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Turbulent Convection in the Laboratory
K.R. SreenivasanNew 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.
Flu id
T+ T
T
D
H
Basic Notation
Q = vertical heat fluxk = thermal conductivity of the fluid
Rag TH
3
Rayleigh number:
Prandtl number:
Pr
Aspect ratio:H
D
S ~ detailed shape ??
Nu depends on…
Nusselt numberNu = Q/(k T/H)
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)]
(exponent close to 1/3)
1010
Ra=102424
106
108
Nu ≈ 5106
Nu ≈ 2.9109
Nu ≈ 2.61010
(almost the same as the extrapolated value)
Kraichnan (1962)“ultimate state”
Plasting & Kerswell (2003)“upperbound”
Seems consistent with Hanasoge, Duvall and KRS (2012)
Convective processes are far from being optimally efficient.
Rag TH
3
Nu = Q/(k T/H)
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
Sun
(from Cheng et al. (2014), modified by me)
0.4 0.5 0.6 0.7 0.8 0.9 1 20.960
0.965
0.970
0.975
0.980
0.985
0.990
0.9954.23x1015<Ra<4.31x1015
Nu(0) adjusted accodring to local slopefor calculating ratio
log-log fit:
Nu=1.019Nucorr
(0)Ro0.024
Nu
/Nu co
rr(0
)
convective Rossby number
Heat transport decreases only modestly with rotation,and this appears true for the conditions of the Sun
exponent: 0.024
Rotating Convection
Nu/Nu0
Rossby number
“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
Large scale circulation
(wind)
the container
large-scale
circulation (“mean wind”)
The “mean wind” breaks symmetry, with its own
consequencesThe mean wind
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.
0 2000 4000 6000 8000 10000
-10
0
10
VM
VM
V(t
), c
m/s
t, sec
Glatzmaier, Coe, Hongre & Roberts, Nature 401, 885-890 (1999)
Geomagnetic polarity reversals
The mean wind…with occasional reversals(KRS, Bershadskii & Niemela, PRE 65, 056306, 2000; Niemela et al. JFM, 2001)
Segment of continuous 120-hour record;
The reversals become more frequent with increasing Ra.
50 cm
J.J. Niemela and KRSJ. Fluid Mech. 557, 411-422 (2006).
Ra = 1.9 x 109 cm
12.5
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.