PH YSICAL REVIEW LETTERS. 10 JULY 1995 ... Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania. 15260 ... Spectral analysis of H(x, y, t) reveals a power-law dependence on spatial scales that is ...
VOLUME
PH YS ICAL REVIEW
75, NUMBER 2
Hydrodynamic
Convection in a Two-Dimensional
X-1. Wu, Department
LETTERS
B. Martin,
10 JULY 1995
Couette Cell
H. Kellay, * W. I. Goldburg
of Physics and Astronomy, University of Pittsburgh, Pittsburgh, (Received 14 November 1994)
Pennsylvania
15260
A two-dimensional Couette cell is formed by drawing a soap film over the annular region between an inner rotating disk and a fixed circular hole. When the inner disk is concentric, a smooth laminar flow is observed for all rotation rates. However, when the inner disk is eccentric, the film thickness H(x, y, t) fluctuates in space and time. Spectral analysis of H(x, y, t) reveals a power-law dependence on spatial scales that is consistent with Batchelor's calculation for turbulent dispersion of a passive
scalar. PACS numbers: 47.20.Ft, 47.27.Cn, 47.90.+a
In the experiment reported here we examine twodimensional Couette flow by studying the variation in thickness H(x, y, t) of a soap film which is convected by the velocity field. It is argued that to a reasonable approximation H behaves like a passive scalar, i.e. , the variations in H do not react back on the velocity field. Our most significant finding is that the power spectrum is —(~H(k, t) )] ~ 1/k . Here H(k„kY,t) given by S(k) [= is the Fourier component of H(x, y, t), ( ) is the time average, and k = k + k . This simple result, predicted nearly forty years ago by Batchelor [1], turns out to be difficult to observe [2,3], as it should appear only at very small, i.e. , for k much greater than the viscous dissipation scale of turbulence, ld = 1/kd. Our findings are consistent with Batchelor's prediction over more than one decade of length scales. Couette flow is a paradigm for studying the transition to turbulence [4]. A 3D Couette cell consists of two concentric coaxial cylinders with fluid contained between them and with the inner cylinder rotating at angular frequency A while the outer cylinder is usually held fixed. The 3D Couette flow ceases to be purely concentric and develops toroidal structures or rolls when A exceeds a critical value A, . On increasing even further, the flow becomes somewhat random and ultimately turbulent. The rolls appear when the Reynolds number Re = R A/v exceeds a critical value Re, —100. Here R is the radius of the inner cylinder and v is the kinematic viscosity of the fluid. Two-dimensional Couette flow should be qualitatively different from its 3D counterpart in that fluid motion along the axis of rotation cannot appear, therefore the characteristic roll structure will likewise be absent. In our 2D Couette cell a soap film is drawn over the space between a thin plastic rotating disk (5 cm in diameter) and a circular hole (10 cm in diameter) made of the same material. The flow in the film is two dimensional, since the components of velocity v are almost fully confined to the plane of the film [5]. The soap solution is made of liquid detergent (3%), glycerol (10%), and distilled water. The kinematic viscosity v of the film, according to earlier measurements [5,6], is of the order of 0. 1 cm /s. ~
0
0031-9007/95/75(2)/236(4) $06.00
When the inner disk is concentric with the hole, the flow is stable. In fact, the film velocity is time independent even for rotation rates as high as 50 Hz, which corresponds to Re = 2 X 10 . This Reynolds number is about 100 times greater than Re„which lends support to a wellknown prediction [7], which shows that 2D Couette flow is linearly stable for all Reynolds numbers. Figure 1(a) is a snapshot of the thickness profile for the concentric flow at Re = 8000. The photograph shows the appearance of the film viewed in reflected white light. The different colors appearing here represent an interference pattern; light of wavelength A is strongly reflected when the film thickness H is an integral multiple of A/2. More interesting results appear when the rotating disk is off-center, as in Fig. 1(b). In this case a single counterrotating vortex is created in the large gap between the hole and the disk. Small particles deliberately injected into the film circulate periodically about the vortex and follow a kidney-shaped