Hollow Vortices Research Articles

Experimental and numerical study on flow dynamics and universal characteristics of ventilated supercavities behind different cavitators

In this study, physical aspects of a ventilated supercavity behind different cavitator geometries such as the hydrodynamic characteristics, distribution of pressure within the cavity, hysteresis phenomenon, and gas leakage mechanism were qualitatively and quantitatively investigated using experimental and numerical methods. For the simulation and tunnel tests, we employed five cavitators, each with different angles (45°, 60°, 90°, 135°, and a 180° cavitator, commonly referred to as a disk cavitator), all sharing the same diameter. The results revealed that the drag force experienced on the cavitator decreased linearly with an increase in the ventilation rate, and a consistent trend was observed for all test cavitator angles. Through experimental measurements, a universal equation has been derived to predict the drag force exerted on a supercavitating vehicle employing a cavitator. In addition, the pressure distribution inside the supercavity was significantly influenced by the angle of the cavitator. The pressure kept almost unchanged in the first half of supercavity; a slight increase in pressure occurred in the remainder of the supercavity. Twin-vortex gas leakage mode was clearly observed. The distance between the two hollow vortices increased significantly, whereas the incline angle of these vortices and the horizontal line changed insignificantly.

Hollow vortex in a corner

Abstract

The corotating hollow vortex pair: steady merger and break-up via a topological singularity

Abstract

The effect of core size on the speed of compressible hollow vortex streets

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

Hollow vortices in shear

An analytical method for determining the shape of hollow vortices in shear flows is presented in detail. In a non-dimensional formulation, it is shown that the problem has one degree of freedom represented by the free choice of the non-dimensionalized speed $\unicode[STIX]{x1D705}$ at the boundary of the vortex. The solutions form two families of shapes corresponding to vortex circulation and shear-flow vorticity having the opposite or same sign. When the signs are opposite, the shape family resembles that described by Llewellyn Smith & Crowdy (J. Fluid Mech., vol. 691, 2012, pp. 178–200) for hollow vortices in a potential flow with strain. As for that flow, there is a minimum value of $\unicode[STIX]{x1D705}$ below which there is no solution as the boundary of the vortex self-intersects, while, when the signs are the same, solutions exist for $0<\unicode[STIX]{x1D705}$.

Vortices in stratified protoplanetary disks

Large scale vortices could play a key role in the evolution of protoplanetary disks, particularly in the dead-zone where no turbulence associated with magnetic field is expected. Their possible formation by the subcritical baroclinic instability is a complex issue due to the vertical structure of the disk and to the elliptical instability.} {In two-dimensional disks the baroclinic instability is studied as a function of the thermal transfer efficiency. In three-dimensional disks we explore the importance of radial and vertical stratification on the processes of vortex formation and amplification.} {Numerical simulations are performed using a fully compressible hydrodynamical code based on a second order finite volume method. We assume a perfect gas law in inviscid disk models in which heat transfer is due to either relaxation or diffusion.} {In 2D, the baroclinic instability with thermal relaxation leads to the formation of large-scale vortices, which are unstable with respect to the elliptic instability. In the presence of heat diffusion, hollow vortices are formed which evolve into vortical structures with a turbulent core. In 3D, the disk stratification is found to be unstable in a finite layer which can include the mid-plane or not. When the unstable layer contains the mid-plane, the 3D baroclinic instability with thermal relaxation is found to develop first in the unstable layer as in 2D, producing large-scale vortices. These vortices are then stretched out in the stable layer, creating long-lived columnar vortical structures extending through the height of the disk. They are also found to be the source of internal vortex layers that develop across the whole disk along baroclinic critical layer surfaces, and form new vortices in the upper region of the disk.} {In three-dimensional disks, vortices can survive for a very long time if the production of vorticity by...

An Experimental Study on the Gas Entrainment in Horizontally and Vertically Installed Centrifugal Pumps

In this work, we have studied how gas accumulates in an industrial centrifugal pump under various steady-state two-phase flow conditions. Thereby, we considered both horizontal and vertical pump installation positions. Phase fractions within the impeller region of the pump have been quantitatively disclosed using high-resolution gamma-ray computed tomography (HireCT) and applying time-averaged rotation-synchronized CT scanning technique. The study was made for inlet volumetric gas flow rates between 0% and 5%. To account for different inlet flow conditions, which are assumed to occur during unwanted gas entrainment by hollow vortices, we produced disperse and swirling gas–liquid inlet flows. In this way, the influence of inlet flow boundary conditions on the pump performance as well as gas fraction distributions and gas holdup within the impeller wheel region could be successfully analyzed and compared with respect to the impeller alignment. It was shown that the installation position offers only a minor effect on the pump performance in comparison to the inlet flow conditions. In addition, for the first time, thin gas films at the pressure side of the impeller wheel blades could be visualized in an industrial centrifugal pump.

