Bulletin of the American Physical Society
64th Annual Meeting of the APS Division of Fluid Dynamics
Volume 56, Number 18
Sunday–Tuesday, November 20–22, 2011; Baltimore, Maryland
Session G21: Vortex Dynamics III 
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Chair: Stefan Llewellyn Smith, University of California, San Diego Room: 324325 
Monday, November 21, 2011 8:00AM  8:13AM 
G21.00001: Evolving geometry of a vortex triangle Vikas Krishnamurthy, Hassan Aref We explore a new description of threevortex motion in terms of the angles, $A, B, C$, in the triangle of vortices, the radius, $R$, of the circumcircle to this triangle, and the position of the circumcenter. We show that the equations of motion for the radius and angles may be expressed in terms of these variables alone, i.e., the evolution of $R, A, B, C$ leads to an autonomous fourdimensional dynamical system. The evolution of the circumcenter, however, depends on the absolute vortex positions. Thus, the motion of three vortices may be regarded as a sum of two components: the motion of the circumcenter (giving the absolute position of the circumcircle) and, superimposed on this, the intrinsic dynamics of the size of the circumcircle and shape of the vortex triangle. Several results from the theory of threevortex motion follow in this new representation and a number of new issues may be investigated. [Preview Abstract] 
Monday, November 21, 2011 8:13AM  8:26AM 
G21.00002: Bilinear relative equilibria of identical point vortices Hassan Aref, Peter Beelen, Morten Br{\O}ns A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, taken to be the $x$ and $y$axes of a cartesian coordinate system, is introduced and studied. In general we have $m$ vortices on the $y$axis and $n$ on the $x$ axis. We define generating polynomials $q(z)$ and $p(z)$, respectively, for each set of vortices. A second order, linear ODE for $p(z)$ given $q(z)$ is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm's comparison theorem, is that if $p(z)$ satisfies the ODE for a given $q(z)$ with its imaginary zeros symmetric relative to the $x$axis, then it must have at least $nm+2$ simple, real zeros. For $m=2$ this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that given $q(z) = z^2 + \eta^2$, where $\eta$ is real, there is a unique $p(z)$ of degree $n$, and a unique value of $\eta^2 = A_n$, such that the zeros of $q(z)$ and $p(z)$ form a relative equilibrium of $n+2$ point vortices. We show that $A_n \approx \frac{2}{3}n + \frac{1}{2}$, as $n\rightarrow\infty$, where the coefficient of $n$ is determined analytically, the next order term numerically. [Preview Abstract] 

G21.00003: ABSTRACT WITHDRAWN 
Monday, November 21, 2011 8:39AM  8:52AM 
G21.00004: New Results on Hollow Vortices Stefan Llewellyn Smith, Darren Crowdy Hollow vortices are vortices whose interior is at rest. We obtain exact solutions for hollow vortices in linear and nonlinear strain. We then investigate the stability properties of these vortices and for the hollow vortex street of Baker, Saffman and Sheffield. [Preview Abstract] 
Monday, November 21, 2011 8:52AM  9:05AM 
G21.00005: Analysis of Reynolds number scaling for viscous vortex reconnection Qionglin Ni, Fazle Hussain, Jianchun Wang, Shiyi Chen A theoretical analysis of viscous vortex reconnection based on scale separation is developed, where the Reynolds number (Re = circulation/viscosity) scaling for reconnection time Trec is derived. The scaling varies from $T_{rec} \sim Re^{1}$ to $T_{rec} \sim Re^{0.5}$, and the direct numerical simulation (DNS) data from Garten [Garten et al, J. Fluid Mech. 426, 1 (2001), cited hereinafter as GW] and Hussain [Hussain et al, Phys. Fluids 23, 021701 (2011) cited hereinafter as HD] collapse well within the range of the asymptotic scalings. Moreover, our analysis predicts two Reynolds numbers, namely, a characteristic $Re_{theory} \in \left[ {O\left( {10^2} \right),O\left( {10^3} \right)} \right]$ for the $T_{rec} \sim Re^{0.75}$ scaling given by HD, and the critical Reynolds number $Re_c \sim O\left( {10^4} \right)$ for the transition after whom the largescale vortex reconnection does no longer occur. [Preview Abstract] 
Monday, November 21, 2011 9:05AM  9:18AM 
G21.00006: Streamline Topology of Helical Fluid Flow Morten Andersen An incompressible velocity field with helical symmetry is investigated. Helical symmetry means the velocity field ``looks'' the same as one moves on a given helix. This means a stream function can be constructed. The focus is on a proper description of helical symmetry and a bifurcation analysis of a flow with helical symmetry  the helical vortex filament. [Preview Abstract] 
Monday, November 21, 2011 9:18AM  9:31AM 
