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Turbulent Rayleigh-Benard Convection A Progress Report

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Turbulent Rayleigh-Benard Convection

A Progress Report

Work done in collaboration with

Eric Brown, Denis Funfschilling

And Alexey Nikolaenko,

supported by the US Department of Energy

Guenter Ahlers

Department of Physics and iQUEST

UCSB

Rayleigh-Benard Convection Cell

Fluid: H2O

Wall: Plexigl.

G = D/L = 1.0

D

Q

Three samples:

Small: D = L = 9 cm

Medium: D = L = 25 cm

Large: D = L = 50 cm

DT = 20 oC : R = 1011

Q = 1500 W

L

DT

R = a g L3 DT / k n

Q

Prandtl No. s = n/k = 4.4

( H2O, 40 oC )

Important process in the

Atmosphere: Weather

Mantle: Continental Drift

Outer core: Magnetic field

Sun: Surface temperature

etc.: ???

Interesting Physics

A central prediction: Heat transport.

Define Nusselt number N = leff / l ; leff= Q / ( DT / L )

Various models were proposed:

Malkus (1954), Priestly (1959): N ~ R1/3

Kraichnan (1962): N ~ R1/2

Castaing et al. (1989)

Shraiman + Siggia (1990): N ~ R2/7

Grossmann and Lohse (2000):

No single power law; Crossover between

two power laws

[Quarterly Journal of

the Royal

Meteorological

Society 85, 415 (1959)]

Define N = leff / l ; leff= Q / ( DT / L )

Q = N l DT / L = heat-current density

Assume that there are power laws and that the

R- and s-dependence of N separates:

N = f( s ) Rg ; R = (a g DT / k n) L3

Q = f( s ) Rg l DT / L

Q = f( s ) (a g DT / k n)gl DT L(3g - 1)

Assume that the heat-current density Q is determined by the BLs and does not depend on the distance between them. Then

3 g - 1 = 0; g = 1/3; N ~ R1/3

R. Krishnamurty and L.N. Howard, Proc. Nat. Acad. Sci. 78, 1981(1981):

Large Scale Circulation (“Wind of Turbulence”)

R = 6.8x108

s = 596

- = 1
cylindrical

slightly tilted

in real time

Movie from the group of K.-Q Xia, Chinese Univ., Hong Kong

connects the BLs and invalidates the simple models.

Q

T1

thermal boundary layer

cold plumes

schematic drawing

of the flow structure

“wind” or

large scale

circulation

( T1 + T2 ) / 2

thermal boundary

layer

hot plumes

T2 > T1

See, e.g., X.-L. Qiu and P. Tong, Phys. Rev. E 66, 026308 (2002)

However, assuming no interaction between

BLs is not needed to get 1/3 !

W.V.R. Malkus, Proc. Roy. Soc. (London) A 225, 196 (1954)

Assume laminar BLs with conductivity l

and DT/2 across each:

Q = N l DT / L = l (DT/2) / l

l = BL “thickness”

l = L / 2N

Assume laminar BLs are marginally stable:

R = Rc = a g l3 DTc / k n = O(103); DTc =DT/2

l ~ (DT/2)-b ~ R-b; b = 1/3; N = L/2 l;

N ~ R1/3

z / l

Wind

direction

x

Experiment: b not = 1/3 and depends on horizontal location !

S.-L. Lui and K.-Q. Xia, Phys. Rev. E 57, 5494 (1998). s = 7.

1975: D.C. Threlfall, J. Fluid Mech. 67, 17 (1975).

1987: F.Heslot, B. Castaing, + A. Libchaber(Chicago), Phys. Rev. A 36, 5870 (1987).

N ~ R0.282

Mixing layer model (bulk, BL, and plume region between them)

ofthe Chicago group

[Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thome, Wu, Zaleski, & Zanetti,

J. Fluid Mech. 204, 1 (1989)]

and of

B.I. Shraiman and E. Siggia, Phys. Rev. A 42, 3650 (1990):

N ~ s-1/7 R2/7; 2/7 = 0.2857… .

S. Grossmann and D. Lohse, J. Fluid Mech. 407, 27 (2000) (GL)

start with the kinetic and thermal dissipation rates

Their volume averages follow from the Boussinesq equations and are given by

GL set each equal to a sum of a BL and a bulk contribution:

They assume that the separate contributions can be modeled using approximations

to the length, temperature, and time scales, e.g.

(assumes laminar BLs, uniform in the

x-y plane, with conductivity l)

(based on Blasius BL model)

etc.

log( s )

log( R )

No simple power laws, but rather cross-overs from a small-R to a large-R

asymptotic region.

Various regions in the R - s plane, depending on which dissipative

term dominates, etc. For s > 1 and large R, IVu pertains.

There eu and eq are both bulk dominated.

