Round Phenomena and Fluid Salients
THE STRUCTURING OF MOVING FLUIDS
Round Phenomena and Fluid Salients
(The Structuring of Moving Fluids) [4]
Michael A. Gorycki, Ph.D. (Revised August 30, 2018)
ABSTRACT
The importance of salients to the structuring of moving fluids in a number of natural and laboratory phenomena has been presented in other web sites. Included in the present web site are simple experiments and/or arguments for some previously discussed or additional phenomena that present as round patterns since I believe these also involve the mechanism of fluid salient formation. These phenomena include; spheroidal (globoidal) cumulous structures, stream upwellings, solar granulation, drop impact patterns with an argument against the importance of surface tension, Hele-Shaw Flow, vortex rings (toroidals) and salient formation, the Cartwheel Galaxy, polygonal radial hydraulic jumps, and the north pole hexagon of Saturn. Flow through pipes and along smoke columns are also discussed and compared. It is also suggestion that fluid salients develop before turbulence, and that turbulence results in a reduction in flow velocity.
INTRODUCTION
As described in this and my other web sites,
[1] Fluid Salients and the Structuring of Moving Fluids
[2] Fluid Salients and the Formation of Beach Cusps
[3] Fluid Salients and Spiral Galaxies and Other Vortices
[4] Fluid Salients and Round Phenomena
[5] Fluid Salients and Linear Structures
[6] Fluid Salients and Stream Meandering
[7] Fluid Salients and the Jet Stream
[8] Fluid Salients and Planar Structures,
a variety of natural or laboratory phenomena can be attributed to what I call the fluid salient mechanism.
I feel that fluid salients are ubiquitous and legion, and the list presented of phenomena generated by the operation of fluid salient formation in [1] was incomplete. Following are several, relating to round phenomena, that I would like to add, along with some already described, that require additional comments. Vortices might be considered a special case of “round” phenomena, but are considered separately in a discussion of hurricanes, tornadoes, whirlpools and spiral galaxies [3]. As a conceit, a number of asides have been included in the discussions of the various phenomena, and it is hoped these will add to the argument. The phenomena discussed here all appear as round/circular/spherical patterns, and radial arrays or toroidals, and they fall under the following section headings. They are:
Spheroidal, Cumulous Formation
Stream Upwellings
Solar Granulation
Drop Impact Patterns-Ink Splotches
Hele-Shaw Flow
Vortex Ring Toroidals and Salient Formation
Vortex Ring Collisions
The Cartwheel Galaxy
Polygonal Radial Hydraulic Jumps
North Pole Hexagon of Saturn
Spheroidal, Cumulous Formation
Repeated observation of video recordings of cumulous structures developing from; clouds, explosions, volcanic plumes, nuées ardentes, structure (building) demolitions, smoky fires, snow avalanches, venting steam/condensate, etc., reveal that secondary and higher order spherically protruding fluid salients develop and impinge upon each other on the surface of a first-order cloud (Fig. 1) <A>.
Fig. 1. A first order volcanic plume exhibiting cumulous structures comprised of second, third, and fourth order fluid salients. This plume is essentially an elongate arrangement composed of spheroidal cumuli. Depicted are about one dozen second order fluid salients. Each is subdivided into a few dozen third order salients, and each of these is comprised of fourth order salients that are not resolvable at this magnification. Note that each order is comprised of salients of about the same size. A zone of retarded-reversed flow separates each salient in each order.
The entire structure increases in size with new, higher-order (smaller), salients appearing on continuously expanding spheroidal surfaces. Their mutual interference results from the development of intervening zones of retarded-reverse flow. Note that within each order, all the fluent salients are approximately the same size. This is also the mechanism for spherical cumulus cloud formation in the atmosphere, which exhibit evenly spaced periodic structures if they are actively forming. These salients are Bénard-like structures that form on a three-dimensional, spherical surface. Well-formed mammatocumulus clouds are Bénard-like, closely spaced planar phenomena in the atmosphere, that slowly grow downward (see Mammatus clouds in Wikipedia). Stratocumulus clouds are similar to mammatus, can be evenly spaced, but are rounded at their crests. The patternless distribution of rain cells in an area of low pressure suggests disorganized but intense fluid salient formation. Interestingly, one may see spheroidal cumulous structure whenever cream is poured into a cup of tea, and in the growth pattern of some plants such as cauliflower. There, the mutual interference of florets, as they spherically compete for space, produces at least three orders of salients and creates the same pattern as seen in cumulus clouds.
Note that if water flows upward and out of a vertical pipe, the jet generates a varying cluster of knobby fluid salients at its uppermost limit. This is considered to be turbulence by most workers <B>.
<B> https://www.youtube.com/watch?v=-xvVgXNAsk8 (go to 1.25 minutes and later)
This water jet is mindful of the second order salients in the volcanic plume seen in Fig. 1.
If the water flow is lessened, the salient structuring disappears, and is replaced by a gentle, laminar flow of the water. This indicates that a critical velocity is necessary for fluid salient structure to develop.
I suggest that the individual projections in these clusters have their origins in the fluid salient mechanism operating as the result of peripheral attenuation against the walls of the pipe. It seems this structuring, with its associated retarded zones and attendant drag may be the cause of the eventual slowing of fluids during their passage through pipes when a critical velocity is achieved. The formation of the fluid salient structuring is constricting, increases the friction against the walls of the pipe, and it is this which results in the slowing of fluid flow when flow is increased beyond laminar. It is called “friction loss” and requires increased pump energy and costs. I suggest that turbulence (sensu stricto) in pipes may not be easily achieved, but only the earlier development of knobby fluid salients. The transition from laminar flow to fluid salients to turbulence is shown in Fig. 3a in “Rising Smoke Columns and Meanders” [5] (Fluid Salients and Linear Structures). The knobby fluid salients here in <B> appear as overrollings in Fig. 3a, [5].
The discussion of laminar flow leading to turbulence in smoke columns [5] describes an intermediate region of repeated overrollings which I again ascribe to the formation of fluid salients when a critical velocity is achieved. These overrollings are free to form in a smoke column because they are not confined to the stricture of a pipe. The widening fluid salient portion then gives rise to turbulence (sensu stricto) as it becomes slowed higher in the smoke column because of the greater volume of the structure formed (again, see Fig. 3a, [5]).
