Axisymmetric Vortex Rings Research Articles

Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring

We explore the flow field around a thin axisymmetric vortex ring steadily translating in an ideal fluid, with vorticity proportional to distance from the axis of symmetry, originally treated by Dyson (Philos. Trans. R. Soc. 184 (1893) 1041). By making a higher-order extension of the method of matched asymptotic expansions in a small parameter ε, the ratio of the core radius to the ring radius R0, an explicit form of the streamfunction is derived, to O(ε3), both inside and outside the core. The pressure and velocity fields are written out in full to the same order. It is shown that the circle of minimum pressure coincides, to O(ε2R0), with the circle of stagnation points viewed from the frame moving with the core.

Statistical mechanics of axisymmetric vortex rings.

We construct maximum entropy states of a collection of interacting uniform (omega/R=const) axisymmetric vortex rings in a semiperiodic bounded volume. Following Miller [Phys. Rev. Lett. 65, 2137 (1990)] and Robert and Sommeria [J. Fluid Mech. 229, 291 (1991)], we obtain an equilibrium measure that preserves all the ideal invariants such as the total energy, total impulse, circulation, and an infinity of Casimirs. The numerical solution for a wide range of total flow energy and for given values of total circulation and total impulse is presented.

The formation of ‘optimal’ vortex rings, and the efficiency of propulsion devices

The formation of an axisymmetric vortex ring by forcing uid impulsively through a pipe is examined. An idealized model of the circulation, impulse and energy provided by the injected plug is developed, and these quantities are equated to the corresponding properties of the class of rings with finite cores described by Norbury (1973). It is shown that, as the length-to-diameter aspect ratio L/D of the plug increases, the size of the core increases in comparison with all the fluid carried along with the ring, until the limiting case of Hill's spherical vortex is reached. For aspect ratios larger than a certain value it is not possible to produce a single ring while conserving circulation, impulse, volume and energy. This implies that the limiting vortex is ‘optimal’ in the sense that it has maximum impulse, circulation and volume for a given energy input. While this matching calculation makes the physical mechanism clear, the L/D ratio that can be achieved in practice is more appropriately taken from the direct experimental measurements of Gharib et al. (1998) who concluded that the limiting value is L/D = 4. This is close to the value found in our calculation.

Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

An investigation of the flow regimes resulting from splashing drops

Numerical simulations have been used to investigate the flow regimes resulting from the impact of a 2.9 mm water drop on a deep water pool at velocities in the range 0.8–2.5 m/s. The results were used to identify the conditions leading to the formation of vortex rings, entrapment of a bubble during cavity collapse and the formation of vertical Rayleigh jets. Bubble entrapment and the associated growth of a thin high speed jet were shown to be the result of a capillary wave that propagates down the walls of the crater resulting from drop impact. Although the existence of a capillary waves is a necessary condition for bubble entrapment, bubbles will only occur when the wave speed and maximum crater size is such that the wave reaches the bottom of the crater before collapse has resulted in the formation of a thick Rayleigh jet. Simulations also clarified the conditions for which drop impact leads to axi-symmetric vortex rings. Results not reported previously, include the observation that a single drop can produce multiple vortex rings and that vortex rings can occur for conditions that lead to broad Rayleigh jets. Based on these results, it was concluded that the formation of vortex rings depends on the time at which vorticity is generated and the nature of its subsequent transport.

Effects of trailing jet instability on vortex ring formation

Numerical simulations of an impulsively started jet were performed in order to investigate the effects of trailing jet instability on axisymmetric vortex ring formation. The predictions were compared to experimental results reported in the literature and to recently published numerical results. The total and vortex ring circulations were found to be in good agreement with both the experimental and the numerical results. The presence of a universal formation time scale was confirmed. The results also highlighted an important interaction between an instability which develops in the trailing jet for large discharge times and the dynamics of the head vortex ring. This interaction accelerates the process by which the vortex ring detaches from the trailing jet and has a significant effect on the vortex ring circulation.

