Vortex rings.

INTRODUCTION

Smokers have been known to sometimes manage, as they exhale their smoke, to form well-structured smoke rings with their lips. The smoke is then propelled away from the mouth, maintaining the same ring structure, for a considerable duration before the configuration finally breaks up and the smoke diffuses randomly into the surrounding air. Baby dolphins have been observed to blow bubble rings in the same way in water (Lundgren and Mansour,1991) and watching these bubbles form can sometimes be a great source of amusement to the baby dolphins. This seemingly persistent configuration in which the smoke or the bubbles travel is what is referred to as a vortex ring.

To a scientist, this phenomenon should signify a very interesting and significant academic problem. From such an observation, any credible engineer working on a particular flowing fluid system should be able to draw comparisons and ask pertinent questions such as: could this phenomenon be present in my system? Is it having an adverse or beneficial effect on my system? Is there a need to research ways in which it could be eliminated, or do we want to find ways of actually introducing or enhancing them in the system.

Such are the issues that have made the study of vortex rings fascinating for researchers over the centuries. Some early observations, way back in the nineteenth century, led to very strong statements in fluid dynamic terms such as: ‘The vortex ring in an ideal fluid is indestructible’ by Thompson (1867). Then there was a suggestion by Kelvin (1880) who using the common cigarette smoker’s smoke-ring example, maintained that the steady smoke-ring configuration is indeed stable.

Much effort has been made to research this field and throughout this report there are references to some existing studies, literature and applications relating to fluid vortices and vortex rings, which may give some idea of the magnitude of their significance in fluid dynamics. Thanks to such efforts, the understanding of vortex rings has, of course, now gone way beyond such early observations, in some cases even appearing to discredit them. For instance, work on the subject by Widnall (1973) and Widnall (1977) have revealed instabilities in vortex rings. However, the gist of the subject still remains (by and large) their apparent stability and their extremely significant effects on flow dynamics. This would have been what prompted Küchemann (1965) to refer to vortices as ‘the sinews and muscles of fluid flow’.

The author has a personal way of picturing this phenomenon: if a flowing fluid could form any kind of compact structure that propagates persistently in the same structure without breaking up, the vortex ring is the realistic structure that would offer optimal compactness and persistence. This is not a definition by any standard – just a personal way of forming a mental picture, and duly stands open to correction.

Potential implications or possible applications of vortices in the geophysical and industrial world are prodigious. Again, one may look at the aforementioned smoke-ring example and relate it to a more serious problem. The smoke keeps its structure and other characteristics for such long durations due to the ring configuration. What would then happen if extremely high-speed winds in the atmosphere, say a couple of hundred miles per hour, managed to configure in similar kinds of structure? It would seem that these high-speed winds would then gain a persistent nature, travel for longer, and keep their characteristics for longer. And, should they meet anything in their way that would not withstand them (not many man-made structures would withstand wind speeds in excess of a hundred miles an hour) they would be destroyed. This is in fact, not far from the description of a tornado or a hurricane if they involve spinning storms – highly characterised by vortex flow. Other examples of vortex rings are the mushroom-shaped clouds formed during major explosions or eruptions, such as volcanoes, as shown in Fig (i) below.

Fig (i) – Courtesy of http://enstrophy.colorado.edu/~mohseni/VortexRing1.html: A volcanic vortex ring formation. Etna Decade Volcano, Sicily, Italy. The ‘steam rings’ are about 200m across.

Vortex rings are also useful in many applications. Akhmetov (1980) shows their use in fighting oil well fires, and, cavitating vortex rings formed via exciting cavitating jets are used in underwater drilling, rock cutting and cleaning (Chahine & Genoux, 1983). They have also proved highly beneficial in the vacuum-cleaning technology as employed by the BHR group in 1999 to develop the ‘Hoover vortex cleaner’¹. In aerospace and aeronautics they are useful in such aspects as the modelling of downburst which is a hazard to an aircraft. They also have more academic applications as suggested by workers like Maxworthy (1972) regarding the possibility of being used as a model elementary excitation or turbulent eddy, to allow understanding of some aspects of such flow as turbulent motion. Further, such examples can be found in Lugt (1978), but it has to be said that the list of potential applications for vortex rings is open to grow with imagination.

The study of vortex rings using laboratory experiments usually involves some method of injecting fluid through an orifice into a (usually) quiescent ambient fluid. By dying the injected fluid, its behaviour as it ejects through the orifice into the ambient fluid can be observed, with vortex rings beginning to form at the exit of the orifice. For most such tests described in the literature, such orifices are circular; thus the vortex formed at its tips forms a circular ring following the geometrical shape of the orifice. Therefore any conclusions and discussions regarding the behaviour and dynamics of such rings have to be assumed to be specific to circular vortex rings. But what if such an orifice was not perfectly circular? What if it was elliptical, square, rectangular or maybe triangular? Such geometrical aspects of the orifice, herein referred to as the nozzle, forms the essence of this particular work. All the tasks within this work will be aimed at achieving this goal.

