sand ripples in an oscillating annular sand–water...

PHYSICS OF FLUIDS VOLUME 11, NUMBER 1 JANUARY 1999

Sand ripples in an oscillating annular sand–water cellM. A. SchererInstitut fur Experimentelle Physik, Otto-von-Guericke-Universita¨t, Postfach 4120,D-39016 Magdeburg, Germany

F. Meloa)

Departamento de Fı´sica de la Universidad de Santiago, Casilla 307, Correo 2 Santiago-Chile, Chile

M. MarderCenter for Nonlinear Dynamics, The University of Texas at Austin, Austin, Texas 78712

~Received 19 March 1997; accepted 22 September 1998!

Sand ripples, familiar from ocean floors, are studied by oscillating an annular sand–water cell.Empirical rules for the onset and disappearance of ripples are uncovered and compared with simpledimensional arguments, and with prior experimental and theoretical work. In addition the dynamicalbehavior of sand ripples are investigated, and the existence of a secondary instability where smalland large wavelengths coexist in space is shown. Finally, by visualizing the fluid flow above thesand ripples the paper describes vortices which are an integral part of the ripple formation and showthat there is a strong coupling between two different types of vortices and the sand ripple patternitself. © 1999 American Institute of Physics.@S1070-6631~99!00601-7#

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I. INTRODUCTION

It is impossible to swim near a sandy beach at the ocside without being struck by the regular pattern of ripplusually a few centimeters in wavelength, that form onocean floor. These have been studied for a long time.

Gilbert showed in 18991 that a surface wave can creasand ripples, because the fluid—even in deep water—iscillating back and forth. This motion can be expressed byelliptical equation.2 The water particles move in ellipsesthe surface, and as one proceeds downwards, the veamplitude of the ellipses vanishes smoothly giving rise tsimple back and forth motion of fluid at the bottom. Darwi3

in 1883 initiated the experimental and theoretical effortmodel the natural situations which give rise to sand rippHe filled a cylindrical vessel with sand and water and pformed rotational oscillations with a jerking motion.

Darwin’s work was followed by that of Ayrton,4 whomade general observations on the development and behof ripples and the water motion above them. She showedripple formation is closely related to the appearance of vtices in the flow. Later Bagnold5 in 1946 studied the behavioof sand ripples quantitatively. By oscillating a pendulum bin still water he found a simple empirical expression for tcritical amplitude of water motion needed for the first distubance of a smooth sand surface. Additional experimeevidence6 suggests that there are two different mechaniswhich operate in a cycle of motion: On the one hand, pticles are trapped in the vortex structure when this strucbegins to be established and on the other hand, the partare carried far away from the point where they were pickup as the flow reverses.

a!Author to whom all correspondence should be addressed; [email protected]

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Manohar7 also reported on ripple formation, but indifferent geometry. Rather than driving water over stationsand, he oscillated a flat tray covered with sand beneathter. In this case, the relative motion of the water with respto the sand layer is equivalent to the one induced by aface wave but with an important difference: An additioninertial force acts on the sand particles.

Recently, Blondeaux and Vittori8 examined the timeevolution of vortices by numerical analysis of the vorticiequation and of the Poisson equation. They found thatoscillatory flow over a rippled bed produces a secondary vtex which influences the initial vortex structure. In additiothey showed that there is a route from a periodic behaviochaos in an oscillatory flow which follows the Feigenbauscenario.9 However, the theoretical aspect of the connectbetween those vortex structures and the process of sandtern formation remains to be understood.

In this paper, we study ripple formation in an annulcell. Although this geometry has some effect upon walength selection, phenomena previously observed in a odimensional channel are reproduced. The advantage oannulus is that it allows us to avoid end effects. We carrymeasurements of ripple onset and the restabilizationripples. Special attention is paid to the visualization of tflow and its relation to the sand pattern. We also reportsecondary instabilities occurring in sand ripples, namellong scale modulation of the amplitude and wavelength aits later evolution to a spatial period doubling bifurcatstate.

The remainder of this paper is divided into three setions. In Sec. II we describe the behavior of our system afunction of various experimental parameters. We measthe transition between a stable flat bed and the developmof vortex ripples and the point where ripples disappear duhigh erosion rates. These results are compared with eail:

© 1999 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp

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59Phys. Fluids, Vol. 11, No. 1, January 1999 Scherer, Melo, and Marder

work. We find that our results are in reasonably good agrment with a prediction of Taylor.5 Moreover we show thathe restabilization of the pattern, i.e., the disappearancripples, is due to inertial effects.

