Betat, André

Frette, Vidar

Rehberg, Ingo

English

Name not available

VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 19998Sand Ripples Induced by Water Shear Flow in an Annular ChannelA. Betat, V. Frette, and I. RehbergInstitut für Experimentelle Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany(Received 27 January 1999)We have studied experimentally the formation and evolution of sand ripples under water shear flowin a narrow annular channel. The amplitude of the sediment ripples vanishes at a critical value of theshear rate. Near this onset, a band of Fourier modes can be described by exponential growth functions.The growth rates vary linearly with the shear rate. On longer times the sand ripples display complexdynamics.PACS numbers: 47.54.+r, 45.70.–n, 92.10.WaOne of the most conspicuous examples of pattern forma-tion in nature is the ripples and dunes formed in sand, eitherby wind, flowing water, or surface waves on water [1,2].In particular, ripple formation induced by water shear flowis well known: everybody is familiar with ripples in mud-flats or on riverbeds. To describe pattern-forming systemslike these near onset the normal procedure is to linearizethe governing equations [3]. However, for sand there areno well-established equations to start from. In order tomake progress, one has to rely on experiments carried outnear the onset of pattern formation.To evade the lack of fundamental equations for sand,two main approaches have been followed in theoreticalwork. The first consists of various continuum descriptions,which can be valid only on length scales significantlylarger than the grain size. Under some model assumptionsit has then been possible to treat ripple formation driven bywater shear flow in terms of linear stability theory [4–8].In the second approach, one attempts to model explicitlythe grain-level events: the formation and motion of ripplesor dunes result from grains that are set in motion by thewater, follow some trajectories, and come to rest furtherdownstream. To mimic this motion, cellular automata areemployed, which seem to catch many of the qualitativefeatures of bedform formation [9–11].Most experimental studies have focused on questionsof importance in engineering [12,13] or on the longtimebehavior [14]. In Refs. [15,16], the early stages of rippleformation were considered. They determined the mostunstable mode, and the ripple length was found to beconsistent with estimates from linear stability analysis [7].However, by merely determining the most unstable mode,one cannot deduce whether linearized theories apply.Furthermore, all these experiments were performed inlong and wide rectangular channels. A serious limitationof these setups is that the system must be fed with bothwater and sand. This might disturb the ripple dynamics,especially close to the threshold where the system is verysusceptible.We have built an experimental setup that has sev-eral advantages compared to previous work: strict massconservation, rotational symmetry that allows unrestricted8 0031-90079983(1)88(4)$15.00use of Fourier techniques, quasi-one-dimensional geome-try, as well as high spatial and temporal resolution. Wefind that a linear description indeed is possible near on-set. While the ripple formation in this respect resemblesother pattern-forming systems, there are also striking dif-ferences that originate from the unusual properties of thesand: the ripple pattern does not disappear as the drivingis reduced, and more complex behavior arises when theamplitude is comparable to the grain size. Finally, wefind a series of intriguing features in the dynamics as theripples are driven into the strongly nonlinear regime.Our experimental setup [see Fig. 1(a)] consists of a15 mm wide and 58 mm high annular channel of 292 mmdiameter formed by a solid inner cylinder and an outercylinder. The channel is filled with glass beads 280–300 mm in diameter to a height of 38 mm. It is loweredinto an aquarium that is filled with water to a height of14 cm. The shear flow is generated with a rotor disc thatextends 4 mm from the top into the channel. The typicalroughness of the sand surface before the start of a run isabout 3 d, where d  290 mm is the mean grain diameter.Six cameras, controlled by a single frame grabber card,FIG. 1. (a) Side view of the experimental setup; (b) the sixcamera images of the sand-water interface are separated byvertical white lines. The images have been expanded in thevertical direction.