V O R T T C K S TN H Y D R A U L I C S *

By Eiizo Levi1

ABSTRACT: Eleventh Hunter Rouse Lecture recounts the writer's 40 years of the­oretical and experimental researches concerning the inception, development, and damaging action of vortices. It was found that a vertical vortex is normally formed in an unstable environment, in the presence of a diverging or converging flat jet that subsequently feeds it by momentum transfer. A tentative explanation of the intermittency of swirls at intakes was obtained proving the theoretical viability of vortices whose intensity varies more or less periodically along their axes. The in­spection of intermittent vortices that develop spontaneously behind a weir or under a sluice gate led to the establishment of a universal Strouhal law, able to predict the frequency of the oscillations that can be excited within a restrained fluid body by a free current flowing along it, and, as a consequence, the frequency of vortex shedding and the structural vibrations that this phenomenon may induce. Appli­cations that define the causes of revetment damages in hydraulic structures are made.

The announcement of this award bestowed on me by the American Society of Civil Engineers has moved me deeply for many reason: First, because it is named after Hunter Rouse, whom I acknowledge and respect as one of the great men of American hydraulics; second, because of your kindly ap­preciation of what I have had the chance to do; and third, because you crossed the border to award a non-American for the first time. Mexican hydraulics owes much to American hydraulics, from the friendly advice on the design and building of hydraulic structures given to us by the experts of your Bu­reau of Reclamation to the excellent teching and warm hospitality that your universities and research institutes have offered to many of our best grad­uates. Let us now add this award as one of the reasons to be grateful to our American colleagues, especially the Hydraulics Division of the American Society of Civil Engineers, to whose assiduous activity on behalf of our professional field all of us are indebted.

INTRODUCTION

I had my first encounter with vortices about 40 years ago. I was then working at the hydraulic laboratory of the Mexican Ministry of Hydraulic Resources. We had been asked to find the cause of the failure of the re­vetment of the stilling basin that received the discharge of a hollow-jet valve at the Alvaro Obregon Dam, in Sonora, Mexico. In the middle of the basin, 52 m long and 5.50 m deep, a big hole almost 1 m deep had been scoured in the reinforced concrete floor. A 1:20 hydraulic model of the valve and the stilling basin was built in the laboratory and supplied with a set of suit-

"Presented at the July 30-August 3, 1990, ASCE Nat. Hydr. Engrg. Conf., held at San Diego, CA.

'Prof. Emeritus, Universidad National Autonoma de Mexico, Cactus 14A, Ped-regal De Las Fuentes, 62550 Jiutepec, Mexico.

Note. Discussion open until September 1, 1991. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on No­vember 7, 1990. This paper is part of the Journal of Hydraulic Engineering, Vol. 117, No. 4, April, 1991. ©ASCE, ISSN 0733-9429/91/0004-0399/$1.00 + $.15 per page. Paper No. 25675.

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FIG. 1. Transport, Settlement, and Swirl Zones in Model of Stilling Basin of Al­va ro Obregon Dam

FIG, 2. Grate of Staggered Cylinders Adopted to Make Flow Uniform and So Avoid Vortex Formation

<l = *Om% Without dlatlpator

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FIG. 3. Curves of Velocity at 1 m (Prototype) Level with and without Grate

ably distributed piezometers. The model was operated with various heads and discharges, but no suspicious over pressures or under pressures were detected.

We scattered fine sand on the water surface of the stilling basin to locate zones of transport and deposit along the bottom. As we expected, pure trans­port occurred in the first half of the basin and settlement in the second (Fig. 1). Between these zones, at the place where the damage occurred, unex­pected horizontal swirling motions appeared. Measurements performed at a distance of 1 m (prototype) from the bottom revealed, at an approximate distance of 16 m from the valve, a sudden halt of the flow, followed by upstream velocities. It was realized afterward that the tank had not been thoroughly cleaned before using and that many fragments of concrete had not been removed, in the hope thai the current would carry them away. Evidently, these fragments were dragged by the swirling motion and had scoured the revetment.

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Our problem was to avoid the occurrence of vortices. This was not easily solved because we were not allowed to submerge the valve because it might then stimulate cavitation. The adopted solution was to split the jet through a grate of staggered cylinders (Fig. 2). Fig. 3 shows how the initial curve of velocities at the 1 m level (dotted line, for a discharge of 40 m3/s) was changed by the presence of the grate (broken line, for 40 m3/s; full line, for 75 m3/s) (Levi and Gonzalez-Escamilla 1957).

