Nous étudions la transition vers le chaos de l’écoulement entre deux plaques verticales différentiellement chauffées.  Nous nous appuyons sur la simulation numérique directe (DNS) pour étudier le comportement spatio-temporel de l’écoulement en faisant augmenter le nombre de Rayleigh.  A partir du régime de conduction pure, l’écoulement devient temporellement mono-périodique,  après avoir traversé trois bifurcations supercritiques dont deux du type de fourche et une du type de Hopf.  Si on restreint le domaine de simulation afin d’admettre une seule longueur critique dans la direction verticale, une cascade de doublement de période est observée. Le comportement temporel de l’écoulement peut être décrit à l’aide d’une itération unidimensionnelle, et correspond physiquement à la déformation périodique d’une structure tourbillonnaire dite en œil de chat. Le régime du chaos temporel est atteint à Ra_infini estimé à 12330. La constante de Feigenbaum est  calculée à partir des premiers doublements de période et un accord relatif est obtenu avec la valeur théorique.  Au-delà de Ra_infini, des fenêtres correspondant à des régimes périodiques apparaissent à nouveau. Quand Ra est augmenté au delà de Ra = 15500, une intermittence est  observée entre deux structures en œil de chat, séparées d’une demi-période.  Les observations de l’écoulement sont cohérentes avec un scénario de crise par fusion, et réminiscentes de cycles hétéroclines structurellement stables dus à la présence d’une symétrie O(2). Le temps caractéristique séparant deux  passages d’une structure à une autre  obéit à une loi exponentielle