© 1999 The American Physical SocietyVOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 1999are placed at angular distances of 60± and cover the entirecircumference of the channel. The spatial resolution isabout 0.7d.At a constant sampling rate fs  1Ts the images ofall six cameras are stored. The boundary between brightsand and darker water is detected, rescaled to account forthe geometrical aberration, and the six pieces are matchedtogether. An example is shown in Fig. 1(b). The linkbetween two images is indicated by white vertical lines.The water is flowing from the left to the right.The boundary curve hx, nTs, where x is the spatialposition and nTs with n [  the elapsed time, is Fourierdecomposed,Hx, nTs a021NXm1ΩamnTs cosµm2pxL∂1 bmnTs sinµm2pxL∂æ, (1)where L is the circumference of the channel and 2N 4112 the number of data points in the boundary curve.The mean ripple amplitude Aay is given by the squareroot of the sum of all Fourier amplitudes as Aay qPNm1a2m 1 b2m.Following earlier work on bedform formation (see, e.g.,Refs. [15,17]) we quantify the applied shear stress by theparticle Reynolds number Re  dutn. This impliesthat the relevant length scale is given by the mean graindiameter d, n is the kinematic viscosity of water at roomtemperature, and ut  n≠U≠y 12 is the unperturbed shearvelocity acting on the particles. ≠U≠y was estimated byassuming a linear profile for the fluid velocity Uy inthe middle of the channel.The left-hand part of Fig. 2 shows the typical longtimeevolution of the boundary curve. The abscissa gives theposition along the channel. In the right-hand part of Fig. 2the first 25 Fourier intensities a2m 1 b2m are plotted.On the horizontal axis of the right-hand part the modesof the Fourier series are given. The shear stress wasRe  2.40. The sampling time was Ts  400 s, but onlyevery eighth curve is plotted. Between two consecutiveboundary curves an offset of 8 mm is added; the offsetbetween two consecutive spectra is 2.5 mm2. After thenucleation of one ripple at a specific position there is afast generation of more ripples downstream of the firstone. This rather rapid dynamics can be observed untilthe whole system is filled with ripples. After about 6 hthe pattern reaches an “equilibrium state” with a rathernarrow spectrum. The ripples drift with a velocity ofapproximately 9 cmd, corresponding to an oscillationperiod of 27 h, in the downstream direction. As is shown inFig. 3, the mean ripple amplitude saturates upon reachingthis state. Interesting and worth noting is the breakup of aseemingly “stable” state after an elapsed time of about 50 has can be seen in Fig. 2(b): m  9 as the dominant modesplits into m  10 and m  6. As shown in the inset of 0  5 10 15 20 25Mode Number50 mm201224364860 0 30 60 90Elapsed Time (h)Position (cm)20 cmFIG. 2. Temporal evolution of the sand-water interface alongthe circumference of the channel (left) and of the intensities ofthe first 25 modes of the Fourier series (right) for Re  2.40.Consecutive data sets are separated 3200 s in time and areplotted above each other with a certain offset. The scale isgiven at the left side of each part of the figure.Fig. 3, a relatively rapid change in the mean amplitude isassociated with this event. The observation of an initialregime of rapid growth followed by a saturation of themean amplitude is in qualitative agreement with previouswork [14,15]. The time needed to reach the equilibriumstate becomes progressively longer as the shear rate isreduced towards threshold.To get deeper insight into the transient state before themean amplitude reaches the plateau we reduced the sam-pling time to Ts  14 s, the highest temporal resolutionwe are able to achieve. The early evolution of the ampli-tudes of three different Fourier modes is shown in Fig. 4.The experiment was performed at Re  2.40, the sameshear stress as for the experiment in Fig. 2. Figures 4(a)and 4(b) correspond to modes in the vicinity of the fastestgrowing mode, which was determined to be m  10, cor-responding to a wave number k  0.69 cm21 (see Fig. 502460 12 24 36 48 60Mean Amplitude Aav (mm)Elapsed Time (h)5.415.35.45.5AavFIG. 