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Hopf Bifurcation with Broken Reflection Symmetry in Rotating Rayleigh-Bénard Convection

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1992 Europhys. Lett. 19 177 (http://iopscience.iop.org/0295-5075/19/3/005) View the table of contents for this issue, or go to the journal homepage for more

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EUROPHYSICS LETTERS

1 June 1992

Europhys. Lett., 19 (31, pp. 177-182 (1992)

Hopf Bifurcation with Broken Reflection Symmetry in Rotating Rayleigh-B6nard Convection. R. E. EcKE(*), FANG ZHONG(*)(§) and E. KNOBLOCH(**) (*) Physics Division and Center for Nonlinear Studies Los Alamos National Laboratory - Los Alamos, N M 87545 (**) Department of Physics, University of Calgornia - Berkeley, C A 94720 (received 5 December 1991; accepted in final form 27 April 1992) PACS. 47.20 - Hydrodynamic stability and instability. PACS. 47.25 - Turbulent flows, convection, and heat transfer.

Abstract. - Experimental observations of azimuthally traveling waves in rotating RayleighBBnard convection in a circular container are presented and described in terms of the theory of bifurcation with symmetry. The amplitude of the convective states varies as fi and the traveling-wave frequency depends linearly on E with a finite value at onset. Here E R/Rc - 1, where R, is the critical Rayleigh number. The onset value of the frequency decreases to zero as the dimensionless rotation rate SZ decreases to zero. These experimental observations are consistent with the presence of a Hopf bifurcation from the conduction state expected to arise when rotation breaks the reflection symmetry in vertical planes of the nonrotating apparatus.

Rayleigh-Bhard convection has been a model system for the study of nonlinear phenomena including bifurcations, routes to chaos and spatial pattern dynamics [1-31. In certain circumstances (see, for example, binary fluid convection a t negative separation ratios [3-5]), oscillatory convection can set in at onset in the form of traveling waves. Here we report an unexpected time dependence in rotating Rayleigh-B6nard convection for a cylindrical convection cell with radius-to-height ratio r = 1. Linear stability analysis for this geometry predicted an azimuthally periodic convective structure localized near the lateral boundary [6]. We have observed flows of this type using optical shadowgraph visualization of the temperature field. It was seen, however, that the azimuthally periodic mode precessed in the rotating frame. In a laterally unbounded domain, theoretical considerations [7] show that a Hopf bifurcation from the conduction state is possible only in low Prandtl number fluids. For the fluid used (water) no oscillations were therefore expected. We show visualization and local temperature measurements that indicate that the bifurcation to time dependence is a Hopf bifurcation and present the relevant theoretical analysis by which the transition can be understood. Previous experiments [8] using water could not have detected this transition,

(§)

Present address: Department of Physics, Duke University, Durham, NC 27706.

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EUROPHYSICS LETTERS

Fig. 1. - Schematic illustration of the convection cell showing the location of the local temperature sensors.

since global heat transport measurements alone cannot distinguish between a stationary state and a uniform traveling-wave state. The experimental system is Rayleigh-Benard convection with rotation about a vertical axis. The convection cell is cylindrical with radius = 5 cm and height d = 5 em, and the convecting fluid is water with a Prandtl number Pr = V / K = 6.4, where v is the kinematic viscosity and K is the thermal diffusivity. Determination of the state of the fluid consists of global heat transport measurements which determine the convective onset, observations of the temperature field using the optical shadowgraph technique to establish the spatial structure of the flow, and two sensors that probe the temperature a t points near the lateral wall. The local probe signal is used to determine the amplitude, frequency, and spatial mode number of the convective traveling wave. A schematic illustration of the convection cell in fig. 1 shows the position of the local probes. The frequency of the traveling wave is determined by the analysis of the time series from either sensor. Knowing the frequency o, the physical angle 6 separating the probes, and the phase difference Ay between the probe signals, the mode number m of the wave can be calculated. Finally the heat transport is determined by the heat input and the temperature difference AT across the cell with corrections included for parasitic heat conduction. The control parameters for this problem are the Rayleigh number R 3 gad3AT/vrc, where g is the acceleration of gravity, x is the thermal expansion coefficient and AT is the temperature difference across the fluid layer, and the dimensionless rotation rate

"!I-----Fig. 2. - Nusselt number

R. E. ECKE

et al.: HOPF

BIFURCATION WITH BROKEN REFLECTION SYMMETRY ETC.

