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Long-wavelength modulation of turbulent shear flows
Physica D 174 (2003) 100–113
Long-wavelength modulation of turbulent shear flows Arnaud Prigent, Guillaume Grégoire, Hugues Chaté, Olivier Dauchot∗ CEA, Service de Physique de l’Etat Condensé, Centre d’Etudes de Saclay, 91191 Gif-sur-Yvette, France Received 31 October 2001; received in revised form 3 June 2002; accepted 10 June 2002
Abstract The reverse transition from turbulent to laminar flow is studied in very large aspect ratio plane Couette and Taylor–Couette experiments. We show that laminar-turbulence coexistence dynamics (turbulent spots, spiral turbulence, etc.) can be seen as the ultimate stage of a modulation of the turbulent flows present at higher Reynolds number leading to regular, long-wavelength, inclined stripes. This new type of instability, whose originality is to arise within a macroscopically fluctuating state, can be described in the framework of Ginzburg–Landau equations to which noise is heuristically added to take into account the intrinsic fluctuations of the basic state. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Wavelength modulation; Turbulent shear flows; Taylor–Couette experiments; Turbulent flow bifurcation
1. Introduction In spite of more than a century of theoretical and experimental efforts, the transition to turbulence in some basic hydrodynamical flows is still far from being fully understood. This is especially true when linear and weakly nonlinear analysis cannot be used. When the basic flow remains stable against infinitesimal perturbations, new solutions with no direct connection to it may arise, triggered by finite amplitude disturbances. In physical space, when such a sub-critical instability drives the system to turbulence, inhomogeneities often develop in the form of fluctuating domains separated by fronts [1]. In the plane Couette [PC] flow [2,3], the flow between two walls moving in opposite direction at equal speed, these domains, also called spots, have a rather complex spatio-temporal intermit∗
Corresponding author. E-mail address: [email protected] (O. Dauchot).
tent dynamics. In the contra-rotating Taylor–Couette [TC] flow [4–6], the flow between two independent contra-rotating coaxial cylinders, one observes, apart from spots, the so-called “spiral” or “barber pole” turbulence, commonly described as a single helical stripe of turbulence winding around the cylinders’ axis. Two major difficulties occur when studying this type of abrupt transition to turbulence. First, when triggering the instability with finite amplitude disturbances, one has to deal with the difficult issues of their reproducibility and the flow “receptivity”. Second, the turbulent domains often typically extend to the border of the system, so that confinement or side-wall effects must be taken into account. Here, we investigate both plane Couette and TC flow in the largest aspect ratio apparatus ever built. Decreasing the Reynolds number, we study the reverse transition from the turbulent to the laminar flow. In both flows, we observe a continuous transition towards a regular pattern made of periodically spaced,
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inclined stripes of well-defined width and alternating turbulence strength. In spite of its conventional aspects, the instability leading to this modulated turbulence is of a new type because it arises within a macroscopically fluctuating state. One consequence of this intrinsically noisy context is that the flow observed just above threshold is characterized by the frequent nucleation of domains of stripes of opposite inclination. Going away from threshold, the typical domain size increases to reach the system size, leaving a regular pattern. In a parallel numerical investigation, we show that a coherent amplitude equation description of this pattern can be given in terms of coupled noisy Ginzburg–Landau equations. After the presentation of our experimental setup, we first describe the typical patterns of interest here and how they take place in the phase diagram of each flow. We then turn to a more quantitative report of our experimental investigation of the TC flow when decreasing the Reynolds number from the homogeneous turbulent flow. Next, we detail our description in terms of noisy amplitude equations. A general discussion concludes the article.
2. Experimental setup Our TC apparatus [7] is made of two independently rotating coaxial cylinders of radii ri = 49.09 ± 0.005 mm and ro = 49.95 ± 0.005 mm. The useful length is L = 380 ± 0.1 mm, and the gap size is d = 0.863 ± 0.01 mm, so that the aspect ratio ΓzTC = L/d = 442 and ΓθTC = π(ri + ro )/d = 362 are large, and the radius ratio η = 0.983 is very close to 1 (see Fig. 1). The flow is governed by the inner and outer Reynolds numbers Ri,o = ri,o Ωi,o d/ν, with Ωi,o the angular velocities, and ν the kinematic viscosity of water. The flow is thermalized by water circulation inside the inner cylinder. At thermal equilibrium the temperature is uniform in space up to 0.1 K and does not vary more than 0.1 K/h. As a result, the accuracy on Ri,o is of the order of 3%. The flow is visualized using a “fluorescent lighting” technique [7] developed for the purpose of this study. The water flow is seeded with Kalliroscope AQ 1000 (6 m ×30 m ×0.07 m
Fig. 1. Schematic drawing of the TC (top) and PC apparatus (bottom).
platelets). The inner cylinder is covered by a fluorescent film and the entire apparatus is UV-lighted. The fluorescent film re-emits a uniform visible lighting, transmitted through the fluid layer: the more turbulent the flow, the brighter it appears. Images and spatio-temporal diagrams (temporal recording of one line along the cylinder axis) are recorded by a CCD camera. Two plane mirrors reflect the two thirds of the flow hidden to the camera so that the whole cylindrical flow can be reconstructed. A second TC apparatus with a slightly larger gap (ri = 48.11 mm; η = 0.963) allows us to perform one point laser Doppler velocimetry (LDV) measurements of the axial and azimuthal velocity components at mid-height and mid-gap between cylinders.
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Our PC apparatus [2] is made of an endless transparent plastic film belt (363.0 cm long, 25.4 cm wide, 0.15 mm thick) driven by two pairs of small guiding rotating cylinders and one pair of large rotating cylinders as shown in Fig. 1. In this study, the guiding cylinders are 1.8 mm apart so that the gap between the wall is δ = 1.5 mm. The belt is also guided by two glass plates 3 mm apart and strictly parallel to the walls. Using δ as a length scale, the spanwise aspect ratio is ΓzPC = 169 and the streamwise aspect ratio ΓxPC = 385. The entire setup is placed in a tank filled with water. The temperature is controlled within 0.5 K but the gap size cannot be controlled with an accuracy better than 0.1 mm, that is 6.5%. Accordingly, the accuracy on the Reynolds number is at best 7.5%. The flow is again seeded with Kalliroscope AQ 1000 and lighted on its whole width with a thin laser sheet. In a laminar flow, the reflected light is steady and rather weak, because of the averaged orientation of the flakes parallel to the laser sheet. In a turbulent flow, the reflected light intensity is larger and fluctuating. Images and spatio-temporal diagrams (temporal recording of one line along the spanwise direction) are recorded by a CCD camera. For the purpose of comparison between the two situations, we define the Reynolds number R equivalently in both flows as the ratio of the viscous time scale to the shear time scale. The length scale h is chosen to be half the gap, as usually done in the plane Couette flow context, so that R
PC
U h2 Uh = = , h ν ν
where U is the belt velocity, and R TC =
ri Ωi − ηro Ωo h2 (ri Ωi − ηro Ωo )h = . (1 + η)h ν (1 + η)ν
3. A long-wavelength pattern in Couette flows Figs. 2 and 3 display snapshots of each flow for decreasing values of the Reynolds number (internal Reynolds number in the TC case). The deceleration rate is as small as 0.1%R s−1 and the flow is let to stabilize before each snapshot is taken. At high enough
Fig. 2. Turbulent spots and stripes along path F (see Fig. 4). R TC = 391 (a), 368 (b), 340 (c), and 331 (d). Each picture displays a 360◦ view of the whole flow (38 cm height and 31.4 cm wide). The vertical lines are dued to the image reconstruction.