3. Evolution of the mean amplitude of the sand-waterinterface for Re  2.40. The inset is a partial enlargement.The solid line denotes the mean value obtained by neglectingthe first 12 h.89VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 19990.00.20.40.60.8A10 (mm)(a)0.00.20.40.6A13 (mm)(b)0.00.20.40.60 5 10 15A18 (mm)Elapsed Time (min)(c)FIG. 4. Three examples for the temporal evolution of theamplitude of a certain mode. The subscripts of each ordinatelabel denote the mode m in the Fourier decomposition [Eq. (1)].The arrows indicate the upper boundary for the exponential fits,which are given by the solid lines. Each figure refers to thesame value of the shear stress, Re  2.40.below). For Figs. 4(a) and 4(b) the amplitude could bewell adapted to an exponential growth function of the formA0 expst, and the growth rates s could be extracted.However, this procedure does not work well for Fig. 4(c).This might either be due to a limited temporal resolution ora systematic deviation from exponential growth for modestoo far above or below the fastest growing mode. Nev-ertheless, we find the exponential growth with time fora band of Fourier modes, as was theoretically predicted[7,18], can be experimentally verified for particle Reynoldsnumbers near threshold, and for modes near the fastestgrowing mode.Figure 5 shows the growth rates obtained for fivevalues of the particle Reynolds number. To comparethe data to the results of linearized theories, we fittedeach set of growth rates for the lower four values of Reto a parabola sk, e  t210 e 2 j20k 2 kc2, wheree  Re 2 RecRec and Rec is the critical particleReynolds number for the onset of ripple formation. Wefound Rec  2.18 6 0.04, kc  0.69 6 0.06 cm21,t0  22 6 3 s, and j0  0.16 6 0.09 cm. The solidlines in Fig. 5 are the result of the fit. The data forRe  2.52 cannot be fitted using those values. Thisindicates that for e  0.16 one is outside of the linearregime. The normalized wavelength ld  300 of thefastest growing mode is in line with the measurementsgiven in Ref. [16].90048120.0 0.2 0.4 0.6 0.8 1.0 1.2 1.430 15 10 8 6 5Growth Rate σ (10-3 /s)Wave Number k (cm-1)Wavelength λ (cm)FIG. 5. The growth rates as a function of the wave number forfive different values of the particle Reynolds number. On thelower horizontal axis the wave number k  2pmL is given,on the upper one the corresponding wavelength l  Lm. Theexperiments were performed at a shear stress of Re  2.28(), 2.30 (), 2.34 (), 2.40 ( ), and 2.52 ( ). The solidlines are parabolic fits to the lower four data sets. The dottedline is the extrapolation for Re  2.52, while the dashed lineindicates an independent parabolic fit to the upper data.Figure 6 shows the characteristic behavior of the equi-librium state mean amplitude (the plateau in Fig. 3) as afunction of Re. The error bars give the standard devia-tion of the temporal fluctuations. For the lowest three val-ues of Re the mean amplitudes are significantly reduced.Note that these values are in the order of a grain size, ascan be seen in Fig. 6 using the right ordinate. An evolu-tion towards a plateau similar to the one shown in Fig. 3is not observed here; instead we observe irregular fluctua-tions. In our interpretation these experiments were belowthe threshold for ripple formation.05102.0 2.2 2.4 2.6 2.80102030Mean Amplitude Aav (mm)Aav/dParticle Reynolds Number Re*2.2 2.539σ (10-3 /s)Re*FIG. 6. The mean amplitude (left ordinate in millimeters,right ordinate in units of the grain size) as a function of theshear stress. The error bars denote the standard deviation. Theinset shows the behavior of the growth rates for modes 8 (),10 ( ), and 13 (). The arrow indicates the upper data pointsincluded in the linear fit.VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 1999In the inset of Fig. 6 the growth