Instability of isolated hollow vortices with zero circulation

Inviscid linear stability analysis and numerical simulations are used to investigate how temporal disturbances evolve in double-annular hollow vortices with an opposite-signed vorticity (the total circulation is zero). Two extrema exist in the vorticity profile and constitute a factor of instability. The dispersion relation is expressed as a simple cubic equation. The results show that the instabilities of vortices are strongly enhanced by the hollow effect of the annular vorticity. In addition, the growth rate of the dominant modes significantly increases with decreasing negative-vorticity thickness. During the initial stage, the dominant unstable modes obtained from simulations are consistent with those obtained from the linear analysis. In nonlinear developments, the flow field stretches out in one direction depending on the motion of the plural vortex pair formed by rolling up the positive and negative vorticities. Once such structures in the vortex are generated, the vortex immediately breaks down and does not become metastable.

Experimental and numerical investigation of ventilated cavitating flow with special emphasis on gas leakage behavior and re-entrant jet dynamics

The objective of this paper is to investigate the ventilated cavitating flow structure by combining experimental and numerical methods. A high-speed camera technique is used to record cavity evolution patterns. The numerical simulation is performed by CFX with a free surface model and a filter-based model, and the gravity effect is considered. The results show when the gas entrainment coefficient Qv is constant, two typical mechanisms of the gas leakage exist at different Fround numbers Fr, namely toroidal vortices mode and two hollow tube vortices mode. With the increasing of Fr, the cavity would transfer from the two hollow tube vortices to the toroidal vortices. Moreover, when the Fr number keeps constant, the enlargement of the cavity causes the gravitational effect to be more significant for the case of larger value of Qv. The detail analysis of re-entrant behaviors is also conducted. One type of re-entrant flow is unsteady with air cluster being periodically rejected at the rear of the cavity. The other type of the re-entrant flow shows that the majority of the cavity is transparent, only the region at the tail of the cavity is nontransparent, due to the re-circulation of water back into the cavity.

Hollow vortices and minimal surfaces

We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.

Analytical solutions for two hollow vortex configurations in an infinite channel

New analytical solutions for a co-travelling hollow vortex pair and a single row of hollow vortices in an infinite channel are presented. These new solutions generalise several known classical solutions for hollow vortices. The mathematical problems to be solved in each geometry are particular types of free boundary problem over a multiply connected domain. We present concise formulae for the conformal mapping determining the shape of the boundaries of the hollow vortices in both channel geometries by employing free streamline theory in combination with the function theory of the Schottky–Klein prime function. Various properties of the solutions are also presented.

An Eigenfrequency Analysis of Mixed Rossby–Gravity Waves on Barotropic Vortices

Abstract In this study, the linearized, f-plane, shallow-water equations are discretized into a matrix eigenvalue problem to examine the full spectrum of free waves on barotropic (monopolar and hollow) vortices. A typical wave spectrum for weak vortices shows a continuous range between zero and an advective frequency associated with vortex Rossby waves (VRWs) and two discrete ranges at both sides associated with inertio-gravity waves (IGWs). However, when the vortex intensity reaches a critical value, higher-frequency waves will be “red shifted” into the continuous spectrum, while low-frequency waves will be “violet shifted” into the discrete spectrum, leading to the emergence of mixed vortex Rossby–inertio-gravity waves (VRIGWs). Results show significant (little) radial wavelike structures of perturbation variables for IGWs (VRWs) with greater (much smaller) divergence than vorticity and the hybrid IGW–VRW radial structures with equal amplitudes of vorticity and divergence for mixed VRIGWs. In addition, VRWs only occur within a critical radius at which the perturbation azimuthal velocity is discontinuous. As the azimuthal wavenumber increases, lower-frequency waves tend to exhibit more mixed-wave characteristics, whereas higher-frequency waves will be more of the IGW type. Two-dimensional wave solutions show rapid outward energy dispersion of IGWs and slower dispersion of VRWs and mixed VRIGWs in the core region. These solutions are shown to resemble the previous analytical solutions, except for certain structural differences caused by the critical radius. It is concluded that mixed VRIGWs should be common in the eyewall and spiral rainbands of intense tropical cyclones. Some different wave behaviors associated with the monopolar and hollow vortices are also discussed.

Hollow vortices, capillary water waves and double quadrature domains

Two new classes of analytical solutions for hollow vortex equilibria are presented. One class involves a central hollow vortex, comprising a constant pressure region having non-zero circulation, surrounded by an n-polygonal array of point vortices with . The solutions generalize the non-rotating polygonal point vortex configurations of Morikawa and Swenson (1971 Phys. Fluids 14 1058–73) to the case where the point vortex at the centre of the polygon is replaced by a hollow vortex. The results of Morikawa and Swenson would suggest that all equilibria for will be linearly unstable to point vortex mode instabilities. However even the n = 3 case turns out to be unstable to a recently discovered displacement instability deriving from a resonance between the natural modes of an isolated circular hollow vortex. A second class of analytical solutions for periodic water waves co-travelling with a submerged point vortex row is also described. The analysis gives rise to new theoretical connections with free surface Euler flows with surface tension and, in particular, with Crapperʼs classical solutions for capillary water waves. It is pointed out that the equilibrium fluid regions found here have a mathematical interpretation as an abstract class of planar domains known as double quadrature domains.

