Pattern dynamics near homoclinic bifurcation in Rayleigh-Bénard convection

0.52 MB·English

by  Pinaki Pal

Pattern dynamics near homoclinic bifurcation in Rayleigh-B´enard convection
Pinaki Pal,1 Krishna Kumar,2 Priyanka Maity,2 and Syamal Kumar Dana3
1Department of Mathematics, National Institute of Technology, Durgapur-713 209, India
2Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur-721 302, India
3CSIR-Indian Institute of Chemical Biology, Jadavpur, Kolkata-700 032, India
We report for the first time the pattern dynamics in the vicinity of an inverse homoclinic bi-
furcation in an extended dissipative system. We observe, in direct numerical simulations of three
dimensional Rayleigh-B´enard convection,a spontaneousbreakingof a competition oftwo mutually
perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as
the Rayleigh number is raised above a critical value. The time period of the cross-roll patterns
diverges, and shows scaling behavior near thebifurcation point. This is an example of a transition
from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a
simple four-mode model that captures thepattern dynamics quitewell.
PACSnumbers: 47.20.Ky,47.55.pb,47.20.Bp
3
Extended dissipative systems driven away from ther- We report, in this article, for the first time the
1
modynamic equilibrium often form patterns, if the driv- possibility of an inverse homoclinic bifurcation in di-
0
2 ing force exceeds a critical value [1]. Competing in- rect numerical simulations (DNS) of three dimensional
stabilities may lead to interesting pattern dynamics, (3D)Rayleigh-B´enardconvection(RBC)inlow-Prandtl-
n
which helps in understanding the underlying instabil- number fluids, and the results of our investigations of
a
J ity mechanism. Several patterns are observed in con- fluid patterns close to the bifurcation. We observe spon-
tinuum mechanical systems, such as Rayleigh-B´enard taneous breaking ofa periodic competition of two mutu-
8
2 systems [2], B´enard-Marangoni systems [3], magneto- allyperpendicularsetsofcrossrollstoonesetofoscillat-
hydrodynamics[4], ferrofluids [5], binaryfluids [6], gran- ingcrossrolls,astheRayleighnumberRaisraisedabove

] ular materials [7] under shaking, biological systems [8], a critical value Ra . The time period of the oscillating
S h
etc. Symmetries and dissipation play a very significant patternsdiverges,andshowsscalingbehaviorintheclose
P
role in pattern selection in such systems [9]. The selec- vicinityofthe transitionpoint. The exponents ofscaling
.
n tion of a pattern is a consequence of at least one broken are asymmetric on the two sides of the transition point.
li symmetry of the system. Unbrokensymmetries oftenin- We also present a simple four-mode model, which cap-
n
troducemultiplepatterns,whichmayleadtoatransition tures not only the pattern dynamics in the vicinity of
[
from local to global pattern dynamics. The gluing [10] the inversehomoclinic bifurcation but also the whole se-

1 of two limit cycles on two sides of a saddle point in the quence of bifurcations observed in DNS quite well over
v phase space ofa givensystem is an example ofa localto a wide rangeof Rayleighnumber in low-Prandtl-number
1 nonlocal bifurcation. It occurs when two limit cycles si- fluids.
4
multaneously become homoclinic orbits of the same sad- The hydrodynamics of RBC in a thin layer of Boussi-
5
dle point. This phenomenon has been recently observed nesq fluid of thickness d, kinematic viscosity ν, thermal
6
inavarietyofsystemsincludingliquidcrystals[11],fluid diffusion coefficient κ, and thermal expansion coefficient
.
