Bulletin of the American Physical Society
67th Annual Meeting of the APS Division of Fluid Dynamics
Volume 59, Number 20
Sunday–Tuesday, November 23–25, 2014; San Francisco, California
Session R18: Vortex Dynamics: Theory |
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Chair: Michael Triantafyllou, MIT Room: 2004 |
Tuesday, November 25, 2014 1:05PM - 1:18PM |
R18.00001: Mathematical models for exotic wakes Saikat Basu, Mark Stremler Vortex wakes are a common occurrence in the environment around us; the most famous example being the von K\'{a}rm\'{a}n vortex street with two vortices being shed by the bluff body in each cycle. However, frequently there can be many other more exotic wake configurations with different vortex arrangements, based on the flow parameters and the bluff body dimensions and/or its oscillation characteristics. Some examples include wakes with periodic shedding of three vortices (`P+S' mode) and four vortices (symmetric `2P' mode, staggered `2P' mode, `2C' mode). We present mathematical models for such wakes assuming two-dimensional potential flows with embedded point vortices. The spatial alignment of the vortices is inspired by the experimentally observed wakes. The idealized system follows a Hamiltonian formalism. Model-based analysis reveals a rich dynamics pertaining to the relative vortex motion in the mid-wake region. Downstream evolution of the vortices, as predicted from the model results, also show good correspondence with wake-shedding experiments performed on flowing soap films. [Preview Abstract] |
Tuesday, November 25, 2014 1:18PM - 1:31PM |
R18.00002: Analytical approach to the energy transfer around elliptic Burgers vortices Hiromichi Kobayashi The energy transfer from large-scale to small-scale around elliptical Burgers vortices is analytically examined. The elliptical Burgers vortex is constructed by the background straining flow to the Burgers vortex, so that the Burgers vortex becomes non-axisymmetric. By taking a filter to the elliptical Burgers vortices, we obtain the filtered velocity field. In large eddy simulation (LES), understanding the energy transfer from resolved-scale to subgrid-scale (SGS), the so-called forward scatter, around the eddy is important. The SGS stress tensor is decomposed to Leonard, cross and Reynolds terms. Those contribution to the energy transfer is discussed. The forward scatter region of the Leonard term appears along the major axis of the elliptic Burgers vortex. For cross and Reynolds terms, the forward scatter regions emerge along the minor axis. The Reynolds term has much smaller intensity than the cross term. [Preview Abstract] |
Tuesday, November 25, 2014 1:31PM - 1:44PM |
R18.00003: The motion of helical vortices Oscar Velasco Fuentes We study the motion of a helical vortex in an inviscid, incompressible fluid of infinite extent. The vortex is a thin tube, of circular cross section and uniform vorticity, whose centerline is a helix of uniform pitch. Ever since Joukowsky (1912) deduced that this vortex is a steady solution of the Euler equations, numerous attempts have been made to compute its self-induced velocity. Here we use Hardin's (1982) solution for the velocity field in order to compute, for any pitch value, the linear and angular velocities of the vortex. Our formulas were verified by direct numerical integration of both the Biot-Savart and Helmholtz equations, and were also found to compare favourably with previous theoretical results. In terms of the vortex capacity to transport fluid, we identified three regimes: a helix of large pitch moves slowly, carrying a large mass of fluid; a thin helix of small pitch moves fast, carrying a small mass of fluid; and a fat helix of small pitch is a moderate carrier itself but it pushes fluid forward along its axis. [Preview Abstract] |
Tuesday, November 25, 2014 1:44PM - 1:57PM |
R18.00004: Helicity conservation in classical vortex knots and links Martin W. Scheeler, Dustin Kleckner, Gordon L. Kindlmann, William T.M. Irvine Vortex knots and links in an ideal fluid remain knotted or linked, ensuring that the topology of the vortex field lines is conserved. For a real fluid, however, this conservation is jeopardized by the presence of reconnection events, which allow vortex tubes to reconfigure their global topology; indeed, it has recently been observed that knotted and linked vortex tubes in classical fluids unknot or untie themselves via a series of these reconnection events. Remarkably, we observe that these reconnection processes conserve a measure of the vortex line topology (helicity) and do so through a geometric mechanism that efficiently transfers this topology across scales. The geometric nature of this topology transfer, along with its recent observation in superfluid vortices, suggests that helicity conservation may be a robust and generic feature of non-ideal flows. [Preview Abstract] |