The selection and the dynamics of the wavelengthdescribed in Sec. III. First, we explore how the wavelenof ripples changes when the frequency is increased graally. We observe that it remains constant in a certain fquency range, but increases drastically at a critical poAbove this critical point the wavelength is still well selecteThe transition depends strongly on the size of the frequestep and displays strong hysteresis. A further increase ofrequency results in a second transition after which the fistate of the system shows two different wavelengths coexing in space. This transition is also hysteretic and relatedsecondary bifurcation of the system.

In Sec. IV we report on the vortex structure in the fluand relate it to particle motion during the process of pattformation. This is done for different Keulegan–Carpennumbers~the ratio of amplitude of motion and ripple wavelength!, as well as for different Reynolds numbers. The retion between the flow and the sand pattern for a state ofcoexisting wavelengths is also investigated.

II. ONSET OF RIPPLE FORMATION

A. Experimental setup

We have carried out experiments in two sets of appatus; a linear channel and an annular channel. The linear cnel is better suited for visualizing the flow above the ripplsurfaces. However, for the purpose of measuring phasegrams, the annular apparatus is preferred because it avend effects which strongly influenced measurements in mprior experimental work in this field. Our work is carried oin a regime where the motions are essentially one dimsional; the implications of this choice are discussed laterthe text.

Our linear channel consists of a plate oscillated by aconnected to an eccentric wheel. The frequencyf and thetotal amplitudeA of these oscillatory motions can be variefrom 0 to 4 Hz and from 5 to 110 mm, respectively.

In order to test the importance of lateral boundary effewe have built four different linear channels for use on ooscillating plate. Their widths are 19, 25, 37, and 75 mwith the length in each case 400 mm. The height of the wain the tank is 160 mm and the height of sand in the channis 2.5 mm. At all frequencies tested, we observe that inchannels the wavelengths and amplitudes of the patternsthe same, except in the larger channels, where grain boaries can appear. These correspond to the interface bettwo domains of equal wavelength and amplitude, but shifby half a wavelength in the direction of the flow. We coclude that the channel width does not play any role inselection of the pattern when the frequency is large enouIn our experiment this is the case whenf is larger than 0.5 Hzin a channel of 20 mm width.

Since all experiments in a linear channel have to dwith the problem that after a few cycles sand is carried frthe middle of the tray to both ends, where it finally drops o


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we have also developed an annular geometry, where nois lost and such end effects are avoided. All our measuments referring to the development and behavior of sripples are performed in the annular channel. The linchannel is only used for the flow visualization describedSec. IV.

The heart of this second apparatus is a sand–waterwhich consists of two concentric cylinders fixed to a bottoplate and closed by a cap. This forms an annular channewhich particles and water are located~Fig. 1!. The sand–water cell is linked with the same rod described previouslyoscillate the cell. A conical mirror, which produces a 4projection of the circular side view, is used for visualizatioof the time evolution of the whole pattern. This side vieprojection is observed from above with a video camera. Tentire apparatus is mounted on a tripod to allow leveling

The width of the annular channel is 20 mm. The oudiameter of the outer cylinder is 178.5 mm, and the cylindare made of Plexiglass whose walls are 12.5 mm thick.sand we use spherical glass particles with a diameterD,varying from 0.001 to 1 mm. However, because cohesplays a significant role in the case of very small partic~they stick together! and because for larger particles the sadiameter exceeds the boundary layer, we only take intocount measurements which were done with the followmean particle diameters: 0.06, 0.11, 0.2, 0.3, and 0.4 mThe density of this material is 2.5 kg/l .

B. Experimental results

In this section we describe the primary transitions of tsystem in a phase space defined by the driving frequencf,the amplitude of motionA, and the particle sizeD. All mea-surements presented in this section are taken in the ann

FIG. 1. Side and top views of oscillating annular sand–water cell.


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60 Phys. Fluids, Vol. 11, No. 1, January 1999 Scherer, Melo, and Marder

channel. All quantities that result from our measuremeand which are presented and discussed after this sectionbe expressed in terms of four dimensionless numbers.10 Thedimensional quantities involved are the fluid densityr, thekinematic fluid viscosityn, the particle densitys, the particlesizeD, the total amplitudeA, the frequency of motionf, andthe gravitational accelerationg.

