Heat transfer in convective

turbulence

Jörg Schumacher

Theoretical Fluid MechanicsTechnische Universität Ilmenau

Germany

Where is Ilmenau?

Johann W. Goethe Karlheinz

Brandenburg

Barrel of Ilmenau

Über allen GipfelnIst Ruh'In allen WipfelnSpürest DuKaum einen Hauch;Die Vögelein schweigen im WaldeWarte nur, baldeRuhest Du auch.

Outline

IntroductionLocal: Boundary layer structureLocal: Temperature and thermal dissipationGlobal: Large-scale flow patternsLocal: Lagrangian fingerprintGlobal: Plume clusters?Outlook

Hydrostatic equilibrium

warmer fluidelement

colder fluidelement

fluid element

Ttop Tbottom

Nusselt number

Rayleigh instabilityRayleigh, Philos. Mag. 1916; Jeffreys, Philos. Mag. 1926;

Schlüter, Lortz & Busse, J. Fluid Mech. 1965

Further bifurcations Transition to convective turbulence

stable unstable

Ra

Heatconductiononly

Heatconduction &convection

unstable branch

Model equations

Rotation (opt.) Buoyancy

Approximations

• Density is linear function of temperature Boussinesq-Approximation

• Flow is incompressible (u much smaller than speed of sound)

Oberbeck, Ann. Phys. Chem. 1879; Boussinesq, Théorie Analytique de la Chaleur, 1903

Rayleigh-Bénard convection

HDimensionless controlparameters

Examples

Semiconductor chips, Ra=106

Buildings, Ra=1012

Atmosphere, Ra=1020

Sun, Ra=1023

Rayleigh-Bénard convection

H

„System response“

Power law for heat transfer (for fixed Pr)

=2/7 or 1/3?Shraiman & Siggia, Phys. Rev. A 1991; Großmann & Lohse, J. Fluid Mech.

2000

Deviation by morethan 100% !!!

Niemela et al., Nature

2000

Different models for the velocity boundary layer

Regions in convective turbulence

Bulk Boundary layer

Mixing zone?

Mixing zone?

Plume

Boundarylayer

thickness:0.01

Turbulenttemperature

field

Mean temperature

Heig

ht

Turbulent

structureMean profile Transport law

„Perturbation“„Response“

Why study mean profiles?

7 Meters

„Barrel of Ilmenau“

Worlds biggest convection experiment

Accessible range in the Barrel

6 9 12 15 18

log(Ra)

log

(R

)

Fitzjarrald (1976)

Castaing (1989)

Niemela (2000)

Barrel of Ilmenau (2000)50000

5000

500

50

R=Size of Experiment

Size of Sensor

Luft

Luft

Helium

Helium

Air

Air

Velocity measurements

LD

A

du Puits, Resagk & Thess, Phys. Rev. Lett. 2007

Laser-Doppler-Anemometry vx

Theoretical model

Resolve

boundary

layers!!

vy

Data

Numerical simulations in cylindrical cell

T =0.00005 HRa=1017

T =0.0014 HRa=1012

T =0.016 HRa=109

Proper resolution of boundary layerslimits accessible Ra

Verzicco & Orlandi, JCP 1996

No-slip boundaries, adiabatic side walls

Second-order finite difference scheme

Temperature & temperature derivativeEmran & Schumacher, J. Fluid Mech., in revision

BL Bulk

BLBulk

Strong vertical dependence of statistics of turbulent temperature field

Deviations from local isotropySchumacher & Sreenivasan, Phys. Rev. Lett. 2003

BLBulk

Return to local isotropy (K41) for temperature fluctuations in bulk Formation of „superconducting core“ in high-Ra turbulence

Niemela & Sreenivasan, Phys. Rev. Lett. 2008

Thermal dissipation rateEmran & Schumacher, J. Fluid Mech., in revision

Significant contribution to dissipation due to temperature fluctuations Does not enter the Großmann-Lohse scaling theory!

Großmann & Lohse, J. Fluid Mech. 2000

„Wind of turbulence“

Mean (114 double frames) andsnapshot

Ra=1011, =2

z

x

Barrel: Visualisation with He bubbles

Geometry dependence of heat transport

10% effect

POD analysis

Time-averaged streamlines

Pair dispersion in turbulent convectionSchumacher, Phys. Rev. Lett. 2008

Lagrangian fingerprint of thermal plumes

L

H

Successively flatter cells

W

Cartesian geometry

Free-slip b.c. in zPeriodic b.c. in x and y

Pseudospectral method

Volumetric 3D-FFT

xy

z

y1. r2c in x

2.

kx kx

3. c2c in y

4.z

ky

5. c2c in z

One more communication step necessary !

kz

3

2

1

0

0

1

2

3

Nx = Ny >> Nz is possible as necessary for our studies

N2 processors can be used 2D processor grid(iproc,jproc)

iproc=2

jproc=2

pencil

D. Pekurovsky (SDSC), p3dfft package

Massively parallel supercomputing

Production jobs on 4096 CPUs

Schumacher & Pütz, Proceedings of PARCO 2007

Effect of rotation

Temperature

Temperature

Ta=0

Ta=2.5 108

Mean temperature and heat transfer

Rotation diminishesfluctuations and thus heat

transport

Rotationincreases

Temperature patternsHartlep, Tilgner & Busse, J. Fluid Mech. 2005; von Hardenberg et al., Phys. Letters A

2008

Plume clusters?

... the road ahead

Scale analysis of plume clusters (image segmentation)

Lagrangian analysis in large-aspect-ratio convectioncells temporal persistence of plume clusters?

Statistical significance of thermal plumes with increasingRayleigh number (BL thickness ~ 1/Nu)

Together with O. Pauluis (NYU): simple extension tomoist convection

condensation at saturation level piecewise linear dependence of buoyancy on S and q dry and moist lapse rates allow to tune stability

Summary

Boundary layers (BL): neither laminar Blasius type nor turbulentPrandtl type Scaling theories of heat transfer

Turbulent temperature fluctuations contribute significantly tothermal dissipation in BL

Lateral dispersion of Lagrangian tracers is Richardson-like

Rotation prohibits formation of horizontal large-scale patterns

Clusters of thermal plumes seem to cause large-scaletemperature patterns in non-rotating case

Thanks to

Mohammad Emran, Jorge Bailon-Cuba (TU Ilmenau)

Ronald du Puits, Christian Resagk, Andre Thess (TU Ilmenau)

Katepalli R. Sreenivasan (ICTP Trieste / UMD College Park)

Roberto Verzicco (U Roma II)

Matthias Pütz (IBM Germany)

Jülich Supercomputing Centre (Germany)