Tuesday, November 25, 2014 1:57PM - 2:10PM |
R18.00005: Generalized Adler-Moser and Loutsenko polynomials and point vortex equilibria Nicholas Cox-Steib, Kevin O'Neil The Adler-Moser polynomials, well-known as generators of rational solutions to the Korteweg-de Vries equation, also have an ``electrostatic interpretation'' in which the zeroes of the polynomials form equilibrium configurations of point vortices of equal and opposite strengths. The Loutsenko polynomials similarly form equilibria of vortices with strength ratio -2. The present work describes a new family of polynomials that generalizes the aforementioned polynomials by introducing a third vortex strength. This new doubly-indexed family has many of the unusual characteristics of the two-strength polynomials such as the presence of continuous parameters and a connection to rational potentials for a second-order partial differential equation, and reproduces the earlier polynomials when the third strength is set to zero. [Preview Abstract] |
Tuesday, November 25, 2014 2:10PM - 2:23PM |
R18.00006: Where Does Vorticity Annihilation Occur? James Schulmeister, Michael Triantafyllou Accounting vorticity, from generation at boundaries, diffusion and convection in the fluid, to finally annihilation, is critical to understanding vortical flows. This study presents a complete accounting of vorticity, including annihilation. Control volume analysis leads to the concept of the vorticity annihilation line, which is the locus of points where vorticity annihilates. This study considers two examples. The first is the flow above a plane that oscillates harmonically in its plane. This flow supports a quasi-steady analytic solution and is driven by diffusion. The second is the flow past a circular cylinder with Reynolds number equal to 10. This flow supports a steady solution that is solved numerically and is driven by both diffusion and convection of vorticity. In both examples, the vorticity is fully accounted from generation to annihilation using the annihilation line concept. [Preview Abstract] |
Tuesday, November 25, 2014 2:23PM - 2:36PM |
R18.00007: Annihilation of strained vortices Yoshifumi Kimura As an initial stage of vortex reconnection, approach of nearly anti-parallel vortices has often been observed experimentally and studied numerically. Inspired by the recent experiment by Kleckner and Irvine on the dynamics of knotted vortices [1], we have studied the motion of two anti-parellel Burgers vortices driven by an axisymmetric linear straining field. We first extend the Burgers vortex solution which is a steady exact solution of the Navier-Stokes equation to a time-dependent exact solution. Then by superposing two such solutions, we investigate the annihilation process analytically. We can demonstrate that during the annihilation process the total vorticity decays exponentially on a time-scale proportional to the inverse of the rate of strain, even as the kinematic viscosity tends to 0. The analytic results are compared with the numerical simulations of two strained vortices with the vortex-vortex nonlinear interaction by Buntine and Pullin [2]. \\[4pt] [1] Kleckner, D. \& Irvine, W.T.M. 2013 Creation and dynamics of knotted vortices. {\it Nature Physics} {\bf 9}, 253258. doi: 10.1038/nphys2560.\\[0pt] [2] Buntine, J.D. \& Pullin, D.I. 1989 Merger and cancellation of strained vortices. {\it J. Fluid Mech.} {\bf 205}, 263-295. [Preview Abstract] |
Tuesday, November 25, 2014 2:36PM - 2:49PM |
R18.00008: Hollow vortices in weakly compressible flows Vikas Krishnamurthy, Darren Crowdy In a two-dimensional, inviscid and steady fluid flow, hollow vortices are bounded regions of constant pressure with non-zero circulation. It is known that for an infinite row of incompressible hollow vortices, analytical solutions for the flow field and the shape of the hollow vortex boundary can be obtained using conformal mapping methods. In this talk, we show how to derive analytical expressions for a weakly compressible hollow vortex row. This is done by introducing a new method based on the Imai-Lamla formula. We will also touch upon how to extend these results to a von-Karman street of hollow vortices. [Preview Abstract] |
Tuesday, November 25, 2014 2:49PM - 3:02PM |
