Core Dynamics of Multi-Armed Spirals in Rayleigh-Bénard Convection

Transcription

1 B. B. Plapp and E. Bodenschatz, page 1 Core Dynamics of Multi-Armed Spirals in Rayleigh-Bénard Convection Brendan B. Plapp and Eberhard Bodenschatz * Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York Last modified Monday, February 05, :57 am PACS numbers: r, Bp, Te Abstract We report experimental observations of the core dynamics of multi-armed, rotating spirals in Rayleigh-Bénard convection for a fluid with Prandtl number near one. In addition to the largescale rotation of the spirals, we found a cyclic core motion within a central area of radius r nλ/2, where n is the number of spiral arms ending in the core and λ is the wavelength of the pattern. The dynamics of the core was much faster than the large-scale spiral rotation. We observed multiarmed spirals for which the two periods were commensurate and others for which they were incommensurate. 1. Introduction Pattern-formation in convection of fluids has been of continuous interest since it was first observed in 1855 by Weber [1]. Over the last 30 years, Rayleigh-Bénard convection of a horizontal fluid layer heated from below and cooled from above has proven itself a paradigm for pattern forming systems [2][3]. No other pattern forming system is as well understood. The stability of parallel rolls as a function of Prandtl number σ [4], the so-called Busse Balloon, was calculated early on [5] and verified in a number of experiments [6][3]. Non-relaxational, spatio-temporal behavior not explainable by the Busse balloon was first observed in 4 He ( σ = 0.8) [7], then in water at σ = 5.7 [8] and σ = 2.5 [9] and later in compressed argon gas with σ = 0.7 [10]. It was shown that large-scale flows driven by roll curvature were very important for the observed non-relaxational behavior [10][11]. Until quite recently all experiments were limited to small aspect ratios, typically Γ< 40, where Γ = L d with L and d being the typical horizontal dimension and the height of the fluid layer, respectively. Recently, by using compressed gases as the fluids, convection cells with large aspect ratios were realized and surprising behavior was found [12][13][14][15][16]. Multi-armed rotating spirals were first observed in an experiment on non-boussinesq convection of compressed CO 2 [12]. In another experiment under Boussinesq conditions, Morris et al. found a spatio-temporal chaotic state of Spiral Defect Chaos (SDC) [13] which is dominated by a perpetual creation and annihilation of spirals. Most surprisingly, the average spatial periodicity (wavenumber) of SDC was located in the middle of the Busse balloon [5]. This led to a number of further investigations with CO 2 as a working fluid [14][15] and experiments with SF 6 gas close to its critical point [16]. Large rotating spirals were first observed theoretically in simulations of a model equation, the modified Swift- Hohenberg (msh) equation [17][18]. It was shown that a coupling of the rolls to a large-scale vorticity field driven by curvature was necessary for spiral rotation. The numerical investigations also suggested that non-boussinesq effects were not essential for the existence of large rotating spirals. SDC was also observed in numerical simulations of the msh equation [19][20] and later in simulations of the full Boussinesq equations [21]. The latter simulations showed that within the SDC-regime straight roll solutions were indeed stable solutions in agreement with the predictions from the Busse balloon. This was later experimentally verified [22]. Recently Cross and Tu [23] suggested that SDC is governed by the invasive dynamics of the spiral-defects. They showed by investigating the nonlinear phase equations that the spiral rotation is in lowest order driven by wavevector frustration and not by mean flow effects. As a first step towards the understanding of SDC we have designed an experiment to investigate individual rotating spirals in convection cells with various sizes. By using specially designed cell boundaries we selected circular convection patterns out of the basin of possible convection structures to investigate individual spirals and their stability in much detail [24][27]. Here, we report the first experimental observations of the core dynamics of multi-armed spirals. 