Transitions between turbulent states in rotating Rayleigh-Benard convection

0.17 MB·English

by  Richard J. A. M. Stevens

Transitions between turbulent states in rotating Rayleigh-B´enard convection
Richard J.A.M. Stevens1, Jin-Qiang Zhong2, Herman J.H. Clercx3,4, Guenter Ahlers2, and Detlef Lohse1
1Department of Science and Technology and J.M. Burgers Center for Fluid Dynamics,
University of Twente, P.O Box 217, 7500 AE Enschede, The Netherlands
2Department of Physics and iQCD, University of California, Santa Barbara, CA 93106, USA
3Department of Applied Mathematics, University of Twente,
P.O Box 217, 7500 AE Enschede, The Netherlands and
4Department of Physics and J.M. Burgers Centre for Fluid Dynamics,
Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
(Dated: January 14, 2009)
9
0 Weakly-rotating turbulent Rayleigh-B´enard convection was studied experimentally and numer-
0 ically. With increasing rotation and large enough Rayleigh number an abrupt transition from a
2 turbulent state with nearly rotation-independent heat transport to another turbulent state with
n enhanced heat transfer is observed at a critical inverse Rossby number 1/Roc ≃ 0.4. Whereas for
a 1/Ro < 1/Roc the strength of the large-scale convection-roll is either enhanced or essentially un-
J modified depending on parameters, its strength is increasingly diminished beyond 1/Roc where it
competes with Ekman vortices that cause vertical fluid transport and thus heat-transfer enhance-
4
ment.
1

PACSnumbers: 47.27.te,47.32.Ef,47.20.Bp,47.27.ek
]
n
y
d Turbulence evolveseither througha sequence ofbifur- was tuned. One sees that sharp transitions, usually re-
- cations, possibly passing through periodic and chaotic garded as characteristic of low-dimensional systems, are
u
states[1]asinRayleigh-B´enard(RB)convection[2]when displayed also by the fully developed turbulent flows.
l
f the Rayleigh number Ra (to be defined below) is in-
. We present both experimental measurements and di-
s creased,orthroughsubcritical bifurcations [3] as in pipe
c rect numerical simulations (DNS) for a sample with di-
i or Couette flow. Once the flow is turbulent, it usually is ameter D equal to L. They cover different but overlap-
s
characterized by large random fluctuations in space and
y ping parameter ranges and thus complement each other.
h time and by a loss of temporal and spatial coherence. Where they overlap they agree very well. Without or
p Fortheturbulentstatecommonwisdomisthatthelarge
withonly weakrotation,forthis systemitis knownthat
[ fluctuations assure that the phase-space is always fully
there are thermal boundary layers (BLs) just below the
1 explored by the dynamics, and that transitions between topandabovethebottomplate,withatemperaturedrop
v potentially different states that might be explored as a approximatelyequalto∆/2acrosseach. Thebulkofthe
6 control parameter is changed are washed out.
system contains vigorous fluctuations, and in the time
1
0 Contraryto the above,we showthat sharp transitions averagea large-scalecirculation (LSC) that consists of a
2 between distinct turbulent states can occur in RB con- single convection roll with up-flow and down-flow oppo-
. site each other and near the side wall.
1 vection[4,5]whenthesystemisrotatedaboutavertical
0 axis at an angular velocity Ω. In dimensionless form the The numerical scheme was already described in refs.
9 angular velocity is given by the inverse Rossby number [7, 8, 9, 10]. The experimental apparatus also is well
0
1/Ro = 2Ω/pβg∆/L. Here L is the height of a cylin- documented [10, 11] and we give only a few relevant
:
v dricalRBsample,β thethermalexpansioncoefficient,∆ details. The sample cell had D = L = 24.8 cm, with
i
X the temperature difference between the bottom and top plexiglas side walls of thickness 0.32 cm and copper top
plate, and g the gravitational acceleration. At relatively and bottom plates kept at temperatures T and T , re-
r t b
a smallRawheretheturbulenceisnotyetfullydeveloped, spectively. The fluid was water. The Rayleigh number
we find that the system evolves smoothly as 1/Ro is in- Ra βg∆L3/(κν) (ν and κ are the kinematic viscosity
≡
creased. However, when Ra is larger and the turbulent and the thermal diffusivity, respectively), Prandtl num-
state of the non-rotating system is well established, we ber Pr ν/κ, and Ro were computed from the fluid
≡
find that sharp transitions between different turbulent properties at the mean temperature T = (T +T )/2.
