J. Plasma Fusion Res. SERIES, Vol. 2 (1999) 28-31

Does the Ultimate State of Thermal Turbulence Exist?
SANo Masaki''2, SEGAWA Takehikor, NAERT Antoine2 and GLAZIER James A.3
tResearch Institute of Electrical Communication, Tohoku University,

2-2-l Katahira, Aoba-ku, Sendai 980, Japan
2Loboratoire de Physique, Ecole Normale Supdieure de Lyon,
46 AIl6e d'Italie, 69364 Lyon Cedex 7, France
3Department of Physics, University of Notre Dame, Notre Dame IN 46556-5670' USA(Received: 16 July 1999 / Accepted: 24 August 1999)

Abstract
It is predicted that the ultimate state of thermal turbulence should exsist in natural convection forlarge enough Rayleigh (Ra) number. In 80's Chicago group [1] found hard turbulent state in thermal
turbulence which is characterized by the scaling relation between the Nusselt number (Nll) and Ra
number as Nu - Ra2t7 . The ultimate state was predicted to have a new scaling relation; Nu - Rattz, but it
is never be observed in any experiment. In this paper, the prediction was tested by using mercury (Hg)for Ra numbers up to 3 order magnitude larger than the predicted value. However, our experiments in Hg
found no eveidence of a transition and suggest the assumptions of thories are invalid. Thus it appears thatthe hard turbulence is the ultimate state.

Keywords:
thermal turbulence, Rayleigh-Benard convection, hard turbulence

What is the character of turbulence in the limit ofhave a new scaling relation between the Nusselt number(the ratio between advective thermal transport andinfinitely large fluid velocities? Is this turbulence
universal? The simplest experimental system to test

diffusive thermal transport in the absence of flow, Na)
and Ra withthermal turbulence is known as Rayleigh-B6nard (RB)convection. The control parameter in RB convection is

Nu - Rat/2.

1. IntroductionQ)

rather than the relation for hard turbulence [l] ofRayleigh (Ra) number,

apLTL3

Ra =-: Kv,(l)Nu - Ra2t7

where d is the coefficient of thermal expansion, r the(3)Such a transition would have important consequences
for all calculations of thermal transport at very high Ra
number. Kraichnan [4] originally suggested this limiting

thermal diffusivity, v the kinematic viscosity, g thegravitational acceleration, t the typical length and AZ

state for turbulence in 1962, with later theories by
Howard [5] and Busse [6]. Cioni et al. [7] in Hg andChavanne et al. [8] in gaseous He have claimed todetect this transition. However, our experiments in Hg,

the temperature difference across the cell.
Theoretical predictions 12,31 that a new, final anduniversal type of turbulence, ultra-hard turbulence,
should exist in RB for very large Ra numbers stimulated
current experiments. The new state was predicted to@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author's e-mail: sano@ riec.tohoku.ac.jp28

Sano M. et al., Does the Ultimate State of Thermal Turbulence Exist?

which reached higher effective Ra, found no evidence ofspectrum.

a transition and suggest that the assumptions of thetheories are invalid. Thus it appears that ultimate

Because Pr = 0.O25 at 2OC in Hg compared to Pr

turbulence is hard turbulence.

= 0.7 for He gas below I atm at 5oK, searching newrange in Hg required only a Ra number of order 108.
Our initial Hg experiments used aspect ratio one half,