Combining these quantities we obtain the four followidimensionless numbers:

Fd5pA f@~s21!gD#21/2, ~1!

Rd5A~p f /n!21/2, ~2!

Rd5pA f Dn21, ~3!

s5s/r, ~4!

whereFd is Froude number,Rd is the Reynolds number,Rd

is the particle Reynolds number, ands is the ratio of sedi-ment to fluid density.

In a first experiment we keepD constant and explore thdevelopment of sand ripples by changingA andf. Starting ata constant amplitude and low frequencies we obtain a stflat bed. In Fig. 2, the view from above of the sand–wacell refers to the annulus around the closed circle, whichsurrounded by the outer cylinder of the cell and finally by t45° projection of the circular side view. As the frequenincreases in small steps we observe that at a certainquency there is a sharp transition between no movemenparticles and the development of vortex ripples as showFig. 2. We call this frequencyf a . At this critical frequencygrains start to move back and forth over the surface. Tlength of the grain path is short; about ten times smaller tA, but quickly increases asf rises pastf a . As time evolves,the rolling grains become organized and give rise to liwavy ridges a few grains high. This kind of structure, whiis called ‘‘rolling grain ripples,’’ has been observe

FIG. 2. Vortex ripples,D50.11 mm, A560 mm, andf 51.53 Hz. Theannulus around the closed circle refers to the view from above on the sawater cell. This is surrounded by the outer cylinder of the cell and finallythe 45° projection of the circular side view.


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previously5 and reported to be stable. However, in our carolling grain ripples never stabilize to a finite amplitudThey always constitute a transient prelude to the formatof full-fledged ripples. The vortex ripple structure arisesdescribed by Bagnold,5 suddenly and with finite amplitudeThe transition is therefore subcritical; when the frequencydecreased below the critical point the ripple pattern remareflecting a strong hysteresis.

An increase of the frequencyf beyond f a leads to asecond transition frequency,f e , where vortex ripples disappear and the bed restabilizes. At restabilization, many pticles move back and forth, their distance of travel is onorder of the amplitudeA, but no persistent structure is observed either in the flow or on the sand bed. By changingamplitude and determining the two transition frequenciesobtain a phase diagram in the plane~f,A!. This phase diagramis shown forD50.11 mm in Fig. 3. For small amplitudesA,f a , and f e are found at higher frequencies, while for largamplitudes the vortex ripple zone is shifted to smaller fquencies. Figure 4 shows the phase diagram forD50.11 mm as an ln plot. By using the best curve fit we o

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FIG. 3. Phase diagram in the plane (f ,A) for D50.11 mm.

FIG. 4. ln plot of the (f ,A) plane forD50.11 mm.


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tain: ln(f a)52.52620.646 ln(A) and ln(f e)52.71620.508 ln(A) for the appearance of vortex ripples and dappearance of vortex ripples, respectively.

It is interesting to ask how strongly the phase diagrdepends upon the particle sizeD. To check this dependencewe have carried out experiments with particle sizes rangfrom 0.001 to 0.5 mm. In Fig. 5, the experimental resultsD50.4 mm are seen. As before, all transition points lietwo different straight lines which define the borders of tvortex ripple zone. Compared with the results forD50.11 mm ~Fig. 4! it is clear that the area where vorteripples appear decreases and that there is a well-definedcal point at small amplitudes, below which no ripples forBelow the critical point, there is a continuous transition frostable flat sand beds to motion of sand particles withouttern formation. ForD50.4 mm this critical point can befound at A59.1 mm andf 54.2 Hz. We find that asDdecreases the amplitudeA of the critical point decreases anits frequencyf increases. Unfortunately, our apparatus limour ability to investigate directly the critical point for smaparticles, because due to mechanical constraints our sycannot be driven at frequencies higher than 4.5 Hz.

For D50.5 mm the amplitude of ripplesA is very smalland it is difficult to determine when ripples appear or disapear. Therefore we do not take these measurements intocount. Further experiments with larger amplitudes of motare necessary to explore this region. For particles of s0.001–0.020 mm we do not observe any ripples. Theseticles first behave in water like a fluid but stick together afsome hours and form a solid surface.