GOALS

The main goal is to investigate the behaviour of geometrically non-circular vortex rings by generating the rings via non-circular shaped nozzles. The different nozzles shapes investigated, with the circular nozzle used for reference, are elliptical, rectangular, triangular and square nozzles, all of the same area. Intuitively it is expected that the vortex formed from such nozzles would, at least initially, follow the outline shape of the nozzle; for example, a rectangular nozzle would produce a rectangular vortex loop. Thus referring to such loops as ‘vortex rings’ in this case could be considered a bit of a misnomer, but in this work they will still be referred to as vortex rings for convenience, with the hope that the reader notes the contradiction.

In the process of achieving this goal, the first task is to conduct a thorough investigation of vortex rings from material that has been published in literature to the aspects of vortex rings that have been employed in technology. This includes a study of the laws of physics and dynamics that give vortices their significance and allow them to be understood and treated as they are in academics. This is presented in the background and literature review (section 1) and theory (section 2) of this work.

The next task, detailed under the experimentation (section 3) of this report, is to utilise the available vortex ring research facility to conduct an experimental work that satisfactorily highlights the major behaviour of non-circular vortex rings. Finally, the observed behaviour and its deviation from circular vortex ring is discussed with respect to theory and background knowledge gained on vortex rings and this gives the results (section 4) of this write-up.

1. BACKGROUND AND LITERATURE REVIEW

The essence of most vortex ring studies, from a scientific view, normally involve a general overview of a typical vortex ring motion and an assessment of how any parts of it actually concern fluid dynamics or science in general. One way of considering an overview of a typical vortex motion is to look at all the stages the ring goes through during its development until decay. To illustrate this, it is perhaps tempting to once again present the smoke ring example: as the smoker exhales, the smoke produced forms a ring at his lips which then travels away from the smoker in the same ring shape for a while before the ring configuration breaks up and the smoke diffuses in a the surrounding air (Fig 1).

Fig 1a) – Courtesy of http://enstrophy.colorado.edu/~mohseni/VortexRing1.html : A smoker blowing a smoke ring.

Some existing works, such as Auerbach (1988), highlight four main stages of development of such vortex motion as:

1. The generation or formation stage
2. The laminar or stable phase

• The unstable or wavy phase

1. The turbulent phase

A ring may not go through all four stages or may skip some stages, depending on the initial conditions. With very low ejection velocities, for example, the ring may decay naturally due to viscosity at the laminar phase (stage (ii)). Very high ejection velocities on the other hand could mean the ring becomes turbulent immediately after formation, bypassing stages (ii) and (iii).

Fig. 1b) illustrates these stages.

Fig 1b) – Courtesy of Lim and Nickels (1995): Image sequence showing the stages of development of a vortex ring. The ring forms at (a) and continues in laminar propagation through (b), (c) and (d), before the formation of a wavy instability at (e) which subsequently results in turbulence at (f).

Other workers prefer to categorise the full range of fluid dynamics problems that present during all these phases of vortex motion. Saffman (1981) takes a view fairly similar to the pattern this particular work follows. This idea is that the entire motion can be formulated as a set of fluid dynamics problems, for example;

• The formation process of the ring at the lips is a vortex sheet dynamics problem

• The motion of the ring in space with respect to time, presents a steady state problem

• The duration of their existence give rise to stability considerations

It follows that most of the literature on vortex ring tends to be specific to particular aspects of vortex-ring problems, and there is an attempt to stick to the pattern of problems listed above in presenting the results of the literature survey. At the end there is a section presenting some literature related to non-circular vortex rings.

1.1 LITERATURE ON THE FORMATION OF VORTEX RINGS

As mentioned before, typical laboratory methods of vortex ring production would involve some method of injecting coloured (dyed) fluid through an orifice into a (usually) quiescent clear ambient fluid. By dying the injected fluid, the formation process of a vortex ring can be observed as it ejects through the orifice into the clear ambient fluid.

Rogers (1858) discussed some vortex ring production techniques, the simplest of which involved allowing a drop of water coloured with food dye to fall from an eye-dropper into a glass of clear water. This simple test highlighted the following features of vortex ring formation:

Energy source for the initial rotational motion

One of the most interesting features of this is that because the initial drop could be very near the surface of the clear water, it effectively has no initial kinetic energy. Yet the ring still forms. This indicates that surface tension could be a vital source of energy for the initial rotational motion. As reported in Shariff and Leonard (1992), for half a centimetre drop, the surface energy is about four times the potential energy.