D. Funfschilling, E. Brown, A. Nikolaenko, and G. A., J. Fluid Mech., submitted.

1.) No power law Fluid Mech.,

2.) 4 parameters of the

GL model were

determined from a fit

to these data

X. Xu, K.M.S. Bajaj, and G. A., Phys. Rev. Lett. 84, 4357 (2000);

G. A. + X. Xu, Phys. Rev. Lett. 86, 3320 (2001)

A.) The important components have been identified: Fluid Mech.,

1.) top and bottom boundary layers

2.) “plumes”

3.) large-scale circulation

B.) The nature of the interactions between boundary layers,

plumes, and large scale circulation is not so clear.

C.) The GL model can be fitted to existing Nusselt data

by adjusting its four undetermined coefficients

D.) Adjustment of a fifth parameter gives reasonably

good agreement with the measured Reynolds numbers

of the LSC.

New Nusselt-Number Fluid Mech.,

Measurements

N / R Fluid Mech., 1/4

Prandtl Number s

R = 1.8x109

s -1/7

R = 1.8x107

X. Xu, K.M.S. Bajaj, and G. A., Phys. Rev. Lett. 84, 4357 (2000);

G. A. + X. Xu, Phys. Rev. Lett. 86, 3320 (2001)

K.-Q. Xia, S. Lam, and S.-Q. Zhou, Phys. Rev. Lett. 88, 064501 (2002).

S. Grossmann and D. Lohse, Phys. Rev. Lett. 86, 3316 (2001).

Foam inside of here Fluid Mech.,

Water cooled Cu top plate

adiabatic side shield

Plexiglas side wall

Joule heated Cu bottom plate

Adiabatic bottom-plate shield

Leveling and support plate

Catch basis

E. Brown, A. Nikolaenko, D. Dunfschiling, and G.A., Phys. Fluids, submitted.

D. Funfschilling, E. Brown, A. Nikolaenko, and G. A., J. Fluid Mech., submitted.

2 % Fluid Mech.,

D. Funfschilling, E. Brown, A. Nikolaenko, and G. A., J. Fluid Mech., submitted.

- The GL model can not reproduce the effective exponent Fluid Mech.,
- = 0.333 ~ 1/3 derived from the Nusselt number data
for R > 1010.

Reynolds-Number Fluid Mech.,

Measurements

0.63 cm Fluid Mech.,

C Fluid Mech., 00

-C04

R Fluid Mech., e = (L / t1)(L / n)

Medium Sample

X.-L. Qiu and P. Tong,

Phys. Rev. E 66, 026308 (2002).

unpublished

Large Sample

A.) The important components have been identified: Fluid Mech.,

1.) top and bottom boundary layers

2.) “plumes”

3.) large-scale circulation

B.) The nature of the interactions between boundary layers,

plumes, and large scale circulation is not so clear.

C.) Models yielding relationships between, the Nusselt number,

Rayleigh number, Prandtl number, and Reynolds number

(of the LSC) are at best good approximations, but for

large R miss important physics.

LSC Reversals Fluid Mech.,

E. Brown, A. Nikolaenko, and G. A., Phys. Rev. Lett., submitted.

T Fluid Mech., i = <T> + d cos( ip/4 + q )

Rotation Fluid Mech.,

Cessation Fluid Mech.,

probability distribution of Fluid Mech., |Dq| for reorientations with d/<d> < 0.25

A.) LSC “reversal” can occur via Fluid Mech.,

1.) rotation of the vorticity vector (“rotation)

2.) shrinking of the vorticity vector, followed by

re-development with a new orientation (“cessation”)

B.) Cessation is followed by re-development of the LSC

in a circulation plane with an arbitrary new orientation,

i.e. P(Dq) ~ constant.

C.) Rotation through an angle Dq has a powerlaw

probability distribution P(Dq) ~ Dq-g with g ~ 4.

D.) Reversals are Poisson distributed.

Shadowgraph Fluid Mech.,

lens

Hot plumes/rolls

near the bottom plate

appear dark

Cold plumes/rolls

Near the top plate

Appear bright

pin hole

and LED ligh

source

beam-splitter

lens

Rayleigh

Bénard cell

2o tilt

mirror

The maximum of the correlation Fluid Mech.,

Function is located at DX, DY

Relative to its origin (center).

Viewed from

Above:

direction of plume

movement and

presumed direction of

circulation of LSF

Lowest point

Speed

Q

Direction of 2 deg. tilt

Highest point

The angle Fluid Mech., qof the plane of the large-scale-flow circulation,

and its time correlation function

R = 7.0 x 108

Assumption: Fluid Mech.,

Near the center of the top and bottom plate plumes/rolls

follow the large-scale flow

Conclusion

Near the center of the top and bottom plate the large-scale flow direction oscillates about the vertical axis of the cell.

This oscillation has the same frequency as the periodic

signals seen by others in measurements at individual points.

The frequency yields a Reynolds number consistent with measurements by other methods and the GL model.

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