path. The coherent motion of the particles on scales down to 1 mm or less established that the flow, though quite chaotic, was not turbulent. The nonsteady circulation causes strong spatiotemporal fluctuations in the interference pattern within the vortex. Images such as the one shown in Fig. 1(b) are suggestive of the chaotic patterns produced by the stirring of differently colored, miscible fluids at low Reynolds numbers. This mixing phenomenon, sometimes called "Lagrangian turbulence, " appears even when the Reynolds number is so small that the nonlinear term in the Navier-Stokes equation can be dropped [8]. The low Reynolds number mixing experiments are described by the mathematics of mapping and the ideas of chaos in systems with a small number of degrees of freedom [8]. Even though our experiments are at high Reynolds numbers, the similarities to the chaotic mixing patterns suggest that an understanding of the observed thickness variations may also invoke such ideas rather than those associated with strong turbulence. In quantitative studies of flow patterns produced in eccentric 2D Couette cells, it is obviously important to measure the local velocity v (x, y, t) in the film. However, it is also interesting, as we will show, to study variations in H, which we write as a spatiotemporal
1995 The American Physical Society
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75, NUMBER 2
PHYSICAL REVIEW LETTERS
10 JvLv 1995
(b)
(a)
FIG. 1. Flow patterns for a concentric (a) and an eccentric (b) setup. The pictures are taken with a white, broad light source The film spans the annular region between The color seen in the picture is due to the interference of light of different wavelengths. a rapidly rotating inner disk and a fixed outer ring. The diameters of the disk and the ring are 5 and 10 cm, respectively. The rotation speed is approximately 20 Hz, which corresponds to a linear velocity of 3 m/s at the edge of the disk. The texture seen in the film (b) resembles that of chaotic advection of dye in a tluid. time- and space-averaged part hp plus the fluctuating part h(x, y, r) If h were t. o behave like a passive dye of concentration c(x, y, t), then h and c would obey the diffusion equation
Bc
+v Vc=DV c,
(1)
Bf where D is the diffusivity of the dye in the fluid. An extension of the work of Bruinsma t9] enables one to show that the thickness of the soap film is described by an equation that is rather similar to that of a passive scalar: Bh + v . Vh = —~V 4 h —hpV . v.
„-
The spatial derivatives in this equation are two dimensional. The constant sc plays the role of the diffusion term in the passive scalar equation. It is a function of both the viscosity and the surface tension of the film and has a magnitude of the order of 10 'P cm4/s. The appearance of V h, instead of the Laplacian, is connected with the fact that the fluid in the film is assumed to obey Darcy's law (velocity proportional to pressure gradient). The second term on the right reflects the fact that the film can fluc-
tuate in thickness, so that the two-dimensional divergence need not be zero. However, since all our measurements were carried out at the center of the large vortex where the liow is dominated by large scale shear, hpV v is much smaller than the convective term v . Vh, so we ignore it. It is important that fluctuations in H do not react back on the velocity field if indeed H is to act like a passive scalar. We therefore examine the equation of motion for v itself. The velocity equation for soap films is essentially the same as the 2D Navier-Stokes equation except that the pressure term is now replaced by a surface elastic term that is a function of h [5], i.e. , C2 Bv Vh + vV v. (3)
+v Vv=—
hp
Here C (= QE jphp) is the velocity of elastic waves on the soap film, and F and p are the elastic modulus and the density of water, respectively. In general, H is not a passive scalar, since fluctuations in H cause an elastic response in the film. Nevertheless, H will act like a passive scalar when the speed of the inner disk is so high