Again, for the smoke column, the incomplete cone increases in diameter as it rises because the rolls incorporate smoke and clear air from both sides of the cone. As a result of shearing, the rolls maintain their thickness as the cone-like ribbons they form extend laterally.
As an aside, widely popular decorative laminar flow fountains <C>
are a relatively new and expensive technology that tries to overcome what is considered to be turbulent flow in pipes by subdividing the water in the pipe through small channels such as mesh, holes, drinking straws, etc. This obviates or diminishes the effect of knobby fluid salient formation in the pipe. The result is a streamlined, smooth, ostensibly laminar flow of water.
Stream Upwellings
Observing the upwellings caused by fluid salients at the surface of a large, swiftly flowing stream or ocean channel <D>
also reveals structuring similar to the knobby fluent salient cluster produced by a vertical jet of swiftly-flowing water <B>. Radially spreading upwellings in the channel, with diameters up to several meters across, usually contain a cluster of several secondary upwellings within their central portions. These similarly spread radially on the surface of the water, and mutually interfere at their peripheries as zones of retarded flow. Third order, and even higher, upwellings may also occur within the second order upwellings. The upwellings are persistent, caused by irregularities in the stream channel which interfere with the horizontal flow of water. In the case of the maelstrom at Saltstraumen, Norway, the channel is narrowed by the island (hence the bridge) and the deeper flow of water is forced upward toward either side of the channel. The upwellings on both sides often rotate due to shearing with the main current flowing through the passage and the water there is lighter in color due to included air bubbles. The lighter water darkens down-current as the included air escapes. Note the white waves rolling toward the central current. This inflow is brought about by the rise in water level of the upwellings on both sides of the channel. Also, more placid upwellings are further away from the channel’s center.
The center passage of the maelstrom at Saltstraumen is free of upwellings, because the water there is deep and free to flow. Note that the upwellings <D>, the water jet <B>, and cumulous structures <A> in general, are all comprised of a first-order and at least a second-order series of fluid salients. The gravitationally induced flat surface of the maelstrom imparts a two-dimensional, pseudo cross-section to the upwellings, somewhat similar to Bénard structuring, and provides a better understanding as to the internal architecture of knobby cumulous phenomena <B> as generated by the fluid salient mechanism.
The development of spheroidal patterns may be the result of a greater rate of expansion of surface, compared to volume. That is, for a cauliflower, it is an excessive growth of florets at the surface of the inflorescence when compared to the growth within. For a fiery explosion, the greater availability of oxygen at the surface of the spheroid permits improved oxidation, heating, and the expansion of burning gases there. A slowly expanding cumulus cloud owes its fluid salient surface pattern to a more gentle, mutual interference with the surrounding air, whereas the hemi-spheroid of salients in a thick vertical jet of water is caused by the mutual lateral interference of water resulting from the effect of circumferential friction at the periphery of the jet in the tube, possibly friction with the air, and the effects of gravity. Under the retarding effect of gravity, this structure will form at the terminus of a powerful, thin, single (laminar flow?) jet of water emanating vertically upward from a thin pipe. The terminus spreads radially in an unopposed fashion to reverse its direction, falling downward as retarded-reverse flow <E> (see 0.33 minutes).
For the moment, we may think of the thin jet as a single Bénard-like “fluid salient” structure. This simple mechanism and its ramifications will be discussed in the next section.
Solar Granulation
In my earlier web site [1], I suggested that Bénard circulation is simply a vertically viewed, foreshortened, nearly two-dimensional (plan view) version of fluid salient motion. Solar granulation studies reveal an example of this structuring on a grand scale <F>.
Each of these cells is similar to the single toroidal formed by a thin jet of water <B>. En masse, they are competing on a grand scale upon what seems the nearly planar surface of the spherical sun. This structuring appears as numerous granules in the sun's photosphere that are each about 1,000 km in diameter. At a greater depth in the sun's convection zone are supergranules, each of which is about 30,000 km in diameter. Both types of granules serve to transfer heat radially from the sun's interior toward its surface, and their respective sizes apparently depend, in part, on the composition and thickness of the zones in which they develop. Heat at the surface is lost radially, primarily from the hot centers of each of the shield-shaped granules (fluid salients). The cooler gas then moves radially from the centers of the granules to the darker intergranular lanes (zones of retarded-reversed flow at the edges of the granules) where it sinks to deeper regions of the sun to become reheated and recirculated. Another example of these can be produced in the laboratory with gold paint dissolved in a shallow layer of acetone. Apparently, as the acetone evaporates, fresh acetone rises to the surface, bringing included pigment particles in it to continue the convection and the Bénard circulation <Ga>.
Note the second and third order salients in some of the primary structures.
A second experiment using silicon oil, aluminum powder, and heat is similar <Gb>.
<Gb> https://www.youtube.com/watch?v=6w8u9YzZXNM (go to 1:40 minutes)
The point to remember is that as previously mentioned, Bénard circulation is a rather shallow version of fluid salient formation. As the cells (salients) develop, they impinge on each other to cause the extensive fluid salient structure. This concept is enhanced by the fact that for beach cusps, stream meanders, spheroidal cumulus clouds, stream upwellings, solar granulation, etc., there is the effect of retarded-reverse flow circulation associated with the salient structuring, suggesting that all the phenomena mentioned here have the same origin.
Drop Impact Patterns-Ink Splotches
In my paper (Gorycki, 1973a), I suggest that the fingering of ink splotches on paper and the corona of a drop impacting a liquid surface are the result of fluid salient formation along a circumferentially expanding (but impeded) toroidal vortex (Fig. 2).
Fig. 2. Asteriated patterns produced by the impact of drops of India ink falling from heights of 25 cm (A) and 180 cm (B) onto a sheet of smooth paper. The salients on the rim of each drop are evenly spaced and suggest the development of fluid salients as the ink spreads centrifugally on the paper (slightly magnified).
The drop expands probably by overrolling and friction with the substrate. This results in an unusual, simultaneous, circumferential, extension/compression at the rim. This is relieved by the formation of evenly spaced salients with intervening zones of retarded flow (Gorycki, 1973a). The direction of rolling of a toroidal can be difficult to visually discern. That is, in the case of ink splotches, whether it is an outward overrolling, or an underrolling as the toroidal expands across the paper, the results will be the same; salient formation. I suggest this direction of rotation may be true for any example of fluid salient formation described here or in my earlier web site.
