Turbulent Thermal Convection

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Turbulent Thermal Convection
J. Niemela ICTP, Trieste
A. Bershadskii ICAR, Jerusalem 91000,
Israel R.J. Donnelly University of Oregon R.
Hwa University of Oregon A. Praskovski NCAR,
Boulder L. Skrbek Institute of Physics ASCR and
Charles University, Prague K.R. Sreenivasan ICTP,
Trieste
Research supported by the National Science
Foundation grant DMR-9529609
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Thermal Convection
a fluid thermal expansion coefficient n fluid
kinematic viscosity k fluid thermal diffusivity
Control parameters for convection
Pr n/k
G D/H
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Global heat flux Nusselt number
Nu ?

measured heat flux
heat flux due to pure conduction
Nu f (Ra Pr G S .. )
Holding (Pr G S ..) constant we expect
Nu m Rab
(possibly with logarithmic corrections and/or
linear combinations of power laws).
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Very high Ra thermal boundary layers at the
upper and lower walls are highly stressed regions
giving rise to plumes
Plume Sparrow, Husar Goldstein J. Fluid Mech.
41, 793 (1970)
The temperature gradient is all at the wall!
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Self-organization of plumes in large scale flows
the complication of lateral confinement.
(from L. Kadanoff, Physics Today, August 2001)
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Phase diagram for helium. For experiments
operating in low temperature gas phase, Ra can be
varied many orders of magnitude.
4.4 K , 2 mbar
5.25K, 2.4 bar
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A cryogenic apparatus for very high Ra (sample
height 1 meter, diameter 0.5 meter)

• Ra (gaDTH3)/(nk) constant(r2aCP). Ra
  increases as r2 in ideal gas regime and as aCP
  near critical point. aCP 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.

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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.
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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.
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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
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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 lt 1014 are
considered to satisfy the Boussinesq
approximation.
The ratio of estimated temperature drops across
the top (DC) and bottom (DH) thermal boundary
layers for the aspect ratio unity experiment of
Niemela and Sreenivasan (2002) following Wu
Libchaber (1989).
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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.
Grossmann/Lohse phase diagram
Niemela Sreenivasan
1/3
Grossmann Lohse
1/5
1/2
Triangles, Chavanne, et al, 2001. Squares,
Niemela, et al, 2000.
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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.

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).
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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 gt 1013) but not so large that
the Boussinesq approximation comes into doubt (Ra
lt 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.
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Plumes and the mean wind using the aspect ratio
unity cell to enhance the organized large-scale
circulation, we study its properties
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.
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A large scale windwith occasional reversals
Segment of continuous 120 hour time series. G
1. The wind oscillates non-periodically between
- VM .
Geomagnetic polarity reversals due to turbulent
convection in the outer core
Glatzmaier, Coe, Hongre and Roberts Nature 401,
p. 885-890, 1999
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Comparing duration time for medium energy solar
flares obtained from the RHESSI satellite to
times corresponding to flow in one direction.
(Bershadskii, Niemela and Sreenivasan (2004)
Reuven Ramaty High Energy Solar Spectroscopic
Imager (RHESSI)
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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.
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Exponential decay tails for durations times of
(a) thermal convection and (b) solar flares at
long times
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Relation to neighboring events requires moving
average moments
For critical clusterization we expect
where the order parameter
Scaling of the moving moments for (a) thermal
wind reversals and (b) solar flares.
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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.
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• Discussion
• Principally because of the large scale modes, G
  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 gt 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.