Sound generation by the interaction of a vortex ring with a rigid sphere

High-Reynolds-number interactions between a vortex ring and a stationary rigid sphere are computed using two Lagrangian particle models. The first is a 3D slender vortex filament scheme which tracks the motion of the filament centerline. The centerline velocity is expressed as the sum of a self-induced velocity and potential velocity added to satisfy potential boundary conditions on the surface of the sphere. The self-induced velocity is computed numerically from the line Biot-Savart integral, which is carefully desingularized so as to provide the correct behavior of the vorticity distribution in the asymptotic thin-core limit. The second model is a particle scheme for the simulation of axisymmetric viscous flow. The scheme is based on discretization of the vorticity field into desingularized vortex elements that are advected in a Lagrangian frame. The velocity of the particles is expressed as the sum of a vortex interaction velocity expressed in terms of a desingularized Biot-Savart law, a potential velocity expressed in terms of the image of vortex elements, and a diffusion velocity. The filament and the particle schemes are applied to compute the passage of axisymmetric vortex rings over a stationary rigid sphere in the high-Reynolds-number limit, and to analyze the far-field sound generated by this interaction. Both models show that as the ring passes over the sphere, its radius increases while its core shrinks due stretching. In the parameter regime considered, the two models yield very close predictions of vortex trajectories and speeds. The filament model indicates that the passage leads to the generation of a pressure spike in the acoustic far-field. Meanwhile, the particle computations reveal that in addition to a pressure spike the far-field sound can also exhibit a substantial high-frequency quadrupole emission. Analysis of the computations reveals that this high-frequency noise emission is due to filamentation within the vortex core. The results also show that the high-frequency quadrupole noise may be dominant, especially when the vortex core is thin and the Mach number is not very small.

Circulation and formation number of laminar vortex rings

The formation time scale of axisymmetric vortex rings is studied numerically for relatively long discharge times. Experimental findings on the existence and universality of a formation time scale, referred to as the ‘formation number’, are confirmed. The formation number is indicative of the time at which a vortex ring acquires its maximal circulation. For vortex rings generated by impulsive motion of a piston, the formation number was found to be approximately four, in very good agreement with experimental results. Numerical extensions of the experimental study to other cases, including cases with thick shear layers, show that the scaled circulation of the pinched-off vortex is relatively insensitive to the details of the formation process, such as the velocity programme, velocity profile, vortex generator geometry and the Reynolds number. This finding might also indicate that the properly scaled circulation of steady vortex rings varies very little. The formation number does depend on the velocity profile. Non-impulsive velocity programmes slightly increase the formation number, while non-uniform velocity profiles may decrease it significantly. In the case of a parabolic velocity profile of the discharged flow, for example, the formation number decreases by a factor as large as four. These findings indicate that a major source of the experimentally found small variations in the formation number is the different evolution of the velocity profile of the discharged flow.

Experiments on stability and oscillatory behavior of planar buoyant plumes

Experiments on the onset of buoyant instabilities leading to periodic formation of vortical structures in planar buoyant plumes of helium and helium/air mixtures injected into quiescent air are reported for a range of nozzle widths (w=20–70 mm), plume fluid densities (pure helium to that approaching air), and velocities at the nozzle exit. First, the plume parameters corresponding to the onset of the oscillatory instability were experimentally determined by varying the nozzle exit velocity for different nozzle widths and plume fluid densities in two different nozzle configurations. These configurations corresponded to a freestanding rectangular nozzle and a rectangular nozzle surrounded by a flat plate in the plane of the nozzle exit. The observed plume behavior in the near field was characterized as nonoscillatory, transitional, or pulsatile. The onset of pulsations in the near field of these buoyant plumes (within a height of two nozzle widths) was best correlated in terms of the plume source Reynolds number and the plume fluid to ambient density ratio. It was also found that the boundary conditions surrounding the nozzle exit had an influence on the onset of plume instability in the near field. Specifically, at a given plume to the ambient density ratio, the plumes with flat plate surround were found to transition to the oscillatory state at a lower value of the threshold velocity and therefore are less stable than the plumes originating from freestanding nozzles. Subsequently, the plume oscillation frequencies were measured as a function of plume width, plume source velocity, and the density ratio for a range of these parameters. The plume oscillation frequency was found to correlate well in terms of the nondimensional parameters, Strouhal number, S=(fw)/Vp, and Richardson number, Ri=[(ρ∞−ρp)gw]/ρ∞Vp2, yielding a correlation S=0.55Ri0.45 determined for 1<Ri<102. This correlation is somewhat different from that of the axisymmetric buoyant plumes, which can be attributed to the differences in mixing rates and the strength of the local buoyancy flux in planar and axisymmetric plumes. The vortical structures formed in the unstable plumes also exhibit several distinct vortex pair modes. The centers of the formed vortex pairs, in general, do not remain colinear and distort with respect to each other when compared with the axisymmetric plume vortex rings, which are toroidal. The convection speeds of the vortex pair centers were also measured and reported in this study.

