scholarly journals Force-enhancing vortex equilibria for two parallel plates in uniform flow

2012 ◽  
Vol 468 (2140) ◽  
pp. 1175-1195 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Stationary Point ◽  
Design Problem ◽  
Flow Problem ◽  
Point Vortex ◽  
Uniform Flow ◽  
Point Vortices ◽  
Multiply Connected Domains ◽  
Parallel Plates ◽  
Single Plate ◽  
Point Vortex Equilibria

A two-dimensional potential flow in an unbounded domain with two parallel plates is considered. We examine whether two free point vortices can be trapped near the two plates in the presence of a uniform flow and observe whether these stationary point vortices enhance the force on the plates. The present study is an extension of previously published work in which a free point vortex over a single plate is investigated. The flow problem is motivated by an airfoil design problem for the double wings. Moreover, it also contributes to a design problem for an efficient wind turbine with vertical blades. In order to obtain the point-vortex equilibria numerically, we make use of a linear algebraic algorithm combined with a stochastic process, called the Brownian ratchet scheme. The ratchet scheme allows us to capture a family of stationary point vortices in multiply connected domains with ease. As a result, we find that stationary point vortices exist around the two plates and they enhance the downward force and the counter-clockwise rotational force acting on the two plates.

scholarly journals A transformation between stationary point vortex equilibria

2020 ◽  
Vol 476 (2240) ◽  
pp. 20200310
Author(s):  
Vikas S. Krishnamurthy ◽  
Miles H. Wheeler ◽  
Darren G. Crowdy ◽  
Adrian Constantin
Keyword(s):  
Finite Number ◽  
Stationary Point ◽  
Point Vortex ◽  
Wide Range ◽  
Finite Sequences ◽  
Free Parameters ◽  
Point Vortex Equilibria ◽  
Isolated Point ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to −1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37 , 1309–1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.

Steady point vortex pair in a field of Stuart-type vorticity

2019 ◽  
Vol 874 ◽  
Author(s):  
Vikas S. Krishnamurthy ◽  
Miles H. Wheeler ◽  
Darren G. Crowdy ◽  
Adrian Constantin
Keyword(s):  
Point Vortex ◽  
Vortex Pair ◽  
Point Vortices ◽  
Incompressible Euler Equation ◽  
New Class ◽  
New Family ◽  
Point Vortex Equilibria ◽  
Similar Class ◽  
Incompressible Euler ◽  
Additional Point

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

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

2017 ◽  
Vol 827 ◽  
pp. 121-154 ◽  
Author(s):  
R. Nelson ◽  
B. Protas ◽  
T. Sakajo
Keyword(s):  
Feedback Control ◽  
Point Vortex ◽  
Basins Of Attraction ◽  
Feedback Stabilization ◽  
Linear Feedback ◽  
Linear Quadratic ◽  
Linear Control Theory ◽  
Single Plate ◽  
Point Vortex Equilibria ◽  
Almost All

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 modelled 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 linearized around these equilibria is 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 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.

scholarly journals Stationary states of identical point vortices and vortex foam on the sphere

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120622 ◽  
Author(s):  
Kevin A. O'Neil
Keyword(s):  
Initial Conditions ◽  
Point Vortex ◽  
Intermediate Energy ◽  
Point Vortices ◽  
Stationary States ◽  
Vortex Sheets ◽  
Identical Point ◽  
Equilibrium Configurations ◽  
Stationary Vortex ◽  
Point Vortex Equilibria

Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary ‘vortex foam’ on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included.

scholarly journals Point vortex interactions on a toroidal surface

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160271 ◽  
Author(s):  
Takashi Sakajo ◽  
Yuuki Shimizu
Keyword(s):  
Symplectic Geometry ◽  
Hamiltonian Formulation ◽  
Point Vortex ◽  
Beltrami Operator ◽  
Point Vortices ◽  
Uniformization Theorem ◽  
Vortex Interactions ◽  
Toroidal Surface ◽  
Point Vortex Equilibria ◽  
Insight Into

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 Laplace–Beltrami 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.

scholarly journals Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 943
Author(s):  
Henryk Kudela
Keyword(s):  
Differential Equations ◽  
Point Vortex ◽  
Optimization Procedure ◽  
Vortex Sheet ◽  
Point Vortices ◽  
Vortex Sheets ◽  
Impermeable Barrier ◽  
Initial Location ◽  
Passive Tracers ◽  
Initial Iterate

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

A versatile taxonomy of low-dimensional vortex models for unsteady aerodynamics

2018 ◽  
Vol 858 ◽  
pp. 917-948 ◽  
Author(s):  
Darwin Darakananda ◽  
Jeff D. Eldredge
Keyword(s):  
Large Scale ◽  
Bluff Body ◽  
Point Vortex ◽  
Unsteady Aerodynamics ◽  
Shear Layers ◽  
Point Vortices ◽  
The Body ◽  
Vortex Sheets ◽  
Proposed Model ◽  
Low Dimensional