Examining Fig. 2 one notes that the asteriated patterns (fluid salients) are quite regular in length, width, and are evenly-spaced, separated by zones of retarded-reverse flow. This is the definition of fluid salients. Some workers would call them “perturbations”, suggesting that the salients shown here or in other studies, where a repeated evenly-spaced structure is seen, is caused by an unknown disturbance to the system. Also, what causes the perturbations in the first place? Perturbations by definition imply a disturbance, disorder, confusion, or disruption; qualities that do not match the orderly, structured array of evenly-spaced periodicities that fluid salients display. Do they refer to a single salient, or to something more complex which produces a large number of evenly-spaced salients? Either way, it is a weak explanation for a dynamic, ordered structuring.
Conventional wisdom holds that coronas produced by drop impacts are simply due to surface tension. This might be a case of trying to fit an established mechanism (surface tension) to a phenomenon, when a newly recognized mechanism (fluid salients) is more appropriate. If surface tension does have an effect, it might be to only affect the shape of the ends of salients, after they have been formed by the fluid salient mechanism. Surface tension may shape the parabolic tips of fluid salients that spread across a solid surface, or it can form separated droplets in the air at the tips of fluid salients when a drop lands.
There is also further discussion of other workers’ experiments [1] in which no salients are produced at an apparently overrolling circular rim as water drops impact and readily spread across a smooth surface. If the drop spreads across a rough surface, however, salients (“perturbations”) do develop at its rim (Fig. 3).
Fig. 3. Blood spatter reduced on a smooth (left) and amplified on a rough (right) surface. (See www.crimescene-forensics.com, Drip stains)
Salvi (2001) states “Perturbations on the rim appear on rough surfaces even at relatively low impact velocities. By increasing this velocity, perturbations develop on the rim even on smooth surfaces.” (page 262). He presents photos on his page 265 of water droplets impacting smooth and rough glass surfaces that show smooth and “perturbed” rims, respectively.
As stated, the friction that is caused by the rough surface apparently inhibits radial expansion of the liquid, thus promoting circumferential extension/compression, and fluid salient formation. This results in the question, should surface tension (rather than fluid salients) also be affected by the smoothness of the surface, or even the speed of spread?
Work by Nagel, Xu, and Zhang (2005) at the University of Chicago describes alcohol drops spreading across a smooth glass surface. If atmospheric pressure is increased from about 1/5 to normal, or if a sequence of gases of greater density is employed, there is a sequence from a smooth rim, to where the thickness of the rim undulates (is perturbed), to droplet formation. Nagel, et al., suggest that “the drag of the surrounding air might play some role in the break-up process.” <H>.
As can be seen, when the pressure is at 140 torr, there are no undulations, but at 760 torr, there are. They suggest that, “As a drop strikes a surface, liquid begins to spread sideways at supersonic speed, creating a shockwave. The shockwave pushes back on the liquid, and if that force is greater than the internal forces holding the drop together, the shockwave will lift the liquid off the surface and create a splash. Reducing the pressure reduces the force exerted by the shockwave.” They also found that “water exhibits the same behavior, but its higher surface tension narrows the range of splash-forming impact velocity and creates a much larger margin for experimental error”, and that “It’s much harder to splash than ethanol”. Thinking of it another way, the undulations may, in fact, derive from the fluid salient mechanism and not rely on a shockwave pushing back on the liquid.
Additionally, they find that “compressible effects in the gas are responsible for splashing in liquid solid impacts.” <I>.
The explanations presented by Nagel, et al., for the formation of undulations in the spreading drop, imply that the liquid merely has a passive rather than the active role suggested for the gas phase by the “drag”, the “shockwave”, or “compressible effects”. I suggest that employing normal air pressure, or a denser gas, the gas inhibits radial expansion at the drop’s periphery. This leads to a hindered circumferential extension/compression (as with the aforementioned smooth versus rough surface experiments and, possibly, including overrolling) and is seen only as an undulating thickness at the rim of the spreading drop below 100 kPa <H>. If sufficient energy is present in the system to overcome the higher gas pressure, the edge of the spreading drop can then become elevated above the substrate (at 100 kPa and 0.276 ms) with further break-up into droplets. That is, higher gas pressure against the spreading drop would induce undulations (salients) to form in liquids by restricting the radial spread of the drop. This suggests that it could be the liquid, rather than the gas that becomes structured.
These arguments thus far speak for the possible action of either surface tension or the operation of the fluid salient mechanism (including overrolling) in the formation of drop impact patterns. The following examples speak for the fluid salient mechanism. Included in my earlier web site [1] are subtle, corona-like images of radially spreading ground level (gas-solid) and upper-level clouds produced during atomic explosions which represent gas-gas interaction. A concentric circular pattern of salients, superimposed on a radial centrifugal pattern, can appear on the surface of a body of water as the result of a helicopter's downdraft (Fig. 4) (gas-liquid).
Fig. 4. Helicopter downdraft on water surface exhibiting radial fluid salient pattern superimposed on circular wave pattern.
An example of a gas-solid interaction is a rocket’s exhaust producing a regular radial fluid salient pattern against the earth’s surface (Fig. 5).
Fig. 5. Centrifugal pattern of salients generated at the periphery of a rocket’s exhaust cloud as it spreads against the earth’s surface where a number of evenly spaced salients are visible at the cloud’s edge. Note also the compressed but uniformly spaced salients making up the vertical portion of the exhaust cloud.
From the same web site [1] there is also a discussion of similar (cumulous) patterns produced if cream is poured directly down the center of a cup of coffee (liquid-liquid). Also, when a stream of milk impinges the bottom of a cup of thick (viscous) fruit syrup (liquid-solid). The same is true at the periphery of an overflowing soup bowl or soup plate if a fixed (but variable) vertical flow of water impinges its center. A similar centrifugal pattern readily appears under schlieren illumination if water from a tap is allowed to flow into a partially filled sink or straight-sided stainless steel pot.
As an aside, a pattern similar to Fig. 5 at ground level is obtained if a teaspoon of salt is thrown into approximately 10 cm of quiescent water in a pot. The falling salt particulates energize the water so that a primary toroidal forms at the bottom of the pot. The toroidal forms an evenly spaced pattern of centrifugal secondary toroidals (fluid salients) comprised of water from the primary salient mixed with the motivating salt grains. Due to the irregular shape of the mass of falling salt, the primary and secondary toroidals are not well-formed.