Analyzing our measurements, it follows that the vortripple zone is defined by two curves:

ln~ f a!5a2b ln~A!, ~5!

ln~ f e!5c2d ln~A!, ~6!

where b50.64860.012,c52.6560.03, and d50.49560.05 are constant and all independent ofD.

By plotting a vs ln~D! ~Fig. 6! we find:

FIG. 5. ln plot of the (f ,A) plane forD50.4 mm.


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wherem53.2360.10 andn50.3160.06.From ~5! it follows that vortex ripples appear at

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wherec15em525.262.7.This can be compared with the work of Taylor5 in which

he derives a condition for the initial motion of particles uder the influence of water which is oscillating horizontallTaylor distinguishes two different cases: laminar and turlent flow, respectively. He shows that the critical frequenf im where the grains first move can be written as follows

~i! laminar flow,

f im;~gg!2/3n21/3A22/3D2/3, ~9!

~ii ! turbulent flow,

f im;~gg!1/3n1/3A22/3D21/3, ~10!

whereg5s/r21.Comparison with our empirical formula shows that t

exponent ofA is in good agreement with Taylor’s result. Thempirical exponent ofD shows that our measurements fallan intermediate regime. However, the exponent 0.31 is cloto the laminar regime. This shows that the force acting onparticles due to the flow is rather a viscous type force. Ttypical value of the Reynolds number in our experimentabout 20. Thus, Taylor’s arguments suggest that the deslizing mechanism of ripple formation is caused by a comptition between the fluid force acting on the particles andopposing packing force.

For the disappearance of ripples we obtain:

f e5c2A20.5. ~11!

Herec25ec514.260.5.As in the case of initiation of vortex ripples, there mu

be a competition between the forces acting on the particlethe transition to restabilization. Because the Reynolds nuber is high, we cannot neglect inertial forces on the particHowever, if these inertial forces are out of phase with respto the motion, they are then able to remove grains fromcrests of the ripples in the part of the cycle where the visc

FIG. 6. a vs ln~D!.


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force is small. This can occur if the inertial force on a sphecal particlep/6s(2p f )2AD3 is larger than the packing forcp/6(s2r)gD3. Balancing both forces we obtain the criticfrequency for restabilization which varies likeA21/2 and isindependent of the particle size. Both facts are in agreemwith our empirical formula~11!. Therefore, we conclude thainertial effects are mainly responsible for the disappearaof the vortex ripples in our experiment.

We also observe ‘‘cross-pattern ripples’’~Fig. 3!. Herethe maxima of the ripples, which always point to the cenof the cell, are crossed by an azimuthal ripple of constamplitude.

This kind of pattern has been found in nature on thebottom of Tatado Beach, Izu Peninsula, Central Japa11

Rubin12 points out that cross-pattern ripples are producedan oscillatory flow. Because this kind of pattern only occuat small amplitudes, we conclude that it is an example ofwell-known brick-pattern ripples where bridges perpendilar to the direction of oscillation form in the trough betwetwo ripples. But these bridges are not linked from troughtrough as in cross-pattern ripples. The bridges are shifteda wavelength and are at the same position every sectrough.

As in the case of cross-pattern ripples, brick-patteripples were also experimentally found at small amplitudof motion by Bagnold,5 Manohar,7 and Sleath.13 Sleathclaims that brick-pattern ripples are formed by horsesvortices. Vittori and Blondeaux14 showed in a theoretical investigation that brick-pattern ripples can be explained byinteraction of two-dimensional and three-dimensional perbations which grow simultaneously.

Hara and Mei propose a centrifugal instability of an ocillatory flow as the mechanism for their formation.15 We donot explore this phenomenon in detail since we have decto focus upon one-dimensional aspects of the ripples.

III. WAVELENGTH DYNAMICS

For studying the behavior of sand ripples we usesame experimental setup described in Sec. II. We meathe amplitude of the oscillatory motionA, the amplitude ofthe ripplesA, and the wavelength of the ripplesL on theouter boundary of the outer cylinder. This means thatlength measurement refer to the projection on the ouboundary of the cell. The error of these measurement60.5 mm.

In this section we describe the dynamical behavior ofamplitudeA and especially of the wavelengthL of the pat-terns. We keep the amplitude of motionA and the size ofparticlesD constant in the well-defined vortex ripple zonwe chooseA560 mm andD50.11 mm.