Reynolds number, both flows are homogeneously turbulent. Decreasing R, there is a threshold below which a periodic structure appears with two preferred opposite inclinations. The pattern occurs simultaneously in the whole flow and does not appear to be triggered by end effects. Close to threshold, nucleation of competing domains of both orientations occur. For lower R, a regular pattern is eventually reached after a transient during which domains, separated by wandering fronts, compete. The oblique stripes have a wavelength of the order of 50 times the gap. The pattern is stationary in the plane Couette flow case, and rotates at the mean angular velocity of both cylinders (see further) in the TC flow case. For even lower Reynolds number, the stripe pattern breaks down, leaving a spatio-temporally
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Fig. 3. Turbulent stripes in plane Couette flow. R PC = 393 (a), 358 (b), and 340 (c). Each picture displays a view of the whole flow (25.4 cm height and 57.7 cm wide).
intermittent regime of turbulent patches evolving in an otherwise laminar flow. The pattern was observed for 340 < R < 415 in the plane Couette flow, and in the so-called “spiral turbulence” region of the phase diagram in the TC flow (see Fig. 4). This phase diagram, obtained by following the now-traditional procedure introduced by Coles [4] is similar to the one obtained by Andereck et al. [6] for a different geometry (η = 0.878 and much smaller aspect ratio): the same regimes are observed, but the thresholds are shifted to higher R values and the sub-criticality is “enhanced” (the laminarturbulent coexistence, “INT” and “SPT”, regions are larger).
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The “spiral turbulence” flow described by Coles and Van Atta [8], later by Andereck et al. [6] is thus to be identified with one wavelength of the regular pattern observed here, in agreement with the azimuthal aspect ratio (Γθ = 55) of their apparatus. We indeed verified that for azimuthal aspect ratios as small as 20 no “spiral turbulence” takes place. Similarly, previous plane Couette flow studies [3,9] could only produce one or two turbulent “spots” taking sometimes the shape of an inclined stripe. As it can be noticed in Fig. 4 the path F defined by Ri = −ηRo , where µ = Ω0 /Ωi = −1, crosses the spiral turbulence regime, in contrast with smaller η experiments where spiral turbulence occurs only below this straight line. We take advantage of this opportunity to develop a full comparison with the plane Couette flow, since in this case both patterns are stationary and governed by only one control parameter. As shown in Fig. 5, the similarity between the two flows goes beyond the above qualitative description. Both patterns have the same Reynolds number range of existence and exhibit the same wavelengths at any R, the only difference being that the azimuthal wavenumber is quantized by the circumference in the TC case. The plane Couette flow experiment being rather difficult to control for such a small gap, we now investigate the transition in more details on data collected solely with the TC apparatus. We are nevertheless confident that most of our findings apply to both flows.
4. A modulation of the turbulence strength In most of the “modulated turbulence” region of the (Ri , Ro ) plane, the azimuthal wavenumber is constant, nθ = 6. It is then convenient to only record the light intensity I along the cylinders’ axis z and analyze the spatio-temporal diagrams I (z, t). Fig. 6 displays typical light intensity profiles I (z) for various Reynolds number. Keeping Ro fixed and decreasing Ri from the fully turbulent to the intermittent regime, a modulation gradually appears along I (z), indicating the continuous nature of the transition. The amplitude and wavelength of the modulation increase until the modulation minima saturate at the light intensity corresponding to
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Fig. 4. Zoom on the region of interest of the experimental phase diagram (η = 0.983). The labels (see [6] for details) stand for AZI: azimuthal flow; SPI and IPS: spiral and inter-penetrating spiral vortices; WIS: wavy inter-penetrating spiral vortices; INT: intermittency; SPT: spiral turbulence; TUR: turbulence. The solid straight lines show the paths along which measurements are conducted.
the laminar flow. Only then can one speak of coexistence of laminar and turbulent domains. At still lower Reynolds numbers, the pattern loses its coherence and the turbulent stripes become “independent” patches. We now investigate through LDV measurements the relationship between the light intensity modulation and the flow itself. In a larger gap TC apparatus (η = 0.963), both the light intensity along z and the axial velocity at mid-gap and mid-height vz (t) are recorded.
Fig. 5. (a) Streamwise/azimuthal wavelengths and (b) spanwise/ axial wavelengths vs. the Reynolds numbers R PC in PC (䊊) and R TC = (Ri + Ro )/2(1 + η)ν along path F in TC (×).
Fig. 6. Axial light intensity profiles in the TC flow, decreasing Ri from 900 (top) to 550 (bottom), at fixed Ro = −1055. The mean value of each profile has been arbitrarily fixed for the purpose of clarity.
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For both the fully turbulent flow (Fig. 7(a)) and the laminar one (Fig. 7(c)), the signal only shows random fluctuations. In the turbulent regime, these fluctuations are much larger than the experimental noise, whose intensity is given by the fluctuations level in the laminar case. In the spiral regime (Fig. 7(b)), in addition to the turbulent fluctuations, the velocity is modulated on a long, well-defined time scale. Performing running average using a time-window intermediate smaller than the period of these oscillations and much longer than the turbulent time scales, reveals that the rms of the averaged velocity exhibits the same oscillations. This frequency of these oscillations is nothing but the mean angular rotation frequency times the azimuthal wavenumber (nθ = 3 in the present geometry). Furthermore, decreasing the inner Reynolds number down from the fully turbulent to modulated turbulence regime, both the average velocity modulation and the velocity fluctuations modulation increase, as does the light intensity modulation: the lighter regions are more turbulent. More specifically, the amplitude of the modulation of the light intensity A shows a linear dependence with the amplitude of the modulation of vrms (Fig. 8). As a result, I (z, t) can
Fig. 7. Local axial velocity time series vz (t) at mid-height and mid-gap. η = 0.963, Ro = 850 and Ri = 2000 (a), 670 (b), and 0 (c). Top: raw signal (in black) and running average of vz (t) over windows of 0.4 s (in white). Bottom: rms of previous average.
Fig. 7(a–c), displays vz (t) for Ro = 850 in the three different regimes identified by the light intensity observation: the fully turbulent flow (Ri = 2000), the spiral regime (Ri = 670) and the laminar flow (Ri = 0).
Fig. 8. Amplitude of the modulation of the light intensity against the amplitude of the modulation of vrms (η = 0.963, Ro = 850 and 640 ≤ Ri ≤ 730).
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favorably be used to investigate the spatio-temporal dynamics of the modulation. Fig. 9 displays the spatio-temporal diagrams I (z, t) for Ro = −850 just below threshold (Ri = 785) (Fig. 9(a)) and in the core of the spiral regime (Ri = 720) (Fig. 9(b)). Close to threshold, domains are constantly nucleated their sizes are distributed exponentially with a characteristic scale which increases rapidly to reach the system size as R is decreased.
Below Ri = R ∗ , no nucleation occurs, and only transient fronts are observed. The basic frequency and wavelength (Fig. 10(a) and (b)) of I (z, t) are easily obtained via Fourier analysis of the spatio-temporal diagrams. The actual angular velocity of the pattern (the measured frequency divided by the azimuthal wavenumber) is equal to the mean angular velocity of both rotating cylinders. The axial wavelength increases while decreasing Ri , as
Fig. 9. Spatio-temporal diagrams and demodulation of the dynamics with (left column, Ri = 785) and without (right column, Ri = 720) domains nucleation for Ro = −850. (a), (b) I (z, t); (c), (d) |A+ (z, t)|; (e), (f) ∂z arg A± .
