Dispersion Relation of Standing Waves on a Vertically Oscillated Thin Granular Layer

Transcription

1 Journal of the Physical Society of Japan Vol. 71, No. 11, November, 2002, pp #2002 The Physical Society of Japan Dispersion Relation of Standing Waves on a Vertically Oscillated Thin Granular Layer Akiko UGAWA and Osamu SANO Department of Applied Physics, Tokyo University of Agriculture and Technology, Koganei, Tokyo (Received April 15, 2002) An experimental study of standing waves on a thin granular layer, which oscillated vertically under atmospheric pressure, was made. We controlled frequencies f and amplitudes of oscillations as well as the layer heights h, and wave lengths were observed by means of a high-speed video camera. Atfirst and second critical acceleration, the granular layer forms standing wave patterns oscillating at half and quarter, respectively, of the excitation frequency. Glass beads of a different diameter were tested, and the data normalized by means of three different scales are compared. Scaling by particle diameter d is relevant near the onset of waves, whereas scaling of either by h or by g=f 2 (g: acceleration of gravity) is appropriate for fully developed waves. At higher frequencies, wave length has a tendency to saturate, reflecting the fact that the effective depth pertinent to the wave formation is confined to a region of a few layers near the upper surface. Dispersion relation is characterized by a power-law whose exponent depends on layer height and acceleration amplitude, which seems to be explained by a shallow water gravity wave of viscous fluid. KEYWORDS: granular layer, vertical oscillation, standing wave, dispersion relation, scaling, fractional exponent, viscous shallow water wave DOI: /JPSJ Introduction Pattern formation on a thin granular layer due to vertical vibration has been extensively studied since the pioneering work by Faraday. 1) Recently many experimental 2 10) and numerical 11 15) investigations have revealed wavy motion on the thin layer. In particular the dependence of the wavelength on frequency f was empirically given by ¼ min þ g eff =f 2 ð1þ by Melo et al., 2) Clément et al., 4) Metcalf et al. 5) experimentally, and by Aoki and Akiyama 11) numerically. This dispersion relation is expected when the standing waves in the granular layer are similar to the surface gravity waves of inviscid water. In the latter, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2h f ¼ tanh ; ð2þ 2 so that g=ð2f 2 p Þ for h, and ffiffiffiffiffi gh =f for h. The former seems to explain the f 2 dependence. The deviation from this dispersion relation at higher frequencies, which is closely related to the presence of a minimum wave length, is attributed to the finiteness of the particle. The minimum wavelength also raises a question on the onsetof standing waves. Melo et al. 2) and Clément et al. 4) showed that they occur when the layer is a few particle diameters thick (N 3). The saturation of the wave length has also been remarked by Melo et al., 2) Clément et al., 4) and Metcalf et al. 5) They showed experimentally that wave length first increases with layer thickness, and then saturates to a constant value at N 7, which results in the deviation of the dispersion relation in thicker layer from that mentioned above. Alternativelyp Clément et al. 4) suggested the dispersion relation = ffiffiffiffi N ¼ ðdþþg =f 2 ( ¼ 7:2 mm, g ¼ 1:05 m/s 2 ), whereas Bizon et al. 12) and Umbanhowar et al. 8) suggested a scaling using layer height, i.e. =h and f p ffiffiffiffiffiffiffi f h=g. In spite of these modifications f 2 dependence remains the same Recently fractional exponent of the dispersion relation is suggested. Sano et al. 10) shows s ffiffiffi! d / f d ; ð3þ g where 2 for lower frequencies but 2=3 for higher frequencies, whereas Umbanhower and Swinney 8) shows h / const þ f ffiffiffi h g s! ; ð4þ where ¼ 1:32 0:03. In order to explain this fractional exponent, the latter considered influence of air viscosity, static charging in non-conducting grains, small aspect ratios, and drag associated with the side walls. However, the first two of these seem to be irrelevant because similar dependence is shown by the experiment under evacuated container using well-treated grains. 5) The other two factors also do notseem likely, because a lotof wavelength measurements with different aspect ratios or different patterns (squares, stripes, etc.) give almost the same values, and also because the influence of the side walls is limited to very short distance due to strong dissipation. In spite of academic interest and industrial importance, the basic understanding of the physical mechanisms underlying the collective behaviors of this material is still inadequate. In order to clarify this mechanism, we shall concentrate on the dependence of wave length on the frequency f, amplitude a and layer height h. In this paper we show experimentally that the exponent depends on layer height and acceleration amplitude. Appropriate scaling of the vibrating granular layer that describes the mesoscopic behavior is discussed. 2. Experimental Apparatus We performed an experiment on the ripples on the surface of granular materials in a vessel, which is vibrated vertically