R18.00009: Theoretical and Experimental Investigation of Subcritical and Supercritical Vortex Flows Martin Bruschewski, Heinz-Peter Schiffer, Sven Grundmann The presented work deals with the subcritical and supercritical behavior of vortex flows. A vortex filament method is proposed for the simulation of these two flow states. The flow is modeled by a continuous distribution of vortex filaments in which the axial velocity component is induced by a helical winding of the filaments. By this method, the three-dimensional steady and incompressible vortex flow in a circular channel with different exit orifices is computed. The reference velocity fields are obtained experimentally by Magnetic Resonance Imaging. As the main outcome it was found that there are two conjugate solutions for every investigated case. The first solution requires all vortex filaments to terminate at the fluid boundaries. It does not depend on the downstream geometry and it therefore represents the supercritical state. For the conjugate solution, some regions contain ring-shaped vortex filaments instead of terminated filaments. The manifestation of these vortex rings depends on the downstream geometry. Hence, the occurrence of vortex rings is considered as an indicator for the subcritical state. The results in terms of the velocity field are in very good agreement to the measured subcritical and supercritical vortex flows. [Preview Abstract] |
Tuesday, November 25, 2014 3:02PM - 3:15PM |
R18.00010: Inverse Problem of Vortex Reconstruction Bartosz Protas, Ionut Danaila This study addresses the following question: given incomplete measurements of the velocity field induced by a vortex, can one determine the structure of the vortex? Assuming that the flow is incompressible, inviscid and stationary in the frame of reference moving with the vortex, the ``structure'' of the vortex is uniquely characterized by the functional relation between the streamfunction and vorticity. To focus attention, 3D axisymmetric vortex rings are considered. We show how this inverse problem can be framed as an optimization problem which can then be efficiently solved using variational techniques. More precisely, we use measurements of the tangential velocity on some contour to reconstruct the function defining the streamfunction-vorticity relation in a continuous setting. Two test cases are presented, involving Hill's and Norbury vortices, in which very good reconstructions are obtained. A key result of this study is the application of our approach to obtain an optimal inviscid vortex model in an actual viscous flow problem based on DNS data which leads to a number of nonintuitive findings. [Preview Abstract] |
Tuesday, November 25, 2014 3:15PM - 3:28PM |
R18.00011: Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows Diego Ayala, Bartosz Protas In this investigation we analyze a family of extreme vortex states which maximize the instantaneous production of enstrophy under Navier-Stokes dynamics on 3D periodic domains. They are found by numerically solving suitably constrained optimization problems and include other well-known flows, such as the Taylor-Green vortex and the ABC flow, as special cases. Initially discovered by Lu \& Doering (2008), these optimal vortex states saturate an analytic upper bound on the rate of growth of enstrophy, indicating that this estimate is in fact sharp. We provide a numerical characterization of the set of initial data for which smooth solutions are guaranteed to exist for all times, thereby offering a physical interpretation of a well-known result of mathematical analysis. The results from high-resolution direct numerical simulations indicate that the flows triggered by these optimal fields produce a larger finite-time growth of enstrophy than the flows obtained from other widely-used initial conditions, such as the Taylor-Green vortex, Lamb dipoles and perturbed anti-parallel vortex tubes. Although numerical in nature, these results illustrate a systematic approach to finding a worst-case initial condition which could lead to the potential formation of a singularity in finite-time. [Preview Abstract] |
Tuesday, November 25, 2014 3:28PM - 3:41PM |
R18.00012: Sadovskii vortex in strain Daniel Freilich, Stefan Llewellyn Smith A Sadovskii vortex is a patch of fluid with uniform vorticity surrounded by a vortex sheet. Using a boundary element type method, we investigate the steady states of this flow in an incompressible, inviscid straining flow. Outside the vortex, the fluid is irrotational. In the limiting case where the entire circulation is due to the vortex patch, this is a patch vortex (Moore \& Saffman, \emph{Aircraft wake turbulence and its detection} 1971). In the other limiting case, where all the circulation is due to the vortex sheet, this is a hollow vortex (Llewellyn Smith and Crowdy, \emph{J.\ Fluid Mech}.\ \textbf{691} 2012). This flow has two governing nondimensional parameters, relating the strengths of the straining field, vortex sheet, and patch vorticity. We study the relationship between these two parameters, and examine the shape of the resulting vortices. We also work towards a bifurcation diagram of the steady states of the Sadovskii vortex in an attempt to understand the connection between vortex sheet and vortex patch desingularizations of the point vortex. [Preview Abstract] |
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