VORTICES FROM DIVERGING FLAT JET

The problem at Alvaro Obregon Dam was solved, but the cause of the vortex motion remained unknown. What bewildered us more was the fact that the swirls had a vertical axis instead of a horizontal one. The latter appeared to be more plausible because the phenomenon could be explained through the simple mechanism of frictional drag by the overhead jet. But what could have been the source of vertical vortices? Leonardo da Vinci wrote, "Frequently, when a wind meets another, forming an obtuse angle, both swirl and stretch together in the form of a huge column." That is, the vertical vortex comes to life because of the mutual rubbing of diversely ori­ented currents; and the same explanation has been repeated again and again by the few people who attempted to explain the phenomenon. But in the present case the flow was practically symmetrical and counteracting flows did not appear to exist, just as in the drain swirls, which spring up in a quiet medium and can be ascribed to earth rotation only when they are of cyclonic dimensions.

About 20 years had to pass before we could observe a fact that offered a valuable hint to clarify the matter. I was preparing a lecture in which I wished to show my students the well-known vibration of a discharging nozzle set mouth-down against the floor. When trying to prepare the experiment, my assistant, Gabriel Echavez, noticed that if he slanted a nozzle so that it dis­charged at an angle, but still at the floor of a tank containing some water, a vertical vortex appeared in front of the nozzle. When repeating the ex­periment with an 8 mm diameter nozzle set at an angle of 45° beneath 15 cm of water (Fig. 4), we observed that no disturbance appeared at the time the flow began. After a few seconds, however, turbulent motions in the form of eddies turning in either direction could be perceived at the free surface. Fig. 5(a) shows their shadows. This eddy system was unstable: either the eddies mixed and mutually destroyed themselves, or one of them was rein­forced by pairing with others that turned in the same direction. In the latter case, the opposite weaker ones were suppressed until only one remained. The surviving eddy began swaying with conical motion, similar to a spinning top on a hard surface [Fig. 5(b)]. Finally, like the top, it raised itself little by little until it stood erect [Fig. 5(c)]. During the entire process, the foot of the swirl remained at the same position in front of the nozzle, apparently at the point where the jet acquired its maximum velocity before spreading. It maintained that position until the air that advanced within its central funnel contacted the jet. Then air bubbles were sputtered all around and the vortex was destroyed [Fig. 5(d)]. After a while, the process began anew.

Measurements substantiated that the jet spreads at about 90° and advances with a speed that is almost inversely proportional to the distance from the nozzle. Thus the jet, wholly flattened, maintains the same thickness every-

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FIG. 4. Vortex Formed over Diverging Jet Issuing from Slanted Nozzle

I N :•-.•

FIG. 5. Subsequent Phases of Life of Vortex Formed over Diverging Jet

where. Since the law of variation of the tangential velocities of the overlying vortex is the same, the similar behavior in the jet was interpreted as cir­cumstantial evidence that the gradient of kinetic energy in both jet and vortex tends to be the same (Levi 1972).

OTHER EXAMPLES OF VORTICES

Before inquiring more deeply into the energetic bond that should exist between the jet and the vortex, we wanted to know if similar circumstances are present in other cases. We made a drain vortex in a large tank containing

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a central orifice at the bottom. Being unable to quantify the velocities that were too low to be gauged by our measuring instruments, we submerged vertically a thin rod to which a set of silk threads had been fastened. Re­peating the plunge at different points, we detected the existence of an almost uniform bottom layer in which the threads pointed always to the orifice, and an upper layer in which they stretched tangentially. The layers were sepa­rated by a narrow intermediate layer where the direction of the threads varied progressively from the radial to the azimuthal direction. In other words, the vortex motion was found to be superimposed on the sinking motion of a convergent jet. A similar association occurs at pump sumps and at inlets, where a swirl arises within a quiet water body that lies upon a current that converges toward the intake. This discovery led us to ask ourselves if the harmful vortices at Alvaro Obregon Dam had been produced and maintained by the expanding flat jet, which resulted from the spreading of the round jet after it had been deflected by the floor of the stilling basin.

In the latter case the configuration appears to be inverted in relation to the previous instances: the jet should then stay overhead and the vortex be­low. This gave me the idea that tornadoes might have a similar origin. Tor­nadoes, as everyone knows, are always associated with a thunderstorm. They travel with the parent cloud, from which they normally hang. In typical con­ditions in the American Great Plains, the parent cloud is guided by a south­erly wind that veers to a westerly aloft at a 3-4 km altitude. An interesting point is that the tornado always develops at the right-hand side of the ad­vancing thunderstorm, never the left.