2. Experimental Setup The experimental setup was similar to the one described in [2]. The main part of the experiment was a stainless steel pressure vessel with a 3.2 cm thick sapphire window at the top. Within this vessel was the convection cell, comprised of a top sapphire plate and a bottom silicon plate, each 1cm thick, 10cm in diameter, and polished to optical quality. The top plate was held from above by a flow distributor ring which also served to direct temperature controlled water across the top surface [2]. To minimize bending of the top plate, both the circulating water and the gas were held at the same pressure. The circulating water bath was kept constant throughout the experi- * eb22@cornell.edu

2 B. B. Plapp and E. Bodenschatz, page 2 paper sapphire silicon gas Figure 1: Schematic of the sidewall design. ment at (20.00±0.05) C and was regulated to ±0.2 mk. The bottom plate temperature was regulated by an electric film heater to ±0.3 mk and was adjusted to vary the temperature difference across the cell. The bottom plate was resting on three short, 0.8 mm diameter steel pins which were mounted at the rim of a 10 cm diameter stainless steel disk. This design had the advantage that it avoided mechanical stresses due to thermal expansion leading to a possible bending of the top and bottom plates. The cell height was adjusted in situ under the working pressure from outside by three screws moving the flow distributor. During this procedure the plates parallelism was monitored interferometrically using the shadowgraph setup described below, except that a 5 mw He-Ne laser coupled into a single mode optical fiber was used as a light source. This produced coherent but speckle-free illumination. The side walls of the cell were made of 6 sheets of paper. The paper was pliant, allowing an adjustment of the cell height. When the paper side walls were not compressed, the fringe pattern fluctuated by about 1/5 of a fringe, suggesting a movement of the top plate due to the pressure fluctuations caused by the impeller pump. The cell height was reduced and the paper sidewalls were compressed until these fluctuations ceased. The interferometric measurements gave a maximal height variation of 1µm over the whole diameter of the assembled cell. The cell s parallelism varied slightly over the 3.5 month duration of the experimental runs, however, it stayed well within the 1µm given above. The cell boundaries were constructed to force convection rolls parallel to the boundaries using a method similar to that in Ref. [12] (as described in Ref. [2]). As shown schematically in Fig. 1, the bottom two paper sheets had smaller apertures [25]. Since the paper was a better thermal conductor than the gas by approximately a factor of three, the two sheets caused a horizontal temperature gradient forcing upwelling hot fluid at the sidewalls. To allow the simultaneous investigation of circular patterns and spirals in cells of different aspect ratios, we designed a cell that had six convection cells with different diameters. A typical shadowgraph picture of the cells with convection patterns is shown in Fig. 2. The cells are numbered in order of decreasing size from 1 to 6. A shadowgraphy system similar to the one described in Ref. [2] was mounted on top of the vessel for visualization of the convective structures. The cleaved end of an optical fiber was used as a point light source in the focal plane of a 60cm telescope objective. For shadowgraphy, a 100W halogen lamp filtered by a dichroic filter was coupled into a multimode fiber. Collimated light passed through the cell and was reflected back through the telescope objective by the silicon bottom plate. The reflected light was collected by a zoom photo lens mounted in front of a CCD camera. The analog CCD camera was connected to a frame grabber for further digital analysis. While passing through the cell, light traversing cold fluid was focused while light traversing warm fluid was defocused, leading to the typical bright and dark shadowgraph pictures. (For details see Ref. [2].) 