Superposition Of Vortices Research Articles

On instability characteristics of isolated vortices and models of trailing-vortex systems

This paper demonstrates the applicability of a two-dimensional eigenvalue problem approach to the study of linear instability of analytically constructed and numerically calculated models of trailing-vortex systems. Chebyshev collocation is used in the 2D eigenvalue problem solution in order to discretize two spatial directions on which non-axisymmetric vorticity distributions are defined, while the third, axial spatial direction is taken to be homogeneous and is resolved by a Fourier expansion. The leading eigenvalues of the matrix discretizing the equations which govern small-amplitude perturbations superimposed upon such a vorticity distribution are obtained by Arnoldi iteration. The present approach has been validated by comparison of its results on the problem of instability of an isolated Batchelor vortex. Here benchmark computations exist, employing classic instability analysis, in which the azimuthal direction is also treated as homogeneous. Subsequently, the proposed methodology has been shown to be able to recover the classic long- (Crow) and short-wavelength instabilities of a counter-rotating vortex-pair basic flow obtained by direct numerical simulation. Finally, the effect on the eigenspectrum of the isolated Batchelor vortex is documented, when the basic flow consists of a linear superposition of such vortices. The modifications of the eigenspectrum of a single vortex point to the potential pitfalls of drawing conclusions on the instability characteristics of a trailing-vortex system by monitoring the constituent vortices in isolation.

Circularly polarized beams and vortex generation in uniaxial media.

We deduce the expressions for the two circularly polarized components of a paraxial beam propagating along the optical axis of a uniaxial crystal. We find that each of them is the sum of two contributions, the first being a free field and the second describing the interaction with the opposite component. Moreover, we expand both components as a superposition of vortices of any order, thus obtaining a complete physical picture of the interaction dynamics. Consequently, we argue that a left-hand circularly polarized incoming beam, endowed with a circular symmetric profile, gives rise, inside the crystal, to a right-hand circularly polarized vortex of order 2. The efficiency of this vortex generation is investigated by means of a power exchange analysis. The Gaussian case is fully discussed, showing the relevant features of the vortex generation.

Superposition of noncoaxial vortices in parametric wave mixing.

In this paper we present a comprehensive study of the dynamics of screw phase dislocations under conditions of noncoaxial parametric three-wave mixing in the pump low-depletion regime. Under such conditions the signal and idler fields couple and so, the fields' properties change through propagation in the nonlinear crystal. We present an analytical model and a comprehensive study of the vortical features of the resulting field. The model is compared with the numerical solutions of the full equations. It is shown that by changing the relative amplitude and phase of the initial fields, one can control the domains where creation and annihilation of vortex-antivortex twins lead to different vortex content. We show that the effects studied here are relevant to a variety of physical systems. In particular, we show that the same phenomena are expected to occur in gyrotropic media and photonic crystals.

Internal gravity wave emission from a pancake vortex: An example of wave–vortex interaction in strongly stratified flows

At small Froude numbers the motion of a stably stratified fluid consists of a quasisteady vortical component and a propagating wave component. The vortical component is organized into layers of horizontal motions with well-pronounced vertical vorticity and often takes the form of so-called “pancake” vortices. An analytical model of such a vortex that is a solution of the Euler–Boussinesq equations at a vanishing Froude number is constructed as a superposition of horizontal two-dimensional Kirchhoff elliptic vortices. This vortex is nonstationary and internal gravity waves are, therefore, excited by its motion. The radiation properties are studied by matching the vortex field with the far internal gravity wave field according to the procedure applied in acoustics to determine vortex sound. The structure of the gravity wave field is completely quantified. By calculating energy and angular momentum fluxes carried by outgoing waves and attributing them to the adiabatic change of the vortex parameters, we calculate the backreaction of the internal gravity waves radiation and show that, as in the case of acoustic radiation by the Kirchhoff vortex, this adiabatic evolution leads to an elongation of the vortex, and its eventual destabilization.

Nonparaxial gaussian beams: 3. Optical vortices

It is shown that an arbitrary nonparaxial laser vortex near a focal plane in the free space should be represented as a discrete superposition of intrinsic vortices possessing elliptical cross sections. The intrinsic vortices, albeit degenerate with respect to the propagation constants, exhibit a characteristic spectrum of singularities and topological phases.

Simulation of homogeneous and incompressible Çinlar flows

We study the simulation of stationary, homogeneous, incompressible and isotropic Çinlar flows on R 2 . The flow is generated by a velocity field obtained by the superposition of vortices of rotation. The arrival time and location of vortices form a Poisson point process. The two stages of the simulation of the flow are the generation of the velocity field and the integration of the particle paths. We generate the velocity field on a bounded domain D exactly. The velocity field on D is fully described by the parameters of vortices that are stored in a stack the size of which is fairly stable at the stationary regime. We obtain the particle path by integrating the flow equation using a fourth order Runge–Kutta method. A range of ratios of the two relevant time scales lead to a variety of particle paths. Under some regimes, the paths are nearly Brownian, under other, the paths are clearly circular with some drift. Finally, we compute single particle dispersion, Lagrangian autocorrelation, and diffusivity estimators through Monte Carlo simulations. The results are useful for fitting the model to real data.