Now, let us recall that prior to the appearance of the tornado, the southerly cold wind runs against a body of warm air and lifts it up, drawing it near the westerly wind that blows aloft. As it contacts the latter, the warm air current veers and flattens, causing the formation of the expanding jet that creates and maintains the tornado (Fig. 6). This description also explains

FIG. 6. Supposed Mechanism of Tornado Formation

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why the tornado is always seen as descending from above. In the same man­ner, the drain vortex has its starting point underneath—in the zone of the converging jet—contrary to the widespread belief that it begins at the free surface, where a dimple usually appears. The point of origin was shown experimentally by Trivellato (1987) who, having put two drops of dye at different levels in the central part of a natural drain vortex, found that the swirling motion first develops in the lower drop.

Let us add another meteorological example. There is a difficulty in the prediction of tropical cyclones because in an environment conducive to their production, some disturbances will become so intense as to grow into cy­clones, while others, which are apparently similar, will not. Such pertur­bations are characterized by an associated flat, almost horizontal, layer that can be located in the upper troposphere and contains an outflow jet usually considered to be mainly produced and maintained by the atmospheric dis­turbance. It was found that when the disturbance is able to develop into a cyclone, the outflow jet diverges; for nondeveloping disturbances it usually takes the shape of a nearly parallel flow (McBride and Zehr 1981).

RELATIONSHIP BETWEEN JETS AND VORTICES

The dynamic relationship between the jet and the vortex still remained to be elucidated. The first idea that came to my mind was that a flat converging or diverging jet is endowed with a momentum flux that does not appear to be associated with corresponding external forces, while inside the vortex there is a force—the centrifugal force—that could well be related to the momentum flux of the jet. That is, the vortex could be fed by the jet through momentum transfer. It seemed expedient to check whether this relationship truly existed. I decided to use the slanted nozzle for this purpose. Since the nozzle discharge was constant but the intensity of the vortex varied contin­uously, we had to measure the maximum value of the latter. We painted a grid on the floor, illuminated it with a parallel beam of white light and observed the evolution of the swirl shadow, whose diameter had been pre­viously related to the intensity of the vortex. The idea was to detect the maximum diameter attained by the shadow for each case. The instability of the phenomenon, however, was such that we failed in our efforts to obtain a consistent result.

Thus we had to resort to a vortex stabilizer composed of guide vanes evenly spaced around a cylinder whose axis passed through the center of the jet. The bottom of the stabilizer was placed a short distance from the floor to avoid disturbing the jet (Fig. 7). A strong, steady vortex appeared at once. Later we discovered that the stabilizer allows the vortex to be formed when the nozzle is placed at 90° to the floor as well as when the nozzle is slanted (Fig. 8). This result could not have been attained without the stabilizer, be­cause the vertical nozzle tends to hinder the development of a coaxial swirl. The vortex thus formed differs from a drain vortex—it is associated with a source-like jet; the drain vortex is related to a sink-like jet. Fig. 9 exhibits the way in which this vortex changes with different depths. Fig. 9(a) shows the jet alone. Fig. 9(b-d) shows how the vortex appears, reducing its strength as the depth increases, since a larger mass must be included in the rotation. This vertical vortex device has subsequently been used for processes of water

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(a)

FIG. 7. Slanted Nozzle with Stabilizer

treatment—mixing, aeration, and flocculation (Levi 1975). The use of the stabilizer facilitated the measurement of the vortex strength

for different nozzle diameters, discharges, and water depths, as well as the evaluation of two parameters, q0 and r0 , which characterize the momentum flux of the jet and the centrifugal force within the vortex. Fig. 10 shows that these quantities are roughly proportional in the vertical nozzle system. Similar results were obtained for the slanted nozzle and the vortices that form behind a sluice gate across a rectangular channel (Levi 1972). These ex­periments confirm the energy connection of the vortex with the jet. Let us guess now how the vortex is started.