3 5 Figure 2: Shadowgraph picture of the convection cells at a reduced control parameter ε = 0.26 and Prandtl number σ = 1.39 We used compressed CO 2 gas as a working fluid. The pressure was held constant with the help of a second pressure vessel filled with liquid CO 2. It was temperature controlled to around 12 C by keeping the pressure measured by an electrical strain gauge [26] constant. With this method we were able to regulate the pressure to a shortterm stability of ±5x10-3 bar. However, over the 3.5 month period of the experiment, the strain gauge used to regulate the pressure drifted by 0.3 bar as determined by later measurements with a Heise Bourdon tube pressure gauge. The Prandtl number for the gas was σ 1.4 and the vertical thermal diffusion time was t v 2sec. We measured the onset of convection by quasistatically increasing the temperature of the bottom plate and thus the temperature difference T. As for any real fluid, one would expect hexagons at the onset of convection and not rolls [12]. Our experimental conditions and the fluid properties were such that we found a narrow range for hexago

3 B. B. Plapp and E. Bodenschatz, page 3 rial properties as obtained by a computer program which is described in Ref. [2]. As mentioned before, due to an unexpected drift of the pressure sensor, the pressure drifted slowly by 0.3 bar over the period of 3.5 months. Eighteen days into the experimental run we measured a critical temperature difference of (2.015 ± 0.005)K and later determined a pressure of (47.2 ± 0.1) bar. Using the method described at the beginning of the previous paragraph we calculated a theoretical temperature difference of (2.02 ± 0.07) K. We obtained equally good agreement for a second measurement after 87 days into the experimental run. We determined a pressure of (47.4 ± 0.1) bar and an experimental threshold of (1.960 ± 0.005)K, which should be compared with the theoretical value of (1.95 ± 0.07)K. The agreement between the experiment and theory is remarkable, showing that the material properties program [2] gives excellent results even at high pressures close to the liquidus line of CO 2. Figure 3: Shadowgraph picture of cell #1 at ε = and σ = 1.41 divided by a background image without convection. nal convection [29]. Just below the onset temperature difference T c, i.e., where the reduced control parameter ε = ( T- T c ) T c < 0, convection rolls parallel to the boundaries were forced, but the center of the cells remained quiescent. At onset, hexagons nucleated in the middle of the largest cell #1. However, none of the smaller aspect ratio cells showed hexagonal convection. We associate the appearance of hexagons with the onset of convection since hexagons (composed of three nonlinearly interacting rolls) are the preferred state at the true onset of convection for an infinitely extended system. To compare the experimentally observed critical temperature difference with the theoretical predictions, we first determined the cell height d and then used the formula for the critical Rayleigh number R c = ( αg T c d 3 ) ( νκ) = 1708 [6], where α, g, ν, and κ are, respectively, the thermal expansion coefficient, the gravitational acceleration, the kinematic viscosity, and the thermal diffusivity. We determined the unknown cell height d by measuring the wavenumber of both the hexagons and the circular rolls in the pattern shown in Fig. 3. Both hexagons and rolls gave the same roll wavenumber. By using the cell sizes determined from the paper cut-outs for reference, we transformed the wavenumber from the measured pixel values into physical units. Identifying the measured wavenumber with the critical wavenumber q c = 3.117/d [6], we then determined the height to be d = (447 ± 3) µm. Using this value and the formula for R c, we calculated the critical temperature difference using the mate- defect Figure 4: Shadowgraph picture of a two-armed spiral in cell #1 at ε = 0.88 and σ = The revolving defects and the spiral s core region are marked. 3. Targets and Spirals core defect A snapshot of a typical convection pattern showing targets and spirals is shown in Fig. 2. Cell #1 was filled with a clockwise rotating two-armed spiral, cell #2 with a counterclockwise rotating one-armed spiral, and cells #3 to #5 with target patterns. Cell #6 showed a one-armed spiral, which was not rotating, apparently pinned by the sidewalls [25]. The experimentally observed target patterns were stationary, while n-armed spirals were rotating. As an exam-