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Fig. 9. (Continued ).
already stated, but depends only slightly on Ro . The axial wavelength at threshold is identical for both Ro values considered.
5. Description in terms of slowly varying amplitudes In order to gather more quantitative information we now apply standard demodulation techniques to spatio-temporal diagrams I (z, t) [10]. The data is timefiltered around the basic frequency ω0 and space-filtered around the two opposite-wavenumber peaks ±k0 corresponding to the inclined stripes of each inclination. In other words, I (z, t) is written in terms of two slowly varying complex fields A+ and A− : I (z, t) = A− (z, t) exp i(ω0 t − k0 z) + A+ (z, t) exp i(ω0 t + k0 z) + c.c.
Fig. 10. (a) Temporal frequency vs. mean angular velocity of both cylinders Ωm in units of viscous time (h2 /ν); (b) axial wavelength λz : (䉫)Ro = −1200; (䊊)Ro = −850.
Fig. 9 shows the output of the demodulation on two typical spatio-temporal diagrams above and below R ∗ . Panels (c) and (d) show the space–time evolution of |A+ |, while the local wavenumber is shown in panels (e) and (f).
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Fig. 11. A2 vs. Ri : (䉫)Ro = −1200; (䊊)Ro = −850.
The square of the mean modulus A ≡ |A+ | + |A− | allows for a first quantitative determination of the threshold of modulated turbulence. Fig. 11 shows that A2 increases linearly as Ri is decreased below R ∗ , as expected for supercritical instabilities. In the nucleation regime, however, this usual variation breaks down, and one observes a dip of A2 , which can be directly linked to the large number of fronts separating domains, where the light intensity is weaker. We thus define the threshold Rc by extrapolating the region of linear variation of A2 in the nucleation regime, going towards A = 0 beyond R ∗ . The nucleation dynamics observed for R ∗ < Ri < Rc and the resulting dip of A is at odds with the ordinary description of weakly nonlinear instabilities [11]. This suggests that the intrinsic fluctuations of the turbulent basic state must be taken into account. We propose an heuristic modification of the usual amplitude equations by transposing the effect of the turbulent fluctuations into an additional constant-strength noise term. In the absence of further indication, we choose this noise term to be additive at the amplitude level, leading to the following Ginzburg–Landau equations governing A− and A+ :
ε = (Rc − Ri )/Rc the reduced distance to threshold, α the noise strength and η± the delta-correlated white noise. Cubic nonlinearities have been chosen to account for the observed supercritical nature of the transition. We now estimate all coefficients of Eq. (1) (and thereby the overall consistency of the present approach) from our experimental data. For each set of parameter values in the range where no nucleation of opposite amplitude domains is observed, several single-wave space–time domains are selected. The variation of the local modulus |A± | (that we shall note |A| when no distinction is required) and local frequency ∂t arg A± with the local wavenumber k = ∂z arg A± is then calculated for all points in each of these space–time domains. As seen in Fig. 12(a), |A|2 (k) exhibits a characteristic parabolic shape,
τ0 (∂t A± ± s0 ∂z A± ) = εA± + ξ02 (1 + ic1 )∂z2 A± − g3 (1 − ic3 )|A± |2 A± −g2 (1 − ic2 )|A∓ |2 A± + αη± ,
(1)
where τ0 and ξ0 are the characteristic scales of the modulations of the amplitudes, s0 the group velocity,
Fig. 12. The complex amplitudes modulation: (a) local modulus and (b) local frequency vs. local wavenumber (Ro = −1200, Ri = 620). Data are taken from a large space–time region where A+ is dominant. For each bin around a given wavenumber value, all the corresponding space–time points are determined. Averages are calculated on each bin.
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which can be written |A|2 (q) = α0 + α2 q 2 , for small q, with q = k − k0 and k0 such that maxk (|A|) = |A(k0 )|. On the opposite, the local frequency is essentially constant and equal to the mean rotation frequency (Fig. 12(b)). On longer time scales, a possible dispersion could lead to a deviation of ω(k) from ω0 . The analysis could not extract a distinct finite value for the slope of ω(k) − ω0 vs. k. Without noise, the later observation would imply c3 = 0 (nonlinear shift), s0 = 0 (no group velocity) and c1 + c3 = 0 (no dispersion). Since the presence of noise is believed not to alter such properties, it is reasonable to consider these coefficients to be zero in the noisy Eq. (1) too. On the contrary, the well known relations α0 = ε/g3 and α2 = −ξ02 /g3 are modified in the presence of noise, as we show below. We performed numerical simulations of Eq. (1) with s0 = c1 = c2 = c3 = 0 and all other coefficients set to one except g2 /g3 = 1.2 > 1 in agreement with the fact that A− and A+ do not coexist (see below). For any reasonable noise intensity, we observe, as ε is varied (see Fig. 13), the same behavior as in the experiment, i.e. nucleation dynamics close to threshold and emergence of a regular pattern for larger ε. In the following, we choose the noise strength α to be used in our further simulations to be that giving the same ε range for the observation of nucleation dynamics. This yields an order of magnitude which we find to be α = 3 × 10−3 . Following the experimental procedure, we recorded the variation of |A|2 with the local wavenumber q in numerical simulations of Eq. (1). The expected quadratic dependence |A|2 = α0 + α2 q 2 is obtained (Fig. 14(a)), coefficient α0 = ε/g3 (Fig. 14(b)), but α2 is not constant (and equal to ξ02 /g3 , as for the noiseless equations) (Fig. 14(c)). Two conclusions follow from these observations: the threshold estimated from measurements performed on the noisy equation (e.g. from the variation of α0 ) is the “real” one, and, similarly, g3 can be determined rather precisely. However, measurements of α2 do not allow to estimate ξ02 . Confirming the general relevance of our approach in terms of noisy amplitude equations, the corresponding experimental measurements show the same behavior: |A|2 depends linearly on q 2 (Fig. 15(a)); α0 does vary linearly with ε (Fig. 15(b)), and α2 varies with ε in
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Fig. 13. Numerical simulation of with (a) and without (b) nucleation dynamics. Both figure display |A+ | − |A− | within a nonlinear gray scale enforcing the fronts. The system size is 256 × 0.5 and the displayed time interval is 256 × 1.6. Initial conditions are two homogenous waves separated by two kicks. The transient time is 1024 time steps. The boundary conditions are periodic. α = 3 × 10−3 , ε = 0.002 (a), ε = 0.047 (b).
a manner similar to Fig. 14(c). Note that α0 = 0 at ε = 0, confirming that the threshold Rc was correctly defined by the variation of A2 , and that the linear variation of α0 allows for an accurate estimate of g3 . In order to estimate τ0 , we performed quench experiments and numerical simulations. For the TC
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Fig. 15. Experimental measurement of |A|2 vs. q 2 : (a) evidence of the linear dependence for various ε (Ro = −1200); (b) α0 and (c) α2 vs. ε. (䉫)Ro = −850; (䊊)Ro = −1200.
Fig. 14. Numerical evaluation of |A|2 vs. q 2 : (a) evidence of the linear dependence for various ε; (b) α0 and (c) α2 vs. ε for the noisy (solid line) and the noiseless (dashed line) equation.
experiments, the system is suddenly brought down from an homogeneous turbulent flow to the desired internal Reynolds number. In numerical simulations, the initial conditions are simply A+ = A− = 0. The subsequent growth rate σ of |A+ | + |A− | (where the average is over space and initial conditions) of the amplitude could provide a rough estimate of τ0 via the noiseless relation σ = ε/τ0 . Fig. 16(a) reveals that the numerical simulations yield the expected linear variation with ε, but with a slightly
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Fig. 17. Experimental measurement of (a) the temporal and (b) the spatial growth rates vs. ε. (䉫)Ro = −850; (䊊)Ro = −1200.
mentally, the linear variation with ε is found, allowing for an estimate of g2 /g3 (Fig. 17(b)).
