Continuous Distribution Of Vorticity Research Articles

Two-dimensional shielded vortices in a shear current

The interaction of shielded vortices, with a continuous vorticity distribution, and a shear current of weak vorticity amplitude but similar velocity compared to the vortex amplitude is numerically investigated in two-dimensional isochoric flows. Different types of axisymmetric shielded vortices, namely, a neutral unstable vortex, a neutral robust vortex, and a non-neutral vortex are considered. The vortices are linear combinations of vorticity layer-modes, which consist of conveniently normalized cylindrical Bessel functions of order 0, truncated by a zero of the Bessel function of order 1. The vortex–current interaction is investigated by superposing initially the vortices at different initial locations along the cross-flow axis in the shear current. The numerical results show that some shielded vortices, as well as the shear current, remain robust while the vortices cross the shear current and reach a stable equilibrium location, which is of the same sign vorticity as its amount of circulation. There exist two unstable equilibrium locations where most of the vortices persist during a relatively short time interval before heading to their stable equilibrium region in the shear current.

On the linear stability of the Lamb–Chaplygin dipole

The Lamb–Chaplygin dipole (Lamb, Hydrodynamics, 2nd edn, 1895, Cambridge University Press; Lamb, Hydrodynamics, 3rd edn, 1906, Cambridge University Press; Chaplygin, Trudy Otd. Fiz. Nauk Imper. Mosk. Obshch. Lyub. Estest., vol. 11, 1903, pp. 11–14) is one of the few closed-form relative equilibrium solutions of the two-dimensional (2-D) Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2-D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time integration of the 2-D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2-D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.

An efficient algorithm for calculating boundary elements in vortex methods

A theorem on the conformal mapping of the flow field with a continuous vorticity distribution is proved. An efficient algorithm of the boundary elements computing based on the theorem is applied in simulation of two-dimensional flow around an elliptic cylinder by the meshless method of Viscous Vortex Domains (VVD). The algorithm makes it possible to significantly reduce (orders of magnitude compared to conventional methods) the time spent on calculating boundary elements and multiply the number of calculation points on the body contour, which is necessary for high resolution of the boundary layer at high values of the Reynolds number. The algorithm can also be used to calculate boundary elements on cylindrical surfaces of arbitrary section and in combination with different vortex methods. The flow around an elliptical cylinder is simulated with a high resolution of the boundary layer for Re numbers from 102 to 105.

Axisymmetric contour dynamics for buoyant vortex rings

On the Benjamin–Lighthill conjecture for water waves with vorticity

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

Hamiltonian description of shear and gravity shear waves in an ideal incompressible fluid

Methods of studying the dynamics of wave disturbances in st;ratified shear flows of an ideal incompressible fluid are considered. The equations governing the motions of interest represent Hamilton equations and are derived by writing the velocity field in terms of Clebsch potentials. Equations written in terms of semi-Lagrangian variables are integrodifferential equations, which make it possible to consider both continuous and discontinuous solutions, as well as the cases where the parameters of the undisturbed medium are step functions. Two dynamic systems are presented. The first, canonical system of equations is most suitable for describing gravity waves in a shear flow in the case where the undisturbed medium is characterized by sharp gradients of density and flow velocity. The simplest model in which disturbances obey this system of equations is the well-known Kelvin-Helmholtz model. The second dynamic system describes, in particular, gravity-shear waves and, in the case of a homogeneous medium, shear waves in a two-dimensional flow. This system is most suitable for studying the dynamics of disturbances in models with sharp gradients of vorticity. On the basis of the approach developed in this study, the problem of the dynamics of disturbances in a flow with a continuous distribution of vorticity in a finite-thickness layer is solved. If the thickness of this layer is small compared to the characteristic wavelength and the gradient of the undisturbed vorticity in this layer is large, the solution has the form of a mode whose frequency is close to the frequency of the shear wave on a vorticity jump that would be obtained by letting the layer’s thickness approach zero. The results obtained allow, in particular, the estimation of the range of validity of finite-layer approximations for models with smooth profiles of flow and density. In addition, these results can be interpreted as the basis for the development of nonlinear aspects of the theory of hydrodynamic stability.

Numerical solutions for a two-dimensional airfoil undergoing unsteady motion

Continuous vorticity panels are used to model general unsteady inviscid, incompressible, and two-dimensional flows. The geometry of the airfoil is approximated by series of short straight segments having endpoints that lie on the actual surface. A piecewise linear, continuous distribution of vorticity over the airfoil surface is used to generate disturbance flow. The no-penetration condition is imposed at the midpoint of each segment and at discrete times. The wake is simulated by a system of point vortices, which move at local fluid velocity. At each time step, a new wake panel with uniform vorticity distribution is attached to the trailing edge, and the condition of constant circulation around the airfoil and wake is imposed. A new expression for Kutta condition is developed to study (i) the effect of thickness on the lift buildup of an impulsively started airfoil, (ii) the effects of reduced frequency and heave amplitude on the thrust production of flapping airfoils, and (iii) the vortex-airfoil interaction. This work presents some hydrodynamic results for tidalstream turbine.