1
dynamical systems [12], biological systems [13], optical α, subjected to an adverse temperature gradient β, is
0
systems [14], and electrical circuits [15], and is a topic governed by the following dimensionless hydrodynamic
3
of current research. The pattern dynamics in the vicin- equations:
1
: ity of a homoclinic bifurcation has, however, not been
v investigated in a fluid dynamical system. ∂tv+(v )v = p+ 2v+Raθe3, (1)
i ·∇ −∇ ∇
X A Rayleigh-B´enardsystem [16, 17], where a thin layer Pr[∂tθ + (v )θ]= 2θ+v3, (2)
·∇ ∇
r ofafluidis heateduniformlyfrombelowandcooleduni- v = 0, (3)
a formly from above,is a classical example of an extended ∇·
dissipative system which shows a plethora of pattern- where v(x,y,z,t) (v ,v ,v ) is the velocity field,
1 2 3
≡
forming instabilities [2], chaos [18], and turbulence [19]. θ(x,y,z,t) the convective temperature field, p the pres-
Low-Prandtl-number [20] and very low-Prandtl-number sure due to convection, and e a unit vector directed
3
convection [21, 22] show three-dimensional oscillatory against the direction of the acceleration due to gravity
behavior close to the instability onset. In addition, g. Lengths, time and temperature are measured in units
the Rayleigh-B´enard system possesses symmetries un- of the fluid thickness d, viscous diffusion time d2/ν, and
der translationand rotation in the horizontalplane that νβd/κ, respectively. We use thermally conducting and
can introduce multiple sets of patterns. The possibil- stress-free boundary conditions which imply that θ =
ityofahomoclinicbifurcationandthe patterndynamics v = ∂ v = ∂ v = 0 at z = 0,1. All the fields are
3 z 1 z 2
in its vicinity are unexplored in three dimensional (3D) assumed to be periodic in the horizontal plane. The
Rayleigh-B´enardconvection. Rayleigh number Ra = αβgd4/νκ and Prandtl number
2
Pr = ν/κ are two dimensionless numbers that decide
the convective flow structures in the fluid. Convection 0.4
appearswhenthereducedRayleighnumberr =Ra/Rac 0.3
with Ra =27π4/4 is raised above unity.
c 0.2
We integrate the full hydrodynamic system (1 -
3) for low-Prandtl-number (Pr 0.025) fluids us- 0.1
≤
ing an object oriented code [23] based on pseudo- 0
spectral method. The vertical velocity v and the
3 −0.1
temperature field θ are expanded as: v (x,y,z,t) =
3
P W (t)eik(lx+my)sin(nπz) and θ(x,y,z,t) = −0.2
l,m,n lmn
P Θ (t)eik(lx+my)sin(nπz). The horizontal ve- −0.3
l,m,n lmn
locitiesv andv havesimilarexpansioninxy planewith
1 2 −0.4
co-sinusoidaldependenceinthez direction. Theintegers

l,m,n can take values consistent with the equation of
continuity (eq. 3) and k = k = π/√2. The size of the
c
periodic cell for DNS is 2√2 2√2 1, and its resolu- FIG. 1: Contour plots of the temperature field at the mid
× ×
tion is 64 64 64. The pattern dynamics is complex plane (z = 0.5) near inverse homoclinic bifurcation as ob-
× ×
at the primary instability in very low-Prandtl-number served in DNS (Pr = 0.01). The upper row (r = 1.076)
fluids [22] due to chaotic flows just above the instabil- shows competition between two sets of oscillating cross rolls.
ity onset. We investigate pattern dynamics as soon as Themiddleandlowerrowsshowtwopossibilitiesofoscillating
cross rolls (r=1.088).
we observethe firstoscillatorypattern closeto the onset
of convection. The simulation was started with random
initial conditions, and it was continued until a steady
state was reached. The steady state values of fields in andthirdrowsofFig.1 showtwopossibilitiesofOCR-II
a simulation were used as the initial conditions for the for r =1.088. We get oscillating cross-rollpatterns with
next simulation. The value of r was increased in small either W101 > W011 orW101 <W011. Twosetsofmul-
| | | |
steps ofsize ∆r (0.0001 ∆r 0.01)andthe numerical tiple solutions, which are connected by rotation about a
simulations were done f≤or sev≤eral values of r. We also vertical axis by π/2, continue until r = 1.120. Further
repeatedseveralruns startingwithrandominitialcondi- increase in r leads to the appearance of two sets of sta-
tions for different values of r and found no hysteresis in tionary cross rolls (CR) at r = 1.121, which is observed
the parameter range considered here. Various observed till r = 1.199. Raising the value of r even further leads
convectivepatternsintheDNSforPr=0.01andPr=0 to a transition from stationary cross rolls (CR) to sta-
are listed in the first two columns of Table I. tionary square (SQ) patterns. The similar sequence is
observedin the limit of Pr 0 (see Table I). The range
→
of competing cross rolls becomes wider as Pr decreases.
TABLE I: Convective patterns computed from DNS in two We have observed the spontaneous breaking of compet-
columns in themiddle and from a model in the last column. ingcrossrollstotwosetsofoscillatorycrossrollsinfluids
with 0 Pr 0.025.