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that the nonlinear term v . V'v in Eq. (3) overwhelms the elastic term (C /hp)V'h. The ratio of these two terms, (U/C) hp/h, is like a Reynolds number which is of the order of 100. Here we have taken C 100 cm/s [10], 20 Hz, and the fractional U = 300 cm/s at A/27r change of the film thickness to be -5%. This estimate implies that H behaves essentially like a passive scalar when the soap film is driven vigorously. One way to study the thickness fluctuations h(x, y, t) is to form images of the type seen in Fig. 2(a) using monochromatic light rather than white light. In general, the intensity variation 1(x, y, t) is periodic with h as H changes through multiples of a half wavelength. However, for small thickness variations, h A, this interference phenomenon will not occur, and I(x, y, r) is proportional to h(x, y, t). Over the spatial scales probed in this experiment cm in Fig. 2), the thickness variations were estimated to be no more than several hundred angstroms, making it safe to take I(x, y, t) ~ h(x, y, t) [11]. In this regime, one can Fourier analyze I(x, y, t) to obtain the instantaneous Fourier components h(k„k~, t) Avera. ging the modulus
=
=
«
(-1
10
Hz
10 JULY 1995
~h(k„kY, t)~ over times long compared to the longest characteristic variation time of h yields the spectral density S(k, k~). This quantity characterizes the strength of the thickness fluctuations on length scales 1/k, and I/k~. A conspicuous feature of our 2D Couette flow is that despite large Re, the flow is not turbulent in the sense that most of the kinetic energy is not in the form of small eddies. The flow, even in the eccentric case, has certain laminar characteristics, evident by the striations seen in Fig. 2(a). The random stretching and folding of H(x, y, t) motivates a comparison of our measurements with dispersion of a passive scalar in the Batchelor viscousdiffusive subrange in a 3D turbulent fluid. Moreover, there are recent experiments which show striking similarities between the small-scale behavior of strongly turbulent flow and creeping but chaotic mixing of a passive scalar [12]. Batchelor [1] has analyzed the behavior of the concentration spectrum E, (k) of a passive dye c in strong and isotropic turbulence, where E, (k) is defined by the equation z(c ) = E, (k)dk. We are interested in Batchelor s prediction for k lying in the viscous-diffusive subwhere k is the wave number above range k~ & k & which fluctuations in c are rapidly dissolved by diffusion. Within this subrange, velocity fluctuations are heavily damped yet strong enough to produce a straining action on the concentration field. When Batchelor's argument is generalized to two dimensions so that E, (k)/2vrk S(k, kY), one expects that for isotropic turbulence S(k„kY)= A/k, where A is proportional to the scalar injection rate g and inversely proportional to Re /2. In our case, g presumably depends linearly on II, giving A ~ I/Re'/ . For our soap film, we use dimensional arguments to estimate that this behavior should occur in the wave number range bounded by k& = 5 cm and k = 200 cm ' for A/2~ = 20 Hz. It is clear from Fig. 2(a) that the thickness variations are far from isotropic, assuring that S(k„k~) will also it is be anisotropic. In deriving the equation 5 ~ assumed that the strain axes are randomized by the flow. In the actual experiment randomization occurs, but is As a result 5 is found to be anisotropic, incomplete. even at our maximum Re = 2 X 10 . For fluctuations of h that are only partially randomized by the flow, and k, k~ S(k, kY) will contain terms such as Consequently, in polar coordinates, the spectral density is given by
f k,
~
20 Hz
k,
35
Hz
k„k,
(a) FIG. 2.
(b)
Flow patterns in an eccentric Couette cell. The pictures (a) are taken at the driving frequencies II/2m = 10 to (upper), 20 (middle), and 35 Hz (lower), corresponding Re = 4000, 8000, and 14000, respectively. The size of the pictures is a small area, 1 X 1 cm, in the center of the large vortex. The right column (b) shows the corresponding time averaged power spectra, ln[S(k, k, )], at I), /2' = 10, 20, and 35 Hz. Displayed are contours of constant S. The long axis of the ellipse in the center of the graph corresponds to the k~ direction. The sharp peaks at 45 to the axes and the cusps along the axes in the contour plots are an experimental artifact due to finite pixel size; they contribute to S(k„k~) only for k
&20cm
238
(4)
f
Here the angular distribution function (0) is peaked at some angle Oo corresponding to the minimum strain direction. Any one-dimensional spectrum measured along for kq ~ a line of constant 0 again yields 5 (x k
k&k
.