All these patterns are quite similar in appearance to the India ink splotches in my first web site [1] and are all situations in which surface tension plays no role in the development of salients. This suggests that the “perturbations” displayed by ink splotches are initiated by the fluid salient mechanism rather than solely invoking surface tension.
Again, if surface tension does play a role in the ink drop experiment (Fig. 2), it is merely to further shape the fluid salients after they have formed, and is not the cause of salient formation. Also, surface tension should not be considered the cause of drop impact patterns simply because they involve a liquid. This argument will be continued in the later section on vortex ring toroidals where a submerged liquid-liquid structure forms fluid salients.
Hele-Shaw Flow
The shape of the flow in Hele-Shaw cells suggests that they are the result of the fluid salient mechanism acting to produce a round phenomena between two flat surfaces that are nearly in contact. A viscous fluid (glycerin) is driven radially out from the center of a less viscous liquid (water) and a circular, evenly-spaced fingering begins to develop at the edge of the viscous fluid that becomes more complex with time (see Hele-Shaw cell videos on YouTube). I suggest that the leading edge of the viscous fluid undergoes a circumferential extension/compression that is relieved by fingering. I find that this pattern is very much the same as that in my discussion of beach cusp formation [2], where a wave, running up the beach face overrolls, thins under the influence of gravity, and extends axially. This induces extension/compression that forms evenly spaced fluid salients and intervening zones of retarded flow capable of forming cusps. In the Hele-Shaw arrangement the salients continue to flow radially (and extend circumferentially) under continued pressure from the center (simulating a protracted drop impact) that produces a frond-like pattern composed of second and third order salients and zones of retarded flow, because of the prolonged and ever widening circular area of the flow.
The same pattern occurs if air invades the glycerin, or if two viscous fluids are used. In hot Hele-Shaw cells a fluid is heated from beneath and forms evenly spaced schlieren (fluid salients in a linear Bènard-like structure) for the same reason.
I suggest that Hele-Shaw flow is analogous to drop impacts, and that it also operates without the aid of surface tension.
Vortex Ring Toroidals and Salient Formation
A series of experiments by Thomson and Newall (1885) describe the formation of vortex ring toroidals by allowing drops of various liquids to descend through other liquids. I unwittingly reproduced some of their work about 130 years later for the purpose of showing the similarity between ink splotches on paper and ink drops falling through water. This, to obviate the supposed action of surface tension in the splotches, and to provide additional proof for the operation of the fluid salient mechanism. My observations, involving only water and aqueous dyes, were sufficient to support an understanding of ink splotches, toroidals and fluid salients. This restriction was fortuitous, because Thomson and Newall employed a large number of disparate fluids but found that “a drop of one liquid only makes a ring when let fall into another liquid, when the two liquids can mix, and, therefore, when the surface tension is very small.”
Thomson and Newall found that toroidals become “corrugated” (Fig. 6a)
(Fig.2 from Biswas, 2018)
Fig. 6a. Side view of vortex ring similar to Thomson and Newall’s Fig. 1 showing the “corrugations” that develop as the toroidal falls through the water.
as they descend through an appropriate liquid and that this is the result of “motion in the liquid into which the drop falls.” Thomson and Newall refer to these corrugations as “inequalities [that] make their appearance: more ink seems to collect in some parts of it than in others, and” “these parts of the ring descend more rapidly than the rest”.
Some recent work (Biswas, 2018) also refers to these repeated structures in the descending toroidal simply as “perturbations”. This suggests that these so-called corrugations, perturbations, inequalities, disturbances or instabilities are reasons often given for evenly-spaced structures seen in moving fluids nature. It is apparently assumed that they are merely passive and not the result of an active, interior hydrodynamic structuring seen in the functioning of the fluid salient mechanism (see a discussion of perturbations after the Introduction to Beach Cusps) [2].
Thomson and Newall correctly show in their Fig. 8 the transition from a simple toroidal to the very complex structures as presented here in Figs. 6b, 6c and 7, but I feel that they, as well as later workers; (1) do not offer an adequate explanation for the complex structure of the primary toroidal seen in Fig. 6a, (2) do not explain the secondary and later orders of toroidals derived from the primary toroidal, (3) nor the increasing number of later orders, (4) why there is a decreasing number of toroidals in succeeding orders, (5) why the succeeding toroidals slow their descent, (6) or why the evenly spaced toroidals develop from evenly spaced, downward trending “corrugations” in a toroidal (“the Splitting up of the Rings”, (Thomson and Newall)).
Several experiments involving ink drop toroidals arose from my initial experiment and are included in this section as evidence of the operation of the fluid salient mechanism. An explanation for Thompson and Newall’s Fig. 1 and its “corrugations” is given in experiment 5 below as well as other, more complex observations they, and later workers, have made.
Simple experiments involving aqueous drops of dye descending through water have been devised for this essay. They support the contention that salients (ink splotches) also form at the periphery of a toroidal formed by a drop of fluid impacting a horizontal solid substrate without the influence of surface tension.
Materials can include various pipettes with squeeze bulbs to produce drops of different sizes of dye such as; food coloring, India ink (Higgins® Fountain Pen India, non-waterproof black ink), a saturated solution of potassium permanganate (KMnO4), or Tintex® black dye. In addition, there are two containers of quiescent water. The first container is a one liter glass cylinder 8.5 cm in diameter and 19.5 cm tall. It is placed on a light box so that the drop of dye, settling through the water, can be easily viewed and photographed from above. It is filled to the brim with water that is allowed to stand for 10 minutes to eliminate currents in the water.
The second container is a standard 5-gallon (20 liter, 20 X 20 X 40 cm) all-glass fish tank. To improve visualization of the descending drops, toroidals and fluid salients, the outside surfaces of the tank, except its front, are coated with white paint. This allows a front view as well as a top view of the structures produced. The inside of the tank is well-illuminated from the top and front. To eliminate bubble formation on the glass walls the tank is nearly filled with degassed water, using cooled water from the hot water tap, or water stored in plastic gallon containers that have been shaken to remove those bubbles that have formed on the inside walls through time. Employing a dust cover, the water in the tank is allowed to come to rest overnight. Its temperature thus reaches nearly ambient conditions, and the water is shielded from moving air currents. Water, held in small containers, can also initially be used for quick tests, since this water is both clean and becomes promptly current free. In all these experiments, any ink introduced to the water will produce fresh currents, density variations, and ink structures that can interfere with subsequent observations. As a result, the water should be changed and left standing before each experiment.