Dynamics of a vortex ring in a rotating fluid

The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier–Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a ‘degree of polarization’.If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown.Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

Streamwise and spanwise vortex interaction in an axisymmetric jet. A computational and experimental study

The near-field of an azimuthally excited round jet was investigated in a combined computational/experimental study. The reaction zones in the jet were visualized using OH Planar-Laser- Induced-Fluorescence (PLIF) diagnostics. Both axisymmetric and azimuthal modes of the jet were excited to stabilize its spatial structure. Three-dimensional flame visualization of the laboratory jet reconstructed from multiple two-dimensional images acquired at constant phase angle, reveal a complex structure of the reaction zone. Time-dependent numerical simulations provided insight into the underlying fluid-dynamical processes leading to this flame structure. Simulations of reactive and non-reactive free jets used a Monotonically Integrated Large-Eddy-Simulation (MILES) approach, multi-species diffusive transport, global finite-rate chemistry and appropriate inflow/outflow boundary conditions. The flow visualizations of the experimental and computational jets strongly resemble each other, revealing tight coupling between axisymmetric vortex rings and braid (rib) vortices. The jet vorticity evolution is dominated by the dynamics of vortex-ring self-deformation induced by the azimuthal excitation imposed at the jet exit, the dynamics of rib vortices forming in the braid regions between undulating vortex rings, and strong interactions between rings and ribs. The observed topological features of the flow are directly related to the nearly-inviscid jet vorticity dynamics. These processes affect the mixing pattern of the jet, resulting in localized regions of high fuel concentration leading to combustion inactive regions in the flame, and other regions with enhanced mixing and a proper air-to-fuel ratio in the flame where the combustion process is intense. The vorticity dynamics and ensuing mixing processes determine the regions of combustion within the flame and thus the overall heat release pattern.

Interactions of three-dimensional vicous axisymmetric vortex rings with a free surface

The unsteady nonlinear interaction of three-dimensional vortices with a free surface is a great challenge in fluid mechanics, which has deep theoretical significance and important practical background. Applying the three-dimensional VOF method, the interactions of three-dimensional axisymmetric vortex rings with a free surface in an incompressible viscous fluid are numerically simulated. The influence of the Froude number and the surface tension are studied and the evolution of the vorticity, the trajectories of the vortex rings and the baroclinic vorticity on the surface are obtained. The results agreed well with the experiments reported in the literature.

Mixedness in the formation of a vortex ring

The mixing process in the formation of an axisymmetric vortex ring is analyzed by direct numerical simulation. After the initial phase, when the impulse is imparted to the fluid, the mixing increases linearly in time with a slope that is directly proportional to the vortex circulation (Γ) and to reactants diffusivity to the power −1/3 (Pe−0.33). In addition, for reacting flows with equivalence ratio (s) different from 1, it has been found that as s decreases the mixing first decreases and then increases. An explanation for this behavior is given.