2 2816 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 A. UGAWA and O. SANO High-Speed Video Camera Function Synthesizer Power Amplifier Fig. 1. a, f Vibration Generator Experimental Apparatus. Glass Beads under atmospheric pressure. Dry glass beads of a diameter d ¼ 0:13 0:05 mm or 0:34 0:08 mm were placed to a given filling height h in a rectangular cell of horizontal cross section 91 8 mm 2 or mm 2. The container was mounted with its principal axis parallel to gravity on an electro-mechanical vibration exciter (EMIC, Model 513-B Shaker and a model 371-A Power Amplifier). The block diagram of our experimental apparatus is shown in Fig. 1. The vertical oscillation of the container z ¼ a sinð2ftþ was given, where the frequency f and amplitude a were specified by a function synthesizer (NF Instrument, Model-1915). We apply the prescribed frequency and amplitude abruptly to the layer, so that no hysteretic behavior was examined. No deformation of the bottom of the container was recognized. The observation was made by means of a high-speed video camera to which a close-up lens was mounted. These images were later reproduced and measurement of their contour was made. 3. Results 3.1 Example of patterns Example of patterns are shown in Fig. 2. The typical sequence of pattern changes which we found under frequency forcing of fixed amplitude a of medium values is as follows: initially flat surface disrupted into ripple patterns for acceleration amplitude 3 < < 7, where ¼ ð2f Þ 2 a=g and g is the acceleration of gravity. These patterns [Fig. 2(a)] repeated with 2T, where T (¼ 1=f ) is the period of external forcing. We call this pattern as f =2-ripple. When was further increased, the latter pattern disappeared and a new ripple pattern which repeated with 4T [ f =4-ripple, Fig. 2(c)] or spike pattern [Fig. 2(d)], in which a sharp peak appears in the otherwise valley region, was created. The relative positions of the spikes differ from the peaks observed at the ridges of the layer for lower (Clément et al. 3) ). Undulation was observed for thicker layers at higher [Fig. 2(e)]. In the following we shall focus our attention to the ripple pattern which repeated with 2T. 3.2 Time sequence of the layer deformation We show the time sequence of the position of the layer bottom and that of the vessel in Figs. 3(i) and 3(ii). The layer with ripples is pushed up by the vessel and moves in the positive z direction. The peaks decay by the gravity [Fig. 3(i)-(a)], due to layer of particles falling with velocity larger than those of the valley regions, where relative motion is suppressed. Next the layer detaches the bottom of the vessel, and exchange of peak and valley regions takes place during free flight[figs. 3(i)-(b)(c)]. Then the layer collides with bottom of the vessel and is pushed up, which makes peak heightmaximum [Fig. 3(i)-(d)]. The period from (a) to (e) is T ð¼ 1=f Þ. This process continues from (e) to (f) with an exception that the positions of the peaks are shifted by one half of the wave length in comparison to those from (a) to (d), which leads to a period of 2T for the pattern cycles. We show the vertical positions of the bottom of the container as well as the lower boundary of the layer in Fig. 3(ii). We divide the period between (a) and (e) into three regions: in the interval t ex from (a) to (b) exchange of peak (i) (a) (b) (c) (d) (e) (f) Z (a) (b) (c) (d) (e) Fig. 2. Typical patterns observed on vertically vibrated thin granular layer: (a) f =2-ripple ( f ¼ 25 Hz, a ¼ 2:17 mm, ¼ 5:46), (b) no wave ( f ¼ 35 Hz, a ¼ 1:75 mm, ¼ 8:64), (c) f =4-ripple ( f ¼ 40 Hz, a ¼ 1:70 mm, ¼ 11:0), (d) spike ( f ¼ 45 Hz, a ¼ 1:55 mm, ¼ 12:6) and (e) undulation ( f ¼ 55 Hz, a ¼ 1:30 mm, ¼ 15:8). Fig. 3. Changes of ripples observed on vertically vibrated thin granular layer: (i) Snapshots of the time sequence of f =2-ripples with f ¼ 27 Hz, a ¼ 2:01 mm, d ¼ 0:34 mm and h ¼ 1:70 mm. Solid arrows show the direction of vessel, whereas dotted arrows show the direction of granular layer. (ii) Relation between the vessel and the lower boundary of the granular layer; the alphabets (a) (f) indicate the corresponding phase, (a) ft ¼ 0, (b) ft ¼ 0:1, (c) ft ¼ 0:5, (d) ft ¼ 0:75, (e) ft ¼ 1 and (f) ft ¼ 1:75.

