Bulletin of the American Physical Society
60th Annual Meeting of the Divison of Fluid Dynamics
Volume 52, Number 12
Sunday–Tuesday, November 18–20, 2007; Salt Lake City, Utah
Session FR: Vortex Dynamics and 3D Vortex Flows IV (Point Vortices) |
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Chair: Bartosz Protas, McMaster University Room: Salt Palace Convention Center 251 F |
Monday, November 19, 2007 8:00AM - 8:13AM |
FR.00001: Energy of Relative Equilibria of Identical Point Vortices Hassan Aref Analytical formulae are derived for the energy of simple relative equilibria of $N$ identical point vortices such as the regular polygons, both open and centered, the collinear states, and the various analytically known configurations consisting of two and three nested regular polygons with or without a vortex at the center. The main result is the following: If the vortices are situated at $z_1, z_2, \dots , z_N$, and if $P(z) = (z-z_1)(z-z_2)\dots (z-z_N)$, then $\prod^N_{\alpha, \beta=1}\! ^\prime (z_\alpha-z_\beta) = \prod^N_{\alpha= 1} P^\prime(z_\alpha)$, where the prime indicates a derivative with respect to $z$. The logarithm of the absolute value of the left hand side is, in essence, the kinetic energy of fluid motion associated with the vortex pattern. This formula is known in the theory of polynomial equations as the formula for the {\it discriminant} expressed as a symmetric function of the roots. Its application to vortex dynamics appears to be new. [Preview Abstract] |
Monday, November 19, 2007 8:13AM - 8:26AM |
FR.00002: The Domm System: Vortex Dynamics of wakes Vasileios Vlachakis, Hassan Aref A significant number of engineering applications involve the dynamics of wakes behind bluff bodies. The dynamical system of four vortices in a periodic strip, all of the same absolute magnitude, two of either sign, is considered here. We refer to this as the {\it Domm system} since Domm was the first to consider it in 1956. To study the stability and understand the nonlinear behavior in the vicinity of the vortex street configurations we expose the vortices to perturbations that preserve momentum and energy. The variables that Domm used in his analysis provide a canonical transformation from the original vortex coordinates, as is evident from later work by Eckhardt and Aref. In effect, a canonical reduction of the four-vortex system from four degrees of freedom to two is achieved. The reduced system still has the Hamiltonian as an integral. Hence, it ``lives'' in a three-dimensional space. Our analysis aims to understand the global dynamics of the system and to relate it to finite-amplitude perturbations of vortex streets and their dynamics. [Preview Abstract] |
Monday, November 19, 2007 8:26AM - 8:39AM |
FR.00003: Vortex Design Problem Bartosz Protas In this investigation we are concerned with a family of solutions of the 2D steady--state Euler equations, known as the Prandtl--Batchelor flows, which are characterized by the presence of finite--area vortex patches embedded in an irrotational flow. We are interested in flows in the exterior of a circular cylinder and with a uniform stream at infinity, since such flows are often employed as models of bluff body wakes in the high--Reynolds number limit. The ``vortex design'' problem we consider consists in determining a distribution of the wall--normal velocity on parts of the cylinder boundary such that the vortex patches modelling the wake vortices will have a prescribed shape and location. Such inverse problem have applications in various areas of flow control, such as mitigation of the wake hazard. We show how this problem can be solved computationally by formulating it as a free--boundary optimization problem. In particular, we demonstrate that derivation of the adjoint system, required to compute the cost functional gradient, is facilitated by application of the shape differential calculus. Finally, solutions of the vortex design problem are illustrated with computational examples. [Preview Abstract] |
Monday, November 19, 2007 8:39AM - 8:52AM |
FR.00004: A gradient method for detecting stable configurations of point vortices Makoto Umeki A Hamiltonian H of point vortices can be given by a sum of functions of relative positions of all pairs of vortices. For square periodic boundary conditions, the function has a minimum at the position the most away. Therefore, if we consider the gradient system corresponding to the Hamiltonian system, it should settle down into a stable configuration which gives the local minimum of H. Numerical examples of tens to a hundred of periodic vortices show patterns very close to triangles. This method is not only applicable to the system of point vortices but also many kinds of problems of optimization. Its relation to trianglar vortex patterns observed in Bose-Einstein condensates of superfluids will be discussed. [Preview Abstract] |
Monday, November 19, 2007 8:52AM - 9:05AM |
FR.00005: 3D Euler in a 2D Symmetry Plane Robert M. Kerr, Miguel Bustamante Initial results from new calculations of interacting anti-parallel Euler vortices are presented. The objective is to understand the origins of singular scaling presented by Kerr (1993) for anti-parallel vortices with different core profiles. It is found that having nearly no negative vorticity in the upper half plane is essential, that analysis of enstrophy/its production is the most robust way to get initial trends, and that depletion of circulation in the symmetry plane is the best indication of when a calculation should be terminated due to lack of resolution. [Preview Abstract] |
Monday, November 19, 2007 9:05AM - 9:18AM |
FR.00006: Flying, swimming, falling...: fluid-solid interactions with vortex shedding Sebastien Michelin, Stefan Llewellyn Smith The interaction between the motion of slender bodies and the fluid around them is at the center of several natural phenomenas. To move in fluids, insects and fishes need to deform their bodies in such a way that the resulting flow around them applies the required force on their body. Unsteady pressure effects are essential here to understand the coupling between the fluid and solid motions. We consider slender solid bodies with sharp edges. At intermediate $Re$, the boundary layers separate because of the presence of the edge and strong vortices are shed. A simplified 2D potential flow model is proposed here. Point vortices with monotonically increasing intensity are shed from the edges of the body to enforce the regularity of the flow on its boundary. The Brown-Michael equation describes the motion of these vortices and enforces the conservation of momentum for the fluid around the vortex. The potential flow is computed using conformal mapping or bounded vortex sheet representation for the solid body. Simple representations of locomotion mechanisms such as flapping flight are proposed using this model. The forces resulting from prescribed flapping motions of rigid airfoils and deformable thin bodies are computed. The question of the free motion of an elastic 1D body will also be discussed. [Preview Abstract] |
Monday, November 19, 2007 9:18AM - 9:31AM |
FR.00007: Lagrangian simulation of bubble entrainment by a vortex ring Zhiwei Wang, Bin Chen In present paper the bubble entrainment by a vortex ring is numerically investigated by Lagrangian-Lagrangian method. The motion of vortex ring is simulated by a three-dimensional vortex filament model, in which vortex rings initially are discretized into individual elements with finite spherical cores placed on the centerline of the ring. The velocity field can be obtained by summing the contribution of all individual elements using the Biot-Savart law. The model has been validated by the comparation with theoretical solution of a circular vortex ring. Then based on the analysis of forces acting on bubbles entrained by a vortex ring, bubble dynamic equation is coupled into the three-dimensional vortex model and then bubble trajectories can be obtained accordingly. The trajectory of single bubble is computed and the well agreement between numerical simulation and the experimental observation validated our coupling model. [Preview Abstract] |
Monday, November 19, 2007 9:31AM - 9:44AM |
FR.00008: Vortex methods for bounded motion on a spherical surface Amit Surana, Darren Crowdy We discuss the motion of both point vortices and uniform vortex patches in bounded domains on a spherical surface. The fluid domains are taken to be ``basins'' having solid walls which act as impenetrable barriers for the flow. A theoretical formulation for the general N point vortex problem will be given. In addition, we describe a versatile numerical algorithm (a generalized ``contour dynamics'' scheme) to compute the motion of vortex patches in domains of arbitrary geometrical complexity on a sphere. [Preview Abstract] |
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