T. Maxworthy

The structure and stability of vortex rings

A series of observations on experimentally produced vortex rings is described. The flow field, ring velocity and growth rate were observed using dye and hydrogen-bubble techniques. It was found that stable rings are formed and grow in such a way that most of their vorticity is distributed throughout a fluid volume which is larger than and moving with the visible dye core. As the vorticity diffuses out of this moving body of fluid into the outer irrotational fluid, it has two effects. It causes some of the fluid, with newly acquired vorticity, to be entrained into the interior of the bubble, while the rest is left behind and accounts for the appearance of ring vorticity in a wake. It was found that the velocity of translation U of these stable rings varies as t −1 , at high Reynolds number, where t is the time measured from the start of the motion at a virtual origin at downstream infinity. A simple theoretical model is presented which explains all of these features of the observed stable flow. Rings of even higher Reynolds number become unstable and shed significantly more vorticity into the wake. Under some circumstances a new more stable vortex emerges from this shedding process and continues with less vorticity than before. Eventually, the ring motion ceases as all of its vorticity is deposited into the wake and is spread by viscous diffusion. Observations of the interaction between two nearly identical rings travelling a common path showed that, contrary to popular belief, rings do not pass back and forth through one another, but that the rearward one becomes entrained into the forward one. Only when the rearward ring has a much higher velocity than its partner can it emerge from the joining process and leave a slower-moving ring behind.

Some experimental studies of vortex rings

A series of experiments designed to reveal the properties of high Reynolds number vortex rings, using flow-visualization and laser-Doppler techniques, has uncovered several interesting and unexpected results. Starting at the beginning of the motion, at a nozzle, and proceeding downstream, these include the following. A formation process that is strongly Reynolds number dependent. The amount of vorticity that appears downstream is very close to that predicted by a simple ‘slug’ model. However flow-visualization studies show that such a model is an oversimplification and that an excess of ring vorticity is probably cancelled by the ingestion of vorticity of opposite sign at the nozzle lip. (iii) A new, bimodal form of vortex-core instability has been observed at moderate but not high Reynolds numbers. Azimuthal inhomogeneities in the breaking of these, and the normal instability waves, create an ‘axial’ flow along the vortex core in the turbulent ring. This axial flow takes the form of a propagating wave that has many characteristics of a solitary wave. It is hypothesized that this axial flow prevents further ring instability. The long-term behaviour of the turbulent ring is marked by dramatic changes in its growth rate, which are probably related to changes in the ‘organization’ of the vortex core. The descriptive turbulent-ring model developed in Maxworthy (1974) is substantially confirmed by these experiments and by observation of ring propagation through a stratified ambient fluid.

Experiments on the Weis-Fogh mechanism of lift generation by insects in hovering flight. Part 1. Dynamics of the ‘fling’

From a series of experiments using simplified mechanical models we suggest certain minor modifications to the Weis-Fogh (1973)–Lighthill (1973) explanation of the so-called ‘clap and fling’ mechanism for the generation of large lift coefficients by insects in hovering flight. Of particular importance is the production and motion of a leading edge, separation vortex that accounts for virtually all of the circulation generated during the initial phase of the ‘fling’ process. The magnitude of this circulation is substantially larger than that calculated using inviscid theory. During the motion that subsequently separates the wings, the vorticity over each of them is convected and combined to become a tip vortex of uniform circulation spanning the space between them. This combined vortex moves downwards as a part of a ring, of large impulse, that is then continuously fed from quasi-steady separation bubbles that move with the wings as they continue to open at a large angle of attack. Such effects are able to account for the large lift forces generated by the insect.

On the origin and evolution of streamwise vortical structures in a plane, free shear layer

A plane, isothermal, chemically reacting mixing layer has been experimentally investigated to analyse the origin and the development of three-dimensional stream-wise vorticity. The results show that early in its evolution, the plane, free shear layer is composed of counter-rotating pairs of streamwise vortices superimposed upon the spanwise ones. This coherent, streamwise vortical structure was found to be the result of the unstable response of the layer to three-dimensional perturbations in the upstream conditions. Depending on the magnitude and location of the upstream disturbances, the location of the transition to three-dimensionality varied. However, the concentrated streamwise vorticity was always seen to form first on the braids between consecutive spanwise vortices and then to propagate into their cores. For the low and moderate Reynolds numbers of this study, it was found that the onset of the so-called ‘mixing transition’ does not necessarily coincide with that of the formation of concentrated streamwise vorticity. These vortices were observed to have a scale, as measured by the size of their cores, somewhat smaller than but comparable with that of the spanwise ones, thus contributing substantially to the entrainment process in the early stages of mixing-layer development.

