on and near the surface, allowing for high resolution streamline tracing. Surface vorticity is determined by enforcing flow tangency constraints at each triangle centroid, zero circulation around each panel perimeter, and the unsteady pressure matching Kutta condition. These constraints result in an overdetermined system that is solved in a least squares formulation. Thus, our method is a second-order surface vorticity boundary element method that combines both solid bodies and wakes in a rigorous and consistent manner. The results of the method are shown to compare favorably to wind tunnel experimental results, including wake profiles, for a rectangular wing in a steady freestream, and for a horizontal axis wind turbine. Finally, we demonstrate the capabilities of our method in the context of strong wake–body interactions by simulating two flying wing aircraft in close formation, with the wake from the leading aircraft impacting the tailing aircraft.Ph.D

Vortex Sheet Simulations of 3D Flows Using an Adaptive Triangular Panel/Particle Method.

In this thesis we present an accurate and efficient algorithm for computing ideal flows using vortex sheets. A vortex sheet is a mathematical model simulating slightly viscous flow in which the vorticity is concentrated on a surface and the viscous effects are small. The sheet surface is represented by a set of triangular panels and each panel contains a set of active and passive Lagrangian particles. The active particles carry vorticity and the passive particles are used for panel subdivision and particle insertion. The method computes the vorticity carried by those particles, and then the induced velocities are computed with a tree-code. As the sheet surface evolves, stretching and twisting occur, hence refinement is needed to maintain resolution. The quadrature and the refinement procedure are local in the sense that they only use information within each panel. The purpose of implementing the locality is to avoid taking derivatives of the flow map, which is difficult because the derivatives grow in amplitude as time progresses. Computations of homogeneous flow, in which vorticity is conserved, are presented. We also present results for slightly stratified flow, in which vorticity is generated baroclinically on the sheet.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/55669/2/hualongf_1.pd

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described