Fig. 16. Numerical estimation of (a) the temporal and (b) the spatial growth rates vs. ε for the noisy (solid line) and the noiseless (dashed line) equation.
different slope. The estimated value of σ thus gives an upper bound for τ0 . Experimentally, the measurement of the growth rate is rather difficult to perform. Nevertheless, a roughly linear behavior is obtained for σ from which, again, an upper bound for τ0 can be deduced (Fig. 17(a)). In a similar spirit, the fronts separating A− and A+ domains can be used to estimate the remaining coefficient g2 . We fitted the profile of |A| against an exponential form (|A| exp(κ|z−zf |), with zf the position of the center of the front, and κ the spatial growth rate. In the noiseless amplitude description, it can be shown that ξ02 κ 2 = ε(g2 /g3 − 1). This linear dependence is again conserved in the noisy equation with only a small decrease of the slope itself compared to the noiseless case (Fig. 17(b)). Experi-
6. General discussion Table(a) in Fig. 18 summarizes the estimated values of the coefficients of Eq. (1) obtained from the experiments conducted at Ro = −850 and −1200. As explained above, only an order of magnitude can be given for ξ0 and the estimate of τ0 is very rough as well. These two coefficients only define the space and time scale of the amplitude modulations. They can thus be set to one by an appropriate normalization of time and space. Similarly g3 is a scaling factor for |A|. The experimental curves |A|(ε), properly rescaled with the numerical values of the coefficients obtained experimentally, collapse on the noiseless universal curve for large enough ε (Fig. 18(b)). Most importantly, we found g2 /g3 > 1, in agreement with the overall observation that the two amplitudes (A+ and A− ) do not coexist. We take these results as a confirmation of the overall relevance of our approach in terms of noisy amplitude equations.
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Fig. 18. (a) Experimental estimates for the coefficients of Eq. (1). (b) Scaled amplitude vs. ε. (䉫)Ro = −1200; (×)Ro = −1050; (䊊)Ro = −850.
Given that the coefficients of the Ginzburg–Landau equations are real, it is possible to write down the potential formulation: τ0 ∂t A± = −
∂V + ξ02 ∂zz A± + αη± , ∂ A¯ ±
with V = −ε(|A+ |2 + |A− |2 ) + + g2 (|A+ |2 |A− |2 ).
g3 (|A+ |4 + |A− |4 ) 2
For ε < 0, V has a single minimum (|A+ |, |A− |) = (0, 0), for ε > 0, V has two (family of) minima √ (|A+ |, |A− |) = ( ε/g3 , 0) and (|A+ |, |A− |) = √ (0, ε/g3 ) and one (family of) saddles (|A+ |, |A− |) = √ √ ( ε/(g2 + g3 ), ε/(g2 + g3 )) (see Fig. 19(a)). Such a potential dynamics in the presence of noise constitutes a good general framework for describing the various regimes of the transition. Here the potential must be taken as local in space. Its equilibrium states describe local states of the spatio-temporal dynamics: for ε < 0 the minimum is a patternless state and for ε > 0 each minimum corresponds to one of the incli-
Fig. 19. Potential description. (a) The potential V for ε < 0 (left) and ε > 0 (right). (b) Schematic picture of the transition: (1) the homogeneous state regime; (2) the noise activated nucleation dynamics phase; (3) the mono-domain phase. The dashed line is the transition line where zig or zag domains reach the system size. The solid line is a typical path followed by the system when varying the Reynolds number (see text for details).
nations of the stripes. The unstable saddle describes the superposition of these two states. Fig. 19(b) displays the various regimes observed when varying the symmetry-breaking parameter ε and the noise intensity α. When ε < 0, the unique stable state of the potential extends over all the space, corresponding to the homogeneous turbulent basic state. When ε > 0, domains of both inclinations compete in space. For small noise intensity (compared to the potential barrier), a domain of one or the other state eventually grows up to the system size and all fronts are eliminated. On the contrary for large noise intensity, a state of opposite inclination can locally be created. In real space, this may lead to the nucleation of a new domain together with an associated pair of fronts. Interestingly at the fronts the total amplitude is not zero. This is related to the local state corresponding to the saddle between the two wells of the potential, which is the superposition of both inclinations states and not the patternless state. From a hydrodynamical point of view, not only the distance to threshold but also the noise intensity, i.e. the strength of the turbulent fluctuations,
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depend on the Reynolds number. When varying R, the system follows a path in the (ε, α) plane as sketched in Fig. 19(b) (solid line). A first transition occurs from the homogeneously turbulent flow to the multi-domains nucleation dynamics phase at Rc (ε = 0), when the underlying potential symmetry is broken. It is followed by a second “transition” marked by the end of nucleation events (ε = ε∗ ). To summarize, we have shown that the so-called “spiral turbulence”, “barber pole turbulence” and consecutive “turbulent spots” commonly observed in TC experiments, together with the turbulent spots of the plane Couette flow, all find their origin in a supercritical long-wavelength modulation which arises within a fully turbulent regime present at higher Reynolds numbers. The resulting pattern of modulated turbulence stripes was found to be identical in both TC and plane Couette flow. A consistent description of the weakly nonlinear behavior of this pattern is given in terms of a coupled real Ginzburg–Landau equations endowed with an additive noise term. The noise term accounts for the nucleation dynamics observed just above threshold, as well as for the unusual variation with ε of various quantities characteristic of the amplitude description. Further work should deal with the search for the precise mechanism leading to the turbulent modulation (at the hydrodynamical level) and thereafter with the identification of both the instability and the noise sources. We believe that direct numerical simulations
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of the Navier–Stokes equations should be able to provide such insights. Finally the phenomenon reported here may well provide a good framework for recent discussions about the role of fluctuations at common pattern-forming instability thresholds [12]. Acknowledgements We thank Wim van Saarloos and Eric Bertin for many fruitful discussions. We also thank the Ginzburg–Landau workshop organizers for having invited us to present and discuss this work. References [1] Y. Pomeau, Physica D 23 (1986) 1. [2] F. Daviaud, J. Hegseth, P. Bergé, Phys. Rev. Lett. 69 (17) (1992) 2511. [3] O. Dauchot, F. Daviaud, Phys. Fluids 7 (2) (1995) 335. [4] D. Coles, J. Fluid Mech. 21 (1965) 385. [5] C. Van Atta, J. Fluid Mech. 25 (1966) 495. [6] C.D. Andereck, S.S. Liu, H.L. Swinney, J. Fluid Mech. 164 (1986) 155. [7] A. Prigent, O. Dauchot, Phys. Fluids 12 (10) (2000) 2688. [8] D. Coles, C.W. Van Atta, Phys. Fluids (Suppl.) S120 (1967). [9] S. Bottin, H. Chaté, Eur. Phys. J. B 6 (1198) 143. [10] V. Croquette, H. Williams, Physica D 37 (1989) 300. [11] M. van Hecke, C. Storm, W. van Saarloos, Physica D 134 (1999) 1. [12] M.A. Scherer, G. Ahlers, F. Hörner, I. Rehberg, Phys. Rev. Lett. 85 (2001) 3754.
