Turbulent Rayleigh-Benard Convection A Progress Report

Work done in collaboration withEric Brown, Denis FunfschillingAnd Alexey Nikolaenko,supported by the US Department of Energy

Guenter AhlersDepartment of Physics and iQUEST UCSB

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Rayleigh-Benard Convection Cell

D

LT

= D/L = 1.0

( H2O, 40 oC )Prandtl No. = 4.4

Three samples:

Small: D = L = 9 cm

Medium: D = L = 25 cm

Large: D = L = 50 cmT = 20

oC : R = 1011

Q = 1500 W

Fluid: H2OWall: Plexigl.

R = g L3 T / Q

Q

Important process in the

Atmosphere: Weather

Mantle: Continental Drift

Outer core: Magnetic field

Sun: Surface temperature

etc.: ???

Interesting Physics

Why is it interesting?

Models

A central prediction: Heat transport.

Define Nusselt number N = eff / effQ / ( T / 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

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

Define N = eff / effQ / ( T / L ) Q = N T / L = heat-current density

Assume that there are power laws and that theR- and -dependence of N separates:

N = f() RR = g T ) L3

Q = f() RT / LQ = f() g T ) T L(3

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

C.H.B. Priestley[Quarterly Journal of the Royal MeteorologicalSociety 85, 415 (1959)]

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

Large Scale Circulation (“Wind of Turbulence”)

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

connects the BLs and invalidates the simple models.

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

R = 6.8x108

= 596

= 1cylindricalslightly tiltedin real time

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

hot plumes

cold plumes

“wind” orlarge scalecirculation

T1

T2 > T1

thermal boundary layer

schematic drawingof the flow structure

thermal boundary layer

( T1 + T2 ) / 2

Q

Q

Assume laminar BLs with conductivity and T/2 across each:

Q = N T / L = T/2) / ll = BL “thickness”l = L / 2N

Assume laminar BLs are marginally stable:R = Rc = g l 3 Tc / O(103); Tc = T/2l ~ (T/2)-~ R; = 1/3; N = L/2 l; N ~ R1/3

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

z / l

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)

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

x

Winddirection

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)of the 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 ~ -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 thex-y plane, with conductivity )

(based on Blasius BL model)

etc.

log( R )

log(

)

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

2. Various regions in the R -plane, depending on which dissipativeterm dominates, etc. For > 1 and large R, IVu pertains. There u and are both bulk dominated.

At large R

else

yield and

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

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)

1.) No power law

2.) 4 parameters of theGL model were determined from a fit to these data

A.) The important components have been identified:1.) top and bottom boundary layers2.) “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 Measurements

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).

N /

R1/

4

Prandtl Number

R = 1.8x109

R = 1.8x107-1/7

Adiabatic bottom-plate shield

Joule heated Cu bottom plate

Plexiglas side wall

adiabatic side shield

Water cooled Cu top plate

Foam inside of here

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 %

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

The GL model can not reproduce the effective exponent= 0.333 ~ 1/3 derived from the Nusselt number datafor R > 1010.

Reynolds-Number Measurements

0.63 cm

15 6 071

2

3

4

5

6

7

0

C00

-C04

Medium SampleLarge Sample

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

unpublished

Re = (L / t1)(L / )

A.) The important components have been identified:1.) top and bottom boundary layers2.) “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

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

Ti = <T> + cos( i/4 + )

Rotation

Cessation

probability distribution of for reorientations with /<> < 0.25

A.) LSC “reversal” can occur via 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() ~ constant.

C.) Rotation through an angle has a powerlaw probability distribution P() ~ with ~ 4.

D.) Reversals are Poisson distributed.

More LSC Dynamics

D. Funfschilling and G. A., Phys. Rev. Lett. 92, 194502 (2004).

Hot plumes/rollsnear the bottom plateappear dark

Cold plumes/rolls Near the top plateAppear bright

Shadowgraph

mirror

pin holeand LED ligh source

beam-splitter

RayleighBénard cell

2o tilt

lens

lens

34

37

t = 0.0s

Correlation Functions:

0.3s

0.6s 0.9s

1.2s 1.5s

t = 0.0s

t = 0.9s

t

YXVelocity

22

+

=

⎟⎠

⎞⎜⎝

⎛

=ΘY

Xarctan

The maximum of the correlation Function is located at X, Y Relative to its origin (center).

Speed

Direction of 2 deg. tilt

Highest point

Lowest pointdirection of plumemovement andpresumed direction ofcirculation of LSF

Θ

Viewed from Above:

The angle of the plane of the large-scale-flow circulation, and its time correlation function

R = 7.0 x 108

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.

Assumption:Near the center of the top and bottom plate plumes/rolls follow the large-scale flow