GAO, Zhenlan

LE QUERE, Patrick

PODVIN, Berengère

SERGENT, Anne

XIN, Shihe

I-Revues

21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
Period-doubling scenario and crisis-induced
intermittency in natural convection between two vertical
differentially heated plates
Zhenlan Gaoa,b, Be´renge`re Podvina, Anne Sergenta,b, Shihe Xinc, Patrick Le Que´re´a
a. CNRS, LIMSI, UPR3251, BP 133, 91403, Orsay Cedex, France
b. Universite Pierre et Marie Curie-Paris 6, 4, Place Jussieu, 75252 Paris, Cedex 05, France
c. CETHIL, INSA de Lyon, 69621 Villeurbanne Cedex, France
Re´sume´ :
Nous e´tudions la transition vers le chaos de l’e´coulement d’air entre deux plaques verticales differen-
tiellement chauffe´es. Le domaine bi-pe´riodique de simulation est restreint a` de petites longueurs de
pe´riodicite´. Le comportement spatio-temporel de l’e´coulement en fonction du nombre de Rayleigh (Ra)
est e´tudie´ par simulation nume´rique directe 3D (DNS). La route vers le chaos se fait par une cascade de
bifurcations sous-harmoniques. Des feneˆtres de re´gimes multi-pe´riodiques et une crise d’e´largissement
de l’attracteur sont e´galement observe´es. Quand Ra augmente, une intermittence se produit corre-
spondant a` une crise par fusion de deux attracteurs. Pour des valeurs de Ra supe´rieures, une feneˆtre
pe´riodique apparaˆıt, suivie ensuite d’un retour a` un re´gime chaotique et intermittent.
Abstract :
The chaotic regime of the natural convection of air between two vertical plates maintained at different
temperatures is studied. The periodic dimensions of the plates are relatively small. Direct numerical
simulation (DNS) is used to study the spatio-temporal behavior of the flow, as the Rayleigh number
(Ra) increases. The flow becomes temporally chaotic through a period-doubling cascade. Chaos then
becomes more developed, and windows of multi-periodic regimes, crises are observed. As the Rayleigh
is further increased, intermittency is observed, and is seen to correspond to an ”attractor-merging”
crisis. For still higher values of Ra, a periodic regime is observed, which then gives way to a fully
chaotic and intermittent regime.
Mots clefs : natural convection ; period-doubling scenario ; crisis-induced intermit-
tency
1 Introduction
The flow between two vertical differentially heated plates can be considered as a simplified model for a
number of industrial applications, such as the double-panned windows, heat exchangers in the reactors.
Transition to chaos of the flow in this configuration has been studied by analytical, experimental, and
numerical means since Batchelor’s pioneering work[1]. Most analyses have been carried out in the
2D configurations[2, 3]. Some 3D studies[4, 5] have been limited to the first few instabilities, when
the Rayleigh number remains close to its critical value. We have recently begun an investigation of
this problem using 3D DNS, where only the dependence of air flow on Ra is considered[6]. It has
been observed that although the first instability of the flow is 2D and steady, the steady flow rapidly
becomes 3D, through a second supercritical pitchfork bifurcation. The flow structures observed consist
of (i) primary rolls which are deformed in the transversal directions, (ii) braids or counter-rotating
vortices playing the role to link the primary rolls, and (iii) two small counter-rotating vortices inside
each primary rolls as shown in Figure 1 (b). After a supercritical Hopf bifurcation, the 3D flow
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
becomes oscillatory. A significant exchange of vorticity is detected between the primary rolls and
braids, which are oscillating out of phase. When the computational domain accommodates only one
primary roll in the vertical direction, a cascade of period-doubling bifurcations is observed as the
Rayleigh number increases, and eventually leads to temporally chaotic flow. For larger domains, this
behavior is superseded by a spatial modulation instability in the vertical direction. In the present
paper, we will focus on the case of a small domain. Similar behavior have been reported in 2D for
laterally heated cavities, as in the case of a vibrational cavity heated from the sides [7], where a
subharmonic cascade, followed by a Pomeau-Manneville type-I intermittency was observed. Chaos has
also been observed in 3D cavities at much higher Rayleigh numbers Ra > 108 [8, 9, 10]. We note that
in our case, periodic boundary conditions in the vertical direction replace horizontal wall boundary
conditions. We first describe the numerical configuration, then show simulation results.
2 Physical model and numerical methods
We consider the flow of air between two infinite vertical plates maintained at different temperatures.
The configuration is represented in Figure 1. The distance between the two plates is D, and the period