Interaction of two-dimensional impulsively started airfoils

Continuous vorticity panels were used to model general unsteady inviscid, incompressible, two-dimensional flows. The geometry of the airfoil was approximated by series of short straight segments having endpoints that lie on the actual surface. A piecewise linear, continuous distribution of vorticity over the airfoil surface was used to generate disturbance flow. The no-penetration condition was imposed at the midpoint of each segment and at discrete times. The wake was simulated by a system of point vortices, which moved at local fluid velocity. At each time step, a new wake panel with uniform vorticity distribution was attached to the trailing edge, and the condition of constant circulation around the airfoil and wake was imposed. A new expression for Kutta condition was developed to study the interference effect between two impulsively started NACA0012 airfoils. The tandem arrangement was found to be the most effective to enhance the lift of the rear airfoil. The interference effect between tidal turbine blades was shown clearly.

Do vortex filaments in a superfluid neutron star produce gravimagnetic forces?

A general analysis of the gravitational dynamics of a medium with a continuous distribution of vorticity indicates that the answer to the question raised in the title is affirmative, contrary to a recent claim.

Numerical Simulation of Hydrodynamics by the Method of Point Vortices

The motion of a two-dimensional incompressible inviscid and homogeneous fluid can be thought of in terms of the gradual evolution of a continuous vorticity distribution, each scalar vortex element interacting with every other by an instantaneous action at a distance law. It is of particular interest that this model can be expressed in Hamiltonian form and that it shows many analogies with similar systems in plasma physics. In addition to the standard mesh techniques, a computational description can be obtained if the continuous vorticity distribution is replaced by a finite set of point vortices interacting through a stream function which satisfies Poisson's equation. The point vortices move in a velocity field given on a Cartesian mesh such that there is a close resemblance to particle models used in plasma simulations. The point vortex model is presented with a calculation on a test model and the sources of numerical errors are explained. Graphical results from several calculations are shown and it is concluded that the point vortex model is useful and versatile for a variety of problems in hydrodynamics as well as in plasma physics.

Controlled capture of a continuous vorticity distribution

We develop a chaotic control and targeting scheme to capture and stabilize a concentrated vortex around a cylindrical body embedded in an open fluid flow. We demonstrate that the point vortex-based control model is also valid in a continuum fluid framework, by simulating the system using a Navier-Stokes-like dynamics.

On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid

This paper describes exact solutions of two-dimensional vortex structures that were published by Chaplygin (1899, 1903) at the turn of the last century, which seem to have escaped the attention of later investigators in this field. Chaplygin's solutions include that of an elliptical patch of uniform vorticity in an exterior field of pure shear and that of a (symmetric or non-symmetric) dipolar vortex with a continuous distribution of vorticity translating steadily along a straight path. In addition, a solution is presented for a non-symmetric vortex dipole moving along a circular trajectory. A concise account of Chaplygin's solutions is given, complemented by a more detailed analysis of some of their relevant properties.

Numerical simulation of flow around a sphere with vortex blobs

A vortex model of a blob method (which is one of the vortex methods proposed before) is modified to take into account the viscous effect. The modified method was applied to simulation of a flow around a sphere using the Boundary Element Method. The surface of the sphere was represented by the distributed source panels. The continuous distribution of vorticity over the surface of the sphere was replaced by a number of vortex blobs. All the blobs were allowed to flow at their own convective velocities, and the boundary layer separation was expected to be naturally represented by their convective motion. As a result of this calculation, the boundary layer separation from the surface was reasonably simulated in the initial stage of a starting flow.

A Comparison of the Contour Surgery and Pseudo-spectral Methods

Two very different numerical methods which have been used to simulate high Reynolds number two-dimensional flows are compared for the first time. One method, the pseudo-spectral method, is fundamentally based on the Eulerian description of fluid flow and the other, the contour surgery method, is inherently Lagrangian. The former makes use of a continuous distribution of vorticity, while the latter, a theoretically inviscid method, makes use of a discrete distribution. A comparison is nevertheless attempted in a model problem wherein the initial vorticity distribution is continuous. We examine the stripping of an initially circular vortex by applied adverse shear in doubly periodic geometry. The constraining geometry causes the flow to become very complex, placing great demands on both computational methods. The surprise is that as few as eight discrete levels of vorticity in contour surgery five results which are quantitatively close to those obtained by the pseudo-spectral method at high resolution. Advantages and shortcomings of both methods are noted.