(One liter glass cylinder)
In the first experiment, the pipette is held vertically above the water surface and a drop of ink is allowed to fall for from 1 to 10 cm so it hits the water symmetrically and without too much force. As the drop descends through the water, it usually forms a single, first order, planar, toroidal lightly connected to the surface by a filament of dye.
Most drops produced convey information, but empirical experimentation provides a better insight into the study. If a too robust pattern has developed; the drop has fallen too far, was probably too large, or was too concentrated and should have been diluted with water. If the toroidal is meager; the drop was too dilute, too small, or had not fallen far enough. As a satisfactory drop descends, the first order toroidal typically can quickly develop about 6 salients which likewise descend at its periphery (Figs. 6b and 6c).
Fig. 6b. Large single drop of India ink at center forming a first order planar toroidal after a slight descent through clear water in a small container. Six evenly spaced, fluid salients, connected by a layer of ink, have formed below the periphery of the toroidal. The edges of any ink layer overroll toward each other(caused by their descent), and their ends form second order toroidals. The first order toroidal can just be seen beneath an inconsequential, thin, translucent layer that this ink forms at the water surface.
Fig. 6c. Same drop as in Fig. 6b, but a few seconds later. Most secondary toroidals each have produced a fluid salient which, under these conditions, has again produced two tertiary toroidals. These toroidals are at a greater radius from the vertical axis and are at a greater depth. Each tertiary toroidal will produce a fluid salient capable of producing the fourth generation of toroidals.
Because of their high density, each of these fluid salients descends as a thick, concentrated filament and each quickly forms a toroidal at its end. Often, with low density drops, the salient is paired, and the pair is connected by a ribbon, the edges of which overroll toward each other in a response to their descent through the standing water. A pair of toroidals form at the end of each edge. These toroidals, in turn, quickly form later, and usually fewer salients as they descend. The salients, in turn, each form a toroidal to continue the process. Importantly, the toroidals produce salients which are evenly-spaced in keeping with the fluid salient mechanism. This process is repeated until the toroidals become light in color, slow moving, with each generation forming fewer salients and toroidals until they stop forming; this, as the density of the overrolling dye approaches that of the water. Some first order toroidals produce a number of single salients, and others form a rosette of evenly spaced second order toroidals with no obvious intervening fluid salient attachment to the first order toroidal.
As stated, the initial density (and viscosity) of the drops may be reduced by dilution with water, and their momentum also varied by the distance the drops fall and how big they are. This varies the number of salients formed from a toroidal, and the number of generations of salients and number of toroidals that can form from any fluid salient. The depth of water controls the number of generations of salients and toroidals that can be displayed in the container. A nice video of a vortex ring exhibiting “corrugations” and a series of salients and later generations of toroidals is given at: Google, “vortex ring Thomson and Newall”, “Vortex Ring”, “Videos”. Also, the pipette should be held vertically so that the drop formed will have a radially symmetrical and vertical axis when it enters the water. The pipette shown in the second half of the video is not held vertically and produces an ink structure that is skewed.
The mechanism involved in the production of salients derives from the primary toroidal falling through the water. Viewed from above, it is centripetally overrolling toward its center and is, therefore, seemingly attempting to advance toward that point. This is caused by the ink drop creating an initial downward motion in the center of the toroidal it has created. This direction of overrolling can be easily seen if Tintex® black dye (which contains visible particles) or food coloring (which is transparent) is used.
Clay, rolled on a table can increase its length, but at the expense of its diameter. The toroidal tends to maintain its minor diameter (its thickness) as it falls, because it also overrolls and incorporates layers of clear water. This incorporation of ambient water is shown by the increasing volume of the ink retaining structures, the decrease in ink density and the number of fluid salients with downward distance travelled. Most of the kinetic energy of an ink toroidal is circular around its thickness, and is parallel to the toroidal’s axis. As a result of this dynamic overrolling, the major circumference of the toroidal is retarded in its growth by the undisturbed, quiescent water it is falling through, and it undergoes circumferential extension/compression. That is, the growth of its major diameter is hindered by the kinetics of the overrolling and cannot increase in size as fast as does its circumference. This generates the formation of the fluid salients and the (“corrugations”) of Thompson and Newall. It also causes the outer diameter of the toroidal to be cramped as it rotates downward toward the inner diameter. This is similar to the overrolling of the zone of retarded flow in the case of beach cusps and their rubber cylinder physical model (see Fig. 2, [2]). This cramping of the primary toroidal is sufficient to form evenly-spaced, downward drifting, fluid salients on the inner surface of the toroidal. These salients, each behaving like the original drop that first fell through the water, then form the later generation of toroidals at their terminal ends.
The number of fluid salients that form on the first toroidal can give rise to two or more toroidals each of which give rise to two or more salients. The average density of each later generation of toroidals becomes less as greater amounts of water are incorporated in them, and friction reduces their angular momentum. Terminal toroidals, which produce no salients, have little energy of rotation and also descend very slowly as their average density matches that of the water. This is why Thomson and Newall found that a lack of surface tension between the liquids was necessary if toroidals were to form and state that “rings are formed only when a liquid is dropped into one with which it can mix.”
(Five gallon fish tank)
In the second experiment, a single drop of India ink is allowed to fall into the center of the fish tank. As can be seen in a side view, the descending toroidal gives rise to ever increasing generations of salients and toroidals which become slower to form as more and more water is incorporated into each structure (Fig. 7).
Fig. 7. The branching structure of repeating toroidal-salient-toroidal sequences originating from a single drop of undiluted dye. It is descending through a tank of quiescent water. Approximately one hundred toroidals have been produced from the initial single toroidal. Note that the ink trail at the top abruptly broadens at the level of the formation of the initial toroidal. This toroidal forms a number of salients which then form the second order toroidals. This process is repeated until the dye reaches the density of the water.