Long-wavelength modulation of turbulent shear flows Arnaud Prigent, Guillaume Grégoire, Hugues Chaté, Olivier Dauchot∗ CEA, Service de Physique de l’Etat Condensé, Centre d’Etudes de Saclay, 91191 Gif-sur-Yvette, France Received 31 October 2001; received in revised form 3 June 2002; accepted 10 June 2002
Abstract The reverse transition from turbulent to laminar flow is studied in very large aspect ratio plane Couette and Taylor–Couette experiments. We show that laminar-turbulence coexistence dynamics (turbulent spots, spiral turbulence, etc.) can be seen as the ultimate stage of a modulation of the turbulent flows present at higher Reynolds number leading to regular, long-wavelength, inclined stripes. This new type of instability, whose originality is to arise within a macroscopically fluctuating state, can be described in the framework of Ginzburg–Landau equations to which noise is heuristically added to take into account the intrinsic fluctuations of the basic state. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Wavelength modulation; Turbulent shear flows; Taylor–Couette experiments; Turbulent flow bifurcation
1. Introduction In spite of more than a century of theoretical and experimental efforts, the transition to turbulence in some basic hydrodynamical flows is still far from being fully understood. This is especially true when linear and weakly nonlinear analysis cannot be used. When the basic flow remains stable against infinitesimal perturbations, new solutions with no direct connection to it may arise, triggered by finite amplitude disturbances. In physical space, when such a sub-critical instability drives the system to turbulence, inhomogeneities often develop in the form of fluctuating domains separated by fronts [1]. In the plane Couette [PC] flow [2,3], the flow between two walls moving in opposite direction at equal speed, these domains, also called spots, have a rather complex spatio-temporal intermit∗
Corresponding author. E-mail address: [email protected] (O. Dauchot).
tent dynamics. In the contra-rotating Taylor–Couette [TC] flow [4–6], the flow between two independent contra-rotating coaxial cylinders, one observes, apart from spots, the so-called “spiral” or “barber pole” turbulence, commonly described as a single helical stripe of turbulence winding around the cylinders’ axis. Two major difficulties occur when studying this type of abrupt transition to turbulence. First, when triggering the instability with finite amplitude disturbances, one has to deal with the difficult issues of their reproducibility and the flow “receptivity”. Second, the turbulent domains often typically extend to the border of the system, so that confinement or side-wall effects must be taken into account. Here, we investigate both plane Couette and TC flow in the largest aspect ratio apparatus ever built. Decreasing the Reynolds number, we study the reverse transition from the turbulent to the laminar flow. In both flows, we observe a continuous transition towards a regular pattern made of periodically spaced,
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inclined stripes of well-defined width and alternating turbulence strength. In spite of its conventional aspects, the instability leading to this modulated turbulence is of a new type because it arises within a macroscopically fluctuating state. One consequence of this intrinsically noisy context is that the flow observed just above threshold is characterized by the frequent nucleation of domains of stripes of opposite inclination. Going away from threshold, the typical domain size increases to reach the system size, leaving a regular pattern. In a parallel numerical investigation, we show that a coherent amplitude equation description of this pattern can be given in terms of coupled noisy Ginzburg–Landau equations. After the presentation of our experimental setup, we first describe the typical patterns of interest here and how they take place in the phase diagram of each flow. We then turn to a more quantitative report of our experimental investigation of the TC flow when decreasing the Reynolds number from the homogeneous turbulent flow. Next, we detail our description in terms of noisy amplitude equations. A general discussion concludes the article.
2. Experimental setup Our TC apparatus [7] is made of two independently rotating coaxial cylinders of radii ri = 49.09 ± 0.005 mm and ro = 49.95 ± 0.005 mm. The useful length is L = 380 ± 0.1 mm, and the gap size is d = 0.863 ± 0.01 mm, so that the aspect ratio ΓzTC = L/d = 442 and ΓθTC = π(ri + ro )/d = 362 are large, and the radius ratio η = 0.983 is very close to 1 (see Fig. 1). The flow is governed by the inner and outer Reynolds numbers Ri,o = ri,o Ωi,o d/ν, with Ωi,o the angular velocities, and ν the kinematic viscosity of water. The flow is thermalized by water circulation inside the inner cylinder. At thermal equilibrium the temperature is uniform in space up to 0.1 K and does not vary more than 0.1 K/h. As a result, the accuracy on Ri,o is of the order of 3%. The flow is visualized using a “fluorescent lighting” technique [7] developed for the purpose of this study. The water flow is seeded with Kalliroscope AQ 1000 (6 m ×30 m ×0.07 m
Fig. 1. Schematic drawing of the TC (top) and PC apparatus (bottom).
platelets). The inner cylinder is covered by a fluorescent film and the entire apparatus is UV-lighted. The fluorescent film re-emits a uniform visible lighting, transmitted through the fluid layer: the more turbulent the flow, the brighter it appears. Images and spatio-temporal diagrams (temporal recording of one line along the cylinder axis) are recorded by a CCD camera. Two plane mirrors reflect the two thirds of the flow hidden to the camera so that the whole cylindrical flow can be reconstructed. A second TC apparatus with a slightly larger gap (ri = 48.11 mm; η = 0.963) allows us to perform one point laser Doppler velocimetry (LDV) measurements of the axial and azimuthal velocity components at mid-height and mid-gap between cylinders.
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Our PC apparatus [2] is made of an endless transparent plastic film belt (363.0 cm long, 25.4 cm wide, 0.15 mm thick) driven by two pairs of small guiding rotating cylinders and one pair of large rotating cylinders as shown in Fig. 1. In this study, the guiding cylinders are 1.8 mm apart so that the gap between the wall is δ = 1.5 mm. The belt is also guided by two glass plates 3 mm apart and strictly parallel to the walls. Using δ as a length scale, the spanwise aspect ratio is ΓzPC = 169 and the streamwise aspect ratio ΓxPC = 385. The entire setup is placed in a tank filled with water. The temperature is controlled within 0.5 K but the gap size cannot be controlled with an accuracy better than 0.1 mm, that is 6.5%. Accordingly, the accuracy on the Reynolds number is at best 7.5%. The flow is again seeded with Kalliroscope AQ 1000 and lighted on its whole width with a thin laser sheet. In a laminar flow, the reflected light is steady and rather weak, because of the averaged orientation of the flakes parallel to the laser sheet. In a turbulent flow, the reflected light intensity is larger and fluctuating. Images and spatio-temporal diagrams (temporal recording of one line along the spanwise direction) are recorded by a CCD camera. For the purpose of comparison between the two situations, we define the Reynolds number R equivalently in both flows as the ratio of the viscous time scale to the shear time scale. The length scale h is chosen to be half the gap, as usually done in the plane Couette flow context, so that R
PC
U h2 Uh = = , h ν ν
where U is the belt velocity, and R TC =
ri Ωi − ηro Ωo h2 (ri Ωi − ηro Ωo )h = . (1 + η)h ν (1 + η)ν
3. A long-wavelength pattern in Couette flows Figs. 2 and 3 display snapshots of each flow for decreasing values of the Reynolds number (internal Reynolds number in the TC case). The deceleration rate is as small as 0.1%R s−1 and the flow is let to stabilize before each snapshot is taken. At high enough
Fig. 2. Turbulent spots and stripes along path F (see Fig. 4). R TC = 391 (a), 368 (b), 340 (c), and 331 (d). Each picture displays a 360◦ view of the whole flow (38 cm height and 31.4 cm wide). The vertical lines are dued to the image reconstruction.