Constructions of some exact vortex superposition solutions

In theoretical ways to solve the N-S equation, we will confront many difficulties. For example, the complex mathematical calculation, the 3D velocity components and the nonlinear terms, etc. We find that from the basis of Stokes-Helmholtz decomposition and after some processing, we can get a method of constructing the vortex superposition solutions. We get a set of exact solutions in different coordinates and we hope these solutions will be helpful to a clearer understanding of the vortex in Fluid Mechanics.

Simulation of turbulent dispersion using Weierstrass modes

A simple fractal model is proposed for the dispersion of passive scalars in an incompressible homogeneous turbulent flow field. The dispersion process is based on a three-dimensional velocity field which is assumed to be a linear superposition of Taylor-Green vortices with wave numbers and amplitudes as those in a Weierstrass function. The chosen velocity field satisfies the continuity equation as well as Kolmogorov's inertial range power law, and has a fractal dimension D between upper and lower length-scale bounds. A cloud of tracer particles is relased into the flow field and dispersed by the fluid motion, the trajectories are numerically integrated from the velocity field. The puffs which result from this process are used to examine some aspects of turbulent dispersion, through comparisons with integrated concentration wind-tunnel measurements. The agreement between numerical and experimental results indicates the significance of the proposed simulation model.

Vortex formation around an oscillating and translating airfoil at large incidences

The starting flows past a two-dimensional oscillating and translating airfoil are investigated by visualization experiments and numerical calculations. The airfoil, elliptic in cross-section, is set in motion impulsively and subjected simultaneously to a steady translation and a harmonic oscillation in pitch. The incidence of the airfoil is variable between 0° and 45° and the Reynolds number based on the chord length is between 1500 and 10000. The main object of the present study is to reveal some marked characteristics of the unsteady vortices produced from the oscillating airfoil set at large incidences in excess of the static stall angle. Another purpose is to examine, in some detail, the respective and combined effects of the major experimental parameters on the vortex wake development. It is shown that, in general, the dominant parameter of the flow is the reduced frequency not only when the airfoil oscillates at incidences close to the static stall angle but also at larger incidences. It is also demonstrated that, as the pitching frequency is increased, the patterns of the vortex wake are dependent on the product of the reduced frequency and the amplitude rather than on the frequency itself. It is noted that the combined effect of a high reduced frequency and a large amplitude can give rise to cyclic superposition of leading-edge vortices from which a gradually expanding standing vortex is developed on the upper surface.

A three-dimensional inverse calculation of an impeller by the finite-pitch actuator duct method. (Sequel report).

A full three-dimensional inverse solution of the impeller blade shape is presented. The inverse method automatically calculates the blade shape which satisfies flow characteristics specified by a designer without trial iteration of flow analysis calculations. The flow is assumed to be inviscid and incompressible. The impeller blades are represented by superposition of vortices and sources on the blade mean surface. By expanding generalized velocity potential in a Fourier series, a set of two-dimensional quasi-Poisson equations in r and z is obtained. These equations are solved with SOR. Numerical examples are shown for the design of axial flow rotors with free vortex swirl. A blade thickness distribution and a pressure difference distribution across the blade are specified, and the blade mean-surface shape is determined by the inverse method. The results are compared with those presented in our previous report using a Fourier-Bessel expansion. The flow analysis for the designed impeller is performed by means of the two-dimensional singularity method and the three-dimensional panel method, and the calculation accuracy of the inverse method is examined.

Oblique vortices on a solid wall and on an interface between two immiscible viscous fluids

A local three-dimensional flow analysis has been made about a singular point at the interface between two immiscible viscous fluids. A generalization has been found of a ‘‘refraction law’’ for dividing streamlines that was derived in a previous study [Phys. Fluids 30, 3647 (1987)] for two-dimensional flows. It is shown here that the inclination angle of a vortex axis with an interface lies in a plane perpendicular to the shear flow that causes the inclination. An axisymmetric stagnation-point flow is tilted in the direction of the shear flow. Superposition of both vortex and stagnation-point flow yields an angle of inclination somewhere between the planes perpendicular and parallel to the shear flow. These results are valid for slip and nonslip surface conditions.

A vortex-street model of the flow in the similarity region of a two-dimensional free turbulent jet

The mean-velocity profiles and entrainment rates in the similarity region of a two-dimensional jet are generated by a simple superposition of Rankine vortices arranged to represent a vortex street. The spacings between the vortex centres, their two-dimensional offsets from the centreline, as well as the core radii and circulation strengths, are all governed by similarity relationships and based upon experimental data.Major details of the mean flow field such as the axial and lateral mean-velocity components and the magnitude of the Reynolds stress are properly determined by the model. The sign of the Reynolds stress is, however, not properly predicted.