Inviscid vortex models have been demonstrated to capture the essential physics of massively separated flows past aerodynamic surfaces, but they become computationally expensive as coherent vortex structures are formed and the wake is developed. In this work, we present a two-dimensional vortex model in which vortex sheets represent shear layers that separate from sharp edges of the body and point vortices represent the rolled-up cores of these shear layers and the other coherent vortices in the wake. We develop a circulation transfer procedure that enables each vortex sheet to feed its circulation into a point vortex instead of rolling up. This procedure reduces the number of computational elements required to capture the dynamics of vortex formation while eliminating the spurious force that manifests when transferring circulation between vortex elements. By tuning the rate at which the vortex sheets are siphoned into the point vortices, we can adjust the balance between the model’s dimensionality and dynamical richness, enabling it to span the entire taxonomy of inviscid vortex models. This hybrid model can capture the development and subsequent shedding of the starting vortices with insignificant wall-clock time and remain sufficiently low-dimensional to simulate long-time-horizon events such as periodic bluff-body shedding. We demonstrate the viability of the method by modelling the impulsive translation of a wing at various fixed angles of attack, pitch-up manoeuvres that linearly increase the angle of attack from $0^{\circ }$ to $90^{\circ }$, and oscillatory pitching and heaving. We show that the proposed model correctly predicts the dynamics of large-scale vortical structures in the flow by comparing the distributions of vorticity and force responses from results of the proposed model with a model using only vortex sheets and, in some cases, high-fidelity viscous simulation.

Flexible Flapping Dynamics of Parallel Elastic Plates in a Uniform Flow: Application to Energy Harvesting Devices

2014 ◽  
Author(s):  
Pardha S. Gurugubelli ◽  
Rajeev K. Jaiman ◽  
Boo Cheong Khoo
Keyword(s):  
Coupled System ◽  
Uniform Flow ◽  
Net Energy ◽  
Elastic Plates ◽  
Parallel Plates ◽  
Mass Ratio ◽  
Bending Rigidity ◽  
Fluid Structure ◽  
Flapping Motion ◽  
Wide Range

A new high-order finite element Coupled Field with Explicit Interface (CFEI) code has been developed for simulating flapping motion of a thin flexible body in a uniform flow with strong add-mass effects. In the first part, we study the flapping dynamics of a single cantilevered plate for wide range of mass ratios and maintaining relatively low bending rigidity through our direct fluid-structure simulations. As a function of mass-ratio, the flapping dynamics reveals three distinct regimes: (i) fixed-point stable; (ii) limit-cycle flapping; and (iii) chaotic flapping. The changes associated with regime transition with increasing mass ratio are analyzed by vortex wake patterns and tip-displacement responses. Dependencies of stability predicted by the theoretical analysis are confirmed by the nonlinear fluid-structure simulations. In the second part, two parallel cantilevered plates will be investigated as a function of spacing between the parallel plates to assess the flapping motion and the net energy transfer. The flow-induced vibrations of this kind of coupled system have a potential to extract energy from the surrounding fluid flow for generation of electric power.

scholarly journals Head-on collisions between two quasi-geostrophic hetons in a continuously stratified fluid

2015 ◽  
Vol 779 ◽  
pp. 144-180 ◽  
Author(s):  
Jean N. Reinaud ◽  
Xavier Carton
Keyword(s):  
Parameter Space ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Point Vortex ◽  
Point Vortices ◽  
Symmetric Pair ◽  
Horizontal Shear ◽  
Azimuthal Mode ◽  
Aspect Ratios ◽  
Vortex Merger

We examine the interactions between two three-dimensional quasi-geostrophic hetons. The hetons are initially translating towards one another. We address the effect of the vertical distance between the two poles (vortices) constituting each heton on the interaction. We also examine the influence of the horizontal separation between the poles within each heton. In this investigation, the two hetons are facing each other. Two configurations are possible depending on the respective locations of the like-signed poles of the hetons. When they lie at the same depth, we refer to the configuration as symmetric; the antisymmetric configuration corresponds to opposite-signed poles at the same depth. The first step in the investigation uses point vortices to represent the poles of the hetons. This approach allows us to rapidly browse the parameter space and to estimate the possible heton trajectories. For a symmetric pair, the hetons either reverse their trajectory or recombine and escape perpendicularly depending of their horizontal and vertical offsets. On the other hand, antisymmetric hetons recombine and escape perpendicularly as same-depth dipoles. In a second part, we focus on finite core hetons (with finite volume poles). These hetons can deform and may be sensitive to horizontal-shear-induced deformations, or to baroclinic instability. These destabilisations depend on the vertical and horizontal offsets between the various poles, as well as on their width-to-height aspect ratios. They can modify the volume of the poles via vortex merger, breaking and/or shearing out; they compete with the advective evolution observed for singular (point) vortices. Importantly, hetons can break down or reconfigure before they can drift away as expected from a point vortex approach. Thus, a large variety of behaviours is observed in the parameter space. Finally, we briefly illustrate the behaviour of tall hetons which can be unstable to an azimuthal mode $l=1$ when many vertical modes of deformation are present on the heton.

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