Turbulent Thermal Convection

J. NiemelaICTP, Trieste

A. Bershadskii ICAR, Jerusalem 91000, Israel R.J. Donnelly University of OregonR. Hwa University of OregonA. Praskovski NCAR, BoulderL. Skrbek Institute of Physics ASCR and Charles University,

PragueK.R. Sreenivasan ICTP, Trieste

1

Research supported by the National Science Foundation grant DMR-9529609

Flu id

T+ T

T

D

H

Pr =

D/HRag TH

3

Thermal Convection

Control parameters for convection

fluidthermal expansion coefficient

fluid kinematic viscosity

fluid thermal diffusivity

Nu

Nu = f (Ra; Pr; S; ….. )

Holding (Pr; S; …..) constant we expect:

Global heat flux: Nusselt number

measured heat flux

heat flux due to pure conduction

Nu = m Ra

(possibly with logarithmic corrections and/or linear combinations of power laws).

Plume Sparrow, Husar & Goldstein J. Fluid Mech. 41, 793 (1970)

Very high Ra: thermal boundary layers at the upper and lower walls are highly stressed regions giving rise to plumes

The temperature gradient is all at the wall!

(from L. Kadanoff, Physics Today, August 2001)

Self-organization of plumes in large scale flows: the complication of lateral confinement.

4.4 K , 2 mbar: 3/ 5.8 10

5.25K, 2.4 bar:9/ 6.5 10

3HTgRa

Phase diagram for helium. For experiments operating in low temperature gas phase, Ra can be varied many orders of magnitude.

C o n v ec tio n C e ll(C ry o g en ic H e liu m G as)

M u ltilay e r In su la tio n

B o tto m P la te(F ix ed H ea t F lu x )

To p P la te(F ix ed Tem p era tu re )

L iq u id H e liu mR ese rv o ir

L iq u id N itro g en R ese rv o ir

C ry o co o le r

4 .5 K

2 0 K

7 7 K

P re ssu re R e lie fan d S en s in g

C e ll F illT u b e

S o ft Vacu u mH ea t tran sfe r

The

rmal

Shi

elds

A cryogenic apparatus for very high Ra (sample height = 1 meter, diameter = 0.5 meter)

Ra = (gTH3)~ constant*CP). Ra increases as in ideal gas regime and as CP near critical point. CP is decades larger than for conventional fluids.

11 decades of Ra possible! Large sample height moves entire range of Ra into turbulent regime and indirectly extends conditions of constant Pr (ideal gas) to higher Ra.

105 107 109 1011 1013 1015 1017

101

102

103

104

=1/2

Dashed line "single mode"

model: Nu=0.084Ra3/10ln(Ra)1/5

(Herring, Roberts, Stewartson,Howard)1 free parameter

Solid (red) line fit to data: Nu=0.124Ra0.309

2 free parameters

Nu

Ra

Observed Nu vs Ra for 1 meter tall sample. The single mode model prediction with one adjustable parameter (its prefactor) fits well over 11 decades of Ra, and as well as (and therefore better than) an arbitrary polynomial with 2 degrees of freedom.

108 1010 1012 1014 1016101

102

103

104 Chavanne et al. 2001, Niemela et al. 2000, present data,

N

u

Ra

Removing 1/2 of the sidewall height to form an aspect ratio 1 cell altered the Nu-Ra relation. Differences exist between other groups as well at high Ra for the aspect ratio 1/2. Such differences may arise through influence of lateral confinement on self organized large scale flows at moderately high Ra.

106

108

1010

1012

1014

1016

1018

0.01

0.1

1

Niemela, et al (2000) Chavanne, et al (2001) Niemela & Sreenivasan (2003)

ideal gas

(T

) -1

Ra

Results at the highest Ra may be influenced by approach to the critical point. For the 1 meter, aspect ratio ½ experiment the critical point can be avoided to considerably higher Ra than for other experiments.

Thermal expansion coefficient normalized by its ideal gas value for various experimental operating points

106 108 1010 1012 1014 10160.85

0.90

0.95

1.00

C/

H

RaThe ratio of estimated temperature drops across the top C and bottom (H) thermal boundary layers for the aspect ratio unity experiment of Niemela and Sreenivasan (2002) following Wu & Libchaber (1989).

A relevant measure of broken symmetry is the ratio of calculated temperature drops across the top and bottom thermal boundary layers. Assuming an arbitrary 5% threshold, Ra < 1014 are considered to satisfy the Boussinesq approximation.

Grossmann/Lohse phase diagram

1/3

1/5 1/2

Niemela & Sreenivasan

Grossmann & Lohse

Predicted power law exponents for Nu-Ra relation in various regions of the Pr-Ra phase space according to Grossmann and Lohse are shown in red. A laminar-to-turbulent transition in the momentum boundary layer is estimated by the two labeled dashed and dotted lines. For the two high-Ra experiments considered here the exponent would change from 1/3 to1/2 to the right of these lines.

Triangles, Chavanne, et al, 2001. Squares, Niemela, et al, 2000.