Our first experiment starts with an initially featurelebed. We increase the frequency in small steps. As voripples appear, we wait for stabilization of the pattern ameasureA and L. Next we drive the system at a slightlhigher frequency. In Fig. 7 the wavelength and the amplituof the ripples are plotted as a function of the frequencycan be seen that besides the transition pointsf a and f e dis-


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cussed previously there are three additional critical frequcies:f b, f g , and f d .

As described in Sec. II, there is a transition from a staflat bed to the vortex ripple zone atf a . The wavelength isconstant in the areaf a< f < f b and equal to half of the totaamplitudeA of the oscillatory motion, while the amplitudeAof the sand ripples decreases slowly. Atf b , wavelength andamplitude both jump about 30%, remaining constanthigher frequencies.A decreases linearly between the bordeof f g and f d , althoughL is still constant. However, at thecritical point f d the system goes to a state in which twdifferent wavelengths, a small oneLs and a large oneL l

~Fig. 8! coexist in space. The amplitudeA vanishes atf e ,where the transition from vortex ripple zone to restabiliztion is located.

To study the dynamics of the adjustment of the wavlength near the sharp transitionf b , we create an initiallywell-selected pattern at a frequency in the interval@ f a , f b#

FIG. 7. Dynamics of wavelength and amplitude for increasing frequenin small steps,D50.11 mm, A560 mm.

FIG. 8. Spatial separation of small and large ripples,D50.11 mm, A560 mm, f 51.6 Hz.


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and drive the system at a frequency in the region@ f b , f g#,where a larger wavelength was found. Figure 9 showsinitial pattern, which was developed by starting with a fetureless bed and driving the system for 100 min at thequency 0.9 Hz. The initial wavelengthL i is well selectedand equal to 32.5 mm. Increasing the frequency suddenlf 51.35 Hz results in the disappearance of four ripples anfinal well-selected wavelength~Fig. 10!. L f is equal to 43mm and close to the value obtained in the previous expment. In Fig. 11, which shows the system after 100 s; nubers indicate where sand ripples will disappear, orderedtime. Figure 11 also illustrates that the wavelengths andplitudes of the ripples nearby the indicated positionsmodulated in space. We observe that any change inwavelength of the pattern is preceded by a long scale molation of the wavelength and the amplitude. Thus, the shchange in the wavelength observed atf b suggests that a long

FIG. 9. Initial state, t5100 min at f 50.9 Hz, D50.11 mm, A560 mm.

FIG. 10. Final state, atf 51.35 Hz, D50.11 mm, A560 mm, 60 minafter the frequency change.


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scale modulation of the pattern becomes unstable atf b . Thisphenomenon has many of the features of the Eckhinstability.16 Moreover it is important to note that the wavelength has to be a fraction of the total length of the annuchannel. This condition might play a significant role in slecting the final wavelength and amplitude of the pattern amight be a reason for sudden changes of both while chanthe driving frequency.

To understand why the system bifurcates at a critifrequency f d , we explore in more detail the area wherebifurcated wavelength exists. Therefore, we carry out rustarting always from the same initial condition which is chsen to be a short and well-selected wavelength. To getsame conditions, we drive the sand–water cell atf52.15 Hz, obtaining a featureless bed. After that we crea modulated bed with 21 or 22 ripples. Finally, the systemaccelerated to the frequency which we wish to examine.

First we measure the relaxation time of the system,plotting the number of ripples in our cell versus time forf51.4 Hz. Figure 12 shows that while there are machanges in the first minutes of the evolution, the system gto a stable number of waves after a time on the order ofmin. We wait at least 3 times this relaxation time to ensuthat the system reaches a stationary state.

Next we measure the local mean wavelength of the ptern in the [email protected],1.8# Hz. The results are presenteda three-dimensional plot in Fig. 13. From this diagram itobserved that the system has one well-defined wavelenglow frequencies. However, at a critical frequencyf d it splitsinto two separate wavelengths. In this case,f g and f d areclose together. This shows that the bifurcation point depeon the history of the system and that it could be related todecrease of the amplitude of the pattern observed atf g .

Near the critical frequencyf d the dispersion of the wavelength is larger than it was in the situation where thereonly one or two separated wavelengths. These fluctuatare the analog of the critical fluctuations observed in

FIG. 11. Long scale wavelength modulations,t5100 s, f 51.35 Hz, D50.11 mm, A560 mm.


