Bulletin of the American Physical Society
63rd Annual Meeting of the APS Division of Fluid Dynamics
Volume 55, Number 16
Sunday–Tuesday, November 21–23, 2010; Long Beach, California
Session GN: Vortex Flows: Inviscid |
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Chair: Hassan Aref, Technical University of Denmark and Virginia Polytechnic Institute and State University Room: Long Beach Convention Center 202C |
Monday, November 22, 2010 8:00AM - 8:13AM |
GN.00001: Relative equilibria of point vortices and the fundamental theorem of algebra Hassan Aref The fundamental theorem of algebra implies that every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted with its multiplicity. This result may be applied to the generating polynomial for a relative equilibrium of point vortices and used to derive differential equations for this polynomial in various situations, e.g., when the vortices are on a line or all on a circle. The derivations thus obtained are quite elegant and compact compared to the corresponding derivations found in the literature. A new formula that provides the basis for application of the fundamental theorem to vortex equilibria is outlined and a number of the further derivations demonstrated. [Preview Abstract] |
Monday, November 22, 2010 8:13AM - 8:26AM |
GN.00002: Hamiltonian-Dirac Simulated Annealing: Application to the Calculation of Vortex States P.J. Morrison, G.R. Flierl A simulated annealing method for calculating stationary states for models that describe continuous media is proposed. The method is based on the noncanonical Poisson bracket formulation of media, which is used to construct Dirac brackets with desired constraints, and symmetric brackets that cause relaxation with the desired constraints. The method is applied to two-dimensional vortex dynamics and a variety of numerical examples are given, including the calculation of monopole and dipole vortex states. [Preview Abstract] |
Monday, November 22, 2010 8:26AM - 8:39AM |
GN.00003: Mathematical modeling of ``2P'' mode vortex wakes Saikat Basu, Mark Stremler, Teis Schnipper, Anders Andersen The ``2P'' mode vortex wake, in which two vortex pairs are generated per shedding cycle, is a commonly occurring wake structure behind oscillating bluff bodies. We will present an idealized model of these wakes that consists of a singly-periodic Hamiltonian system of four point vortices. The system is made integrable with an imposed spatial symmetry that is motivated by the experimentally observed wake structure. This model generalizes our previous work by allowing for unequal vortex strengths in the shed pairs. Comparisons with experimental wakes generated by a flapping foil in a flowing soap film show that this model can be used to characterize the vortex trajectories in ``2P'' mode wakes and to estimate the experimental vortex strengths. [Preview Abstract] |
Monday, November 22, 2010 8:39AM - 8:52AM |
GN.00004: Stability of icosahedral configurations of point vortices on a sphere Vitalii Ostrovskyi, Paul Newton Using icosahedron as the initial geometric distribution of point vortices on a sphere we show existence of icosahedral relative equilibrium configurations. To characterize these configurations we apply method based on finding the fixed points of the nonlinear dynamical system governing the $N(N-1)/2$ equations for interparticle distances. Obtained equations give sufficient conditions for the relative equilibria and lead to a problem of finding solutions to $A \vec{\Gamma} = 0$, where $\vec{\Gamma} \in {R}^N$ is the vector of vortex strengths, and $A \in {R}^{M \times N}$ is a rectangular, non-normal ($AA^T \ne A^T A$) `configuration' matrix determined by the particle positions. Using singular value decomposition of $A$ we prove that for icosahedron the $Nullspace(A)$ is 7 dimensional. Vertex and edge stabilizers, as subgroups of icosahedral symmetry group, are used to build the set of symmetric icosahedral configurations with non-negative strengths. Using exact solution of equations of motion we prove stability of vortex pair configurations. Energy-momentum method is used to study stability of symmetric icosahedral relative equilibria. To prove instability of some of the configurations we show that the matrix of linearized system has eigenvalues with positive real parts. Using the stability results we build an example of linear superposition of stable configurations which gives unstable configuration. [Preview Abstract] |
Monday, November 22, 2010 8:52AM - 9:05AM |
GN.00005: The motion of singularities in potential flow Stefan Llewellyn Smith In the first paper on vorticity, Helmholtz discussed infinitesimal rectilinear filaments, and Kirchhoff subsequently derived the equation of motion of point vortices. This equation can be viewed as the statement that the translational velocity of the point vortex is obtained by removing the leading-order singularity due to the point vortex when computing its velocity. I review the arguments used to obtain this result and discuss their history and limitations. I then examine the extension of these ideas to other kinds of singularities and give some examples. [Preview Abstract] |
Monday, November 22, 2010 9:05AM - 9:18AM |