G21.00007: The constrained dipole dynamical system Andrew Tchieu, Eva Kanso, Paul Newton In this presentation the notion of a \textit{constrained dipole} is introduced as a pair of equal and opposite strength point vortices (i.e. a vortex dipole) separated by a finite distance such that this distance remains constant throughout the time evolution of the vortex dipole. The current formulation unambiguously defines a selfpropelling velocity for the dipole. Equations of motion for $N$ constrained dipoles interacting in an unbounded inviscid fluid are derived from the modified interaction of $2N$ independent vortices subject to the constraint that the intervortex spacing of each constrained dipole remains constant. There are no farfield assumptions for the dipoledipole interactions. We discuss the dynamics of the system of equations in the context of uncoordinated selfpropelled motion in a perfect fluid and give examples of interactions of two, three, and many constrained dipoles. Dipoles are found to collide with one another in abreast formations. Alternatively, the dipoles are conducive to remaining in formation when placed in staggered and diamond formations. Interestingly, equilibria are also formed from these aforementioned interactions, and we investigate the polygonal configurations of constrained dipoles and their subsequent stability. It is found that the $N=3$ case is linear stable, whereas $N>3$ is linearly unstable. [Preview Abstract] 
Monday, November 21, 2011 9:31AM  9:44AM 
G21.00008: Coupling of a 3D vortex particlemesh method with a finite volume nearwall solver Y. Marichal, T. Lonfils, M. Duponcheel, P. Chatelain, G. Winckelmans This coupling aims at improving the computational efficiency of high Reynolds number bluff body flow simulations by using two complementary methods and exploiting their respective advantages in distinct parts of the domain. Vortex particle methods are particularly well suited for free vortical flows such as wakes or jets (the computational domain with non zero vorticity is then compact and dispersion errors are negligible). Finite volume methods, however, can handle boundary layers much more easily due to anisotropic mesh refinement. In the present approach, the vortex method is used in the whole domain (overlapping domain technique) but its solution is highly underresolved in the vicinity of the wall. It thus has to be corrected by the nearwall finite volume solution at each time step. Conversely, the vortex method provides the outer boundary conditions for the nearwall solver. A parallel multiresolution vortex particlemesh approach is used here along with an Immersed Boundary method in order to take the walls into account. The nearwall flow is solved by OpenFOAM\textsuperscript{\textregistered} using the PISO algorithm. We validate the methodology on the flow past a sphere at a moderate Reynolds number. [Preview Abstract] 
Monday, November 21, 2011 9:44AM  9:57AM 
G21.00009: Resonant instability in twodimensional vortex arrays Paolo LuzzattoFegiz, Charles H.K. Williamson In this work, we examine conditions for the development of an oscillatory instability in twodimensional vortex arrays. By building on the theory of Krein signatures for Hamiltonian systems, and considering constraints owing to impulse conservation, we show that a resonant instability (developing through coalescence of two eigenvalues) cannot occur for one or two vortices. We illustrate this deduction by examining available linear stability results for one or two vortices. Our work indicates that a resonant instability may, however, occur for three or more vortices. For these more complex flows, we propose a simple model, based on an elliptical vortex representation, to detect the onset of an oscillatory instability. We provide an example in support of our theory by examining three corotating vortices, for which we also perform a linear stability analysis. The stability boundary in our model is in good agreement with the full stability calculation. In addition, we show that eigenmodes associated with an overall rotation or an overall displacement of the vortices always have eigenvalues equal to zero and $\pm \mathrm{i}\Omega$, respectively, where $\Omega$ is the angular velocity of the array. These results, for overall rotation and displacement modes, can also be used to immediately check the accuracy of a detailed stability calculation. [Preview Abstract] 
Monday, November 21, 2011 9:57AM  10:10AM 
G21.00010: 3D Characterization of Transmitral Vortex using Defocusing Digital Particle Image Velocimetry Ahmad Falahatpisheh, Brandon Dueitt, Niema Pahlevan, Arash Kheradvar In this study, we have experimentally characterized the 3D vortex passing through a physiologically relevant model of mitral valve using Defocusing Digital PIV (DDPIV). The valve model was made of soft silicone with diameter of $25mm$, similar to the adult mitral valve. The mitral model possesses a large anterior and a small posterior leaflet that results in asymmetric formation of transmitral vortex. A pistoncylinder mechanism drives the flow and travels to produce a range of $L/D$ from $2$ to $6$. We have characterized the shape of the 3D vortex forming through the Dshaped orifice of a mitral valve using DDPIV technique. The evolution of the vortex has been illustrated for different stroke ratios. [Preview Abstract] 
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