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It is found that small and large wavelengths arranmostly in groups ofLsL lLsL l . This grouping can be disturbed byLsLs or L lL l defects or can be almost homogneous as seen in Fig. 8.

By plotting the ratior of the average short wavelengLs to the average long wavelengthL l as a function off, inthe region where the wavelength broadening is very smwe get a linear dependence ofr on f. This linear dependenccan be used to determine the bifurcation pointf d , wherermust be equal to 1. From the best curve fit we obtainr

FIG. 12. Number of ripples in the cell vs time after decreasing the frequefrom f 52.15 to 1.4 Hz.

FIG. 13. Wavelength distribution for different frequencies,A560 mm, D50.11 mm.


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53.4321.71f . For r 51 we obtain f d51.42 Hz. The de-pendence off d on the amplitude of motion and particle sizhas not been investigated.

IV. FLOW VISUALIZATION

In the previous two sections we studied the developmand behavior of sand ripples in an oscillating sand–wacell. Our results raised questions about the mechanismpattern formation and stabilization. Because of the complity of the behavior we observed, it appears necessary toplore the coupling between fluid motion and sand waves

By injecting Kalliroscope into the fluid we visualize thflow structure of the vortices over a rippled surface. Becathe structure is extremely complex at high frequencies,explore the vortex structure at low frequency. The mainsualization problems at high frequency are related to thethat while driving our system in the vortex ripple zone, tflow has small scale fluctuations and the thickness ofviscous layer is too small to observe any structure closethe modulated bed. In our opinion valuable insight canobtained by visualizing the flow at low frequency, whetime and length scales are accessible to our experimeresolution. Moreover, the results presented here can be cpared with recent numerical simulations of the flow.17 Un-fortunately, because of the high computational effortvolved, the numerical simulation for the actual conditionsripple formation have not been performed yet. Nevertheleresearchers such as Hara, Mei, and Shum17 agree that we canimprove our understanding of the problem by performiexperiments at low frequencies. At this point, to supportprevious discussion, we present one of our principal concsions: By observing the vortex structure at a frequency whvortex ripples actually appear, it is observed that the mflow contains basically the same features as the flow atfrequencies.

A. Experimental setup

To visualize the flow above the sand ripples we uselinear channel of length 430 mm, width 20 mm, and heig170 mm. This cell is open on both sides, so that the flmotion is not influenced by lateral walls. To prevent safrom dropping off at the ends of the cell, we build bordersshape of a perpendicular triangle. These borders are aside where they hinder outflow of the sand perpendiculathe bottom and 23 mm in height. The slope of the trianrelative to the bottom plate is 13°. Since end effects dramcally modify the instability threshold and wavelength of thpatterns, we do not carry out a quantitative comparisontween the results obtained here and those obtained in pous sections.

We use spherical glass particles of 0.11 mm in diameThe flow is visualized by injecting Kalliroscope into the wter and images are captured by a video camera whicstationary in the laboratory frame.

B. Experimental results

We study the flow over a rippled surface by running osystem at low Reynolds numbers. We choosef 50.05 Hz

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and amplitudeA567 mm. The pattern belongs to thKeulegan–Carpenter numbera50.71, wherea is defined asA/(2L). ~The pattern is created by driving the system af50.9 Hz andA567 mm. This results in a wavelength oL547 mm with an amplitude ofA58 mm.)

In Figs. 14~a!–14~f! we present relevant stages of halfcycle of motion. Figure 14~a! shows the image at the startinpoint of the cycle. Two different vortices can clearly be sewhich were created during the previous cycle. First, thera circular shaped vortex at the left side of the ripple, whwe call the ‘‘catapulted vortex’’ and second a different votex at the right side. We call this vortex the ‘‘n vortex,’’because it appears like a small Greek lettern which fell overto the right.