2. Search for the New Regime

one, and two cylindrical cells to rcach Ra - 2 x l0e. Our

Convective turbulence is inevitably anisotropic andnon-homogeneous at large length scales. Two types ofboundary layers are important. Near the top and bottom
container walls, flow velocity vanishes due to the non-results were compatible with the classical Nu vs Ra
scaling exponent of 217 = 0.285 [3]. The break in theaspect ratio two data at Ra - 2 x lO5 results from apattern competition instability in the flow [10]. For
small Ra - 106-108, strong steady bulk mean flow in theaspect ratio I cell, reduces the measured scaling
exponent to 0.25 [14]. The exponent increases towards
2n with increasing Ra.
These measurements appear to invalidate both ofthe mechanisms proposed for the onset of ultra-hardturbulence. The thermal and viscous boundary layers
cross below Ra - 105, so we should definitely have seen
the transition if the boundary crossing theory werecorrect. The boundary layer thickness argument also
seems to fail: in Hg, after the viscous boundary layercrosses the thermal they shrink together with a nearly
constant ratio and a scaling exponent of -0.20 + 0.02
different from the theoretical value of -l12 [4].
Cioni et al. in a larger Hg cell of aspect ratio one,slip condition which creates an highly sheared viscousboundary layer. The temperature profile becomes linear
near the walls, because only diffusion transports heatsince advection is suppressed. The time averaged
temperature in the bulk is constant and equal to theaverage of the top and bottom plate temperatures due tostrong mixing by the turbulent flow. The time averaged
profile is steep and linear in the thermal boundary layer.
The thickness of the thermal boundary layer, which is
the inverse of the temperature gradient, limits the grossheat transport across the turbulent cell. Thus Na is
proportional to the ratio between the cell height and thethickness of the thermal boundary layer.
The theories predicting Nu - Rau2 make one of two
assumptions. The first is that, since the thickness of theviscous boundary layer varies as - Ra-rt2, it becomes
negligible at very high Ra. Thus heat is advected bybuoyant structures (e.9. plumes or thermals) whichmove at the free fall velocity, i.e., since viscous forces
are negligible compared with inertial forces in thisregime, the thermals accelerate as if they were free,undamped particles, subject to the buoyancy force
produced by gravity. Since the free fall velocity, V
scales as - (ogLTL) - Rattz, the heat flux scales as Nu Rattz. The other argument is that when the viscousboundary layer becomes thinner than the thermalboundary layer (about Ra = lDta for He [9] and Ra = 105
for Hg t7,l0l), the thickness of the viscous boundary
layer limits the heat flux, which also gives Nu - Ratt2.
The critical Ra for the transition, Ra".i, is predicted todepend on the Prandtl number of the fluid (the ratio ofthermal and viscous dissipation, Pr = rclv) as R4"1 Pra [2).
Shraiman and Siggia predict Re"n - 106-108 for Hg[2,3]. Cioni et al. proposed modifications to the theory
which predicted Ra",;, = 5 x lOs in Hg [7].

measured up to Ra - 5 x lOe t7l. They claimed rhat theNu vs Ra scaling exponent increased above 217 at asingle point at the very top of their Ra range, but could

not measure the new value. Chavanne et al. studied
aspect ratio one-half RB convection in gaseous andliquid He. In gas near the gas-liquid critical point they
saw an increasing Na vs Ra scaling exponent above Ra- l01r [8] and measured an approximate power law ofNu - Pro'o72Rdo38e10.005. Chavanne et al. claimedthat
the new exponent was compatible with l/2 scaling due

to a logarithmic correction.

3. Experimental Setup
Our current experiment uses an aspect ratio one
half cylinder (30cm x 60cm). The top and bottom plates
are solid copper 5cm thick, coupled on top to coolingwater via 126 vertical, thermally anchored coolingpipes. Heating is supplied by four resistance coils
soldered to bottom plate and driven by four regulated
DC power supplies (Takasago, GPl10-30 x 4) with anaccuracy of 0.0l%o and a maximum combined heat

The Chicago experiments [8,11,121 on turbulence

supply of approximately l2,000watts. Cooling isin RB convection in low temperature gaseous He, saw

supplied by two building air-conditioning units (capacity- l0r4 but-l5000Watts) with PID temperature regulation

did see a suggestive change in the temperature power

controlled by a flow rate throttle. Temperature control ofno indication of a shift in Na scaling up to Ra

29

Sano M. et al., Does the Ultimate State of Thermal Turbulence Exist?

numb.L fluid, and the estimated Reynolds number (Re)is 5 x 10s [l4], which is higher than that of Chavanne e/al. Evet at the highest Ra numbers, the data show noindication of an increase in power law.