m t b
states occur, with different heat-transfer properties and The Nusselt number Nu λ /λ was determined from
eff
≡
different flow organizations. Similar sharp transitions the effective thermal conductivity λ = QL/∆ (Q is
eff
between different states were reported recently for tur- the heat-current density) and the conductivity λ(T )
m
bulent flows in liquid sodium [6], where the increase of of the quiescent state. Eight thermistors, labeled k =
the magnetic Reynolds number beyond a certain thresh- 0,...,7, were imbedded in small holes drilled horizon-
old led to the spontaneous creation of a mean magnetic tally from the outside into but not penetrating the side
field and where sharp bifurcations between different tur- wall [12]. They were equally spaced around the circum-
bulent states were observed when a control parameter ference at the horizontal mid-plane (z = 0). A second
2
and third set were located at z = L/4 and z = L/4. 1.08
− 0.055
SincetheLSCcarriedwarm(cold)fluidfromthebottom
(top)plateup(down)thesidewall,thesethermistorsde-
tected the location of the upflow (downflow) of the LSC 0) 1.06 L0.047
byindicatingarelativelyhigh(low)temperature. Tode- u ( λ/u
N
termine the orientation and strength of the LSC, we fit
/ 1.04
the function )
Ω 0.037
0.1 0.5 1
Tf,k(z =0)=Tw,0+δ0cos(kπ/4 θ0); k =0,...,7 (1) u ( 1/Ro
− N 1.02
separately at each time step, to the eight temperature
readingsT (z =0)obtained fromthe thermistors atz =
k
0. Similarlyweobtainedθ , δ , andT for the toplevel 1
t t w,t 0 0.2 0.4 0.6 0.8 1
at z = L/4. At z = L/4 only the mean temperature
1/Ro
−
T was used in the current work.
w,b
In Ref. [10] we explored Nu as a function of Ra, Pr,
FIG.1: Theratio Nu(Ω)/Nu(Ω=0) asfunction of1/Rofor
and Ro in a large parameter regime, ranging towards
Ra = 4×107 and Pr = 6.26. Open black squares indicate
strong rotation (1/Ro 1) and from small to large Pr.
≫ the numerical results. The numerical error is approximately
Here wefocus onPr 4 7(typicalofwater)andweak
rotation (Ro > 1) to≈stud−y the transition from the non- 0T.h2%e twhihckicnhesiss oinfdtihcaetekdinebmyatthice tsoizpeaonfdthbeotstyommboBlLs.s bInasseetd:
rotating state∼at 1/Ro=0 towards the rotating case for on themaximum rmsazimuthal (uppersymbols: red (black)
different Ra. fortop(bottom)BL)andradial(lowersymbols: blue(green)
Westartwithnumericalresultsfortherelativelysmall for top (bottom) BL) velocities. The vertical dashed lines in
Ra=4 107 whichisnotaccessiblewiththecurrentex- both graphs represent 1/Roc and indicates the transition in
× boundary-layer character from Prandtl-Blasius (left) to Ek-
perimental apparatus because L is too large (and thus
man (right) behavior.
Ra too smallto be accessiblewith reasonable∆). Those
simulationswheredoneonagridof65 129 129nodesin
× ×
theradial,azimuthalandverticaldirections,respectively,
allowingforasufficientresolutionofthesmallscalesboth data are consistent with no heat-transfer modification
inside the bulk of turbulence and in the BLs adjacent to as compared to the non-rotating case. For this larger
the bottom and top plates where the grid-point density Ra one sees that experimental and numerical data (now
wasenhanced[8,13]. ThesmallRaallowedforverylong based on a resolution of 129 257 257, see [10]) agree
× ×
runs of 2500 large-eddy turnover-times 2L/U (where U extremelywell. InRef.[13]datafromDNSwerereported
is the free-fall velocity U = √βg∆L) and thus excellent onthe relativeNusseltnumber forhigher Rayleighnum-
statistics. Fig.1showstheratioofNu(Ω)inthepresence ber, Ra = 1 109 and Pr = 6.4, which – as we now
×
ofrotationtoNu(Ω=0)asfunctionof1/Ro. Thisratio realize retrospectively – showed a similar transition also
increases rather smoothly with increasing rotation. This at 1/Roc 0.4.
