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Hopf Bifurcation with Broken Reflection Symmetry in Rotating Rayleigh-Bénard Convection
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1992 Europhys. Lett. 19 177
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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.
178
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.
180
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.
181
(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 ~ < <and
1 g may be expanded in a Taylor series:
U = Ea
+ alal2a + ... .
(4)
Writing a = A exp [i@],the onset of a steady-state instability is described by equations of the
form
A =&A+orA3+
...,
d =o.
(5)
The second equation is a consequence of neutral stability of the pattern with respect to
rotations.
Now suppose that the apparatus rotates with a small angular velocity SZ. This has the
effect of weakly breaking the reflection symmetry but not the rotation symmetry. The
coefficients in eq. (4) consequently acquire nonzero imaginary parts:
U = (E
+ iSZ6)a + ( a + inp)la12a +
.")
(6)
where now E , 6, a, are all functions of Q2 that can, in principle, be calculated from the
hydrodynamical equations [ll]. In terms of the real variables we now have
+ aA3 + ".,
d = Q(6 + PA2 + ...).
A = &A
(7)
(8)
Equations (7), (8) are the usual normal-form equations for a Hopf bifurcation. Thus what was
a steady-state bifurcation in the nonrotating system has become a Hopf bifurcation in the
rotating one. The only difference is that d is here identified with the rate of change of the
azimuthal phase, i.e. it is the precession frequency cop in the rotating frame. The field 0 takes
the form
0 = %{AexpLi(m4 +
t)]fm (r,z ) }
+ ... .
(9)
The bifurcation is thus to a rotating wave[12]. A supercritical steady-state pattern with
182
EUROPHYSICS LETTERS
amplitude A 2 = - & / a+ 0(t2),
a < 0, precesses with the frequency
(
o p = Q6 -
"1
a& + 0 ( & 2 ) .
(10)
Thus the theory predicts that (near onset) the precession frequency depends linearly on E and
has a finite intercept oo= 652. Since generally 6(0) f 0, oo= 6(0)52 O(Q3), and vanishes
linearly with 52. These predictions are fully corroborated by the experiments. In particular
we find from the experiments that 6(0) = 0.1 and P(O)/a(O) = - 5. These values are limited
somewhat by our inability to probe close to 52 = 0. We are in the process of improving on this
limitation by increasing the aspect ratio by a factor of 2.5.
We conclude that the onset of Rayleigh-B6nard convection in rotating cylindrical
containers with radius-to-height ratio r = 1 is a Hopf bifurcation. The instability gives rise to
rotating (traveling) waves with a preferred direction of precession selected by the broken
reflection symmetry, and determined by the sign of 6(0). The precession develops
continuously as 52 is increased. The accompanying theory shows that provided m f 0 a Hopf
bifurcation to rotating waves is expected regardless of the Prandtl number of the fluid. An
important question for future investigation concerns the dependence of the above results on
the cell aspect ratio: does the traveling-wave frequency go continuously to zero as r increases
at fured 52, or is there a sharp transition at a finite aspect ratio? Finally, multiple wave
interactions at finite E should also provide an interesting subject for future study.
+
***
We acknowledge the assistance of T. SULLIVAN
and V. STEINBERG
and useful discussions
with G. AHLERS.We also thank R. GOLDSTEIN,
I. MERCADER
and M. NET for allowing us to
show their computational results prior to publication. This work was funded by the United
States Department of Energy and by the University of California Nuclear Science Fund
Programs for Institutional Collaborative Research.
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