3 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 A. UGAWA and O. SANO 2817 and valley regions occurs; in the meantime t g from (b) to (d) peaks grow; and during the interval t d from (d) to (e) peaks decay. We found the necessary condition on the formation of ripples, i.e. appearance of ripples for free flighttime t g & T=2, which is similar to Clément et al. 3) 3.3 Dispersion relation The wave length of the ripples depends on f, a and h. Glass beads with different sizes were tested, and all the data are plotted in Fig. 4. The ordinate is the normalized wave length ¼ =h, whereas the abscissa is the normalized frequency f p ffiffiffiffiffiffiffi ¼ f h=g. These data points are best fitted by / f with 1:47 over the normalized frequency range 0:11 f 0:64. Dispersion relation for f =4-ripples is similar to f =2-ripples, which has nearly the same exponent but is shifted upward by Wave length seems to saturate athigher f. Closer look of the exponent with fixed acceleration amplitude reveals that the former depends on the latter, which is shown in Fig. 5. Smaller values of (shown by open squares ) correspond to the data in the region 0:64 < f in Fig Onset of waves and dependence of wave length on layer height As we mentioned in 3.3 wave length seems to saturate at higher f. In order to clarify this point, we examine the dependence of wave length on the layer height. The same data are normalized by particle diameter d and free flight length g=f 2, and are plotted in Figs. 6 and 7, respectively. Figure 6 shows the dependence of ^ =d on ^h h=d. The normalized wave length saturates to a constant value at larger layer heights, which shows that the effective depth pertinent to the wave formation is confined to a region of a few particle diameter near the upper surface. The data seems to be fitted by ^ ¼ C 1 ð ^h ^h c Þ for ^h h^ c. Here C 1 depends on frequency and particle size, but critical layer height ^h c seems to be the same within the accuracy of our measurement, which suggests that the scaling by particle diameter d is relevant near the onset of waves. On the other f /2-patterns f /4-patterns log(λ /h) ( ± 7) ( ± 2) log( f (h/g) 1/2 ) Fig. 4. Dispersion relation. Fig. 6. Dependence of the wave length on the layer height normalized by particle diameter d α Γ= a(2πf ) 2 /g Fig. 5. Dependence of on λ( f 2 /g) d=0.34mm f =20Hz f =25Hz f =30Hz h( f 2 /g) d=0.13mm f =20Hz f =25Hz f =30Hz f =35Hz f =40Hz Fig. 7. Dependence of the wave length on the layer height normalized by g=f

4 2818 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 A. UGAWA and O. SANO hand, has rather large variations with f, d and h. To make clear the latter point, we examine the dependence of wave length ~ f 2 =g on normalized layer height ~h hf 2 =g. Figure 7 is the results, which shows that all data points fall into a single curve ~ ¼ C 0 þ C 2 ~h exceptnear the onset region. Introduction of some constant C 0 reflects the presence of cutoff wave length near the onset of waves. This results also suggests that the scaling by g=f 2 is appropriate for fully developed waves. 4. Discussion We shall briefly consider the meaning of the normalization of. In 3.4 we have introduced two normalizations, which are associated with the choice of different length scale d and g=f 2. By combining the particulate aspect and macroscopic behavior mentioned above, we have either ~ ¼ ~ m þ C 1 ð ~h ~h c Þ þ C 2 ~h ; ð5þ or, g 1 ^ ¼ ^ m þ C 1 ð ^h ^h f 2 c Þ g 1 þ C 2 ^h ; ð6þ d f 2 d where ~ ¼ f 2 =g and ~h ¼ hf 2 =g, ^ ¼ =d, ^h ¼ h=d, ^ m ¼ g ~ m =f 2 d and ^h c ¼ g ~h c =f 2 d. The same dispersion relation (5) or (6) is described by ¼ ~ m f þ C f ð f 2 ~h 2 c Þ þ C 2 f ; ð7þ 2ð1 Þ if normalizations ¼ =h and f pffiffiffiffiffiffiffi pffiffi ¼ f h=g ð¼ h ~ Þ are used. In order to fit our data in Fig. 6 with eq. (6), we require 0 <<<1 and ^h c ¼ constant, the latter of which is associated with the particulate characteristics that the onset of waves is mainly determined by the number of layers (in our case it is between 2 and 3). By the assumptions of constant ^h c and weak dependence of ~ m on f, we can explain the