Trapped vortices in multiply connected domains

Nelson, Rhodri; Sakajo, Takashi

doi:10.1088/0169-5983/46/6/061402

Cited by 4 publications

(8 citation statements)

References 15 publications

(26 reference statements)

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“…Based on a mathematical framework for point vortex dynamics [10] on 2D Riemannian surfaces, Dritschel & Boatto [11] have recently investigated point vortex dynamics on 2D surfaces conformal to the unit sphere. Point vortex dynamics in multiply connected planar domains [12] is applied to many physical and engineering problems such as an ocean flow [13] and an efficient force-enhancing wing design [14]. The stability of Kármán vortex streets [1] and exotic vortex wakes behind an oscillating bluff body [15] are examined with point vortex models with a periodic boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Point vortex interactions on a toroidal surface

Sakajo¹,

Shimizu²

2016

Proc. R. Soc. A.

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Owing to non-constant curvature and a handle structure, it is not easy to imagine intuitively how flows with vortex structures evolve on a toroidal surface compared with those in a plane, on a sphere and a flat torus. In order to cultivate an insight into vortex interactions on this manifold, we derive the evolution equation for N-point vortices from Green's function associated with the LaplaceBeltrami operator there, and we then formulate it as a Hamiltonian dynamical system with the help of the symplectic geometry and the uniformization theorem. Based on this Hamiltonian formulation, we show that the 2-vortex problem is integrable. We also investigate the point vortex equilibria and the motion of two-point vortices with the strengths of the same magnitude as one of the fundamental vortex interactions. As a result, we find some characteristic interactions between point vortices on the torus. In particular, two identical point vortices can be locally repulsive under a certain circumstance.

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Section: Introductionmentioning

confidence: 99%

Point vortex interactions on a toroidal surface

Sakajo¹,

Shimizu²

2016

Proc. R. Soc. A.

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“…We begin by stating the governing equations for both the single plate and Kasper Wing systems and following this survey the stability of some of the equilibria these systems admit. A detailed derivation of these equations is presented in Nelson & Sakajo (2014), whereas the stability and non-linear robustness of various configurations is examined in Nelson & Sakajo (2016). Here, attention is restricted to configurations which will be studied in the controlled setting.…”

Section: Flow Models In the Uncontrolled Settingmentioning

confidence: 99%

“…where a is a scaling constant such that |dW/dz| → U as |z| → ∞ (see Nelson & Sakajo, 2014); 2. flow due to a single point vortex…”

Section: Governing Equationsmentioning

confidence: 99%

“…However, closedform solutions are not possible for the Kasper Wing system (M = 2) and a numerical approach is required. As in Nelson & Sakajo (2014, 2016, a Brownian ratchets scheme is used to solve equations (10) to within a prescribed numerical tolerance. Further details regarding such Brownian ratchets schemes can be found in Newton & Chamoun (2007).…”

Section: Governing Equationsmentioning

confidence: 99%

“…Given that flow configurations with trapped vortices tend to be unstable, the purpose of the present study is to propose and validate practical stabilization strategies. Flow configurations will be based on those of Saffman & Sheffield (1977) for the single plate case and those of Nelson & Sakajo (2014) for the Kasper Wing case. We will first provide a control-oriented characterization of the stability of these flow equilibria and will then design a Linear-Quadratic-Gaussian (LQG) compensator for their stabilization.…”

Section: Introductionmentioning

confidence: 99%

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Linear feedback stabilization of point-vortex equilibria near a Kasper wing

2017

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This paper concerns feedback stabilization of point vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper Wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modeled by the two-dimensional incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink-source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearised around these equilibria are both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the Linear-Quadratic-Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding of this study is that Kasper Wing configurations are in general not only more controllable than their single plate counterparts, but also exhibit larger basins of attraction under LQG feedback control. The feedback control is * Email address for correspondence: rnelson@ic.ac.uk 1 then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, except in a few isolated cases, introducing a vorticity shedding mechanism enhanced the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.

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Shape calculus for vortex patch equilibria and its application to lattice configurations

Uda

2018

Journal of Computational and Applied Mathematics

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Trapped vortices in multiply connected domains

Cited by 4 publications

References 15 publications

Point vortex interactions on a toroidal surface

Point vortex interactions on a toroidal surface

Linear feedback stabilization of point-vortex equilibria near a Kasper wing

Shape calculus for vortex patch equilibria and its application to lattice configurations

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