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M

FIG. 8. Vertical Nozzle with Stabilizer

I FIG. 9. Vertical Nozzle with Stabilizer; Subsequent Phases of Vortex Formation when Increasing Depth

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INITIATION OF VORTEX

There are vortices whose mechanism is easy to understand because it de­pends on friction. For example, friction causes the permanent vortex that occurs in a vertical slot cut in a channel wall. Friction is not the cause of the initiation of the vortices with which we are concerned, and therefore they are more difficult to understand. The following is the way I like to imagine it. When an expanding or converging roughly plane jet develops under a still water body, an unstable situation must occur because the kinetic energy of the still water is small compared to that of the jet. Although tur­bulence develops in the stagnant water, its energy is low compared to the energy of the jet. Only a suitable vortex can establish equilibrium, but its formation is not easy. A swirl has a sense of rotation, and the jet, being symmetrical, has no means to favor one direction over another. The delay in the vortex inception is due to the lack of a trigger to make it turn one way or the other. Only an intentional or fortuitous breakdown in the sym­metry of the established field of forces can start it. But when this happens, strengthening occurs at once until the vortex reaches an intensity commen­surate with the momentum flux of the jet.

INTERMITTENCY OF INLET VORTICES

Let us now discuss another topic, the intermittency of inlet vortices. As daily experience shows, it is unusual for such vortices to maintain a constant

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800

400

D = 61 cm

O d = 2 . 540 cm

A d - 2 . 0 6 3 cm

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FIG. 10. q0 as Function of r0 for Vertical Nozzle and Stabilizers of Different Di­ameters D

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intensity unless they are very strong. Sometimes they undergo small fluc­tuations; sometimes they appear, strengthen to a maximum, decrease, and disappear, only to reappear and develop again with the opposite rotation. It is a puzzling phenomenon, because it seems to imply a concomitant fluc­tuation of the available energy, which is far from the truth. Hydraulic trea­tises refrain from mentioning such behavior because there are no models able to justify it.

Recently, I found a clue for the interpretation of this phenomenon. In September 1988, an extremely strong hurricane, Gilbert, struck the coasts of Mexico. Beach erosion and floods resulting from it were so destructive that the Mexican Institute of Water Technology started a research program focused toward the improvement of our ability to predict hurricane arrival and diminish the damage. To acquire some understanding of the peculiarities of hurricanes as vortex motions, I examined diagrams showing vertical dis­tributions of maximum azimuthal velocities within Gilbert. I was impressed by the variation of velocity not only through time, but especially in relation to height.

We are accustomed to believe that a vortex is bidimensional; that is, that the particles lying on an ideal cylinder centered at the vortex axis revolve all with the same speed. Seeing that this was not the case with Gilbert, it occurred to me that perhaps other natural swirls possess a similar structure. Therefore, I decided that it was first necessary to investigate whether in the case of vortices a velocity that varies with height is compatible with the fundamental momentum equations of fluid mechanics. Starting from the equation of vorticity in steady flow, I tried to integrate it under the hypoth­esis that the velocity is simultaneously a function of the distance r from the vortex axis and of the ordinate z measured along the axis. The integration was unexpectedly easy. The vorticity appeared to vary sinusoidally with z, and a new integration showed the same to be true for the tangential velocity. As a function of r, it came about that the velocity could be expressed by means of two modified Bessel functions: one that approximates, for small values of r, the free vortex; and another that approximates the forced vortex, the combination of which composes the Rankine combined vortex. I con­cluded that a sinusoidal variation of the vortex velocities in relation to the height is perfectly compatible with the fundamental equations of fluid dy­namics. As a matter of fact, more complex vertical profiles may result as linear combinations of different sinusoids (Levi 1988).

Now, regarding the intermittency, let us suppose that an axially sinusoidal vortex is formed at an intake with its axis directed along the current. The particles of the vortex will follow helicoidal trajectories resulting from the composition of their rotational and translational motions. The vortex will progress with the main flow, and the free surface at the inlet will be suc­cessively intersected by its variable cross sections, giving the impression of an intensity fluctuation from a maximum to zero; then, changing the sense of rotation, to a minimum, and so on. The verification of this hypothesis requires laser measurements of the velocities at a fixed section of a trans­parent pipe when an intermittent vortex is visible at the inlet. I haven't yet had the opportunity to perform the test. Perhaps someone else will be in­terested in it. Meanwhile, I believe that the model could be used as a math­ematical approach for handling the intermittency.