variations of the minimum wave length ^ m with f,as well as systematic shift of the onset condition in terms of h ~ in Fig. 7 (i.e., hc ~ increases as f or d increases). When the layer height is increased beyond the critical value, wave length ~ increases firstly like ~h and then like ~h, although the transition between the latter two is not so clear. However by comparing the expression (7) with Fig. 4, we can immediately identify that the tail region (i.e., f > 0:64 with smaller exponent 0:5 in Fig. 5) corresponds to the third term of eq. (7) with ¼ðþ2Þ=2 0:75, which reflects the macroscopic behavior of thicker granular layer. Note that in the case of hydrodynamic dispersion relation on the inviscid shallow water wave, this exponent is 1=2. Difference of for the granular layer from that for the water wave will probably be remedied by considering the effective layer height in the former case. Clémentand Labous 15) also discussed the dispersion relation, based on their numerical simulation by event-driven method for hard spheres with velocity dependent restitution coefficient. They showed ^ ^h relation (Fig. 8 of ref. 15) similar to our results, and gave an empirical bestfitby a straightline. However frequency is not given so that quantitative comparison of their results with ours cannot be made. On the other hand, Umbanhowar and Swinney 8) also remarked the presence of a kink in the dispersion relation in f diagram, and they identified it as grain mobility transition, whose physical mechanism, using their statement, remains to be understood. The same may hold in our case, or it may be associated with strong dissipative nature of the system. In fact, if we take into account of the viscosity in the shallow water wave of depth h, the surface elevation and horizontal velocity u in the x direction are given by : Here we take a frame of reference which is fixed to the container, so that the vertical acceleration ð2f Þ 2 a sinð2ftþ is applied to the layer in contact with the bottom of the container. The latter, however, is described by a potential ð2f Þ 2 a sinð2ftþz, which is added to the pressure term. Then the equation of motion in the x direction is unchanged, and the external forcing serves only to maintain a stationary state in our formulation. The present treatment is different from other works, who introduced an effect of viscosty either by taking account of the diffusion effect that relaxes layer thickness gradient in the layer, 12) or by reducing the Boltzman Enskog equation for inelastically colliding hard spheres to continuum equation of motion. 13) Dispersion relation in our case is k 2! 2 ¼ gh i! ; ð10þ where p! ¼ 2f, so that / 1=f for f 1 and / 1= ffiffiffiffi f for f 1. In the present case k or! takes a complex value, and wave motion will attenuate without external forcing. The exponent 1=2 is very close to our data in the tail region. On the other hand we can not identify the 1=f 2 region in Fig. 4, which mightbe realized for very small f. The latter corresponds to very thin layers according to our normalization, in which velocity fluctuation of an individual particle becomes so large that coherent layer motion, if any, is easily destroyed. In the intermediate region, in which granular layer shows both microscopic behavior reflecting the motion of the constituent particles and macroscopic cooperative behavior, dispersion relation seems to be characterized by a certain power law. Comparison of the results of Fig. 4 with eq. (7) shows 1:47 and ¼ ð þ 2Þ=2 0:27 for overall frequencies in this region, although the exponent of which varies with acceleration amplitude. Note that the numerical value ¼ 1:32 0:03 for 3 shown by Umbanhowar and Swinney 6) seems to agree with our corresponding result, in spite of different experimental conditions, i.e. the former being obtained under evacuated container. The details on this process, as well as the relation to grain mobility, are left for our future investigation. Acknowledgement This work is partially supported by Grant-in-Aid for Scientific Research (C). ð8þ ð9þ

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