Convective DNS Model ≤ ≤
Figure 2 shows the details of a transition from global
patterns r(Pr = 0.01) r(Pr= 0) r(Pr = 0) to localpatterndynamicsfor Pr=0.01. The divergence
OCR-I 1.010 - 1.0835 1.0049 - 1.0708 1.010 - 1.0953 of the time period of oscillatory patterns close to the
transition point is displayed in Fig. 2a. The amplitude
OCR-II 1.0836 - 1.1200 1.0709 - 1.1315 1.0954 - 1.1584
of the largest Fourier mode of OCR-I patterns decreases
CR 1.1210 - 1.1990 1.1316 - 1.2005 1.1585 - 1.2519
linearly with the increase of r and shows two possible
SQ 1.2000 - 1.4200 1.2006 - 1.4297 1.2520 - 1.5128
values just above the transition(r =r ) point (Fig. 2b).
h
The appearance of two amplitudes signifies a transition
The first ordered state for Pr = 0.01 appears in the fromanonlocaltoalocalpatterndynamics. Thescaling
formofacompetitionoftwomutuallyperpendicularsets oftimeperiodτ ofoscillatingpatternonbothsidesofthe
ofoscillatorycrossrolls(OCR-I)whichcontinuestoexist transitionpointisdisplayedinFig.2c,showingasymme-
untilr =1.0835. ThefirstrowofFig.1displaysthe pat- try. Thetime periodτ ofcompetingpatternsscaleswith
tern dynamics for r=1.076. Two sets of cross rolls, one ǫ r rh as ǫ−0.115 before transition and as ǫ−0.033
≡ | − |
with W > W and another with W < W , after transition. The scaling behavior of the time period
101 011 101 011
appea|r per|iodi|cally.| The patterns ap|pear |as s|quare|s of OCRsuggests the transitionto be inversehomoclinic.
when the amplitudes of the two sets of cross rolls be- Simulations [21] of low-Pr RBC with no-slip boundaries
comeequal. Thecompetitionrepresentsaglobalpattern areknowntoshowrelaxationoscillationofpatterns. The
dynamics. Asr israisedaboveacriticalr =1.0835,the homoclinic instability may be accessible to experiments,
h
competing cross rolls (OCR-I) spontaneously break into if performed in square containers, in this regime.
two possible oscillatorycrossrolls (OCR-II). The second We now construct a simple low dimensional model to
3
17(a) r > r h 17(b) 1.5 1.5
r < r (a) DNS (b) Model
τ127 h (W)101max127 τlog()0.51 6.367ε−γ1 τlog()0.51 5.368ε−γ1
−02.0 15 −0.0075 r−0rh 0.0075 0.015 12.0 7 1.075 1.0r8 1.085 1.09 −04 −3 log(ε)−2 1.829−ε1−γ2 −04 −3 log(ε)−2 2.095−ε1−γ2
1.25(c) 4.907ε−0.115 15 15
3.511ε−0.033 (c) r = 1.0708 (DNS) (d) r = 1.071 (DNS)
τlog 1 W101105 W101105
0.75
0 0
t t+20 t+40 t t+20 t+40
Dimensionless time Dimensionless time
0.−54 −3.5 −3 −2.5 −2 −1.5
log(ε) 30 30
(e) r = 1.0953 (Model) (f) r = 1 .0954 (Model)
FIG. 2: Scaling behavior near inverse homoclinic bifurcation W10120 W10120
(Pr=0.01)asobtainedfromDNS:(a)Thedivergenceofthe 10 10
dimesionless time period τ of OCR patterns close to the bi-
0 0
furcation. (b)ThevariationofmaximaofFouriermodeW101 t Dimensti+on20less time t+40 t Dimensti+on20less time t+40
with the reduced Rayleigh number r near the bifurcation,
showing the spontaneous transition from a global oscillation
(star ∗) to two possible local oscillations(cross ×) at the ho-
moclinicpoint. (c)Thetimeperiodτ ofOCRpatternsscales
FIG. 3: Comparison of results of the model with those from
with ǫ≡|r−rh|. The scaling exponents are different before DNS for Pr = 0: Scaling of the dimensionless time period
and after thebifurcation.
τ of the oscillating patterns with ǫ ≡ |r−rh| before (solid
curve) and after (dashed curve) the bifurcation, as observed
in DNS (a) and the model (b). The exponents from DNS
analyzepatterndynamicsneartheinversehomoclinicbi-
are: γ1 = 0.086 and γ2 = 0.158, while those from the model
furcation. Forthispurposewetakethelimitofvanishing
are: γ1 = 0.092 and γ2 = 0.124. Temporal variation of the
Prandtl number (Pr 0). As the temperature field is
→ Fourier mode W101 computed from DNS before (c) and after
slavedto the vertical velocity, the number of modes rep- (d) the homoclinic bifurcation. Temporal variation of the
resentingtheeffectivedynamicsisexpectedtobesmaller modeW101 computedfromthemodelbefore(e)andafter(f)
in this limit. We begin with the standardGalerkin tech- thebifurcation.