Reynolds number, both flows are homogeneously turbulent. Decreasing R, there is a threshold below which a periodic structure appears with two preferred opposite inclinations. The pattern occurs simultaneously in the whole flow and does not appear to be triggered by end effects. Close to threshold, nucleation of competing domains of both orientations occur. For lower R, a regular pattern is eventually reached after a transient during which domains, separated by wandering fronts, compete. The oblique stripes have a wavelength of the order of 50 times the gap. The pattern is stationary in the plane Couette flow case, and rotates at the mean angular velocity of both cylinders (see further) in the TC flow case. For even lower Reynolds number, the stripe pattern breaks down, leaving a spatio-temporally
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Fig. 3. Turbulent stripes in plane Couette flow. R PC = 393 (a), 358 (b), and 340 (c). Each picture displays a view of the whole flow (25.4 cm height and 57.7 cm wide).
intermittent regime of turbulent patches evolving in an otherwise laminar flow. The pattern was observed for 340 < R < 415 in the plane Couette flow, and in the so-called “spiral turbulence” region of the phase diagram in the TC flow (see Fig. 4). This phase diagram, obtained by following the now-traditional procedure introduced by Coles [4] is similar to the one obtained by Andereck et al. [6] for a different geometry (η = 0.878 and much smaller aspect ratio): the same regimes are observed, but the thresholds are shifted to higher R values and the sub-criticality is “enhanced” (the laminarturbulent coexistence, “INT” and “SPT”, regions are larger).
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The “spiral turbulence” flow described by Coles and Van Atta [8], later by Andereck et al. [6] is thus to be identified with one wavelength of the regular pattern observed here, in agreement with the azimuthal aspect ratio (Γθ = 55) of their apparatus. We indeed verified that for azimuthal aspect ratios as small as 20 no “spiral turbulence” takes place. Similarly, previous plane Couette flow studies [3,9] could only produce one or two turbulent “spots” taking sometimes the shape of an inclined stripe. As it can be noticed in Fig. 4 the path F defined by Ri = −ηRo , where µ = Ω0 /Ωi = −1, crosses the spiral turbulence regime, in contrast with smaller η experiments where spiral turbulence occurs only below this straight line. We take advantage of this opportunity to develop a full comparison with the plane Couette flow, since in this case both patterns are stationary and governed by only one control parameter. As shown in Fig. 5, the similarity between the two flows goes beyond the above qualitative description. Both patterns have the same Reynolds number range of existence and exhibit the same wavelengths at any R, the only difference being that the azimuthal wavenumber is quantized by the circumference in the TC case. The plane Couette flow experiment being rather difficult to control for such a small gap, we now investigate the transition in more details on data collected solely with the TC apparatus. We are nevertheless confident that most of our findings apply to both flows.
4. A modulation of the turbulence strength In most of the “modulated turbulence” region of the (Ri , Ro ) plane, the azimuthal wavenumber is constant, nθ = 6. It is then convenient to only record the light intensity I along the cylinders’ axis z and analyze the spatio-temporal diagrams I (z, t). Fig. 6 displays typical light intensity profiles I (z) for various Reynolds number. Keeping Ro fixed and decreasing Ri from the fully turbulent to the intermittent regime, a modulation gradually appears along I (z), indicating the continuous nature of the transition. The amplitude and wavelength of the modulation increase until the modulation minima saturate at the light intensity corresponding to
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Fig. 4. Zoom on the region of interest of the experimental phase diagram (η = 0.983). The labels (see [6] for details) stand for AZI: azimuthal flow; SPI and IPS: spiral and inter-penetrating spiral vortices; WIS: wavy inter-penetrating spiral vortices; INT: intermittency; SPT: spiral turbulence; TUR: turbulence. The solid straight lines show the paths along which measurements are conducted.
the laminar flow. Only then can one speak of coexistence of laminar and turbulent domains. At still lower Reynolds numbers, the pattern loses its coherence and the turbulent stripes become “independent” patches. We now investigate through LDV measurements the relationship between the light intensity modulation and the flow itself. In a larger gap TC apparatus (η = 0.963), both the light intensity along z and the axial velocity at mid-gap and mid-height vz (t) are recorded.
Fig. 5. (a) Streamwise/azimuthal wavelengths and (b) spanwise/ axial wavelengths vs. the Reynolds numbers R PC in PC (䊊) and R TC = (Ri + Ro )/2(1 + η)ν along path F in TC (×).
Fig. 6. Axial light intensity profiles in the TC flow, decreasing Ri from 900 (top) to 550 (bottom), at fixed Ro = −1055. The mean value of each profile has been arbitrarily fixed for the purpose of clarity.
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For both the fully turbulent flow (Fig. 7(a)) and the laminar one (Fig. 7(c)), the signal only shows random fluctuations. In the turbulent regime, these fluctuations are much larger than the experimental noise, whose intensity is given by the fluctuations level in the laminar case. In the spiral regime (Fig. 7(b)), in addition to the turbulent fluctuations, the velocity is modulated on a long, well-defined time scale. Performing running average using a time-window intermediate smaller than the period of these oscillations and much longer than the turbulent time scales, reveals that the rms of the averaged velocity exhibits the same oscillations. This frequency of these oscillations is nothing but the mean angular rotation frequency times the azimuthal wavenumber (nθ = 3 in the present geometry). Furthermore, decreasing the inner Reynolds number down from the fully turbulent to modulated turbulence regime, both the average velocity modulation and the velocity fluctuations modulation increase, as does the light intensity modulation: the lighter regions are more turbulent. More specifically, the amplitude of the modulation of the light intensity A shows a linear dependence with the amplitude of the modulation of vrms (Fig. 8). As a result, I (z, t) can
Fig. 7. Local axial velocity time series vz (t) at mid-height and mid-gap. η = 0.963, Ro = 850 and Ri = 2000 (a), 670 (b), and 0 (c). Top: raw signal (in black) and running average of vz (t) over windows of 0.4 s (in white). Bottom: rms of previous average.
Fig. 7(a–c), displays vz (t) for Ro = 850 in the three different regimes identified by the light intensity observation: the fully turbulent flow (Ri = 2000), the spiral regime (Ri = 670) and the laminar flow (Ri = 0).
Fig. 8. Amplitude of the modulation of the light intensity against the amplitude of the modulation of vrms (η = 0.963, Ro = 850 and 640 ≤ Ri ≤ 730).
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favorably be used to investigate the spatio-temporal dynamics of the modulation. Fig. 9 displays the spatio-temporal diagrams I (z, t) for Ro = −850 just below threshold (Ri = 785) (Fig. 9(a)) and in the core of the spiral regime (Ri = 720) (Fig. 9(b)). Close to threshold, domains are constantly nucleated their sizes are distributed exponentially with a characteristic scale which increases rapidly to reach the system size as R is decreased.
Below Ri = R ∗ , no nucleation occurs, and only transient fronts are observed. The basic frequency and wavelength (Fig. 10(a) and (b)) of I (z, t) are easily obtained via Fourier analysis of the spatio-temporal diagrams. The actual angular velocity of the pattern (the measured frequency divided by the azimuthal wavenumber) is equal to the mean angular velocity of both rotating cylinders. The axial wavelength increases while decreasing Ri , as
Fig. 9. Spatio-temporal diagrams and demodulation of the dynamics with (left column, Ri = 785) and without (right column, Ri = 720) domains nucleation for Ro = −850. (a), (b) I (z, t); (c), (d) |A+ (z, t)|; (e), (f) ∂z arg A± .