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UNIVERSAL STROUHAL LAW

A last topic that I would like to discuss with you is the frequency of vortex shedding from streambeds and some unforeseen consequences resulting from its study. Many of us know the boils that sporadically appear on river sur­faces. They are the outbursts of vertical swirls ejected from the bottom of the bed, called kolks, that are responsible for the uplift of bed material that substantially contributes to the bulk of the sediment that is entrained by the current (Fig. 11). Analyzing boil measurements taken in the United States and the Soviet Union, Jackson (1976) concluded that the frequency of their emission showed kolks to be turbulent bursts from the wall, structures that have been studied exhaustively during the 1960s, because they are regarded as the most important contribution to the creation of turbulence within the flow. An experimental study performed in India (Rao et al. 1971) led to an unexpected discovery: the interval of time between successive wall bursts, besides obviously depending on the thickness 8 of the boundary layer, does not depend on an inner property such as friction velocity, as first suggested, but on an external one, the velocity U of the free stream. For the mean period of appearance of the bursts f, the tests gave the approximate formula f ~ 6b/U. Jackson discovered that almost the same formula seems to be valid for river boils when taking the river depth as 8, and inferred from this that the parent kolks are nothing more than turbulent bursts of the wall.

If you set a weir across a rectangular channel, you can observe the quasi-periodic formation of a pair of erect counterrotating vortices behind it, which, when attaining their maximum strength, stretch so much that they surmount the crest of the weir. Inspired by Jackson's paper, I decided to verify that those swirls could also be interpreted as turbulent wall bursts; since the weir swirls are fixed in space, they promised to be much more manageable for the study of kolk phenomenon than the unpredictable kolks themselves. Hav­ing produced the swirls in the laboratory with different channel widths, weir heights, and discharges, we were able to confirm that the value of TU/d (U being the approach velocity, and d the upstream depth) was close to 6 in this case also. Should we have concluded that they too are turbulent bursts?

What made me cautious about adopting Jackson's opinion was the great

FIG. 11. River Kolk and Boil

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difference in size among the huge kolks and weir swirls and the extremely small wall bursts. Was the identity of a frequency law enough to consider them different aspects of the same phenomenon? I tried another experiment: instead of a weir, a sluice gate was set across the channel, forcing the current to pass below. The mean periodicity of swirls formed at the corners was tested and compared with the upstream velocity and depth. Although they can't be considered wall vortices for the simple reason that they do not touch the wall, 6 as an approaching value for TU/d appeared again!

The result was puzzling indeed. Suddenly, I realized that if the period T is replaced by the frequency/, the foregoing relationship states that the value of the Strouhal number, fd/U is about 0.16. This reminded me of the result of well-known experiments by Roshko (1954). Having measured the fre­quency of alternate vortex shedding behind different kinds of bluff two-di­mensional obstacles, he found it to be characterized by fd/U ~ 0.16, pro­vided that one takes the wake width as d (Roshko 1954). Subsequently, I was able to find in the technical literature many other cases in which a Strou­hal number of 0.16 (or sometimes its double, 0.32) had appeared: fluctua­tions in wall cavities, wakes of round bodies and vortex-breakdown bubbles, flapping of plane jets and puffing of round ones, refilling of cavitation bub­bles, and revolution of autorotating wings. In those instances, U was the external or (in the last case) fall velocity, and d was the body or wake width (or in the case of round jets, the nozzle diameter) (Levi 1983). It was evident that we were in the presence of a Strouhal law of universal validity. But why?

The answer to this question turned out to be much simpler that what ap­peared at first thought. All those phenomena concern oscillations that take place inside a body of fluid whose motion is restrained, being produced by an external free current. Let us idealize such a body as a simple harmonic oscillator, with d its maximum displacement and / its frequency. The spe­cific mechanical energy of the oscillator is K^df)2; the available specific kinetic energy (due in our case to the outer current) is U2/2. Equating these values, one gets

fd 1 - = — = 0.159 (1) V 2ir

that is, the famous value 0.16. Local oscillations governed by this law, if convected by the flow, will look to a stationary observer as undulatory per­turbations whose wavelength according to Eq. 1 is

U \ = - = 2nd (2)

/ Evidence of the validity of Eq. 2 for surface and internal shock waves, as well as for dune and meandering wave lengths, has been detected (Levi 1983).