nique to derive a low-dimensional-model [22]. We ex-
pand the vertical velocity v and the vertical vorticity
3
Z ( v)e3 suchthattheessentialmodestodescribe Runge Kutta (RK4) method. The second and third
≡ ∇× ·
two sets of mutually perpendicular rolls, cross rolls, and columns of Table I summarize the results obtained from
thenonlinearinteractionbetweenthemareretained. We DNS for Pr =0 and the model, respectively. The model
keep five velocity modes W101, W011, W211, W121, and captures the sequence of bifurcations quite accurately in
W112, and two vorticity modes Z110 and Z112. The hy- a wide range of r as observed in DNS. The difference
drodynamicequationsareprojectedonthesemodes. We in the lower and the upper bounds for the range of r
thenadiabaticallyeliminate modesW112, Z110 andZ112. for any solution computed from the model and DNS is
This leads to a simple four-mode model given by, within 6%.
X˙ = µ X+X X A(a X+a Y)+Y Y A(a X+a Y) Figure3givesthe comparisonofresultsobtainedfrom
1 1 2 1 2 1 2 3 4 the model and those from DNS for Pr = 0 near homo-
+ a [X2Y ,X2Y ]T +a [X Y2,X Y2]T,
5 2 1 1 2 6 1 2 2 1 clinic bifurcation. The time period τ of the competing
Y˙ += bµ2[YX2+YX,X1X22YA](Tb1+Xb+[Xb2YY)2,+XY1YY22]AT(,b3X+b4Y(4)) ctrraonsssitrioolnls(sscoalildescwurivthe)ǫan≡d|ars−ǫ−rγh2|aafsteǫr−tγh1ebterfaonrseittiohne
5 2 1 1 2 6 1 2 2 1
(dashedcurve). Thevaluesofthescalingexponentγ ob-
1
where X = [X ,X ]T [W ,W ]T, Y = [Y ,Y ]T tained from the DNS (Fig. 3a) and the model (Fig. 3b)
1 2 101 011 1 2
[W ,W ]T, A = [0≡ 1;1 0], µ = 3π2(r 1)/≡2 are 0.086 and 0.092respectively. Two exponents are dif-
121 211 1
−
and µ = π2(135r 343)/98. The coefficients are: ferentshowingasymmetry in scalingbehavior. DNS and
2
−
a = 3/100, a = 31/3000, a = 209/30000, a = the model yield γ equal to 0.158 and 0.124respectively.
1 2 3 4 2
− −
63/60000, a = 47/1500, a = 1/200, b = 93/700, Figures3c&dshowthetemporalvariationoftheFourier
5 6 1
− −
b = 67/7000, b = 7407/70000,b = 3969/700000, mode W obtained from DNS before and after the bi-
2 3 4 101
− − −
b = 928/7000,and b = 3816/70000. The superscript furcation. The similar behavior is observedin the model
5 6
−
T denotes transpose of a matrix. The model is valid for (fig.3e, f). The modelrevealsthatthe unstable SQ pat-
r <343/135 (i.e., µ <0). terns exist as saddle fixed points for 1 r < 1.252 but
2
≤
The model is integrated using standard fourth order become stable at r = 1.252. The competing cross rolls
4
(OCR-I)breakintotwopossiblesetsofOCR-IIwhenthe rolls into one set of oscillatory cross rolls occurs close
amplitude of OCR-I oscillation touches a saddle square. to the bifurcation point. The time period of oscillation
This confirms the transition from OCR-I to OCR-II as showsasymmetricscalingonthetwosidesofthebifurca-
an inverse homoclinic bifurcation. The model includes tionpoint. We havealsoconstructedasimple four-mode
very few modes, and therefore shows higher values of modelwhichcapturesaccuratelythesequenceofbifurca-
the Fourier mode W . The time periods of oscillating tionsincludingthepatterndynamicsnearthehomoclinic
101
patterns obtained from the model and DNS are in good bifurcation. Themodelwithdifferentvaluesofthecoeffi-
agreement both before and after the transition. cientsa andb maybeusefultostudypatterndynamics
i i
We have investigated the possibility of an inverse ho- onsquarelatticesinotherextendedsystemswithsimilar
moclinic bifurcation and associated pattern dynamics in symmetries.
RBC. The spontaneous breaking of a competition be- We have benefitted from fruitful discussions with J.K.
tweentwomutuallyperpendicularsetsofoscillatorycross Bhattacharjee, H. Pharasi, L.K. Dey, and D. Kumar.
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