Acoustically Induced Periodic Vortex Shedding At Sharp Edged Open Channel Ends: Simple Vortex Models

Simple flow models proposed by Howe [1975 Journal of Fluid Mechanics 71 , 625-673] and Disselhorst and van Wijngaarden [1980 Journal of Fluid Mechanics 99, 293-319] are used to describe the periodic vortex shedding induced at a two-dimensional channel end with infinitely thin, sharp edges by a harmonically varying acoustic flow with zero mean flow. For low Strouhal numbers a quasi-stationary theory can be used, while for high Strouhal numbers the problem can be simplified to that of vortex shedding at a semi-infinite plate. Our experiments show fair agreement with both limiting theories. A single vortex theory will be shown to be useful in the high Strouhal number limit, although a correction for the pressure force acting at a connecting line between the edge and the vortex is necessary when the vortex is increasing in strength. A correction in terms of a Magnus force applied to the single vortex does, however, influence the acoustic energy balance. A correction to the theoretical formula derived for a continuous distribution of vorticity due to the action of external forces is proposed. For intermediate Strouhal numbers, both the quasi-stationary and single vortex models fail to describe the flow at the channel end accurately. A single panel method proposed in this paper is a compromise between the single vortex description and the much more accurate but time-consuming panel method. The vortex shedding process described by the single panel method is compared with flow visualization experiments performed by Diselhorst and van Wijngaarden. Fair agreement between the single panel theory and experiments is obtained.

Numerical Analysis of Cross-Flow Fan by a Clouds-in-Cells Method Combined with the Finite-Element Method.

A suitable simulation scheme for simulating the internal flow of a cross-flow fan was developed. In this scheme, the impeller is treated as an actuator disk which restricts the flow direction and supplies vorticity to the flow. The continuous vorticity distribution is replaced by a finite set of point vortices and calculated by the CIC (clouds-in-cells) method which is combined with FEM instead of FDM (finite-difference method) so that nonstructual meshes can be used. We applied this scheme to simulate the internal flow of a room air conditioner incorporating a cross-flow fan, and the results show good agreement with the experimental data. It is of particular interest that this model is effective to simulate the secondary vortex which sometimes appears at the position opposite to the main vortex on the impeller.

Computation of Three-Dimensional Potential Flow Using Surface Vorticity Distribution

An iterative method is developed using incompressible potential flow theory to compute three-dimensional velocity and pressure distributions on the surface of thick wings. The mathematical formulation is illustrated for a semi-infinite circular cylinder having a hemispherical tip. The procedure starts from a specified twodimensional vorticity distribution on the entire body and converges in three iterations to the three-dimensional distribution on the tip which merges smoothly with the two-dimensional one further inboard. The finite-element approach used here gives a continuous distribution of vorticity which is directly equivalent to the surface velocity distribution. The present method involves repeated use of the Biot-Savart law to relax the surface vorticity strength in successive iterations; whereas the more commonly used integral equation formulation requires solving a large matrix which is usually not well behaved for the surface vorticity model. A modification of the method is applied to compute the nonlifting flow on a semi-infinite NACA 0012 wing with a halt-bod}-ofrevolution tip and a NACA 0012 wing with an aspect ratio of three. The chordwise pressure coefficient distributions are presented for representative spanwise locations for all of the cases.

The calculation of large reynolds number two-dimensional flow using discrete vortices with random walk

The numerical calculation of two-dimensional rotational flow at large Reynolds number is considered. The method of replacing a continuous distribution of vorticity by a finite number, N, of discrete vortices is examined, where the vortices move under their mutually induced velocities plus a random component to simulate effects of viscosity. The accuracy of the method is studied by comparison with the exact solution for the decay of a circular vortex. It is found, and analytical arguments are produced in support, that the quantitative error is significant unless N is large compared with a characteristic Reynolds number. The mutually induced velocities are calculated by both direct summation and by the “cloud in cell” technique. The latter method is found to produce comparable error and to be much faster.

Numerical simulation of hydrodynamics by the method of point vortices

The motion of a two-dimensional incompressible inviscid and homogeneous fluid can be thought of in terms of the gradual evolution of a continuous vorticity distribution, each scalar vortex element interacting with every other by an instantaneous action at a distance law. It is of particular interest that this model can be expressed in Hamiltonian form and that it shows many analogies with similar systems in plasma physics. In addition to the standard mesh techniques, a computational description can be obtained if the continuous vorticity distribution is replaced by a finite set of point vortices interacting through a stream function which satisfies Poisson's equation. The point vortices move in a velocity field given on a Cartesian mesh such that there is a close resemblance to particle models used in plasma simulations. The point vortex model is presented with a calculation on a test model and the sources of numerical errors are explained. Graphical results from several calculations are shown and it is concluded that the point vortex model is useful and versatile for a variety of problems in hydrodynamics as well as in plasma physics.