height and depth of the plates are Lz and Ly respectively. The temperature difference between the
two plates is ∆T . The direction x is normal to the plates, the transverse direction is y, and the gravity
g is opposite to the vertical direction z.
(a) (b)
Figure 1 – (a) Study domain (b) Q-criterion visualization of flow structure at Ra = 11500, Q = 0.12
The fluid properties of air are the kinetic viscosity ν, thermal diffusivity κ, and thermal expansion
coefficient β. Four nondimensional parameters characterizing the flow are chosen in the following way :
the Prandtl number Pr =
ν
κ
, the Rayleigh number based on the width of the gap between the two
plates Ra = gβ∆TD
3
νκ
, and the transverse and vertical aspect ratio Ay = Ly/D and Az = Lz/D,
respectively. Only the Rayleigh number is varied in the present study. The Prandtl number of air is
fixed to 0.71. The transverse aspect ratio is set to be Ay = 1, the vertical aspect ratio is set to Az = 2.5,
which corresponds to the critical wavenumber kc ∼ 2pi/2.5 obtained by the stability analysis[2, 3, 6].
2.1 Equations of motion
The flow is governed by the Navier-Stokes equations within the Boussinesq approximation. D, ∆T ,
κ
√
Ra/D are chosen as the references for length, temperature and velocity, respectively. The nondi-
mensionalized equations are :
∇ · −→u = 0 (1)
∂−→u
∂t
+−→u · ∇−→u = −∇p˜+ Pr√
Ra
∆−→u + Prθ˜−→z (2)
∂θ˜
∂t
+−→u · ∇θ˜ = 1√
Ra
∆θ˜ (3)
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
with Dirichlet boundary conditions at the plates
−→u (0, y, z, t) = −→u (1, y, z, t) = 0, θ˜(0, y, z, t) = 0.5, θ˜(1, y, z, t) = −0.5 (4)
and periodic conditions in the y and z directions. Here t denotes time, −→u = (u˜, v˜, w˜) is the velocity
vector, p˜ is the pressure, θ˜ is the temperature. The equations (1)-(4) admit an O(2)×O(2) symmetry.
One O(2) symmetry corresponds to the translation in the transverse direction y and the reflection
y → −y, while the other corresponds to the translations in the vertical direction z and a reflection
that combines centrosymmetry and Boussinesq symmetry : (x, z, T )→ (−x,−z,−T ).
2.2 Numerical methods
A spectral code [11] developped at LIMSI is used to carry out the simulations. The spatial domain is
discretized by the Chebyshev-Fourier collocation method. The projection-correction method is used
to enforce the incompressibility of the flow. The equations are integrated in time with a second-order
mixed explicit-implicit scheme. A Chebyshev discretization with 40 modes is applied in the direction
x, while the Fourier discretization is used in the transverse and vertical directions. 30 Fourier modes
are used in the transverse direction y for Ay = 1, while 60 Fourier modes are used in the vertical
direction z for Az = 2.5. Convergence of the spatial discretization has been established [6]. We run
our simulations by following a branch of stable solutions. An instantaneous flow realization in the
mono-periodic regime at Ra = 11300 is taken as the initial condition for the initial run. For each
simulation, the Rayleigh number is increased by small increment of 2, compared to the previous one.
The data is sampled when the asymptotic chaotic regime has been reached, that is to say transient
effects have faded after long time numerical integrations (about 104 nondimensional time units). At
the end of each run, the asymptotic solution obtained at a ginve Rayleigh number is used as the initial
condition for the simulation at the next higher Rayleigh number.
1.2 1.21 1.22 1.23 1.24 1.25
x 104
0.454
0.456
0.458
0.46
0.462
0.464
Ra
te
m
pe
ra
tu
re
0.25 0.3 0.35 0.4 0.45 0.5
0.25
0.3
0.35
0.4
0.45
0.5
(a) (b)
Figure 2 – (a) : bifurcation diagram obtained by using the local maxima of the temperature timeseries
at the point (0.038 0.097 0.983). Ra ∈ [12000, 12500] ; (b) : first return map obtained by using the
local maxima of the Fourier mode k = 1 timeseries, calculated from the temperature distribution on
the vertical line x = 0.0381 and y = 0.5.
3 Results
3.1 Period-doubling cascade
After having experienced two supercritical pitchfork bifurcations at Ra = 5708 and Ra = 9980 through
which the flow becomes 2D steady then 3D steady, a supercritical Hopf bifurcation occurs at Ra =
11270, as the flow becomes temporally mono-periodic. As the Rayleigh number increases, a sequence
of period-doubling bifurcations is observed, which leads to temporal chaotic flow [6]. A bifurcation
3
21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
Table 1 – Summary of period-doubling bifurcations
Bifurcations 2i → 2i+1 Local critical Ra2i→2i+1 Estimated Feigenbaum constant δ˜
0-1 11270
1-2 12068.09
2-4 12258.42 4.193
4-8 12305.76 4.020
8-16 12316.72 4.321
diagram of Figure 2 (a) is constructed by the using local maxima Tn of the temperature timeseries at