The single ink drop produces a complex, repeatedly branching, downward pointing salient structure with a toroidal forming at each tip. It somewhat resembles a baroque chandelier. If the water is 20 cm deep, about nine generations of toroidals can form in this experiment depending on the density and size of the original drop. Up to 100 individual toroidals may form from the original primary toroidal. Given enough space, the number of salients produced by progressively later generations of toroidals will decrease to zero. When a toroidal’s density approaches that of water, a fluid salient will terminate in a final toroidal, and no later toroidals or salients will be generated. The end result of this experiment is that the entire structure becomes quiescent, slowly elongates downward, and eventually disappears in a few hours as it succumbs to random currents induced by all the descending toroidals, and is further eliminated by Brownian movement. Again, surface tension plays no role in this falling drop experiment.
It is important to note that those toroidals, having a much greater density than the water, continue to fall through the water. With smoke columns no later generation of the toroidal-like structures occurs because the smoke overrolls have a low density, and, as ribbons, are open at either end allowing them to expand laterally without the circumferential extension/compression that complete, circular toroidals can exhibit. They can only replicate themselves vertically until, encompassing enough ambient air, they cool, loose buoyancy, collide with themselves, and produce the overlying zone of real (sensu stricto) turbulence (see Fig. 3a, [5].
The first and second experiments described here, mimic that of a high-impact drop hitting a solid substrate, except that after the first ink toroidal has formed in the water, there is no solid substrate to hinder all the subsequent toroidals generated, and no influence due to surface tension in forming the so-called “corrugations”. Each overrolling toroidal is hindered in fully increasing its diameter as it descends through the water since it produces subsequent, higher order salients (“corrugations”) that descend more quickly because they are initially compact and still composed of a liquid denser than water.
In the case of a drop impacting a solid, impeding substrate, the “corrugations” spread radially in a planar fashion. Also, an elevated corona is often produced suggesting a diversion of energy from a toroidal to a corona. As a consequence, I again would suggest that with high-impact drops hitting a solid substrate, the “corrugations” and coronas are simply fluid salients that do not penetrate the substrate but expend all their energy at, and rising above, the impacted surface. I suggest they are induced without surface tension effects except those which merely modify the shapes of the distal portions of the salients which break free of the tips to escape the pull of surface tension. Also, if the solid substrate is smooth instead of rough, the “corrugations” do not appear because apparently there is no overrolling, no extension/compression and therefore, no salient formation.
A third experiment, nearly identical to the second, involves use of a drop of ink that has been diluted with seven parts water. Again, the drop forms a single toroidal, which also produces salients, but they are fewer in number when compared to the second experiment. The salients again descend in the tank to form toroidals at their tips, but this process quickly goes to completion after only about four generations of toroidals because the diluted drop’s density was initially low. That is, the third experiment is not as prolonged as the second since this first order toroidal generated here initially has less rotational energy and its average density, as well as that of subsequent toroidals, more quickly approaches that of water.
In a fourth experiment, a more energetic toroidal, comprised of a saturated solution of sodium chloride and ink or KMnO4 was employed. The salt greatly increases the dye’s density. Using a larger orifice pipette, a larger drop of fresh dye is released from a height of three cm. Because of its high density and large size, the drop very quickly sinks through the water forming a toroidal, which overrolls very quickly, and both thins and increases in diameter. As it descends, a number of evenly-spaced bulges form on the toroidal, but do not form salients because its rapid overrolling and descent causes the bulges (rudimentary salients) to quickly become entrained in the toroidal before each bulge can produce its own secondary toroidal. This rapid descent, overrolling, and bulging, quickly causes the system to deteriorate because of induced turbulence, without the chance of forming the branching structure seen here in Figs. 6a, 6b, 6c and 7. A dye trail also connects the water surface with the descending toroidal.
Of interest here is that the tips of the fluid salients, moving with sufficient energy, form a toroidal shape when unhindered by a substrate or mutual interference. Bénard structure salients, which are basically toroidal (see Solar Granulation above), form over a large area as they extend vertically, impinge a planar interface, and mutually interfere at their edges. Also, Simpson's (1972) shadowgraphs of gravity current lobes and clefts (double curling, elongate, mutually interfering, salients formed by a liquid flowing beneath a liquid) seem to be a stretched-out version of the more symmetrical Bénard-like structure and are similar to my tilting trough fluid salients (liquid-solid) (see Fig. 3 in my earlier web site [1]).
In the fifth experiment, to repeat Fig. 6a, I drastically diluted the ink so that a single toroidal was produced that could not form any later generations of salients of toroidals. It merely exhibited undulations at its rim (“corrugations”) which are nascent fluid salients. This feeble toroidal also maintained its diameter, being unable to expand against the quiescent water that contained it. This toroidal is very much like the uppermost ring seen in the physical model in Fig. 9 (Planar Fluid Salients [8]) and the preceding Fig. 8 for toroidals in a Taylor-Couette device. These toroidals, however, do not expand because the constraints of the glass cylinder in which they are contained. Also, an ink drop falling on heavy paper will produce fluid salients that move radially outward on the paper because they are not constricted by a glass cylinder, or are not falling through still water (see Taylor-Couette Flow [8] and Drop Impact Patterns-Ink Splotches above). It should be appreciated that the structures seen in Figs. 6b and 6c represents an evolving variation of the 2 dimensional drop impact pattern onto a sheet of paper seen here in Fig. 2, but in a 3 dimensional form. The fifth experiment is somewhat similar to the fourth in that both do not produce toroidals or well-formed salients.
Simpson's clefts are also similar to Taylor vortices, which form between vertical rotating glass cylinders, in that both salient structures similarly curl in the same way on their long edges. Also, my density current salients (see Fig. 9 in my earlier web site [1]) extend linearly down a planar surface of finite length, and these straight currents are also similar to Taylor vortices, which are produced without a leading edge of fluid salients, but continuously traverse a cylindrical surface as long as the cylinder is rotating. A number of workers have discussed density currents but have not given a cause for their structuring (see “Density Currents” in Fluid Salients and Planar Structures [8].
Experiments with Couette flow produce Taylor vortices if the angular velocity is increased. I consider these to be fluid salients [5], [8] that can give way to wavy vortex flow and the onset of turbulence if the Reynolds number is exceeded. This sequence from laminar flow, followed by fluid salients, and finally turbulence, has already been discussed in more detail [1].
Vortex Ring Collisions
Toroidals are well known in the literature, and much has been written on colliding vortex ring experimentation <J>.