179

Fig. 3. - Stability diagram in R us. 52 parameter space. Solid line shows the prediction of linear stability calculations for the laterally infinite system (Chandrasekhar [7])and the dashed line indicates linear analysis for asymmetric states in a r = 1 cylindrical container with insulating sidewall boundary conditions and Pr = 6.7 [ll].Data show the convective onset (0) and the onset of noisy time dependence (0).

Fig. 4. - Shadowgraph image of the m = 5 state for 52 = 2145 and E = 2.6. The entire pattern precesses in the rotating frame at constant velocity.

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EUROPHYSICS LETTERS

R = RDd2/v, where RD is the physical angular rotation frequency and d 2 / v = 2711 s. We define a reduced bifurcation parameter E = (R- R,(Q))/R,(R), where R,(Q) is the critical value of R a t fEed Q. From the measurement of heat transport (see fig. 2) the onset values R,(Q) are determined much more accurately than is possible using shadowgraph visualization which is limited to E > 0.3. Centrifugal effects are negligible for this experiment, since the ratio of centrifugal-to-gravitational forces, Sa: r/g, is 0.005. Rotation generally suppresses the onset of convection but in small-aspect-ratio cylindrical cells, the onset is substantially shifted to smaller values of R, (see fig. 3) because the onset state is an azimuthally periodic state localized near the lateral boundaries [9,10] instead of the spatially homogeneous planform assumed in the theory for a laterally unbounded system. Figure 4 shows a state with 5-fold periodicity (azimuthal wave number m = 5) for Q = 2145 and E = 2.6. This particulr state is quite far above onset and the structure has grown into the central region but the azimuthally periodic sidewall structure is clearly visible. The entire structure precesses uniformly in the rotating frame. Other states with m = 3 , 4 , 6 and 7 have been observed for different values of Q and/or initial conditions [9-111. An interesting property of these states is that they propagate in the rotating frame, always in a direction opposite to the rotation direction. The question then arises whether this transition to time dependence is a Hopf bifurcation or not. We begin with our experimental characterization of the transition to such a precessing wave. Using the local sensors we have determined the Rayleigh number dependence of its amplitude and frequency. The mode amplitude varies as fi (see fig. 5a)) and the Nusselt number, which is expected to behave like the square of the amplitude, varies linearly with E , fig. 5b). Further, the frequency varies linearly with E and has a finite intercept coo a t onset 1 " " I " " I ' ~ ~ '

21 "

E

I*[, 0

,

I,,

0.0

,

, , I,,, , ,,,

,c;,

,I

0.05 0.10 0.15 &

Fig. 5 . - Plot of the a) amplitude, b) Nusselt number, and c) frequency, U ,ws. E close to onset. The linear dependences of the frequency and Nusselt number and the square-root dependence of the amplitude indicate a Hopf bifurcation. The E = 0 intercept of w is denoted ooand is shown vs. Q in d). The behavior of wo is consistent with a linear relationship for Q < 100. The precession frequency calculated from linear theory [ll]is shown for comparison (--). The discontinuities in the theoretical curve reflect changes in the preferred azimuthal wave number ( 0 r =1).

R. E. ECKE

et al.:

HOPF BIFURCATION WITH BROKEN REFLECTION SYMMETRY ETC.

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(see fig. 5c)), clearly indicating a forward Hopf bifurcation. Finally, in fig. 54, we plot the frequency at onset wo 'us. SZ. Although we cannot accurately determine the precession for SZ < 150 (for this rotation rate and for the cell depth of 5 em, ATc = 5mK) the data are consistent with wo vanishing with Q like coo = 652, where 6 is a constant. We now show that in a rotating circular container one expects a Hopf bifurcation from the conduction state whenever the instability breaks azimuthal symmetry (i.e. has a nonzero azimuthal wave number m).In such a system any field (e.g., temperature perturbation from the conduction state) may be written near onset in the form

@(r,# , x , t ) = %{a(t)exp [im41fm(r,x ) }

+ . . .,

m f 0,

(1)

where (r,4,x ) are cylindrical coordinates,f, (r,x ) is the eigenfunction of the mode m, and a(t) is its complex amplitude. When the container is nonrotating and the boundary conditions are homogeneous in 4 the equation satisfied by a must commute with the symmetries

4 + 4 + q50 : a reflection 4 +. - 4:

rotations

a exp [imdol,

(2)

a +E .

(3)

--$

It follows that U = g( 1 a I 2 , E) a, where the function g is forced by the reflection symmetry to be real. Near onset ~ <