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 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 Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices

2018 ◽  
Vol 30 (9) ◽  
pp. 096603 ◽  
Author(s):  
Konstantin V. Koshel ◽  
Jean N. Reinaud ◽  
Giorgio Riccardi ◽  
Eugene A. Ryzhov
Keyword(s):  
Fixed Point ◽  
Point Vortex ◽  
Vortex Pair ◽  
Point Vortices

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 Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 908
Author(s):  
Perla Celis ◽  
Rolando de la Cruz ◽  
Claudio Fuentes ◽  
Héctor W. Gómez
Keyword(s):  
Survival Data ◽  
Maximum Likelihood Estimates ◽  
New Class ◽  
New Family ◽  
Gamma Distributions ◽  
Special Cases ◽  
Log Normal ◽  
Positive Support ◽  
Positive Distributions ◽  
Family Of Distributions

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution.

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 new class of quaternary generalized cyclotomic sequences of order 2d and length 2pm with high linear complexity

2018 ◽  
Vol 16 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Longfei Liu ◽  
Xiaoyuan Yang ◽  
Bin Wei ◽  
Liqiang Wu
Keyword(s):  
Finite Fields ◽  
Linear Complexity ◽  
Periodic Sequences ◽  
New Class ◽  
New Family ◽  
Generalized Cyclotomic Classes ◽  
Cyclotomic Sequences

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudo-random properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. In this paper, we construct a new family of quaternary generalized cyclotomic sequences with order [Formula: see text] and length [Formula: see text], which generalize the sequences constructed by Ke et al. in 2012. In addition, we determine its linear complexity using cyclotomic theory. The conclusions reveal that these sequences have high linear complexity, which means they can resist linear attacks.

scholarly journals Extended Odd Fréchet-G Family of Distributions

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Suleman Nasiru
Keyword(s):  
Maximum Likelihood Method ◽  
Real Data ◽  
Likelihood Method ◽  
Data Sets ◽  
Data Set ◽  
New Class ◽  
New Family ◽  
Rate Functions ◽  
The Given ◽  
Family Of Distributions

The need to develop generalizations of existing statistical distributions to make them more flexible in modeling real data sets is vital in parametric statistical modeling and inference. Thus, this study develops a new class of distributions called the extended odd Fréchet family of distributions for modifying existing standard distributions. Two special models named the extended odd Fréchet Nadarajah-Haghighi and extended odd Fréchet Weibull distributions are proposed using the developed family. The densities and the hazard rate functions of the two special distributions exhibit different kinds of monotonic and nonmonotonic shapes. The maximum likelihood method is used to develop estimators for the parameters of the new class of distributions. The application of the special distributions is illustrated by means of a real data set. The results revealed that the special distributions developed from the new family can provide reasonable parametric fit to the given data set compared to other existing distributions.

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