Effects of the initial impact shape of the drop

Thomson and Newall (1885) found that if the dropping height was varied, the depths to which the ring penetrated the water oscillated. Because the optimum dropping height corresponded to the free-fall distance of the drop in one period of vibration, it was suggested that a drop that was slightly oval at impact could be significant in determining the vortex strength. Later, Rodriguez and Mesler (1988) presented photographs of the shape of the drops at impact showing that the rings penetrated most when the drops were vertically elongated and vice versa, clarifying an earlier view by Chapman and Critchlow (1967). In the paper, the time sequence photographs of ring generation show that the shape of the crater made on the surface by the forming ring for the two cases (when the ring elongates vertically and when it elongates horizontally) are remarkably different and this somehow affects the ring’s strength and stability.

Interaction between the drop fluid and the clear fluid

Seigler and Mesler (1990) provided photographs of thin films of air trapped between the drop and the clear fluid which formed tiny air bubbles that were subsequently enclosed in the vortex ring. From the argument of surface-tension force, it may seem the connecting neck would widen, thus detering further contact between the fluid masses. This was the view taken by Chapman and Critchlow (1967) for the drop impact problem. The model problem however, illustrated that other points of contact may occur between the fluid masses, trapping the intervening fluid bubbles.

1.2 LITERATURE ON THE MOTION OF VORTEX RINGS

To understand the motion of vortex rings, it is common to make some assumptions regarding the nature of the flow and find theories that would help describe the flow under those assumptions. Workers like Widnall and Bliss (1971) and Moore and Saffman (1975) approach this from a dynamic point of view: they assume invicid flow dynamics and employ a force balance argument for the forces acting on the curved vortex filament, and how this leads to a translation that can be quantified in terms of velocity, etcetera. Others, like Moore (1980), look at the problem from a kinematic angle, and consider the kinetic energy of a thin-cored steadily translating, invicid vortex ring.

The force balance argument is the approach taken in this particular work. One of the difficulties that could be encountered with this approach is the way of measuring, quantifying or calculating some forces. For instance, the direct calculation of vortex line tension could be difficult and might have to be obtained by comparison with a known case. Saffman (1992) uses the case of a hollow vortex as given by Hicks (1884) since this is a case where there are no additional forces due to the rotation at the core or swirl, etcetera.

1.3 LITERATURE ON THE STABILITY OF VORTEX RINGS

The main instabilities of the ring are normally in the form of azimuthal waviness formed around the circumference of the ring that was initially laminar and stable. These waves then grow and in most cases, eventually lead to the ring collapsing in a turbulent blob. As mentioned in the introduction, early studies assumed that vortex rings were almost always stable until the works of Krutzsch (1939) indicated the formation of unstable azimithal waves around the circumference of the rings. Maxworthy (1972) observed similar instabilities and since then, this phenomenon has been widely investigated, for example by Widnall and Sullivan (1973) and Widnall and Tsai (1977), Leiss and Didden (1976) and Auerbach (1988).

There have been attempts to explain these occurrence from varying points of view; for example, Widnall and Sullivan (1973) consider the behaviour of an invicid vortex filament with a thin core, and examine the effect of a small perturbation to this thin core on the induced velocity field of the ring. From this, they deduced that such perturbations could grow, die out or even remain unchanged with time and that different perturbation wavelengths would grow differently.

1.4 LITERATURE ON NON-CIRCULAR VORTEX RINGS

Compared to the research on circular vortex rings, work relating to non-circular rings is extremely rare, but not entirely absent as some examples of vortex rings ensuing from non-circular nozzles have been described in the literature. Most studies seem to investigate heat combustors and engine mixing jets; for example the work by Ho and Gutmark (1987), Haven and Kurosaka (1997) and Gutmark and Grinstein (1999). In such cases, the fuel injected through the jet has to mix with air for combustion to occur. It is understood that the better the mix, the more efficient the engine. Therefore, it is important to control the mixing for optimal outcomes. This requires knowledge of entrainment properties of the vortices formed by the mixing fluids. This analogy could also be extended to the need to control the burning of unwanted exhaust gasses from engines or other combustion processes, or sending (blowing) unwanted waste gasses from chimneys to higher altitudes where they pose less threat of harm.

Similarly, the mixing and entrainment issues arise in other applications such as in the pneumatic conveyance of bulk solids, where the solid particles have to be entrained in moving fluid (air) to be transported. As this kind of transport is usually through pipes, there are several aspects of fluid dynamics that could determine the continued suspension of the transported particles throughout their journey. In such cases, with respect to this work, the region just after the exit would be of greater interest because in effect, the study is of how the exit geometry affects the flow dynamics in terms of vortices of the flow, which, as mentioned, is a major factor in entrainment and mixing characteristics. A nozzle shaped configuration known as a ‘vortex mixing nozzle’ has been employed by Vortex Ventures inc.¹ to perform functions such as blending, emulsifying liquids, suspending solids in a liquid or dissolving polymers, powder and granular materials in a liquid environment. This, they claim, is due to the relatively improved mixing and entrainment characteristics of the vortices formed by such non-circular configurations. The general essence of such investigations is, in most cases, specific to the entrainment properties of the formed vortices and turbulence.