On the instability of three-dimensional flows of an ideal incompressible fluid

In a recent paper Lifschitz and Hameiri [Phys. Fluids A 34 (1991) 2644] demonstrated how the stability of general three-dimensional steady flows of an ideal incompressible fluid with respect to short wavelength perturbations can be studied via the geometrical optics method. The flow is unstable if for some stream line the transport equation associated with the wave which is advected by the fluid has a sufficiently rapidly growing solution. In the present paper we show that under certain conditions this equation can be solved explicitly. Using this fact we demonstrate that all axisymmetric flows such that the corresponding poloidal velocity has a point of stagnation, particularly all axisymmetric vortex rings, are unstable. We also describe explicitly exponentially growing solutions of the transport equation for three-dimensional flows with stretching.

Collision of a vortex pair with a contaminated free surface

Collision of a viscous, two-dimensional vortex pair with a contaminated, free surface is studied numerically. The Froude number is assumed to be small, so the surface remains flat. The full Navier–Stokes equations and a conservation equation for the surface contaminant are solved numerically by a finite difference method. The shear stress at the free surface is proportional to the contamination gradient, and simulations for several values of the proportionality constant (W), as well as Reynolds numbers, have been performed. The evolution is also compared with full-slip and no-slip boundaries. As the vortices approach the surface, the upwelling between them pushes the contaminant outward, reducing the amount directly above the vortices, and leading to a clean region for low W. As W is increased the clean region becomes smaller, and eventually no clean region is formed. Except for very low W, the contaminant layer leads to the creation of secondary vortices, causing the original vortices to rebound in a similar way as vortices colliding with a no-slip boundary. For one case, the numerical results are compared with experimental measurements with satisfactory results. Computations of a vortex pair colliding obliquely with a contaminated surface and head-on collision of axisymmetric vortex rings are also presented.

An experimental study of the molecular mixing process in an axisymmetric laminar vortex ring

Results are presented from an experimental study of the molecular mixing of a dynamically passive conserved scalar quantity in an axisymmetric laminar vortex ring. The experiments are based on highly resolved laser-induced fluorescence imaging measurements of the scalar field ζ(x,t) in the diametral plane of the ring, from which the evolution of the molecular mixing rate field ∇ζ⋅∇ζ(x,t) can be directly examined. In particular, the structure and dynamics of the mixing process are addressed during the three characteristic stages in the ring evolution, namely, (i) the ring generation stage, (ii) the ring pinch-off stage, and (iii) the asymptotic stage of the ring. Results show a layering of the mixing process in which the diffusional cancellation term ∇(∇ζ):∇(∇ζ) plays a major role in setting the overall mixing rate achieved. The scalar field measurements are also used to extract detailed information about the underlying velocity field in the ring.

Vortices interacting with nonslip walls: Stirring and mixing of primary and secondary vorticity

The interaction of vortices with fixed boundaries is an important problem in many areas of practical interest. During the approach of the vortices to the wall, vorticity is created at the wall, and then mixing and stirring of primary and secondary vorticity lead to different scenarios that depend on the Re number. This problem is relevant to bursting in turbulent boundary layers, which is responsible for turbulence production. In the near-wall region of turbulent flows, vortex stretching plays a fundamental role. As a prelude to attempting the solution by direct simulation of 3-D Navier–Stokes equations, it is convenient to describe the mechanism associated with 2-D or axisymmetric flows. In the present study, axisymmetric vortex rings and vortex dipoles approaching a nonslip flat wall have been considered. In both cases, experimental results based on flow visualization are available.1,2 The purpose of the present paper is to do a numerical simulation with the same conditions used in the real experiments and follow the flow evolution for a much longer time than has been possible in such experiments. Both for axisymmetric and 2-D cases the numerical simulation shows, as revealed by the experiments, that, at the initial stage, very thin layers of vorticity of sign opposite to that of the primary vortex are generated at the wall. This mechanism is almost independent of Re. This thin and intense layer immediately becomes unstable and rolls up creating a secondary vortex. This vortex interacts with the primary vortex and different flow structures are generated, depending on Re. The location where these structures form depends, in large measure, on whether vortex stretching is present or not. Particularly relevant to the bursting phenomenon is the case at intermediate Re (e.g., Re=1600). Here, a new dipole is created near the wall after multiple rebounding and pairing of secondary vortices. This new structure has sufficient strength to move itself away from the wall. We conclude that mixing and stirring of vorticity are the relevant mechanisms influencing the whole process in a two-dimensional simulation. This ejection of a new structure was previously noticed in the interaction of vortex rings with a wall. From flow visualization, Walker et al., at Re=Γ/ν>3500, observed the occurrence of azimuthal instabilities and, immediately after that, they noticed a new vortex ring was rapidly ejected away from the wall.2 From these observations, those authors argued that the ejection was due to the azimuthal instabilities. In the present paper, axisymmetric calculations have been done in the same range of numbers used in the experiment of Walker et al.2 From the time evolution of the vorticity distribution, we argue that vortex pairing is the main mechanism that contributes to the generation of the new ring rather than azimuthal instabilities. The numerical simulations clearly show that this new ring is created above the primary ring and not at its interior.