4 B. B. Plapp and E. Bodenschatz, page 4 ple, we have shown a clockwise rotating two-armed spiral in Fig. 4. Multi-armed spirals showed two types of periodic behavior. First, the most obvious was the slow largescale rotation, which can be described by outward travelling waves which are annihilated by defects revolving on circular trajectories [12][23]. In Fig. 4 these defects and their propagation direction are marked. Second, we observed a much faster spatio-temporal periodic behavior in a radial region of approximate size r nλ/2, where n is the number of spiral arms and λ is the wavelength of the pattern. This region is demarcated in white in Fig. 4 and is referred to in the remainder of the text as the spiral s core. We observed both clockwise and counterclockwise rotating spirals. If the sign of the topological charges of all defects in the example shown in Fig. 4 were flipped, a counterclockwise rotating spiral would result. We obtained spirals by two experimental procedures: (i) quasistatically increasing ε, and (ii) jumping from below the onset of convection to positive ε. (i): After passing the onset of convection, all cells initially developed target patterns, which with increasing ε changed their wavenumbers by annihilating rolls in the center of the target. With a further increase of ε, the target moved off-center [28], and the concentric pattern was compressed on one side and dilated on the other. In the compressed region, one skewed varicose instability [10] occurred, causing a roll pair to tear and produce two defects. One defect migrated to the center and the other moved out to a radius and revolved around the center, forming a single-armed spiral. The value of ε at which the target instability occurred increased with decreasing aspect ratio. With a further increase in ε, the single-armed spirals moved off-center in a manner very similar to the target instability. However, the occurrence of a skewed varicose event in the compression region led to the migration of the dislocations to the dilated region, further pushing the core to one side and leading to more skewed varicose instabilities. If the dilation became sufficiently large, cross roll instabilities occurred [10]. After a complicated spatio-temporal dynamics, rolls grew perpendicularly to the sidewalls, destroying the circular symmetry. The pattern then evolved into a PanAmlike structure as shown in Fig. 5. The PanAm patterns have been the subject of a number of earlier investigations [10][11][14]. At large ε, PanAm and SDC competed in a manner similar to that described in Ref. [14]. In another scenario the system retained its circular symmetry and single three- or four-armed spirals were formed. No stable two-armed spirals were observed. The fourarmed spiral was stable over a large ε-region [24]. It was destroyed by the development of a single-armed spiral in the core region which pushed the remaining three core defects to the side. A number of skewed varicose and Figure 5: PanAm-like convection pattern in cell #2 at ε = 0.88 and σ = cross-roll instabilities followed, finally leading to PanAm/ SDC patterns. The breakup of three-armed spirals followed a qualitatively similar process. (ii): Alternatively, ε was increased in a jump from below onset of convection (ε i < 0) to above (ε f > 0). Starting from the same value below the onset of convection, ε i = -0.05, we observed targets for 0 < ε f < 0.13 and a preference for spirals in the range 0.13 < ε f < Jumps to 0.83 < ε f < 1.08 led to two- or three-armed spirals, SDC, or PanAm patterns. A detailed discussion of the dynamics of single-armed spirals and the comparison with the predictions by Ref. [23] is given in [27]. Here, we restrict our discussion to the details of the core motion of multiarmed spirals. 4. Two-armed Spirals A typical picture of a two-armed rotating spiral is shown in Fig. 6. Two defects were bound in the center while two other defects of opposite topological charge revolved on a circular trajectory around them. As mentioned above, we obtained two-armed rotating spirals by control parameter jumps. We did not observe stable two-armed rotating spirals by quasistatically increasing the temperature difference. We observed many different positions of the two outer defects with respect to each other; however, we never observed the defects to be aligned along one radius. If the two defects were opposite each other, the spiral rotated around the geometric center. In the other cases the spiral s core was pushed slightly off-center towards the side opposing the revolving defects. In Fig. 7 the fast time evolution of the core is shown for