After the publication of these results in the ASCE Journal of Engineering Mechanics in 1983, a challenging discussion was presented by two Canadian oceanographers who objected to the universality of the proposed Strouhal law because it did not seem to be fulfilled by eddies and gyres generated by vortex shedding behind prominent land features in the Gulf and Estuary of St. Lawrence. In fact, the real periods observed by them appeared to be greatly underestimated from Eq. 1. For instance, the period of gyres (240

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hr) shed at the Gaspe Peninsula was calculated by the formula to be only 26 min. I suspected that the length taken by them as d (in this case 150 m) was incorrect. Solving for d in Eq. 1 and substituting T and U for the mea­sured values, I obtained the apparently exaggerated value of d = 82.5 km. But then, resorting to a map, I found that this is nearly the distance between Cape Gaspe and the facing Anticosti Island, an indication that all the body of water within the strait is set in oscillation. By similar geographical con­siderations the remaining cases presented in the discussion were also easily conciliated with the universal Strouhal law (Levi 1984).

OTHER APPLICATIONS OF STROUHAL LAW

The Strouhal law is useful in several applications of hydraulics. In fact it leads to predictions of the frequency of the vibrations that a water current can excite in a structure that hinders its motion. When this frequency is obtained by direct measurement, the computed width d points out which fluid body is the cause of the trouble (Levi et al. 1988).

About 20 years ago, a failure similar to the one at Alvaro Obregon Dam occurred, but on a much greater scale. The floor of the stilling basin of the main spillway of the Netzahalcoyotl Dam, designed for 6,000 m3/s, failed during the first two weeks of operation, with about half of that discharge (Fig. 12). The floor of the basin had been protected by huge concrete slabs with horizontal dimensions of 12 m X 12 m and a thickness of 2 m, each of them anchored to the underlying rock with twelve 1.25 in. (3.2 cm) steel bars. The slabs had been cast in situ and the joints between them packed with bituminous fill. As a result of the breakdown, most of the slabs were removed and overturned, and their broken anchorages showed signs of ten­sion failure. Previous evaluation of the uplift forces to which the slabs might be subjected, including the loads due to wave motion and local macrotur-bulence effects, did not predict any risk for the revetment as it had been originally designed.

103.7 m

meters ">ne removed

FIG. 12. (a) Longitudinal Section of Main Spillway of Netzahualcoyotl Dam; (b) Plan View Showing Damage at Floor of Stilling Basin

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Model tests revealed that the first to be lifted was an upstream slab. The failure was preceded by oscillations of the slab around an axis normal to the direction of the flow. The oscillation amplitudes increased with the dis­charge, but their frequencies did not appear to depend on it. The failure occurred suddenly, always responding to more or less the same discharge threshold. It was also proven that no failure would have occurred if the gaps between the floor slabs had remained closed. Finally, it was shown that this kind of failure is not associated directly with a hydraulic jump, but depends on the presence of a high-velocity jet underneath. In fact, the same kind of failure was found also likely to occur in any ordinary high-speed water chan­nel.

The experimental evidence concerning the fact that the uplift is preceded by increasingly strong oscillations pointed to the possibility of flow-induced vibrations and offered a good chance to check the previous conjectures. It was found that in addition to the presence of a high-velocity current over the slab, the existence of a thin layer of water under the slab was necessary and that the overlying jet and the underlying fluid layer be interconnected. Pressure oscillations develop in this layer with a frequency that appears to be predictable by the Strouhal law if U is taken as jet velocity and d as the length of the slab. These oscillations, whose intensity increases with the discharge, can cause the slab to vibrate with the same frequency; and the vibration creates zones of increasingly intense pulsating pressure alternately under the up- and downstream ends of the slab. As a result, it seems that the failure occurs as follows. When the discharge reaches a level able to force the upstream vertical face of the slab to protrude enough to produce a separation of the flow over nearly all of the upper face, an underpressure occurs under the full extent of the underside and pushes the slab up, thus enabling the current to overthrow it. When out of place, the slab disturbs the current and causes neighboring slabs to be dislodged. Because of the chaotic flow that ensues, more and more slabs are uprooted until some sort of equilibrium is reached. By then a great part of the floor revetment has been destroyed (Levi and Del Risco 1989).

CONCLUSION

Usually the engineer looks for the "how" and "how much" and the phys­icist for the "why." In effect, the engineer considers practical technological problems and applications to be primary. This is good and right because, even to those who are interested in basic research, it points out the most important topics and orients investigation. However, I am convinced that sometimes it is expedient to look, in addition to the "how" and "how much," for the "why" of things, since this can bring up results not only interesting by themselves, but also of practical usefulness.

APPENDIX. REFERENCES

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Levi, E. (1975). "Application of jet-driven vortices to water and waste treatment

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Levi, E., and Del Risco, E. (1989). "Search for the cause of high-speed-channel revetment failures." J. Performance Constr. Fac, ASCE, 3(2), 125-136.

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