Dynamics of vortex interaction with a density interface

We present results from an experimental and numerical investigation into the dynamics of the interaction between a planar vortex pair or axisymmetric vortex ring with lengthscale a and circulation Γ encountering a planar interface of thickness δ across which the fluid density increases from ρ1 to ρ2. Similarity considerations indicate that baroclinic generation of vorticity and its subsequent interaction with the original vortex is governed by two dimensionless parameters, namely (a/δ) A and R, where A ≡ (ρρ2 − ρ1)/(ρ2 + ρ1) and R ≡ (a3g/Γ2). For thin interfaces (δ [Lt ] a), the interaction is governed only by the parameters A and R. Furthermore, in the Boussinesq limit (A → 0), the dynamics are governed solely by the product AR and the interaction is entirely invertible with respect to the initial locations and direction of propagation of the vortices. We document details of the interaction dynamics in the Boussinesq limit over a range of the parameter AR. Results show that, for relatively small values of AR, rather than the vortex simply rebounding at the interface, its outermost layers are instead successively ‘peeled’ away by baroclinically generated vorticity and form a topologically complex backflow in which the ring fluid, the light fluid and the heavy fluid are intertwined. For larger values of AR the vortex barely penetrates the interface, and our results suggest that in the limit AR → ∞ the interaction with a density interface becomes similar to the interaction at a solid wall. We also present results for thick interfaces in the Boussinesq limit, as well as larger density jumps for which the density parameter A enters as a second similarity quantity. Comparison of the experimental and numerical results demonstrate that many of the features of such interactions can be understood within the context of inviscid fluids, and that inviscid vortex methods can be used to accurately simulate the dynamics of such interactions.

Vortex shedding behind rings and discs

The wake structure of discs and bluff rings has been investigated experimentally in a wind tunnel. The rings have an inner diameter di, and an outer diameter do and are classified according to the parameter (do + di)/(do − di) = d/w. the ratio of mean diameter to ring width. As d/w → ∞ the flow approaches that around a two dimensional bluff body whereas as d/w tends to unity the body approaches a solid disc. A distinct change in the vortex shedding pattern is found around d/w = 5. Below this critical value velocity fluctuations in the wake have a weak periodic component which is 180° out of phase across a diameter of the body. Above d/w = 5. regular and coherent axisymmetric vortex ring shedding is observed with shedding occurring alternately from the inner and outer circumferences of the bluff body. Flow visualization and conditional averaging of hot-wire data are used to investigate the vortex structure.

A global branch of steady vortex rings.

Abstract : A steady vortex ring of prescribed strength and propagation speed can be described in terms of a Strokes stream function psi. A flux constant k measures the flow through the center of the axisymmetric vortex ring. For k = 0, Hill in 1984 found an explicit solution for the semi-linear elliptic equation satisfied by psi. In this paper it is shown that there is an unbounded, closed, connected branch of solutions emanating from Hill's vortex in the space of pairs (k, psi). (Author)