5 B. B. Plapp and E. Bodenschatz, page 5 the slower large-scale spiral rotation; already beyond a radial distance r λ the pattern was dominated by the large-scale rotation. For the two-armed rotating spiral shown in Fig. 4, where the revolving defects were adjacent to each other, we found τ L = (274.1 ± 0.5) t v and a core period of τ C = (8.4 ± 0.2) t v with t v = 2.03 sec, i.e., τ L /τ C = (32.6 ± 0.8). The core motion was similar to that shown in Fig. 7. Figure 6: Shadowgraph picture of a two-armed, clockwise rotating spiral in cell #1 at ε = 0.98 and σ = Figure 8: Shadowgraph picture of a three-armed, counterclockwise rotating spiral in cell #2 at ε = 0.79 and σ = Figure 7: Shadowgraph pictures of the spatio-temporal evolution of the core of the two-armed rotating spiral shown in Fig. 6. The pictures are spaced 0.77 t v apart which, within the 1/30 sec resolution of the CCD camera, corresponds to 1/8 of a period. the two-armed spiral of Fig. 6. The core region is marked schematically by a white circle in picture #1. The core cycled with a period of τ C = (6.14 ± 0.01) t v, where t v = 2.04 sec. In comparison, the large-scale rotation period was τ L = (223.0 ± 0.5) t v. The core period was a factor τ L / τ C = (36.3 ± 0.1) faster than the spiral itself; the core motion was not commensurate with the large-scale spiral rotation. The core appeared to be quite independent from 5. Three-armed Spirals The shadowgraph image in Fig. 8 shows a counterclockwise rotating three-armed spiral. In a manner analogous to the two-armed spiral, three bound defects were in the spiral s core and three defects of the opposite topological charge revolved around them so as to eliminate the spiral waves propagating from the center. We also observed stable three-armed spirals in which the revolving defects were not aligned as symmetrically. The spirals cores showed a fast spatio-temporal oscillation in which the three bound defects alternately touched each other. The fast motion was restricted within a radius r 3λ/2. We observed stable rotating three-armed spirals by control parameter jumps. In one case we observed a threearmed spiral in cell #2 that was the result of the instability of a one-armed spiral when increasing ε quasistatically. This spiral unwound over the time 1300 t v, with two of the outer defects moving to the center. However, the unwinding three-armed spiral also showed qualitatively the same core motion. In Fig. 9 the time evolution of the core in Fig. 8 is shown over one period. We found for the core τ C = (4.8 ± 0.2) t v

6 B. B. Plapp and E. Bodenschatz, page Figure 9: Shadowgraph pictures of the time evolution of the spiral s core of the counterclockwise rotating three-armed spiral shown in Fig. 8. The pictures are spaced 0.44 t v apart Figure 11: Shadowgraph pictures of the time evolution of the spiral s core of the counterclockwise rotating four-armed spiral shown in Fig. 10. The pictures are spaced 2.65 t v apart. Prandtl number was σ = Figure 10: Shadowgraph picture of a four-armed, counterclockwise rotating spiral in cell #2 at ε = 0.78 and σ = and for the large scale rotation τ L = (391 ± 2) t v, giving a ratio of τ L /τ C = (81 ± 3). In this case, t v = 2.06 sec and the 6. Four-armed Spirals We observed four-armed spirals both as a result of the one-armed spiral instability and of control parameter jumps. Four defects were bound in the core and four defects with opposite topological charge revolved around the core so as to eliminate the outward travelling spiral waves. A snapshot of a four-armed spiral is shown in Fig. 10. We also observed rotating four-armed spirals where the revolving defects were not aligned symmetrically and the core motion was qualitatively similar. In the case of the counterclockwise rotating four-armed spiral shown in Fig. 10, the core s fast spatio-temporal