the point (0.038 0.097 0.983), which is located in the boundary layer near the hot wall. It is found to
be quite similar to that of one-dimensional maps, like the logistic map[13]. 1-D maps with a quadratic
maximum are expected to follow a Feigenbaum scenario [12], and this is what we find in the simulation.
The local critical Rayleigh numbers Ra2i→2i+1 (i = 0, 1, 2, 3, ...) corresponding to the onset of a 2
i-
periodic regime are estimated from linear extrapolation. From these estimations, approximations for
the Feigenbaum constant are calculated and listed in Table 1. Some agreement with the theoretical
value δ = 4.66920161.... [13] is observed. Using the theoretical value for the Feigenbaum number, the
chaotic regime is estimated to be reached around Ra ∼ 12320. For higher Rayleigh numbers, the chaos
continues to develop as shown in the bifurcation diagram. Several periodic windows with numbers of
cycles 12, 10, 6, 5, etc, are observed. For example, a large ’period-6 windows’ (a period-3 windows in
each band) appears at Ra = 12350 in the bifurcation diagram. It then undergoes a period-doubling
cascade in which orbits of period 3× 2m are successively produced This cascade once again leads the
flow to a chaotic behavior. For still higher Rayleigh numbers, the attractor abruptly widens into two
large bandes similar in size to that before the stable ’period-6 ’ orbit came into existence. Finally, the
two large bands merge at around Ra = 12380 and form a large single chaotic band. In this chaotic
regime, all the points fall on a single curve with 3 branches in the first return map at Ra = 12380
as shown in Figure 2 (b), which is similar to the case of Lorenz map [14]. For this range of Ra, the
spatial structure remains similar to that observed in the mono-periodic regime[6], but is observed to
pulsate in time in a quasi-regular fashion.
3.2 Crisis-induced intermittency
The bifurcation diagram for the range Ra ∈ [12400, 12600] is represented in Figure 3(a). A rectangular
area filled with points abruptly appears on the top-right corner of the figure for Ra = 12546, which is
the sign of a crisis [15]. Above this Rayleigh number, intermittent behavior is observed, as is indicated
in Figure 3 (b) which represents the temporal evolution of the first Fourier mode k = 1 of the
temperature distribution on the vertical line (x = 0.0381 and y = 0.5). This evolution is characterized
by switches between chaotic states occurring randomly. Streamlines plots in Figure 4 (a)-(c) show
that the switches correspond to vertical shifts of the flow structure by a half-wavelength Az/2. This
symmetry is apparent in the phase portrait (Figure 4 (d)). The disappearance and formation of flow
structures at two locations separated by half a wavelength Az = 2.5 is reminiscent of heteroclinic
connections between diametrially opposed fixed points in systems with O(2) symmetry. Structurally
stable heteroclinic connections between fixed points or periodic solutions have been shown to exist
in the systems with O(2) symmetry[16, 17]. However, we are not aware of equivalent theoretical
results for heteroclinic connection between strange attractors. As Ra is further increased beyond
about Ra = 13000, a new periodic regime is observed (Figure 5 (a)). When Ra is further increased to
Ra = 14000, the temporal behavior of the flow becomes chaotic again, and a new intermittent regime
is observed as indicated by the timeseries of Fourier mode Re(Tˆ (1)) in Figure 5 (b).
4 Conclusions
The dynamics of the natural convection of air between two vertical plates maintained at different
temperatures have been studied in a domain of small periodic dimensions. Temporal chaos occurs
4
21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
1.24 1.245 1.25 1.255 1.26
x 104
0.44
0.445
0.45
0.455
0.46
0.465
0.47
Ra
te
m
pe
ra
tu
re
0 1000 2000 3000 4000 5000 6000−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Time
R
e
(Tˆ
(1
))
(a) (b)
Figure 3 – (a) : Bifurcation diagram for Ra ∈ [12400, 12600] ; (b) : Real part of the temporal evolution
of the Fourier mode Tˆ (k) (k = 1) calculated on the line (x = 0.0381 and y = 0.5), Ra = 12600.
X
Z
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
X
Z
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
X
Z
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
−0.04 −0.03 −0.02 −0.01 0 0.01 0.02−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Re(Tˆ (2))
R
e
(Tˆ
(1
))
(a) t=8500 (b) t=8525 (c) t=8540 (d)
Figure 4 – (a)-(c) : flow streamlines at different instants on the plane y = 0.5, Ra = 12600 ; (d) :
Phase portraits by temporal evolution of the Fourier modes Tˆ (k) calculated on the line (x = 0.0381
and y = 0.5). Abscissa : Re(Tˆ (1)) ; ordinate : Re(Tˆ (2)).
5000 5200 5400 5600 5800 6000−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Time
R
e
(Tˆ
(1
))