Fully compressible simulations of the baroclinic instability in protoplanetary disks

The occurrence of the baroclinic instability in protoplanetary disks is investigated using fully compressible numerical simulations. We study the role of disk stratification and heat transfer in the development of the instability. Vortices are found to form whatever the importance of the stratification but to grow and persist only when the disk is not stably stratified. In the presence of thermal diffusion the baroclinic instability leads to hollow vortices that decays into transient vortical structures.

Hollow vortices and wakes past Chaplygin cusps

By using analytic tools for 2D rotational inviscid flow, the stagnation points of Pocklington hollow vortices are replaced by Chaplygin cusps, that is, by regions of fluid at rest. By solidifying the bounding free streamlines, solid bodies are obtained along whose walls adverse pressure gradients are avoided. These results are relevant to the theory and practice of control of separated flow at high Reynolds number. Examples are presented pertinent to single bodies and cascade of bodies which trap hollow vortices or generate open hollow wakes.

Translating hollow vortex pairs

A new derivation, and representation, of the classical solution for a translating hollow vortex pair originally discovered by Pocklington in 1895 [H.C. Pocklington, The configuration of a pair of equal and opposite hollow straight vortices of finite cross-section, moving steadily through fluid, Proc. Cambridge Philos. Soc. 8 (1895) 178–187] is presented. The derivation makes use of a special function known as the Schottky–Klein prime function. The new representation of the hollow vortex pair facilitates an investigation of the linear stability properties of this configuration, something apparently not previously studied in the literature. We show here that the hollow vortex pair is linearly stable to infinitesimal irrotational perturbations provided the area of the two vortices is below a critical value.

Analytical solutions for von Kármán streets of hollow vortices

New analytical solutions are presented for steadily translating von Kármán vortex streets made up of two infinite rows of hollow vortices. First, the solution for a single row of hollow vortices due to Baker et al. ["Structure of a linear array of hollow vortices of finite cross-section," J. Fluid Mech. 74, 469 (1976)] is rederived, in a modified form, and using a new mathematical approach. This approach is then generalized to find relative equilibria for both unstaggered and staggered double hollow vortex streets. The method employs a combination of free streamline theory and conformal mapping ideas. The staggered hollow vortex streets are compared with analogous numerical solutions for double streets of vortex patches due to Saffman and Schatzman [“Properties of a vortex street of finite vortices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 285 (1981)] and several common features are found. In particular, within the two different inviscid vortex models, the same street aspect ratio of approximately 0.34–0.36 is found to have special significance for the equilibria.

Hollow wakes past arbitrarily shaped obstacles

An analytical solution is presented for steady inviscid separated flows modelled by hollow vortices, that is, by closed vortex sheets bounding a region with fluid at rest. Steady flows past arbitrary obstacles protruding from an infinite wall are considered. The solution is similar to that of the vortex patch model; it depends on two free parameters that define the size of the hollow vortex and the location of the separation point. When a sharp edge constrains the separation point (Kutta condition), the solution depends on a single parameter. As with the vortex patch model, families of growing vortices exist, which represent the continuation of desingularized point vortices. Numerical results are presented for the flows past a semicircular bump, a Ringleb snow cornice and a normal flat plate. The differences from the previous results found in the literature are analysed and discussed with the present solutions for the flow past a normal flat plate.

Generating hypergeometric laser beams with a diffractive optical element

We derive explicit analytical relations to describe paraxial light beams that represent a particular case of the hypergeometric (HyG) laser beams [J. Opt. Soc. Am. A25, 262-270 (2008)JOAOD60740-323210.1364/JOSAA.25.000262]. Among these are modified quadratic Bessel-Gaussian beams, hollow Gaussian optical vortices, modified elegant Laguerre-Gaussian beams, and gamma-HyG beams. Using e-beam microlithography, a binary diffractive optical element capable of producing near-HyG beams is synthesized. Theory and experiment are in sufficient agreement. We experimentally demonstrate the ability to rotate dielectric microparticles using the bright diffraction ring of a HyG beam.

Convergence of a cylindrical liquid shell and the formation of a bore in a rotating fluid

This paper presents the results of experimental studies of a collapsing cylindrical cavity (the convergence of a liquid shell) in a rotating fluid as well as the formation and propagation of a jump (bore) at the interface. The basic parameters of the liquid shell dynamics for a pulsed one-dimensional load are estimated using the equation of cylindrical cavity pulsation in an unbounded fluid. The theoretical model of a rotationally symmetric hydraulic jump moving along the free surface of a hollow vortex is constructed. The jump is simulated by a discontinuous solution of the equations in the long-wave approximation for tornado-like and hollow vortices. For comparison with the experimental data, basic theoretical results are obtained for flows in a hollow vortex with constant circulation and axial velocity varying along the radius of the rotating liquid shell.