7 B. B. Plapp and E. Bodenschatz, page 7 oscillation is shown in Fig. 11. During one large-scale rotation period τ L the core cycled (15.00 ± 0.04) times; the core was almost synchronized to the large-scale rotation. The large-scale rotation period was τ L = (636.0 ± 0.5) t v, where t v = 2.08 sec, and the core period was τ C = (42.4 ± 0.1) t v. We are currently investigating this interesting state further. 7. Conclusion We have presented the first results on the core dynamics of spirals in Rayleigh-Bénard convection in fluid with Prandtl number 1.4. Our cell wall design forced rolls parallel to the boundaries leading to the selection of circular patterns. Within a certain temperature difference range we found stable multiarmed rotating spirals. We reported on the intriguing dynamics of the spirals cores. Surprisingly, for the observed spirals, the core motion was not necessarily synchronized with the large-scale spiral rotation. Acknowledgments We have benefitted from discussions with G. Ahlers, W. Bäuml, D. Egolf, and W. Pesch, as well as from the contributions by the technical personnel of the Laboratory of Atomic and Solid State Physics. We are very grateful to R. Ragnarsson and C. Santangelo, who provided essential programming for the data analysis and the operation of the experiment. The apparatus was based on the original design used at the University of California at Santa Barbara [2]. This work was supported by the National Science Foundation under Contract No. DMR B.B.P. acknowledges support from the Department of Education and E.B. from the Alfred P. Sloan Foundation. [11] Croquette, V., Le Gal, P., Pocheau, A. and Guglielmetti, R., Europhys. Lett. 1, 393 (1986). [12] Bodenschatz, E., debruyn, J. R., Ahlers, G. and Cannell, D. S., Phys. Rev. Lett. 67, 3078 (1991). [13] Morris, S. W., Bodenschatz E., Cannell, D. S. and Ahlers, G., Phys. Rev. Lett. 71, 2026 (1993). [14] Hu, Y., Ecke, R. E. and Ahlers, G., Phys. Rev. E 48, 4399 (1993), Phys. Rev. Lett. 74, 391 (1995) and Phys. Rev. E 51, 3263 (1995). [15] Ecke, R. E., Hu, Y., Mainieri, R. and Ahlers, G., Science 269, 1704 (1995). [16] Assenheimer, M. and Steinberg, V., Phys Rev. Lett. 70, 3888 (1993) and Nature 367, 347 (1994). [17] Bestehorn, M., Frantz, M., Friedrich, R., Haken, H., and C. Pérez- García, Z. Phys. B 88, 93 (1992). [18] Xi, H.-W., Gunton, J. D. and Vinals, J., Phys. Rev. E 47, R2987 (1993). [19] Xi, H.-W., Gunton, J. D. and Vinals, J., Phys. Rev. Lett. 71, 2030 (1993). [20] Bestehorn, M., Frantz, M., Friedrich, R. and Haken, H., Phys. Lett A 174, 48 (1993) [21] Decker, W., Pesch, W. and Weber, A., Phys. Rev. Lett. 73, 648 (1994). [22] Plapp, B. B. and Bodenschatz, E., unpublished. [23] Cross, M. C. and Tu, Y., Phys. Rev. Lett. 75, 834 (1995). [24] Plapp, B. B. and Bodenschatz, E., in preparation. [25] The step widths varied as follows: cell #1: 1.2mm to 0.9mm; #2: 0.5mm to 0.4 mm; #3: 1.1mm to 0.6mm; #4: 0.8mm to 0.5mm; #5: 0.8mm to 0mm; #6: 0.4mm to 0mm. With a cell height of 447 µm, the cells radial aspect ratios Γ=r/d were: #1: 48.4; #2: 38.8; #3: 28.4; #4: 23.8; #5: 19.1; #6: We excluded from this report cells #5 and #6, for which the radial variation significantly influenced the pattern evolution. [26] Super TJE, Sensotec, Columbus, OH. [27] Plapp, B. B., Bäuml, W., Egolf, D. A., Pesch, W. and Bodenschatz, E., in preparation. [28] This is similar to the target instability; see, for example, Newell, A. C., Passot, T. and Souli, M., J. Fluid Mech. 220, 187 (1990). [29] For this experiment the non-boussinesq number P 0.7, which is much smaller than P 3.4 of the prior experiment on non- Boussinesq convection [12]. Hexagons were replaced by rolls for ε > References [1] Weber, E. H., Ann. Phys. Chemie 94, 447 (1855). [2] For a comprehensive recent review on gas convection experiments, see debruyn, J. R. et al., Apparatus for the Study of Rayleigh- Bénard Convection in Gases under Pressure, submitted to Rev. Sci. Inst. [3] For a comprehensive review on pattern formation, see Cross, M. C. and Hohenberg, P. C., Rev. Mod. Phys. 65, 851 (1993). [4] The Prandtl number σ = t κ t ν, where t κ = d 2 κ is vertical thermal diffusion time, t ν = d 2 ν the viscous relaxation time, and d, κ, and ν are the cell height, the thermal diffusivity, and the kinematic viscosity, respectively. [5] Clever, R. M. and Busse, F. H., J. Fluid Mech. 65, 625 (1974). [6] See, for example: Getling, A. V., Uspekhi Fizicheskikh Nauk 161 no-9, 1 (1991) (Sov. Phys.Usp. 34, 737 (1991)). [7] Ahlers, G. and Behringer, R. P., Phys. Rev. Lett. 40, 712 (1978) and Behringer, R. P. and Ahlers, G., J. Fluid Mech. 125, 219 (1982). [8] Ahlers, G., Cannell, D. S. and Steinberg, V., Phys. Rev. Lett. 54, 1373 (1985). [9] Heutmaker, M. S. and Gollub, J. P., Phys. Rev. A 35, 242 (1987). [10] Croquette, V., Contemp. Phys. 30, 153 (1989) and references therein.