While the rippled surface turns from left to right, thenvortex moves above the crest pushing the catapulted voaway. At vt50.16p the catapulted vortex [email protected]~b!#. Then vortex also changes its shape: The higher wis contracted and now almost perpendicular to the bottThe remaining part of then vortex continues to turn over avt50.4p until the next ripple arrives at high speed, capulting this structure up its slope. This process creates avortex @Fig. 14~d!, atvt50.77p] which finally moves to thenext ripple crest@Fig. 14~e!, at vt50.9p]. Figure 14~f!shows the image at the end of the cycle and is the miimage of Fig. 14~a!. Again the same catapulted vortex anthen vortex are observed. This scenario is in agreement wthe argument of Hara and Mei,18 who predicted that an os

FIG. 14. Vortex structure during half a cycle,f 50.05 Hz, D50.11 mm, A560 mm, a50.71; ~a! vt50, ~b! vt50.16p, ~c! vt50.4p, ~d! vt50.77p, ~e! vt50.90p, ~f! vt5p.


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cillating flow over periodic ripples of finite slope developoscillating vortices. Moreover it is remarkable that the vortdynamics mirrors the scenario of the particle movement.sual inspection of images during ripple development shothat there are two bursts of particles as the ripple crestsing one-half oscillation of the pattern, namely atvt50.14p and vt50.74p. After the bursts the particles filthe trough as a cloud. By comparing images, it is obviothat vortex dynamics and particle dynamics are coupled.

Hence vortices are responsible for raising sand up theside to the crest. In opposition to this destabilizing effect tincreases the slope of the ripples, there is the effect of gity; grains fall down the hills if the slope is too steep. Thequilibrium between these effects leads to a well-selecstructure.

To summarize the dynamics of the vortices we hamade a movie which consists of 110 frames during one coplete cycle. There we look at the space–time evolutiontwo different horizontal cuts:~a! right at the crest and~b! inthe region of the catapulted vortex. In Fig. 15, we seespace–time evolution for these two different cuts. Itclearly observed that the phase of the motion of the rippand the motion of the vortex is shifted byp. This resultshows that there exists a mean flow by which the vortexadvected. This flow, induced by the bed motion, travelsposite to the ripple motion, in order to conserve fluid ma

Moreover we find a relation between the characterislength of the pattern and the vortex structure. It can beduced from Fig. 15 that the total distance traveled byvortex D ~in half a cycle! plus the total amplitude of theoscillatory motion is close to twice the ripple wavelength.additionD is also close to half of the wavelength which givL52A/3. In the case of Fig. 14, the vortex created at oripple enhances the slope of the next nearest neighbor ripi.e., it acts on a length scale of two wavelengths. If theconditions were not satisfied the vortex would erodestructure and thus adjust the wavelength.

It should be noticed that other scenarios are possible;instance for small amplitude ripples, vortex advection is nobserved. Small amplitude ripples refer to the pattern

FIG. 15. Space-time evolution of the vortex structure for one oscillationtwo different horizontal cuts:~a! right above the crest,~b! in the regionwhere the catapulted vortex develops,f 50.5 Hz, D50.11 mm, A560 mm.


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served in the experiments represented in Fig. 7, whereL5A/2 at low frequency. In this caseD50 and the wave-length is equal to one-half of the total amplitude of the ocillatory motion. For the same experiments, the transitfrom ripples with L5A/2 ~small amplitude ripples! toripples ofL equals about 2A/3 ~large amplitude ripples! sug-gests that the distance traveled by the vortex, which depeon ripple amplitude, selects the ripple wavelength.

Next we investigate the situation in which a small andlarge ripple coexist alternatively in space. We create sucstate by driving our linear sand–water cell atf 51.25 HzandA567 mm.

This gives us a pattern where ripples with small wavlength and amplitude and large wavelength and amplitalternate spatially. We describe relevant stages of rippduring one half cycle. The cycle is illustrated in Fig. 16 whthe ripples start to move from left to right. In Fig. 16~a! tothe left above each large ripple a large cell arises. Thesehave the shape of an overturned parabola, with the eye ocell built by the catapulted vortex. On the right-hand sidesee an vortex. Thisn vortex can also be found near smallripples, although at smaller sizes. But, no catapulted vois seen at the left-hand side of the small ripples. The thfollowing images show a familiar process:~b! the largenvortex destroys the catapulted vortex;~c! then vortex is cata-pulted by the small ripple;~d! catapulted vortex arrives at thcrest of the large ripple and feeds the eye of the parabcell.

The development of the vortex structure during this ohalf cycle and the existence of the catapulted vortex andn vortex shows that the process which leads on the one h

FIG. 16. Vortex structure for spatially separated small and large walengths, f 50.05 Hz, D50.11 mm, A567 mm; ~a! vt50, ~b! vt50.31p, ~c! vt50.69p, ~d! vt5p.