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Fig. 9. (Continued ).
already stated, but depends only slightly on Ro . The axial wavelength at threshold is identical for both Ro values considered.
5. Description in terms of slowly varying amplitudes In order to gather more quantitative information we now apply standard demodulation techniques to spatio-temporal diagrams I (z, t) [10]. The data is timefiltered around the basic frequency ω0 and space-filtered around the two opposite-wavenumber peaks ±k0 corresponding to the inclined stripes of each inclination. In other words, I (z, t) is written in terms of two slowly varying complex fields A+ and A− : I (z, t) = A− (z, t) exp i(ω0 t − k0 z) + A+ (z, t) exp i(ω0 t + k0 z) + c.c.
Fig. 10. (a) Temporal frequency vs. mean angular velocity of both cylinders Ωm in units of viscous time (h2 /ν); (b) axial wavelength λz : (䉫)Ro = −1200; (䊊)Ro = −850.
Fig. 9 shows the output of the demodulation on two typical spatio-temporal diagrams above and below R ∗ . Panels (c) and (d) show the space–time evolution of |A+ |, while the local wavenumber is shown in panels (e) and (f).
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Fig. 11. A2 vs. Ri : (䉫)Ro = −1200; (䊊)Ro = −850.
The square of the mean modulus A ≡ |A+ | + |A− | allows for a first quantitative determination of the threshold of modulated turbulence. Fig. 11 shows that A2 increases linearly as Ri is decreased below R ∗ , as expected for supercritical instabilities. In the nucleation regime, however, this usual variation breaks down, and one observes a dip of A2 , which can be directly linked to the large number of fronts separating domains, where the light intensity is weaker. We thus define the threshold Rc by extrapolating the region of linear variation of A2 in the nucleation regime, going towards A = 0 beyond R ∗ . The nucleation dynamics observed for R ∗ < Ri < Rc and the resulting dip of A is at odds with the ordinary description of weakly nonlinear instabilities [11]. This suggests that the intrinsic fluctuations of the turbulent basic state must be taken into account. We propose an heuristic modification of the usual amplitude equations by transposing the effect of the turbulent fluctuations into an additional constant-strength noise term. In the absence of further indication, we choose this noise term to be additive at the amplitude level, leading to the following Ginzburg–Landau equations governing A− and A+ :
ε = (Rc − Ri )/Rc the reduced distance to threshold, α the noise strength and η± the delta-correlated white noise. Cubic nonlinearities have been chosen to account for the observed supercritical nature of the transition. We now estimate all coefficients of Eq. (1) (and thereby the overall consistency of the present approach) from our experimental data. For each set of parameter values in the range where no nucleation of opposite amplitude domains is observed, several single-wave space–time domains are selected. The variation of the local modulus |A± | (that we shall note |A| when no distinction is required) and local frequency ∂t arg A± with the local wavenumber k = ∂z arg A± is then calculated for all points in each of these space–time domains. As seen in Fig. 12(a), |A|2 (k) exhibits a characteristic parabolic shape,
τ0 (∂t A± ± s0 ∂z A± ) = εA± + ξ02 (1 + ic1 )∂z2 A± − g3 (1 − ic3 )|A± |2 A± −g2 (1 − ic2 )|A∓ |2 A± + αη± ,
(1)
where τ0 and ξ0 are the characteristic scales of the modulations of the amplitudes, s0 the group velocity,
Fig. 12. The complex amplitudes modulation: (a) local modulus and (b) local frequency vs. local wavenumber (Ro = −1200, Ri = 620). Data are taken from a large space–time region where A+ is dominant. For each bin around a given wavenumber value, all the corresponding space–time points are determined. Averages are calculated on each bin.
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which can be written |A|2 (q) = α0 + α2 q 2 , for small q, with q = k − k0 and k0 such that maxk (|A|) = |A(k0 )|. On the opposite, the local frequency is essentially constant and equal to the mean rotation frequency (Fig. 12(b)). On longer time scales, a possible dispersion could lead to a deviation of ω(k) from ω0 . The analysis could not extract a distinct finite value for the slope of ω(k) − ω0 vs. k. Without noise, the later observation would imply c3 = 0 (nonlinear shift), s0 = 0 (no group velocity) and c1 + c3 = 0 (no dispersion). Since the presence of noise is believed not to alter such properties, it is reasonable to consider these coefficients to be zero in the noisy Eq. (1) too. On the contrary, the well known relations α0 = ε/g3 and α2 = −ξ02 /g3 are modified in the presence of noise, as we show below. We performed numerical simulations of Eq. (1) with s0 = c1 = c2 = c3 = 0 and all other coefficients set to one except g2 /g3 = 1.2 > 1 in agreement with the fact that A− and A+ do not coexist (see below). For any reasonable noise intensity, we observe, as ε is varied (see Fig. 13), the same behavior as in the experiment, i.e. nucleation dynamics close to threshold and emergence of a regular pattern for larger ε. In the following, we choose the noise strength α to be used in our further simulations to be that giving the same ε range for the observation of nucleation dynamics. This yields an order of magnitude which we find to be α = 3 × 10−3 . Following the experimental procedure, we recorded the variation of |A|2 with the local wavenumber q in numerical simulations of Eq. (1). The expected quadratic dependence |A|2 = α0 + α2 q 2 is obtained (Fig. 14(a)), coefficient α0 = ε/g3 (Fig. 14(b)), but α2 is not constant (and equal to ξ02 /g3 , as for the noiseless equations) (Fig. 14(c)). Two conclusions follow from these observations: the threshold estimated from measurements performed on the noisy equation (e.g. from the variation of α0 ) is the “real” one, and, similarly, g3 can be determined rather precisely. However, measurements of α2 do not allow to estimate ξ02 . Confirming the general relevance of our approach in terms of noisy amplitude equations, the corresponding experimental measurements show the same behavior: |A|2 depends linearly on q 2 (Fig. 15(a)); α0 does vary linearly with ε (Fig. 15(b)), and α2 varies with ε in
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Fig. 13. Numerical simulation of with (a) and without (b) nucleation dynamics. Both figure display |A+ | − |A− | within a nonlinear gray scale enforcing the fronts. The system size is 256 × 0.5 and the displayed time interval is 256 × 1.6. Initial conditions are two homogenous waves separated by two kicks. The transient time is 1024 time steps. The boundary conditions are periodic. α = 3 × 10−3 , ε = 0.002 (a), ε = 0.047 (b).
a manner similar to Fig. 14(c). Note that α0 = 0 at ε = 0, confirming that the threshold Rc was correctly defined by the variation of A2 , and that the linear variation of α0 allows for an accurate estimate of g3 . In order to estimate τ0 , we performed quench experiments and numerical simulations. For the TC
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Fig. 15. Experimental measurement of |A|2 vs. q 2 : (a) evidence of the linear dependence for various ε (Ro = −1200); (b) α0 and (c) α2 vs. ε. (䉫)Ro = −850; (䊊)Ro = −1200.
Fig. 14. Numerical evaluation of |A|2 vs. q 2 : (a) evidence of the linear dependence for various ε; (b) α0 and (c) α2 vs. ε for the noisy (solid line) and the noiseless (dashed line) equation.
experiments, the system is suddenly brought down from an homogeneous turbulent flow to the desired internal Reynolds number. In numerical simulations, the initial conditions are simply A+ = A− = 0. The subsequent growth rate σ of |A+ | + |A− | (where the average is over space and initial conditions) of the amplitude could provide a rough estimate of τ0 via the noiseless relation σ = ε/τ0 . Fig. 16(a) reveals that the numerical simulations yield the expected linear variation with ε, but with a slightly
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Fig. 17. Experimental measurement of (a) the temporal and (b) the spatial growth rates vs. ε. (䉫)Ro = −850; (䊊)Ro = −1200.
mentally, the linear variation with ε is found, allowing for an estimate of g2 /g3 (Fig. 17(b)).