An experimental investigation of swirling jets

The ‘plug’ flow emerging from a long rotating tube into a large stationary reservoir has been used in an experimental investigation of centrifugally unstable swirling jets. A moderate Reynolds number, Re = 1000, was studied extensively, and swirl numbers, S, the ratio of nozzle exit rotating speed to the mean mass axial velocity, were in the range 0–1.1. Four regimes were covered: non-swirling jets with S = 0, weakly swirling jets in the range 0 S c2 ,w hereSc1 =0 . 6a nd Sc2 =0 .88. Particular attention was paid to the dominant role of the underlying vortical flow structures and their dynamic evolution. Kelvin–Helmholtz (K–H) instability in the axial shear layer, generated by the axial velocity, leading to vortex ring formation, dominated non-swirling and weakly swirling jets. After the introduction of rotation, the combined axial and azimuthal shear layers became unstable to a modified form of K–H instability. For weak swirl, tilted vortex rings evolved downstream. For strongly swirling jets, vorticity in the azimuthal shear layer, formed by the azimuthal or swirl velocity, grew and became comparable with vorticity in the axial shear layer. The flow lost its axisymmetry and strong helical waves with m = +2 or +3 replaced vortex rings as the dominant vortex structure before vortex breakdown. After breakdown, strong helical waves with m = +1 and +2 coexisted, with m = +1 being dominant. Here, m is the azimuthal wavenumber, and the positive sign represents the counter-winding direction for helical waves. In the present experiments, the phase speed of each helical wave was the same as the rotation rate of the nozzle exit, in both direction and absolute value. Based on a spectral study of the flow fields in natural, forced and transient experiments, we suggest that the helical wave m = +2 for strongly swirling jets before vortex breakdown and m = +1 after breakdown were self-excited/globally unstable, a behaviour identified as a super-critical Hopf bifurcation. Also it is suggested that the onset of the unstable global mode m = +1 after vortex breakdown was related to the existence of a region of local absolute instability in the wake region of the breakdown structure. This self-excited oscillation first dominated the inner shear layer between the central reverse flow and the conical jet, giving the breakdown the appearance of a spiral one, and then grew and incorporated the outer shear layer between the jet and the ambient fluid. After that, the wake region around the stagnation point served as the wave maker and imposed its frequency of oscillation on the whole flow. An attempt is made to reconcile these results with existing theory and experiments.

Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation

Vortex breakdown of nominally axisymmetric, swirling incompressible flows with jet-and wake-like axial velocity distributions issuing into a semi-infinite domain is studied by means of direct numerical simulations. By selecting a two-parametric velocity profile for which the steady axisymmetric breakdown is well-studied, issues are addressed regarding the role of three-dimensionality and unsteadiness with respect to the existence, mode selection, and internal structure of vortex breakdown, in terms of the two governing parameters and the Reynolds number. Low Reynolds numbers are found to yield flow fields lacking breakdown bubbles or helical breakdown modes even for high swirl. In contrast, highly swirling flows at large Reynolds numbers exhibit bubble, helical or double-helical breakdown modes, where the axisymmetric mode is promoted by a jet-like axial velocity profile, while a wake-like profile renders the flow helically unstable and ultimately yields non-axisymmetric breakdown modes

Miscible displacements in capillary tubes. Part 1. Experiments

Experiments have been performed, in capillary tubes, on the displacement of a viscous fluid (glycerine) by a less viscous one (a glycerine-water mixture) with which it is miscible in all proportions. A diagnostic measure of the amount of viscous fluid left behind on the tube wall has been found, for both vertical and horizontal tubes, as a function of the Peclet (Pe) and Atwood (At) numbers, as well as a parameter that is a measure of the relative importance of viscous and gravitational effects. At values of the average Pe greater than 1000 a sharp interface existed so that it was possible to make direct comparisons between the present results and a prior experiment with immiscible fluids, in particular an effective surface tension at the diffusing interface could be evaluated. The effect of gravity on the amount of viscous fluid left on the tube wall has been investigated also.

On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions

The paper shows how trains of nonlinear, dispersive waves can be produced by allowing a region of mixed fluid, with a potential energy greater than its surroundings, to collapse towards its equilibrium state. The number of waves and their amplitude depend on the properties of the mixed region and of the ambient stratification. Three different geometrical configurations have been chosen and while each gives qualitatively the same results the form taken by the generated waves and the final equilibrium shape of the mixed region depend critically on these geometrical factors. The internal waves produced by this mechanism are related to waves produced in natural systems and it is shown that the observations support at least one possible explanation for those found in the oceans and planetary atmospheres.

Turbulent vortex rings

We consider the motion of the mass of fluid ejected through a sharp-edged orifice by the motion of a piston. The vorticity formed by viscous forces within the separated flow at the sharp edge rolls up to form a concentrated vortex which, after a development period, consists of a core of very fine scale turbulence surrounded by a co-moving bubble of much larger scale turbulence. This bubble entrains outer fluid, mixes with it, and deposits the majority into a wake together with some small fraction of the total vorticity of the ring. Enough fluid is retained to account for the slow growth of the whole fluid mass. A theory which takes account of both the growth process and the loss of vorticity is proposed. By comparison with experimental measurements we have determined that the entrainment coefficient for turbulent vortex rings has a value equal to 0.011 ± 0.001, while their effective drag coefficient is 0.09 ± 0.01. These results seem to be independent of Reynolds number to within experimental accuracy.

The viscous spreading of plane and axisymmetric gravity currents

Measurements of the spreading rates of gravity-driven currents at both the surface and the bottom of a fluid layer of different density are reported. For the case of a constant inflow rate the spreading relations are derived by estimating the order of magnitude of the forces involved. After an initial balance between gravity and inertia forces the final spreading phase is governed by the balance between gravity and viscous forces. For the latter flow regime, measurements in plane and axisymmetric flow geometries agree well with the spreading relations for gravity currents with a no-slip boundary. The proportionality factor, which is not predicted from this model, is then determined from the measurements and a good agreement is found with the theoretical value derived in the accompanying paper by Huppert (1982).