Turbulent Convection
in the Laboratory
K.R. Sreenivasan
New York University
September 5, 2014
Gänseliesel
• Interior convection is an important ingredient
of solar physics
• I have been working on laboratory convection
for many years
• And have always thought controlled laboratory
experiments might shed some light particularly
on interior convection
• Although you are all experts on the subject, I
will explain some laboratory experiments and
computer simulations which may have some
bearing on your expertise.
Basic Notation
T
Nu depends on…
H
Rayleigh number: Ra 
Fluid
Prandtl number:


Pr 

Aspect ratio:  
T+T
D
Nusselt number
Nu = Q/(k T/H)
gTH 3
D
H
S ~ detailed shape ??
Q = vertical heat flux
k = thermal conductivity of the fluid

(exponent close to 1/3)
Niemela, Skrbek, KRS & Donnelly, Nature 404, 837 (2000)
Slightly revised: Niemela & KRS, J. Low Temp. Phys. 143, 163 (2006)
[Pioneers: Threlfall (Cambridge); Libchaber, Kadanoff and coworkers (Chicago)]
Nu ≈ 2.61010
1010
Ra 
108
gTH 3

Nu = Q/(k T/H)
Plasting & Kerswell (2003)
“upperbound”
Nu ≈ 2.9109
Kraichnan (1962)
“ultimate state”
Nu ≈ 5106
(almost the same as
the extrapolated value)
106
Seems consistent with
Hanasoge, Duvall and KRS (2012)
Convective processes are far
from being optimally efficient.
Ra=1024
Urban et al. (2014)
See also: Roche et al. (2010)
Chillá & Schumacher (2012)
It is disappointing that we still don’t know
with confidence the heat transport law at
high Rayleigh numbers even in the simple
case of Rayleigh-Bènard convection
Data on Rotating Convection
our data
(from Cheng et al. (2014), modified by me)
Sun
0.995
15
Nu/Nu0
15
4.23x10 <Ra<4.31x10
Nu(0) adjusted accodring to local slope
0.990
for calculating ratio
Nu/Nucorr(0)
0.985
0.980
0.975
exponent: 0.024
0.970
log-log fit:
0.024
Nu=1.019Nucorr(0)Ro
0.965
0.960
0.4
0.5
0.6 0.7 0.8 0.9 1
2
convective Rossby number
Rossby number

Rotating Convection
Heat transport decreases only modestly with rotation,
and this appears true for the conditions of the Sun
“Giant Convection Cells Found on the Sun”--title of a Science paper
“Large-scale toroidal cells a challenge to
theories of the Sun”---a website declares
The mean wind
The “mean wind” breaks
symmetry, with its own
consequences
Large scale
circulation
(wind)
largescale
circulation
(“mean
wind”)
the container
For
convection in
a round
cylinder, the
mean wind
precesses
freely.
For convection
in a cubic box,
the mean wind
is constrained
along a
diagonal.
The mean wind…with occasional reversals
V(t), cm/s
(KRS, Bershadskii & Niemela, PRE 65, 056306, 2000; Niemela et al. JFM, 2001)
10
VM Segment of continuous
120-hour record; 1
0
VM
-10
0
2000
4000
6000
8000
10000
t, sec
Geomagnetic polarity reversals
Glatzmaier, Coe, Hongre & Roberts, Nature 401, 885-890 (1999)
The reversals become more frequent with increasing Ra.
J.J. Niemela and KRS
J. Fluid Mech. 557, 411-422 (2006).
12.5
50 cm
Ra = 1.9 x 109
4H12.5cm
Aspect ratio effect
Summary remarks
High-Rayleigh-number convection
experiments tantalize us with quantitative
connections to the convection processes in
the Sun: heat transport law, large-scale
convection cells, rotation, etc.
Alas, the connections seem to become
weaker upon scrutiny, but there are
reasons to be optimistic.