106 108 1010 1012 1014 1016

1

10

Pr

Ra

Triangles, Chavanne, et al (2001); squares, Niemela, et al (2000); solid circles, Niemela & Sreenivasan, 2003. The various lines plot the locus of points in Pr-Ra where the momentum and velocity layers have equal thickness. Solid line: Malkus (2001); dashed line; Niemela & Sreenivasan (2003); dot-dashed line: Shraiman & Siggia (1994).

Estimations for unity thickness ratio of momentum to thermal boundary layers —a necessary criterion in order that turbulent momentum layers could enhance the heat transfer through the thermal boundary layer, leading to a heat transfer scaling exponent 1/2. Of all experiments to date only the one-meter experiment appears to simultaneously satisfy this criterion and the presence of a turbulent momentum layer.

9 10 11 12 13 140.03

0.04

0.05

0.06

0.07

0.080.09

0.1

Chavanne et al. 2001, (+0.03) present data, Niemela et al. 2000, (-0.02)

Nu

/Ra1/

3 (a.

o.)

log Ra13.0 13.5 14.0

0.03

0.04

0.05

0.06

0.070.080.09

0.1

Nu

/Ra1/

3 (a.

o.)

log Ra

Since it is unlikely that turbulent boundary layers play an important role in laboratory experiments in determining the heat transfer, we might expect to see a limiting slope of 1/3 at high enough Ra that the organized large scale flow is weak (Ra > 1013) but not so large that the Boussinesq approximation comes into doubt (Ra < 1014). This seems to be consistent locally with the experimental observations over this narrow range.

Nu normalized by Ra to the 1/3 power for various high Ra experiments. Right: an expanded view of the region to the right of the vertical dotted line in the figure on the left.

Plumes and the mean wind: using the aspect ratio unity cell to enhance the organized large-scale circulation, we study its properties

50 c

m w

w = 4.4c md = 1.27c m

0 2000 4000 6000 8000 10000 12000

-20

0

20

40

60

80

(t)

V(t)

sig

nal

time (sec)

Maximizing the correlation between temperature sensors gives the magnitude and direction of the large scale circulation, which exhibits occasional and sudden reversals. The velocity V(t) above is displaced upwards by 60 units for clarity but in actuality oscillates about zero.

250-micrometer NTD-doped Ge sensors are placed in various positions in the flow. Pairs of vertically-spaced sensors placed near the sidewall allow measurement of the vertical component of the large scale velocity.

0 2000 4000 6000 8000 10000

-10

0

10

VM

VM

V(t

), c

m/s

t, sec

A large scale wind…with occasional reversals

Glatzmaier, Coe, Hongre and Roberts Nature 401, p. 885-890, 1999

Geomagnetic polarity reversals due to turbulent convection in the outer core

Segment of continuous 120 hour time series. The wind oscillates non-periodically between +- VM .

Comparing duration time for medium energy solar flares obtained from the RHESSI satellite to times corresponding to flow in one direction.

Reuven Ramaty High Energy Solar Spectroscopic Imager(RHESSI)

(Bershadskii, Niemela and Sreenivasan (2004)

PDFs of duration times for (left) the maintenance of one direction of the wind in confined thermal convection and (right) medium energy solar flares. Both exhibit the same power law scaling with -1.1 exponent, also consistent with self-organized criticality.

Exponential decay tails for durations times of (a) thermal convection and (b) solar flares at long times

Relation to neighboring events requires moving average moments:

1

0)(

1 rni

ir ir

For critical clusterization we expect ppr

~ where the order parameter 1r

Scaling of the moving moments for (a) thermal wind reversals and (b) solar flares.

Critical exponents derived in the two cases of thermal wind and medium energy solar flares reveal remarkable close values. For high energy flares there is no critical scaling at all, as also shown, Such flares are completely random explosive events. The lifetimes of medium energy flares, however, are presumed to be intimately linked to the underlying hydrodynamic structure of the solar convection zone.

Discussion

Principally because of the large scale modes, is an important variable in turbulent thermal convection. Such self organized flows may depend intimately on the details of the boundary conditions, creating difficulties in interpreting data. Hence, large aspect ratio experiments are necessary and are currently in progress.

The observation of an “ultimate regime” with exponent ½ power law scaling of the Nu-Ra relation is unlikely to be observed in experiments and the available data do not support it. Conversely, an asymptotic exponent of 1/3 appears to be consistent with observations over a limited range of Ra high enough to avoid complications of confinement effects, yet reasonably satisfying the Boussinesq approximation.

While the large scale flow complicates analysis of the heat transfer, it is an interesting phenomenon in itself and the distribution of duration times for its flow in any particular direction reveals power-law scaling and critical clusterization, similar to other complex phenomena such as solar flares, and to self organized criticality.

Ultimately, a high Ra experiment under ideal gas (constant Pr) and Boussinesq conditions would be desirable---with a 5 meter tall sample, Ra > 1015 could be achieved in the laboratory under these ideal conditions. Such an experiment would also satisfy necessary criteria for observing the so-called “ultimate regime”.