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to an unbifurcated state and on the other hand to a bifurcstate is similar. In spite of this, there are some differencethe unbifurcated state: First, a large cell exists, whichcaused by the catapulting of then vortex by the small ripple.Second, a large ripple cannot create a catapulted vortexlifting the smalln vortex. From this observation we concludthat the lifting of the largen vortex limits the dimensions othe small ripple. Because the large ripple is not forcedcatapult an vortex it has a larger dimension. This means ththe n vortices control the dimensions of the ripples. Againfeedback effect is responsible for a stable pattern. Howethis observation does not explain the onset of the bifurcat

Blondeaux and Vittori9 showed theoretically that therexists a transition process in an oscillatory flow overrippled surface which leads from periodic behavior to chaThey observed that an increase of the Reynolds numbesults in an infinite sequence of period doublings of the vtical velocity above the ripple crest which follows thFeigenbaum scenario.

In their model two parameters are fixed: First, rippslope e50.17, which is defined ase5A/L and second, adimensionless wave numberk50.19 wherek52pd/L andd5@n/(p f )#21/2 is Stokes boundary layer thickness.

To examine the period doublings, they increasedReynolds numberRd , which is defined by~2! and measuredthe vertical velocity at a distance 2.25 timesd above theripple crest. They found that a first pitchfork bifurcation apears atRd514, a second atRd532, and a third atRd

535. For values larger thanRd537, they predict chaoticbehavior of the system. We have carried out experimeunder the condition above, however the period doubling snario was not observed. We believe that in our experimesmall disturbances by end effects might completely obscthe phenomenon.

The question of whether or not a temporal route to chhas a spatial analog was also investigated by AmritkarGade.19 They showed that a wavelength doubling bifurcatican appear in coupled map lattices. We think that to gebetter idea of whether coexistence of small and large walengths in an oscillating sand–water cell is caused by suctransition, more experiments in cells of very large aspratios have to be carried out. In a linear channel end effemight be too important for visualizing the second or thibifurcation. A large circular channel is more suitable for thpurpose.

V. CONCLUSIONS

We have studied the development and behavior of sripples in an oscillating sand–water cell. The mechanisinvolved in this pattern forming process were investigatedvisualizing the vortex structure of the flow above the ripplsurface.

In Sec. II we showed that the problem can be reducfrom two dimensions to one dimension by working in a suficiently narrow channel. By using a closed annular chaninstead of a linear open one we avoid end effects. Thislows us to explore different regimes more accurately andrelatively short time. We showed that the vortex ripple zocan be defined by two limiting borders. These limits are co

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sistent with simple predictions resulting from the equilibriuof forces acting on the particles and dimensional argumeFinally we predicted a critical point at small amplitudesmotion and high frequencies, below which no ripples dvelop.

In Sec. IV we focused on the dynamical behavior of twavelength. We showed that while exploring the vortripple zone by increasing the frequency in small steps,get three new transition points. The first is a sudden increin wavelength and amplitude of the ripples, a linear decrein the amplitude and finally, we obtain a critical frequencwhere there is a spatial separation between small and lripples. We relate these critical points to long scale modutions of the pattern; they become unstable, destroying scells and allowing a final adjustment of the wavelength.

By visualizing the flow above the modulated bed, wshowed that these transitions are also related to the adveof the vortices. We suggested a resonance condition betwthe amplitude of motion, the ripple size, and the distantraveled by the vortex as a generic mechanism responsfor wavelength selection. When the resonance conditionot satisfied vortices erode the structure, modifying its walength. Finally, the coexistence of small and large cells inpattern is associated with a transition from small wavelenripples to large wavelength ripples.

ACKNOWLEDGMENTS

Most of this work was performed at the Center for Nolinear Dynamics at Austin~CNLD!, Texas. F.M. and M.A.S.would like to thank CNLD for its hospitality. We owe mucto P. Umbanhowar, G. Lewis, D. Holland, and all the mebers of the German exchange student group 1992/93 fBayerische Julius-Maximilians-Universita¨t Wurzburg forstimulating discussion. The work of F.M. was supportedpart by DICYT, Direccion de Investigacio´n Cientıfica y Tec-


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nologica de la Universidad de Santiago de Chile; M.Aacknowledges the Deutsches Bafo¨G-Amt for financial sup-port.

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