<J> me.nus.edu.sg/dept/limtt/flowgallery.html (see Video Gallery and Flow Gallery, “Head-on Collision of Two Vortex Rings”, (a) Re=1000). Also see “Leapfrogging of Vortex Rings”.
An interesting experiment by Lim and Nickels (1992) shows the head-on collision of two smoke vortex rings (a red and a blue one) that produces a pair of expanding laminae that form at right angles to the collision, at the interface. A ring of 18 evenly spaced secondary toroidals forms at the periphery of the toroidal and move radially away, and I suggest that they form at the ends of short-lived salients, and that they travel along the circular membrane, similar to those seen in Fig. 2, which extend from the edge of the circular ink splotch. The smoke toroidals have a density similar to the air through which they travel which allows them to travel in any direction without being effected by gravity. This conveniently allows the secondary toroidal to be propelled horizontally but, unlike the water experiments performed here, they do not produce secondary toroidals until after their energizing collision with each other. Each forms as in the ink drop experiment (Fig. 6a and 6b), but the strength of the collision cannot produce more than the secondary set of toroidals. Note also, that in Lim and Nickels’ experiment <J>, the 18 secondary toroidals are really each composed of two halves. Each set of halves is produced by inhibited circumferential expansion at, and near, the edge of each lamina. The combined momentum of the primary toroidals is sufficient to produce the circular paired laminae which increase their diameters to a limit where the waning peripheral expansion is met with equal resistance from the ambient air.
In <J>, fluid salients are not apparent, but seemingly produce the halves of the secondary toroidals at appropriately evenly-spaced locations. This concept is supported by the fact that each of the toroidal halves of one color are on the same side as the primary toroidal which gave rise to them, and their intermediary lamina. The toroidals enlarge as they simultaneously open upon collision with the surrounding water, and their open faces move radially away from the impact axis. It should be remarked that this collision of evenly spaced primary toroidals (possibly forming salients) produces evenly-spaced secondary toroidals. These are at the rim of an actively expanding “drop” (the mirror imaged laminae), takes place at relatively slow speed, and also without the aid of surface tension.
Due to the operation of the fluid salient mechanism, friction against the motionless air in the tank induces a pronounced extension/compression near the edges of the paired membranes. This is evidenced by their even, undulatory structures seen at the end of the video <J>.
The “leapfrogging” video shows two toroidals exchanging places by having their centers proceed their peripheries (to underroll). This is the same rotation seen in the colliding smoke toroidals. However, I suggest that underrolling proceeds just after the collision.
Using laser transection, Perry and Lim (J. Fluid Mech. V. 88) provide evidence of what I call overrolling, apparently due to shearing, in a smoke column <J> (see Flow Gallery, “Positively Buoyant Jet”: “Cigarette Smoke” and “Laboratory Produced” and “Multiple Pictures of a Single Vortex Ring”), (see also [5] for a more complete discussion of smoke columns).
I suggest that the fluid salient mechanism can be called upon to explain the structuring of a falling drop of dye (Figs. 6b and 6c), regardless of the variations of the experimental parameters used, and also to explain the changing structure of the two vortex rings in collision. Once the primary toroidals are produced in the falling drop of dye, one or more fluid salients are generated which then produce later toroidals. Most papers merely describe the structures, but not how they are produced.
As an aside, the cylindrical shape of the first container (Figs. 6a and 6b) causes the orientation of the ink droplets (and the water) to remain surprisingly stationary and unchanging if the cylinder is rotated about its vertical axis. This is performed simply by holding the cylinder steady, and rotating one’s body. It acts as an inertial compass to indicate the amount of rotation it, and the observer, have undergone, irrespective of compass direction or location in 3D space as the cylinder is moved from place to place. More importantly, there is no evidence of turbulence. Laminar flow obviously is restricted to a very thin layer at the glass wall and bottom of the container. It is very different from the fourth water vortex experiment in [3] involving the production of an aqueous gyroscope. These are two complementary phenomena. In the case of the aqueous compass, the water is immobilized as evidenced by the stationary dye droplets resting on the bottom even though the cylinder is rotated or moved through 3D space. On the other hand, the water vortex of the aqueous gyroscope is rotating rapidly, but it is constrained from motion in 3D space by the conservation of angular momentum.
The Cartwheel Galaxy
A consideration of the Cartwheel Galaxy is of interest here because it has a number of points of correspondence to the above described dye drop experiments. The galaxy apparently is a planar toroidal formed by the collision of a small galaxy (the drop) with the center of a larger galaxy (the water surface) at <K>, <L>, <M>.
Gas and dust from the impact form the outer ring, a toroidal one-half a quintillion miles in diameter. This toroidal is the result of overrolling or underrolling, depending on the observer's point of view. The near radial symmetry of the outer ring with the central galaxy suggests a face-on collision. The radial spokes of the Cartwheel Galaxy are slightly spiraled suggesting, possibly, a drawing out of rotating material from the impacted galaxy. The arms are fairly evenly spaced, and extend from the impacted galaxy to the toroidal. They may be analogous to the dye trail which extends from the drop impact to the first toroidal in the first experiment keeping in mind that we are dealing here with gas being structured in a vacuum rather than two interacting fluids. Fairly discreet bright star clusters in the outer ring may be seen at the ends of the spokes. I suggest they may have formed due to circumferential extension/compression, the result of the fluid salient mechanism. They also appear to be the result of star condensation due to higher gas density, caused by inpouring of gas from the spokes. There is also a discussion of spiral galaxies and fluid salients in [3].
As an aside, another round pattern of bright spots may be seen in Supernova 1987A where a collapsing star imploded and rebounded after the collapsing layers hit the core (Fig. 8a).
Fig. 8a. Hourglass shape of supernova’s released envelope. It increases with distance, time, and energy from the equator of the former star.
This seems to have produced a doughnut shape (toroidal) that is concentric, probably, with the star’s former equator, and oriented perpendicularly to its polar axis. This resulted in the formation of a ring of bright spots on the former star’s equatorial plane (see Google (Supernova 1987A Decoded) (Fig. 8b).
Fig. 8b. Supernova 1987A at various times showing the increase in brightness of the hot fingers aligned with its former equator suggesting locations of evenly-spaced fluid salients brought about by mutual gravitational attraction of collapsing matter that makes up the ring.