≈
increase is thought to be due to the formation of the To characterize the flow field, we numerically calcu-
Ekman vortices which align vertically and suck up hot lated the rms velocities averaged over horizontal planes
(cold) fluid from the lower (upper) BLs (Ekman pump- and over the entire volume, respectively. The maximum
ing) [10, 13, 14, 15]. This is supported by the change in rmsazimuthalandradialvelocitiesnearthetopandbot-
characterofthekineticBLnearthebottomandtopwalls tom wall have againbeen used to define the thickness of
basedonthemaximumroot-mean-square(rms)velocities the kinetic BL, which is shown in the inset of Fig. 2
intheazimuthaldirection. For1/Ro<0.5theBLthick- for Ra = 2.73 108 and Pr = 6.26. The critical in-
ness (based on the rms azimuthal v∼elocity) is roughly verse Rossby nu×mber clearly distinguishes between two
constantorevenslightly increases,andfor1/Ro>0.5it regimes: one with a constant BL thickness (in agree-
behavesaccordingtoEkman’stheoryanddecreas∼eswith ment with the presence of the LSC and the Prandtl-
increasing rotation rate, see the inset in Fig. 1. As the Blasius BL) and another one with decreasing BL thick-
turbulence is not yet fully developed, a smooth transi- ness for 1/Ro > 0.38. The scaling with rotation rate
tionbetweenthenon-rotatingandtherotatingturbulent is in agreement∼with Ekman BL theory λ /L Ro1/2.
u
∼
states is observed. Further support can be found in Ref. [16] where similar
Bothnumericalandexperimentalfindingsareverydif- data for the kinetic BL thickness have been plotted for
8 9
ferentforthelargerRa=2.73 10 andPr=6.26where Ra=1 10 and Pr =6.4. For 1/Ro>1/Ro the nor-
c
× ×
the turbulence of the non-rotating system is well devel- malized(bythevaluewithoutrotation)volume-averaged
oped. InFig.2oneseesthatnowthereisacriticalinverse vertical velocity fluctuations w strongly decrease, in-
rms
Rossby number 1/Ro 0.38 at which the heat-transfer dicating that the LSC becomes weaker, see Fig. 3. The
c
≈
enhancement suddenly sets in. For weaker rotation the decrease in normalized volume averaged vertical velocity
3
1.08 1.1

0.045
)
) 1.06 0.025 0
u( 0 λ/Lu ( ms 1
N 1.04 −1/2 wr
) / ) /
Ωu( 1.02 0.0051 0−1 11/0R0 o 101 Ω( s 0.9
N m
wr
1
0.8
0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1
1/Ro 1/Ro 1/Ro
6F.I2G6.. 2R:eNduso(lΩid)/cNiruc(leΩs:=ex0p)efroimr eRnata=l d2a.t7a3(×Tm10=8 a2n4d◦CPran=d FfoIrGR.a3:=Th4e×n1o0rm7 a(lleizfte)daanvderRagaed=rm2.7s3ve×rt1ic0a8l(vreiglohcti)tiaesswfurnmcs-
tionof1/Ro. Theblacklineindicatesthenormalized volume
∆ = 1.00K). Open black squares: numerical results. The
averaged value of wrms. The red and the blue line indicate
experimental error coincides approximately with the symbol
thenormalized horizontally averaged wrms at theedge ofthe
size and the numerical error is approximately 0.5%. Inset:
thermal BL based on the slope at respectively the lower and
Thickness of the kinetic BL. For dashed vertical lines and
upperplate. The vertical dashed lines again indicate thepo-
inset: see Fig. 1.
sition of 1/Roc.
fluctuations coincides with a significant increase of the
horizontalaverageattheedgeofthethermalBLsindicat- latter case no Nu enhancement is thus observed.
ing enhanced Ekmantransport(see alsoinsets in Figs.1 At higher Ra=9.0 109 (where ∆ is larger and tem-
×
and 2). These averages provide additional support for perature amplitudes can thus more easily be measured)
the mechanism of the sudden transition seen in Nu and and Pr = 4.38 an even more complex situation is re-
indicate an abrupt change from a LSC-dominated flow vealed, as seen in Fig. 4 (here DNS is not available be-
structure for 1/Ro < 1/Ro to a regime where Ekman cause it would be too time consuming). We find that
c
pumping plays a progressively important role as 1/Ro now Nu(Ω)/Nu(Ω=0) (Fig. 4a), after a slightincrease,
increases. first decreases, but these changes are only a small frac-
Our interpretation for the two regimes is as follows: tionofapercent. ThenNuundergoesasharptransition
OncetheverticalvorticesorganizesothatEkmanpump- at 1/Roc,2 = 0.415 (vertical dotted line in Fig. 4a) and
ing sucks in the detaching plumes from the BLs, those beyond it increases due to Ekman pumping. Compari-
plumes are no longer available to feed the LSC which sonwithFigs.1and2showsthatthe transitionofNuis
consequently diminishes in intensity. A transition be- not strictly at a constant 1/Roc, but that Roc depends
tween the two regimes should occur once the buoyancy weakly on Ra and/or Pr.