rates of modes 8, 10,and 13 are given for a subrange 2.1–2.6 in the particleReynolds number. The solid lines represent linear fitsto these growth rates. The point of intersection withthe abscissa is interpreted as the instability thresholdfor the specific Fourier mode. The onset of instabilityfor the fastest growing mode (mode 10) is Re  2.17.Assuming that the lowest three values in Fig. 6 are belowthe threshold the onset must be in the range Re  2.14–2.18. Based on this agreement between values for theonset determined in two different ways we believe thatthe initiation of sand ripples can be described in terms ofa linearized theory. Additional experiments with differentgrain sizes indicate that the critical Reynolds numberincreases linearly with grain diameter. These results willbe published elsewhere [19].The breakdown of a linear description is evident forRe . 2.40 (arrow in the inset of Fig. 6), correspondingto e  0.11. Above this value there is a more pronouncedincrease in the mean amplitude, an indication that one canno longer neglect nonlinear contributions. From Re 2.65 (e  0.22) the curve again flattens. This might becaused by the finite height of the channel. Alternatively,this might be the beginning of the restabilization of theunstructured sandbed, which from previous experimentswas found to occur for high values of the Reynoldsnumber.When comparing our results to related studies, it shouldbe noted that both the theoretical predictions of Ref. [7]and the experimental results given in [16] apply to atiltable channel with an open water surface, and that thereis clear evidence that the situation for closed conduits isdifferent from the one observed in open channels [8].In conclusion, the evolution of sand ripples follows thebehavior seen for a wide class of pattern-forming systems.For the early stages in the evolution the ripple amplitudeis sufficiently small that single Fourier modes can betreated as noninteracting, which allows a description interms of linear stability theory. As the ripple amplitudeincreases one observes nonlinear effects, like saturationof the mean amplitude, “birth” and “death” of ripples,nonmonotonic and correlated evolution of the Fouriermodes. These phenomena will be discussed in more detailelsewhere. Thus, despite the striking similarity with betterunderstood pattern-forming systems [3], an approach usinga continuum theory is questionable: There must be acrossover for length scales comparable to the grain size.We thank G. Straßburger, M. Schröter, and M. A.Scherer for experimental help and clarifying discussions.We are grateful for support from Deutsche Forschungsge-meinschaft.[1] R. A. Bagnold, The Physics of Blown Sand and DesertDunes (Methuen, London, 1941).[2] M. S. Yalin, River Mechanics (Pergamon Press, Oxford,England, 1992).[3] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65,851 (1993).[4] J. F. Kennedy, J. Fluid Mech. 16, 521 (1963).[5] F. Engelund, J. Fluid. Mech 42, 225 (1970).[6] J. Fredsøe, J. Fluid Mech. 64, 1 (1974).[7] K. J. Richards, J. Fluid Mech. 99, 597 (1980).[8] H. Nakagawa and T. Tsujimoto, J. Hydr. Eng. 110, 467(1984).[9] H. Nishimori and N. Ouchi, Phys. Rev. Lett. 71, 197(1993).[10] B. T. Werner and D. T. Gillespie, Phys. Rev. Lett. 71, 3230(1993).[11] W. Landry and B. T. Werner, Physica (Amsterdam) 77D,238 (1994).[12] D. B. Simons, E. V. Richardson, and W. L. Haushild,J. Hydraul. Div. Am. Soc. Civ. Eng. 88, 57 (1962).[13] G. Parker, J. Hydraul. Div. Am. Soc. Civ. Eng. 101, 211(1975).[14] J. H. Baas, Sedimentology 41, 185 (1994).[15] S. E. Coleman and B. W. Melville, J. Hydr. Eng. 120, 544(1994).[16] S. E. Coleman and B. W. Melville, J. Hydr. Eng. 122, 301(1996).[17] A. J. Raudkivi, J. Hydr. Eng. 123, 58 (1997).[18] B. M. Sumer and M. Bakioglu, J. Fluid Mech. 144, 177(1984).[19] A. Betat, Ph.D. thesis, Universität Magdeburg, 1999;A. Betat, V. Frette, and I. Rehberg (to be published).91

Sand Ripples Induced by Water Shear Flow in an Annular Channel

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Sand Ripples Induced by Water Shear Flow in an Annular Channel

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