In the experiment, the spectral densities S(k, k~) were measured in the center of the large vortex using a video camera under monochromatic light. To minimize motion-
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smearing, which affects resolution on small scales, a fast shutter speed of 10 s was used. The power spectra were calculated for each image using the fast Fourier transform with a Hanning window, and S was then obtained by an ensemble average. Figure 2(b) shows contour plots of In[S(k„,kY)] at Re = 4000, 8000, and 14 000. The axes k, and k~ are, respectively, perpendicular and parallel to the time averaged velocity gradient near the center of the vortex. They were determined at small Re where the principal axes are easily identified. We note that the contours of constant S appear to be approximately elliptical. Figure 3 is a log-log plot showing S vs k for Re = 8000, where the direction of k is along 0 = 45 . The generalized Batchelor prediction, Eq. (4), yields S ~ 1/k along this line. Drawn in the figure is a line with the slope —2. There is reasonable agreement between this theory and the measurements over a decade in k. This was found to be the case for all Re examined. If our measurements at the largest k were to extend into the range k k we would expect to find S(k) ~ exp( —i~k /n) with rx being the local strain rate. However, due to experimental limitations, S(k) cannot be measured reliably for large k. We were also unable to measure the dependence of A on Re because lighting conditions changed as A was varied. Examinations of S along other cuts with different 0 were also found to be consistent with the prediction of Eq. (4). In particular, the spectral density could be mimicked by a form S(k„k~)—1/(k, + yk ). Thus S(k, k~) ix f(H)/k with f(0) = 1/(cos 0 + y sin 0). This equation interpolates between the isotropic and anisotropic cases as y is varied. In this experiment y was found to be — 0.5. To summarize, we have studied two different flows in a 2D Couette cell. In the concentric case the flow is remarkably stable up to the maximum attainable Reynolds number Re = 10 . On the other hand, in the eccentric case, the film thickness shows spatiotemporal fluctuations.
induced
)
1010 1
09
108 107 106
00
105 104 10o
101 k
102
(1/cm)
FIG. 3. Scaling form of the spectral density S(k) along 0 = 45 . The solid line with a slope of —2 is a guide to the eye. The Reynolds number is 8000.
10 JUr v 1995
One finds that h(x, y, t) has the appearance and the behavior of a passive scalar in oscillatory, nonturbulent flow. A simple calculation, containing some strong assumptions, suggests that the thickness spectrum (~ h ) should behave like a passive scalar in the viscous-diffusive regime of turbulent flow. Measurements of this spectrum are in approximate agreement with Batchelor's prediction, generalized to the case of strongly asymmetric flows. Since the vorticity field in two dimensions obeys the same equation as a passive scalar, it remains an intriguing possibility that our observed patterns may reflect the dynamics of the vorticity field in eccentric 2D Couette flow. We thank R. Almgren, R. Bruinsma, J. D. Crawford, D. Jasnow, D. Lohse, C. W. Van Atta, and J. Wang for stimulating discussions. We have also profited from an exchange with J. M. Ottino and G. Metcalfe. This research is supported in part by the NASA under Grant No. NAG8-959 and the NSF under Grant No. DMR~
8914351.
*Present address: CPMOH, Universite de Bordeaux I, 351 Cours de la Liberation, 33405 Talence Cedex, France. [1] G. K. Batchelor, J. Fluid Mech. 5, 8 (1958). [2] M. Holzer and E. D. Siggia, Phys. Fluids 6, 1820 (1994). [3] O. M. Phillips, in Turbulence and Stochastic Process (Royal Society, London, 1991). [4] H. L. Swinney and J. P. Gollub, in Hydrodynamic Instabii ities and the Transition to Turbulence (Springer-Verlag, New York, 1985), 2nd ed. [5] Y. J. Couder, J. Phys. Lett. 42, 429 (1981); Y. Couder, J. M. Chomaz, and M. Rabaud, Physica (Amsterdam) 37D, 384 (1989). [6] M. Gharib and P. Derango, Physica (Amsterdam) 37D,
406 (1989). [7] P. D. Drazin and W. H. Reid, in Hydrodynamic Stability (Cambridge, New York, 1981), Sec. 17.2. [8] J. M. Ottino, in The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge University Press, New York, 1989). [9] R. Bruinsma, Physica (Amsterdam) (to be published). [10] The elastic constant F. in soap films is a strong function of surfactant concentration. For concentrations well above the critical micellar concentration, which is the case in our experiment, E drops sharply and is of the order of 1 dyn/cm or less. [See, for instance, A. Prins, C. Arcuri, and M. Van den Tempel, J. Colloid Interface Sci. 24, 84 (1967).] For our soap films with thickness of the order of a micron, the velocity of the elastic wave is C = 100 cm/s. [11] The small variation in h was established by making measurements in white light and analyzing the maximum fluctuations in color over the several mm region of the film, in which the power spectra of the thickness fluctuations were analyzed. Here the linear approximation further requires the measurements be made far from the maxima or the minima of the reflected intensity. [12] K. B. Southerland, R. D. Frederiksen, W. J. A. Dahm, and D R. Dowling, Chaos, Solitons and Fractals 4, 1057
(1994).
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