the top plate is accurate to 27o of the total temperature
difference (AZ) at the maximum power. Top plate
temperature ranges from l5oC to 42"C and bottom plate

temperature from 18oC to 130'C. The maximum

The Nu vs Ra curve is a highly averaged
characterization of the fluid flow. In particular,

variation of Prandtl number occurs at maximum power
when it is 0.024 at the top plate and 0.018 at the bottomplate. The horizontal temperature non-uniformity acrossthe plate diameter is much less than l7o. The size andpower capacity were dictated by our need to search
three decades higher in Ra number than the highest

transitions which change the temperature histograms andpower spectra (about which the theory is mute) might
not change the Nu vs Ra exponent. Figure 2 compares
the power spectra and histograms for temperature timeseries taken for Ra = 1.85 x 10e and Ra = 5.14 x 1010.
Except for the expected increase in inertial scaling range

value predicted for the transition so that we could
definitively determine whether the transition exists.
A thermal shield surrounds the cell. It can betemperature matched to the cell to eliminate radiativeand convective heat losses. Without the thermal shield,
heat loss from the bottom plate is less than 7Vo at thehighest Ra and much lower for smaller Ra. With theand shrinking width of exponential decay in thetemperature histogram, the two curves correspondexactly to each other. The small asymmetry in thehistogram shape results from the location of thetemperature probe above the center of the cell, l5 cm
from the cold top plate [10]. Again we see no sign of aqualitative change in the turbulence.

thermal shield the worst case heat loss is less than 0.17o.
Thermal conduction through the cell walls is negligible.
The power supplied by the heater thus gives the heat
flux to an accuracy of better than l%o.

4. Experimental Results
In Fig. I we show the Nu vs Ra curve for the0

combined data for our four experiments. From Ra = 2 x
105 to 8 x l0r0 the Nusselt numbers lie on top of eachother with a constant scaling exponent of 0.29 + 0.01.
This Ra number is the highest yet achieved in a low PrdlEo3o(L(a)-10-20-30-40-50-60-70
0.01

0.1

110Frequency (Hz)105

2,o104103

101010'lor

Er 102

lott

Ra101

Fig. 1 Nusselt number as a function of Rayleigh number

100

for: circles, large aspect ratio one half cell;-0.3crosses, small aspect ratio 1 cell; squares small
aspect ratio two cell; bullets, small aspect ratio
one half cell. The solid line line is Nu - 8a0285. Thediscontinuity in the small aspect ratio two data isa result of a pattern competition instability. The
reduced slope for small Ba for the aspect ratio
one data results from strong bulk circulation.

-o.2 -0.1 00.1 0.2 0.3

T - <T> (1/AT)
Fig. 2 (a) Power spectra of the temperature fluctuations.

(b) Histograms of the temperature fluctuations.
Thin line: Fa = 1.85 x 10s. Thick line: Fa = 5.14 x1010.

30

Sano M. et al., Does the Ultimate State of Thermal Turbulence Exist?

t5l L.N. Howard, Annu. Rev. Fluid Mech. 4, 473

5. Conclusion(re12).

Chavanne et al. clearly see an interesting effect ingaseous He - an increase of Na at very high Ra - but
the transition is gradual and never shows a clear powert6l F.H. Busse, Adv. Appl. Mech. 18,77 (1978).
[7] S. Cioni, S. Ciliberto and J. Sommeria, J. Fluid

Mech.335, rrr (1997).
t8l X.Chavanne, F. Chilla, B. Castaing, B. Hebral, B.
Chabaud and J. Chaussay, Phys. Rev. Lett. 79,3648 (1997); Physics Today, Physics News oflaw, certainly not Ratt2. One complication in interpretingtheir results is that Pr variations near the critical point.

In Hg, in which these complicating factors do notcontribute (and for substantially higher Re), we see noevidence of a transition. Based on these results it1997.

t9l A. Belmonte, A. Tilgner and A. Libchaber, Phys.

appears that classical hard turbulence holds forarbitrarily high Ra, up to the limit where shocks andRev. E 50,269 (1994).
[0] T. Segawa, A. Naert and M. Sano, Phys. Rev. E 57,557 (1998).
ll1lF. Heslot, B. Castaing and A. Libchaber, Phys.
Rev. A 36, 5870 (1987).

molecular granularity become important.

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