Bulletin of the American Physical Society
2005 58th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 20–22, 2005; Chicago, IL
Session LG: Vortex Dynamics IV |
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Chair: Jamey Jacob, University of Kentucky Room: Hilton Chicago Williford A |
Tuesday, November 22, 2005 8:00AM - 8:13AM |
LG.00001: Adiabatic Control of a Pair of Elliptical Vortex Patches Dmitri Vainchtein, Igor Mezic, Luca Cortelezzi We discuss an adiabatic method of controlling the interaction of a pair of elliptical vortex patches, including forcing or preventing merging. The idealized actuator is a point vortex of time-varying strength positioned at the joint center of vorticity. It represents a rotating rod. The sensors are realistic and measure the fluid velocity at two given locations. An observer is derived in order to reconstruct the internal state of the system, which is used by the controller to predict the strength of the actuator. The adiabatic controller, through the actuator, applies small perturbations to the system in order to modify the interaction of the vortex pair. We show that the controller can be made more efficient by leveraging the internal dynamics of the system, in particular, by keeping the perturbations is phase with the fast variable of the nominal system. The Hamiltonian structure of the control field is crucial to prove the controllability and the Hamiltonian structure of the nominal system leads to significant extension of the reachable domain. We perform a set of numerical simulations to confirm the analytical results. [Preview Abstract] |
Tuesday, November 22, 2005 8:13AM - 8:26AM |
LG.00002: Multiscale three-dimensional vortex pairs of the Poiseuille flow Victor A. Miroshnikov In three dimensions, the Boussinesq-Rayleigh series solutions of the unsteady Navier-Stokes equations are computed by symbolic programming and evaluated by parallel computing. For generating functions, which are bounded together with their derivatives, the absolute convergence of the series solutions is shown by converting the differential recurrent relations into tensor recurrent relations and using the comparison and ratio tests. A pyramidal structure of four-dimensional tensors of derivatives, which is employed in the tensor recurrent relations, is obtained by induction. It is shown that the general solutions away from boundaries are nonlinear superpositions of the Stokes flow, the Bernoulli flow, the Couette flow, and the Poiseuille flow, which are unsteady, three-dimensional continuations of the classical flows at high Reynolds numbers. The general solution for the Poiseuille flow is specified by periodic generating functions, which model mixing in one-, two-, and three- dimensional flows away from boundaries. The emergence and interaction of multiscale vortex pairs are treated mathematically due to the existence of multi-valued general solutions for streamlines of the Poiseuille flow at high Reynolds numbers. [Preview Abstract] |
Tuesday, November 22, 2005 8:26AM - 8:39AM |
LG.00003: Exact Ellipsoidal Hamiltonian Reductions of Euler's Equations P.J. Morrison, N.R. Lebovitz, J.A. Biello There exist special initial conditions that result in simplified yet exact reductions of ideal fluid systems. Examples include point vortex dynamics, Kirchoff and Kida elliptical vortex patch dynamics, quasigeostrophic ellipsoidal dynamics, and equations that describe the dynamics of Riemann's self-gravitating ellipsoids. All of these systems except the last have been directly obtained from the Hamiltonian (noncanonical Poisson bracket) description of Euler's fluid equations [1], and the subject of the present work is to do this for the Riemann ellipsoids. We begin with the Poisson bracket for the compressible fluid and project it to obtain a bracket for incompressible dynamics. We then reduce by considering dynamics restricted to velocity and density moments. It is seen that this reduction is exact, and that the Poisson bracket obtained produces the finite degree-of-freedom Hamiltonian system that describes Riemann's ellipsoids.\newline \newline [1] P.J. Morrison, Rev. Mod. Phys. {\bf 70}, 467 (1998). [Preview Abstract] |
Tuesday, November 22, 2005 8:39AM - 8:52AM |
LG.00004: Hamiltonian Description of Riemann Ellipsoids via Dirac's Bracket N.R. Lebovitz, P.J. Morrison, J.A. Biello The only known exact solutions of the Euler equations for an asymmetric, self-gravitating mass are the Riemann ellipsoids. They are characterized by motions of uniform vorticity in a rotating reference frame, and are governed by a finite-dimensional system, i.e., a system of ordinary differental equations. A noncanonical Hamiltonian description in terms of a finite number of moments of velocity and density was first written down by Rosensteel, for the case when the fluid is unconstrained by the assumption of incompressibility. We show that this system can be obtained systematically by a moment reduction from the compressible-fluid bracket. We then employ the Dirac- bracket procedure to incorporate the constraint of incompressibility in this finite-dimensional system and show that the resulting bracket precisely confirms the moment-reduction obtained directly from an incompressible-fluid bracket, as described in the companion paper. [Preview Abstract] |
Tuesday, November 22, 2005 8:52AM - 9:05AM |
LG.00005: Decaying two-dimensional turbulence in a circular container Kai Schneider, Marie Farge We present direct numerical simulation of two-dimensional decaying turbulence in a circular container with no--slip boundary conditions. Starting with random initial conditions the flow rapidly exhibits a self--organization into coherent vortices. We study their formation and the role of the viscous boundary layer on the production and decay of integral quantities. The no--slip wall produces vortices which are injected into the bulk flow. The self-organisation of the flow is reflected by the transition of the initially Gaussian vorticity probability density function (PDF) towards a distribution with exponential tails. Due to the presence of coherent vortices the pressure PDFs become strongly skewed with exponential tails for negative values. [Preview Abstract] |
Tuesday, November 22, 2005 9:05AM - 9:18AM |
LG.00006: Stereo-PIV measurements of a swirling flow in a straight duct with downstream contraction Benjamin Leclaire, Laurent Jacquin, Jean-Charles Abart, Robert Soares, Didier Soulevant The generating conditions of a high-Reynolds swirling jet are addressed. The setup used involves a rotating honeycomb, followed by a transparent duct of constant cross-section and a final converging nozzle. Traditionaly, the contraction is expected to reduce turbulence in the exit plane, thereby leading to a smooth upstream condition for the subsequent jet instabilities or breakdown. We find that this only pertains to the lowest rotation rates since developed turbulence progressively invades the nozzle exit plane as rotation increases. We therefore investigate the dynamics of the duct flow preceding the jet by means of stereo-PIV measurements in the duct portion of constant cross-section. The Reynolds number of the flow is fixed and the control parameters are the swirl number and the contraction ratio of the downstream nozzle. PIV reveals that at high swirl numbers, the flow exhibits a complex behaviour which may explain the turbulence found in the exit plane. [Preview Abstract] |
Tuesday, November 22, 2005 9:18AM - 9:31AM |
LG.00007: Transitions in an enclosed swirling flow J.M. Lopez Transitions of the flow in an enclosed cylinder driven by the constant rotation of an endwall, from steady axisymmetric flow to aperiodic flow characterized by intermittent bursting dynamics where all the spatial and spatio-temporal symmetries have been broken, are studied numerically. The problem is controlled by two parameters, the Reynolds number and the cylinder aspect ratio. We vary the Reynolds number, fixing the aspect ratio at a value where the primary bifurcation of the axisymmetric steady state is to an axisymmetric periodic flow. The final transition to weak turbulence, however, is governed by a non-axisymmetric branch of rotating waves, which is the primary mode at lower aspect ratios, and the various branches of modulated rotating waves associated with subsequent bifurcations from the rotating wave. We study in detail the spatio-temporal characteristics of the various states encountered along the way, and how the symmetry of the problem impacts on the transition dynamics. [Preview Abstract] |
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