These spots are fairly well evenly-spaced suggesting circumferential extension/compression due to the fluid salient mechanism. Also, the explosion seems to have produced structures analogous to the double laminae bounded by the ring of later formed toroidals (Lim and Nickels (1992) (see above)). However, with Supernova 1987A, the two mirror image laminae are moving away from each other due to the explosion.
Polygonal Radial Hydraulic Jumps
As an aside, the diameters of smooth laminar circular hydraulic jumps presented by Bush and Aristoff (2003), become larger as the flow (Reynolds number) of a vertical jet of water, encountering a flat surface, increases <N>.
A further increase in flow induces a “turbulent” circular jump with a crown which I contend are really nascent fluid salients (Figure 3). A multi-polygonal regime can also be produced which does not exhibit turbulence. They apparently are produced by the interaction of the two superimposed, horizontal fluid laminae created in the system. The lower is formed by the vertical jet impinging a horizontal surface under the influence of gravity and pressure. It flows radially outward from its center. The upper is formed by gravity and rests upon the horizontal surface of the lower lamina. It flows radially inward towards its center. As the upper lamina overrides the lower, overrolling occurs at the step’s periphery. Because of the horizontal, planar structuring of the laminae, fluid salients are constrained to form within the step as the result of extension/compression caused by the energetic, inward overrolling of the upper lamina overriding the radially outward moving lower lamina.
It appears that the sharp angle that points out (from the center) between each “side” of a polygon is the location of a zone of retarded-reverse flow since the sides of the polygons shown in Figures 4-6 are convex and moving inward toward the center. They are salient-like in shape, structure and function. The clover regime in Figures 7-12 is caused by the production of overrolling fluid salients being forced radially outward by the increase in speed of the lower lamina. Here, the sharp angle that points in (towards the center) between each lobe of the jump, is the location of a zone of retarded-reverse flow. Therefore, the movement of the salients can be either toward the structure’s center in the polygon regime or away from it in the clover regime. The reverse is true for the intervening zones of retarded-reverse flow. The difference between the two regimes appears to be the thickness and strength of either lamina. The number of evenly spaced polygonal sides produced in the jumps appears to be controlled by the amount of inward overrolling at the step and peripheral compression of the jump, both a function of the jet’s strength and the operation of the fluid salient mechanism. This centripetal fluid salient structuring of the step also appears to result from a greater depth of the target layer, height of water in the step, and its proximity to the nozzle.
These polygonal radial hydraulic jumps are different from the bunched drapery pattern that a water surface affects as it flows down a circular drain as seen in some dam spillways (see <E> [8] www.youtube.com/watch?v=OBH4takFrT0)
North Pole Hexagon of Saturn
Saturn’s polar hexagon can be seen when its northern hemisphere is illuminated by the sun during its summer. Experimentation suggests that it forms in an area of turbulent flow between two different rotating fluid bodies having dissimilar speeds. It has also been proposed that the hexagon is a narrow jet stream which circles the north pole and that friction with slower clouds on either side of it creates eddies which push it into a wave-like shape as it goes around. This has been shown in an experimental study in which the eddies are very obvious <O>.
However, see also Saturn’s hexagon, Wikipedia, and “Saturn’s Unique Hexagon - YouTube”. These images show surface atmospheric gases over lower level gases, both rotating counter-clockwise. I suggest that the centripetal fluid salient mechanism might be responsible for the formation of Saturn’s hexagon. It would be caused by the centripetal influx of slowly poleward moving, upper level material over the lower central surface gases within the hexagon which are moving equatorward. This would create a constrained overriding of the upper gases similar to that caused by the underriding of the water from the vertically falling fluid jet seen in polygonal radial hydraulic jumps. This leads to the formation of the salients and pentagon seen in Figure 6 in <N>. The upper level gases would become centripetally compressed and form the hexagon. The jet stream, seen at the hexagon is probably caused, like the polar jet stream on earth, by the poleward motion of Saturn’s upper atmospheric layer over the lower and influenced by the Coriolis effect. In this scenario, Saturn’s jet stream could be merely passive and not be a cause of the hexagon. I suggest, also, that the formation of eddies in the experiment at <O> may be the result of the hexagon’s interfering with the circling fluid since the hexagon’s corners make room for the eddies. That is, the eddies are the result, not the cause of the hexagon’s formation.
As an aside, the geometric whirlpools described by Tomas Bohr of the Technical University of Denmark (see Google “Tomas Bohr Rotating Polygons”) look similar to the north pole hexagon of Saturn or the polygonal radial hydraulic jumps. That is, due to the formation of fluid salients, these polygons seem to form by the centripetal compression of the upper level fluid forced up the side of the container where it tries to override the centrifugal force of the underlying fluid as it moves toward the container’s center.
CONCLUSIONS
Much of what is presented here is quite disparate, but I feel that it is still a cohesive argument for the fluid salient mechanism. Add to it my discussions of other fluid salient phenomena and a unifying, comprehensive mechanism seems self-evident.
Questions, comments and criticism are welcomed and may be addressed to me at: gorycki@yahoo.com
REFERENCES
Biswas, D, 2018, Investigating the Perturbed Geometries of Vortex Rings in Free Fall Through Another Liquid: PDF [physics.flu-dyn] 21 May 2018.
Bush, J. W. M. and Aristoff, J. M., 2003, The Influence of Surface Tension on the Circular Hydraulic Jump: J. Fluid Mech., v. 489, p. 229-238.
Gorycki, M. A., 1973a, Sheetflood Structure: Mechanism of Beach Cusp Formation and Related Phenomena: J. Geol., v. 81, p. 109-117.
Lim, T. T. and Nickels, T. B., 1992, Instability and Reconnection in the Head-on Collision of Two Vortex Rings: Nature, v. 357 p. 225-227.
Lim, T. T. and Nickels, T. B., 1995, Vortex Rings. In Fluid Vortices Editor, Sheldon I. Green, Kluwer Academic Publishers.
Salvi, R., 2001, The Navier-Stokes Equations. CRC Press
Simpson, J. E., 1972, Effects of the Lower Boundary on the Head of a Gravity Current: J. Fluid Mech., v. 53, p. 759-768.
Thomson, J. J. and Newall, H. F., 1885, V. On the Formation of Vortex Rings by Drops Falling into Liquids, and Some Allied Phenomena: Proc. R. Soc. Lond., v. 39, p. 417-436.
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