Fig. 16. Numerical estimation of (a) the temporal and (b) the spatial growth rates vs. ε for the noisy (solid line) and the noiseless (dashed line) equation.
different slope. The estimated value of σ thus gives an upper bound for τ0 . Experimentally, the measurement of the growth rate is rather difficult to perform. Nevertheless, a roughly linear behavior is obtained for σ from which, again, an upper bound for τ0 can be deduced (Fig. 17(a)). In a similar spirit, the fronts separating A− and A+ domains can be used to estimate the remaining coefficient g2 . We fitted the profile of |A| against an exponential form (|A| exp(κ|z−zf |), with zf the position of the center of the front, and κ the spatial growth rate. In the noiseless amplitude description, it can be shown that ξ02 κ 2 = ε(g2 /g3 − 1). This linear dependence is again conserved in the noisy equation with only a small decrease of the slope itself compared to the noiseless case (Fig. 17(b)). Experi-
6. General discussion Table(a) in Fig. 18 summarizes the estimated values of the coefficients of Eq. (1) obtained from the experiments conducted at Ro = −850 and −1200. As explained above, only an order of magnitude can be given for ξ0 and the estimate of τ0 is very rough as well. These two coefficients only define the space and time scale of the amplitude modulations. They can thus be set to one by an appropriate normalization of time and space. Similarly g3 is a scaling factor for |A|. The experimental curves |A|(ε), properly rescaled with the numerical values of the coefficients obtained experimentally, collapse on the noiseless universal curve for large enough ε (Fig. 18(b)). Most importantly, we found g2 /g3 > 1, in agreement with the overall observation that the two amplitudes (A+ and A− ) do not coexist. We take these results as a confirmation of the overall relevance of our approach in terms of noisy amplitude equations.
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Fig. 18. (a) Experimental estimates for the coefficients of Eq. (1). (b) Scaled amplitude vs. ε. (䉫)Ro = −1200; (×)Ro = −1050; (䊊)Ro = −850.
Given that the coefficients of the Ginzburg–Landau equations are real, it is possible to write down the potential formulation: τ0 ∂t A± = −
∂V + ξ02 ∂zz A± + αη± , ∂ A¯ ±
with V = −ε(|A+ |2 + |A− |2 ) + + g2 (|A+ |2 |A− |2 ).
g3 (|A+ |4 + |A− |4 ) 2
For ε < 0, V has a single minimum (|A+ |, |A− |) = (0, 0), for ε > 0, V has two (family of) minima √ (|A+ |, |A− |) = ( ε/g3 , 0) and (|A+ |, |A− |) = √ (0, ε/g3 ) and one (family of) saddles (|A+ |, |A− |) = √ √ ( ε/(g2 + g3 ), ε/(g2 + g3 )) (see Fig. 19(a)). Such a potential dynamics in the presence of noise constitutes a good general framework for describing the various regimes of the transition. Here the potential must be taken as local in space. Its equilibrium states describe local states of the spatio-temporal dynamics: for ε < 0 the minimum is a patternless state and for ε > 0 each minimum corresponds to one of the incli-
Fig. 19. Potential description. (a) The potential V for ε < 0 (left) and ε > 0 (right). (b) Schematic picture of the transition: (1) the homogeneous state regime; (2) the noise activated nucleation dynamics phase; (3) the mono-domain phase. The dashed line is the transition line where zig or zag domains reach the system size. The solid line is a typical path followed by the system when varying the Reynolds number (see text for details).
nations of the stripes. The unstable saddle describes the superposition of these two states. Fig. 19(b) displays the various regimes observed when varying the symmetry-breaking parameter ε and the noise intensity α. When ε < 0, the unique stable state of the potential extends over all the space, corresponding to the homogeneous turbulent basic state. When ε > 0, domains of both inclinations compete in space. For small noise intensity (compared to the potential barrier), a domain of one or the other state eventually grows up to the system size and all fronts are eliminated. On the contrary for large noise intensity, a state of opposite inclination can locally be created. In real space, this may lead to the nucleation of a new domain together with an associated pair of fronts. Interestingly at the fronts the total amplitude is not zero. This is related to the local state corresponding to the saddle between the two wells of the potential, which is the superposition of both inclinations states and not the patternless state. From a hydrodynamical point of view, not only the distance to threshold but also the noise intensity, i.e. the strength of the turbulent fluctuations,
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depend on the Reynolds number. When varying R, the system follows a path in the (ε, α) plane as sketched in Fig. 19(b) (solid line). A first transition occurs from the homogeneously turbulent flow to the multi-domains nucleation dynamics phase at Rc (ε = 0), when the underlying potential symmetry is broken. It is followed by a second “transition” marked by the end of nucleation events (ε = ε∗ ). To summarize, we have shown that the so-called “spiral turbulence”, “barber pole turbulence” and consecutive “turbulent spots” commonly observed in TC experiments, together with the turbulent spots of the plane Couette flow, all find their origin in a supercritical long-wavelength modulation which arises within a fully turbulent regime present at higher Reynolds numbers. The resulting pattern of modulated turbulence stripes was found to be identical in both TC and plane Couette flow. A consistent description of the weakly nonlinear behavior of this pattern is given in terms of a coupled real Ginzburg–Landau equations endowed with an additive noise term. The noise term accounts for the nucleation dynamics observed just above threshold, as well as for the unusual variation with ε of various quantities characteristic of the amplitude description. Further work should deal with the search for the precise mechanism leading to the turbulent modulation (at the hydrodynamical level) and thereafter with the identification of both the instability and the noise sources. We believe that direct numerical simulations
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of the Navier–Stokes equations should be able to provide such insights. Finally the phenomenon reported here may well provide a good framework for recent discussions about the role of fluctuations at common pattern-forming instability thresholds [12]. Acknowledgements We thank Wim van Saarloos and Eric Bertin for many fruitful discussions. We also thank the Ginzburg–Landau workshop organizers for having invited us to present and discuss this work. References [1] Y. Pomeau, Physica D 23 (1986) 1. [2] F. Daviaud, J. Hegseth, P. Bergé, Phys. Rev. Lett. 69 (17) (1992) 2511. [3] O. Dauchot, F. Daviaud, Phys. Fluids 7 (2) (1995) 335. [4] D. Coles, J. Fluid Mech. 21 (1965) 385. [5] C. Van Atta, J. Fluid Mech. 25 (1966) 495. [6] C.D. Andereck, S.S. Liu, H.L. Swinney, J. Fluid Mech. 164 (1986) 155. [7] A. Prigent, O. Dauchot, Phys. Fluids 12 (10) (2000) 2688. [8] D. Coles, C.W. Van Atta, Phys. Fluids (Suppl.) S120 (1967). [9] S. Bottin, H. Chaté, Eur. Phys. J. B 6 (1198) 143. [10] V. Croquette, H. Williams, Physica D 37 (1989) 300. [11] M. van Hecke, C. Storm, W. van Saarloos, Physica D 134 (1999) 1. [12] M.A. Scherer, G. Ahlers, F. Hörner, I. Rehberg, Phys. Rev. Lett. 85 (2001) 3754.