force, causing the LSC, and the Coriolis force, causing The LSC amplitudes δ0 and δt determined from fits of
Ekman pumping, balance. The ratio of the respective Eq. (1) to the sidewall-thermometer readings are shown
velocity scales is the Rossby number. For Ro 1 the in Fig. 4b as solid symbols. Consistent with the results
≫
buoyancy-driven LSC is dominant, whereas for Ro ≪ 1 reported in Ref. [12], δt < δ0 when there is no rota-
the Coriolis force and thus Ekman pumping is stronger. tion (1/Ro = 0). This inequality disappears as 1/Ro
The transition between the two regimes should occur at increases. Bothamplitudes firstincreaseby nearlya fac-
Ro=O(1), consistent with the observed Roc ≈2.6. toroftwo. At1/Roc,1 ≃0.337,wherethetwoamplitudes
One wonders of course why the transition in between have just become equal to each other, they begin to de-
the two regimes is sudden (in Nu) for Ra = 2.73 108 creasequite suddenly and remainequalto eachother up
and less abrupt for the smaller Ra = 4 107 show×n in to the largest 1/Ro. The transition at 1/Roc,1 is indi-
×
Fig. 1. Our interpretation is that for that relatively low catedby the leftmost verticaldotted line in Figs.4band
Rayleigh number the flow is not yet fully turbulent so c. Atthatpointtherealsoisatransitionrevealedbythe
that spatial coherences [5] exist throughout the cell. For vertical temperature difference ∆T = 2 [T T ]
w w,b w,t
1/Ro< 1/Ro the vortical structures have time to form alongthesidewallasseeninFig.4cwhich×shows∆−T /∆
c w
andsu∼rviveforawhileintheLSC-inducedwindforlower as a function of 1/Ro. Consistent with the initially en-
Ra, and provide some weak Ekman pumping and thus hancedLSCamplitudesδ0andδt,theseresultsfirstshow
enhanced Nu, while they are easily swept away by the a reduction of the thermal gradient as the LSC becomes
much stronger wind of the LSC for higher Ra. In the more vigorous, but then reveal an increase due to en-
4
process. First, up to 1/Roc,1, the time-averaged LSC
) amplitudes, such as δ0 /∆, nearly double in value (see
01.02 h i
u( Fig.4b)andtherebyreducetheverticalthermalgradient
N
Ω) / (a) eanlohnagnctehdeawcaclulm(sueleatFioign.o4fcp).luBmeeysoanndd1v/oRrtoicc,e1st,hwehreichiscaon-
1.00
(
u incideswithanincreaseoftheBLthicknessnearonsetas
N 0.02
shownbythesimulationsatlowerRa(seeinsetsinFigs.1
∆ and 2). This accumulation detracts from the driving of
δ>/t (b) the LSC but the flow is not yet organized into effective
r <0.01 Ekman vortices. This organization sets in at 1/Roc,2,
o
∆ leads to Ekmanpumping, and enhances Nu and reduces
>/ the strength of the LSC as supported by the volume-
0
δ< averageofwrms,seeFig.3(forlowerRa). Thissequence
0.00 of events is altered as Ra (or presumably also Pr) is
0.075
changed, but it is remarkable that for fully developed
∆
/ w tlaurrbtoultehnotsReBobcseornvveedctiinontusrhbaurlpenbtiffluorwcastiinonlsiqoucidcusrodsiimumi-
T
∆ (c) by [6]. These resemble the characteristic series of bifur-
0.065
cations well-known from low-dimensional chaos.
0.0 0.2 0.4 0.6 0.8 1.0
Acknowledgements: We thank R. Verzicco for provid-
1 / Ro
>
0 1 ing us with the numericalcode and T. Mullin for discus-
δ
) / w 0 sNioantiso.nTalheSecxiepnecreimFeonutanldwatoiroknwtahsrosuupghporGteradnbtyDthMeRU0.S7-.
T (d)
− 02111andthenumericalworkbytheFoundationforFun-
T –1
< ( 0 2 Θ − Θ 4 6 damental Research on Matter (FOM) and the National
0 Computing Facilities (NCF), both sponsored by NWO.
FIG. 4: Results for Ra = 9.0×109 and Pr = 4.38 (Tm =
◦
40.00 C, ∆ = 16.00 K). (a): Nu(Ω)/Nu(Ω = 0) vs. 1/Ro.
The error bar is smaller than the size of the symbols. (b):
Solidsymbols: time-averagedLSCamplitudeshδ0i/∆(z =0, [1] H. G. Schuster, Deterministic Chaos (VCH, Weinheim,
circles) and hδti/∆ (z =L/4, squares) as afunction of1/Ro. 1988).
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to the temperature data. (c): Vertical temperature variation Fluid Mech. 32, 709 (2000).
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5
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