3000 3500 4000 4500 5000 5500 6000−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time
R
e
(Tˆ
(1
))
(a) Ra = 13000 (b) Ra = 14000
Figure 5 – Temporal evolution of Re(Tˆ (1)) at two different Ra.
5
21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
through a sequence of period-doubling bifurcations. In physical space, it corresponds to the modulated
pulsation of three-dimensional spatially localized structures, which consist of distorted transverse rolls
connected by secondary vortices or braids. Estimations of the Feigenbaum constant from the first
few bifurcations are reasonably close to the expected theoretical value. A bifurcation diagram as Ra
is increased was constructed for the temperature evolution of a point in the flow. Several features
common to the characteristics of one-dimensional maps were identified, such as periodic windows and
instances of interior crises. As Ra is further increased, a crisis-induced intermittency is observed, as the
structures are shifted vertically by half a wavelength. A periodic window occurs for higher Rayleigh
numbers, which then gives way to intermittent, chaotic behavior again. The intermittent behavior
observed suggests that heteroclinic connections could be occurring between strange attractors in a
system with O(2) symmetry.
Re´fe´rences
[1] Batchelor, G.K. 1954 Heat transfer by free convection across a closed cavity between vertical
boundaries at different temperature Quart. Appl. Math 12 209-233
[2] Ruth, D.W. 1979 On the transition to transverse rolls in an infinite vertical fluid layer - a power
series solution Int. J. Heat Mass Transfer 22 1199-1208
[3] Bergholz, R.F. 1978 Instability of steady natural convection in a vertical fluid layer J. Fluid Mech.
84 743-768
[4] Chait, A., Korpela, S. 1989 The secondary flow and its instability for natural convection in a tall
vertical closure J. Fluid Mech. 200 189-216
[5] Clever, R., Busse, F. 1996 Tertiary and quarternary solutions for convection in a vertical fluid layer
heated from the side Chaos, Soli., Frac. 5 163-170
[6] Gao, Z., Sergent, A., Podvin, B., Xin, S., Le Que´re´, P. On the transition to chaos of natural
convection between two infinite differentially heated vertical plates Phys. Rev. E submitted
[7] Lizee, A., Alexander, J.I.D 1997 Chaotic thermovibrational flow in a laterally heated cavity Phys.
Rev. E 56 4152-4156
[8] Xin, S., Le Que´re´, P. 1995 Direct numerical simulation of two-dimensional chaotic natural convec-
tion in a differentially heated cavity of aspect ratio 4 J. Fluid Mech. 304 87-118
[9] Paolucci, S., Chenoweth, D.R. 1989 Transition to chaos in a differentially heated vertical cavity J.
Fluid Mech. 201 379-410
[10] Janssen, R.J.A., Henckes, R.A.W.M. 1995 Influence of prandtl number on instability mechanismes
and transition in a differentially heated square cavity J. Fluid Mech. 290 319-344
[11] Xin, S., Le Que´re´, P. 2002 An extended Chebyshev pseudo-spectral benchmark for the 8 :1
differentially heated cavity Int. J. Num. Meth. in Fluids 40 981-998
[12] Eckmann, J.P. 1981 Roads to turbulence in dissipative dynamical systems Rev. Modern Physics
53 643-654
[13] Feigenbaum, M.J. 1980 Universal behavior in nonlinear systems Los Alamos Sciences 1 4-27
[14] Lorenz, E.N. 1963 Deterministic nonperiodic flow J. Atm. Sci. 20 130-141
[15] Grebogi, C., Ott, E., Romeiras, F., Yorke, J. 1987 Critical exponent for crisis-induced intermit-
tency Phys. Rev. A 36 5365-5380
[16] Melbourne, I., Chossat, P., Golubitsky, M. 1989 Heteroclinic cycles involving periodic solutions
in mode interactions with O(2) symmetry Proc. Roya. Soc. Edinburgh : Sec. A Math. 113 315-345
[17] Armbruster, D., Guckenheimer, J., Holmes, P. 1987 Heteroclinic cycles and modulated travelling
waves in systems with O(2) symmetry Physica D. 29 257-282
6


Period-doubling scenario and crisis-induced intermittency in natural convection between two vertical differentially heated plates

National audienceThe chaotic regime of the natural convection of air between two vertical plates maintained at differenttemperatures is studied. The periodic dimensions of the plates are relatively small. Direct numericalsimulation (DNS) is used to study the spatio-temporal behavior of the flow, as the Rayleigh number(Ra) increases. The flow becomes temporally chaotic through a period-doubling cascade. Chaos thenbecomes more developed, and windows of multi-periodic regimes, crises are observed. As the Rayleighis further increased, intermittency is observed, and is seen to correspond to an ”attractor-merging”crisis. For still higher values of Ra, a periodic regime is observed, which then gives way to a fullychaotic and intermittent regime

Period-doubling scenario and crisis-induced intermittency in natural convection between two vertical differentially heated plates

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