GN.00006: An accurate and efficient method to compute steady uniform vortices P. Luzzatto-Fegiz, C.H.K. Williamson Steady uniform vortices represent a widely used approximation in a broad range of contexts, ranging from dynamics of plasmas to geophysical flows. Surprisingly, computing steady uniform vortices still presents several challenges, since vortex boundaries may develop high-curvature regions, which can be prohibitively expensive to resolve. Further to this, flows can bifurcate to lower-symmetry states, which may be difficult to compute reliably. Currently, one must choose between affordable relaxation methods and more reliable approaches based on Newton iteration. However, while the first cannot resolve flows without symmetry (Dritschel 1985), the second are unaffordable for vortices with high-curvature regions (Saffman \& Szeto 1980; Elcrat {\it et al}. 2005). Hence it is typically impossible to compute a family of steady vortices in its entirety. We overcome these limitations by introducing a new discretization, based on an ``inverse-velocity map'', which makes Newton iteration affordable for vortices with high-curvature boundaries. By employing our numerical method in conjunction with the IVI-diagram stability approach (LF\&W {\it PRL} 2010), we explore the full bifurcation structure of several classical flows, including elliptical vortices, co-rotating and counterrotating vortex pairs, and vortex streets. We have also successfully employed our discretization for other fluid problems, such as steep gravity waves. [Preview Abstract] |
Monday, November 22, 2010 9:18AM - 9:31AM |
GN.00007: A Framework for Linear Stability Analysis of Finite--Area Vortices Bartosz Protas, Alan Elcrat In this work we are interested in the linear stability of 2D solutions of the Euler equations which are steady in the appropriate frame of reference and feature compact regions with constant vorticity embedded in an otherwise potential flow. We argue that, since the evolution of such systems is governed by equations of the free--boundary type, the {\em shape calculus} is a natural framework for differentiation of such governing equations. We derive a general equation characterizing the evolution of area--preserving perturbations of the boundary. While for vortex regions with arbitrary shapes the perturbation equation needs to solved numerically (e.g., using spectral Fourier--Galerkin method), we show that for a circular boundary (i.e., the Rankine vortex) the problem can be solved analytically yielding the classical stability results due to Kelvin. We will also present stability calculation obtained numerically for more general vortex shapes and will discuss generalizations of this approach. [Preview Abstract] |
Monday, November 22, 2010 9:31AM - 9:44AM |
GN.00008: Instability of Point Vortex Leapfrogging Laust Toph{\O}j, Hassan Aref The dynamics of interacting point vortices on the unbounded plane can be chaotic if the number of vortices is at least four. The chaotic dynamics is governed by the existence of unstable structures in the phase space, [Toph{\o}j \& Aref, Phys. Fluids, \textbf{20}, 093605 (2008)]. Such structures may be hyperbolic fixed points of the dynamical system, or unstable periodic orbits. Chaos arises as the system is repeatedly repelled by these structures, bouncing back and forth between them. The leapfrogging motion of two vortex pairs possessing a common axis of symmetry is an example of an integrable periodic motion of a four-vortex system. The stability of this periodic motion has been studied numerically by Acheson [Eur. J. Phys. \textbf{21}, 269 (2000)] whose results indicate instability for some but not all parameters. We discuss the stability of leapfrogging, using methods from Floquet theory. Analogies will be drawn to instabilities of the von K{\'a}rm{\'a}n vortex street that can cause the vortex street to break up into vortex pairs moving away from the central axis. [Preview Abstract] |
Monday, November 22, 2010 9:44AM - 9:57AM |
GN.00009: Lagrangian trajectories in Lissajous vortices Sergio Cuevas, Aldo Figueroa, Eduardo Ramos We report Particle Image Velocimetry experiments in a rectangular container with a shallow layer of an electrolyte in which a vortex flow is driven by Lorentz forces produced by the field of a permanent cylindrical magnet and two alternate electric currents perpendicular to each other. Currents are injected through two pairs of parallel electrodes located at the container walls but avoiding short circuit. Due to the harmonic forcing in perpendicular directions, the system is excited analogously to the kinematic Lissajous figures although in the fluid case convective and viscous effects are present. In the creeping flow limit, an analytical solution is obtained so that the Lagrangian trajectories can be integrated. A full numerical solution that accounts for cases where non-linear effects are important is also used in the analysis. Lagrangian trajectories based on analytical, numerical and experimental results are compared for different values of amplitudes, frequencies and relative phases of the electromagnetic forcing. [Preview Abstract] |
Monday, November 22, 2010 9:57AM - 10:10AM |
GN.00010: A Robust Numerical Method for Integration of Point-Vortex Trajectories in Two Dimensions Spencer Smith, Bruce Boghosian The venerable 2D point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. Point-vortex dynamics are described by a relatively simple system of nonlinear ODEs which can easily be integrated numerically using an appropriate adaptive time stepping method. As the separation between two vortices relative to all other inter-vortex length scales decreases, however, the computational time required diverges. Accuracy is usually the most discouraging casualty when trying to account for such vortex motion, though the varying energy of this ostensibly Hamiltonian system is a potentially more serious problem. We solve these problems by a series of coordinate transformations: We first transform to action-angle coordinates, which, to lowest order, treat the close pair as a single vortex amongst all others with an internal degree of freedom. We next, and most importantly, apply Lie transform perturbation theory to remove the higher-order correction terms in succession. The overall transformation drastically increases the numerical efficiency and ensures that the total energy remain constant to high accuracy. [Preview Abstract] |
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