2. THEORY

2.1 THEORY OF THE FORMATION PROCESS

If a section of the forming smoke ring in the example was to be viewed from the side through its cross section, it may be easier to picture a situation where the fluid elements forming the trail of smoke did not initially propagate in a straight line, as would be expected, but ended up in a curved path due to a shearing tendency at the exit. On the other hand, subsequent fluid elements would still continue to puff out, pushing the first ones from behind. With all the pressure from behind, the initial fluid elements that had started on a curved path, would have to continue moving along the curved path. Eventually, they would end up tracing a rather circular path which will then keep coiling within itself as it tends towards the centre of this circle. It is the quantity contained in this circle that then propagates forward lineally as a whole, and it is the quantity contained in this circle that is referred to as a vortex. Its overall size when fully rolled up would be referred to as the vortex core diameter.

To clarify, it should be noted that initially only a cross sectional slice of the smoke ring is used to describe one vortex motion. However, it can easily be seen (theoretically, at least) that if a series of the vortices are formed uniformly following the shape of the smoker’s lips as he/she exhales, they would form a ring-like vortex structure propagating away from the smoker. Hence the name vortex rings, and the overall size of this ring forms the ring diameter. According to Saffman (1981), the vortex ring phenomenon encapsulates the whole range of vortex motion problems.

From this description, it would seem the major factor that triggers the whole roll-up process is the fact that the initial fluid elements ended up in a rather curved path due to the shearing tendency. This shearing tendency can be explained using the ideal flow theory where the fluid in contact with the nozzle forms a no-slip boundary layer while fluid at the centre is at maximum velocity. As it exits from the nozzle, the fluid at the centre attempts to carry on at this maximum velocity, but at the same time is held back by the slower fluid at the edge of the boundary. This leads to the initial shearing tendency that starts the roll-up process as illustrated in Fig 2.1

Fig. 2.1: Shearing tendency at an exit leading to the start of vortex roll-up

It is such theories of ideal flow that enable the quantification of the characteristics of this rolling-up process.

2.11 IDEAL FLOW

Ideal flow in fluid dynamics can be described as that which occurs in some hypothetical fluid which is assumed to exhibit no viscosity or compressibility and, in addition for liquids, no surface tension or vaporisation. It is an idea that seems very far from a real flow but such conditions have been essential in enabling the development of most significant mathematical models and relations in fluid motion.

In considering the above smoke ring example from a side cross-section through which the ring could be cut, the dynamics of the ring can be considered in terms of a two-dimensional ideal flow. Now, to give some quantitative measures of what has been described, one would first consider a single fluid element. The following could then be said about what happens to this single element as the smoker exhales:

• The fluid element may undergo a translational motion

• The fluid element could undergo a rotational motion

• The fluid element may be subjected to deformation

Most probably the fluid element would experience all of the above simultaneously. Illustrations of these are in Appendix 1 together with a full derivation of the quantities they describe which are explained below.

VORTICITY

Accepting that in a fluid motion translation, rotation and deformation are likely to occur simultaneously, it is shown in Appendix 1 that a relationship can be derived that relates them to the angular velocity of the fluid elements. This relationship, known as vorticity, is usually denoted by the symbol, , and is defined as:

(2.11.1)

Where;

, is the angular velocity of the fluid elements about their mass centre

In three dimensional flow, would be just one of the three components of angular velocitythus vorticity would be equal to.

It is worth re-iterating that equation 2.11.1 has only been obtained on condition that the fluid element undergoes rotation as well as translation. Such flow is referred to as rotational flow, and it follows that without rotation, the vorticity expression 2.11.1, would be equal to zero. That is, if the motion of the fluid particles were purely translational and their distortions symmetrical, they would satisfy the condition:

(2.11.2)

This is referred to as irrotational flow.

CIRCULATION

Following from the definition of vorticity, is another equally significant quantity known as circulation. Because the fluid element is rotating within the rotational flow, there must be a resultant peripheral velocity. But because the centre of rotation is unknown, the convenient relationship would be of the sum of products of velocity and distance round the contour of the element. Such a sum is the line integral of velocity around the element, and is referred to as circulation. It is normally denoted by – that is:

It is shown in Appendix 1, following on from the derivation of vorticity, that the circulation around a contour is the sum of the vorticities in the area of the contour. This is known as Stokes’ theorem, and can be stated mathematically for a general case of any contour C, as illustrated in Fig. 2.11 below, as:

Fig 2.11: Illustration of the circulation around a contour