Local wall heat flux in turbulent Rayleigh-Bénard convection

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Local wall heat flux in turbulent Rayleigh-Bénard convection

Ronald du Puits Ilmenau University of Technology

Department of Mechanical EngineeringPOB 100565, D-98693 Ilmenau, GERMANY

contact: [email protected]

Technicians:Vigimantas Mitschunas, Klaus Henschel, Helmut Hoppe

Financial support: German Research Foundation Federal Ministry of Education and ResearchThuringian Government

Collaboration:André Thess, Jörg Schumacher (Ilmenau)

Philippe Roche (Grenoble)

EUROMECH Colloquium #520, High Rayleigh number convection, Les Houches, 24.-29.01.2010

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Outline

1. Motivation to measure the local wall heat flux

2. Heat flux sensor and measurement technique

3. Local heat flux and plume dynamics

4. Local heat flux and the large-scale circulation

5. Conclusion and outlook

1. Motivation

2. Measurementtechnique

3. Plume dynamics

4. Large-scale circulation

5. Conclusion


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1. Motivation

1. Motivation


Rayleigh-Bénard convection reflects a large number of natural and technical flow phenomena.

Atmosheric flows Oceanic flows

Earth core convection Star convection

3. Plume dynamics


5. Conclusion


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1. Motivation


Characteristics of those flows:

2. Small thickness compared with lateral extent large aspect ratio

e.g.

Earth atmosphere: 4000Pacific Ocean: 3000Outer earth core: 10Sun‘s convective zone: 6

HL

1. Extremely high level of turbulence large Rayleigh number

e.g.

Earth atmosphere: 1020

Pacific Ocean: 1021

Outer earth core: > 1020 Sun‘s convective zone: 1023

/3HTgRa

1. Motivation

3. Plume dynamics


5. Conclusion


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1. Motivation


Requirements for experiments to model geophysical flows:

1. high Rayleigh numbers

2. large aspect ratios

Actual high Rayleigh number experiments:

1. Triest (I), Ramax = 1017, = 0.5…4, liquid He

2. Grenoble (Fr), Ramax = 1015, = 0.5…1, liquid He

5. Hongkong (CN) , Ramax = 1014, = 1….20, water

3. Goettingen (D) , Ramax = 1015, = 0.5…1, SF6 (15bar)

6. Ilmenau (D) , Ramax = 1012, = 1…100, Air

4. Santa Barbara (US) , Ramax = 1014, = 0.28…6, various liquids

1. Motivation

3. Plume dynamics


5. Conclusion


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1. Motivation


1. Prediction of the global heat flux in terms of Nu = f(Ra, Pr).

2. The boundary layers play a crucial role for the global heat transport .

1. MotivationThe best known geometry is the = 1 cell of cylindrical shape.

For high Ra convection we know many features of the flow.

3. Plume dynamics


5. Conclusion


q3. The boundary layers interact with the bulk

by thermal plumes.

4. The dynamics of the large-scale circulation, one single role exists, cessations, rotations, reversals can occur, mean plane oscillatesin angular direction, sloshing mode

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1. Motivation


1. Motivation

3. Plume dynamics


5. Conclusion


Can we simply transfer this knowledge to flows at higher aspect ratio?

q

local heat flux sensor

What can we learn from local heat flux measurements at the surface of the heating and the cooling plate?

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2. Experimental facility


Large-scale experimental facility `Barrel of Ilmenau‘

www.ilmenauer-fass.de

H <

6.3

0 m

D = 7.15 m

1. Motivation

3. Plume dynamics


5. Conclusion


Main features:

1. High Ra numbers up to Ra = 1012

2. Continuously variable aspect ratio ≈ 1…100

DNS

small barrel

full-size barrel

Accessible parameter range

1

10

100

5 7 9 11 13

log Ra

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2. Heat flux sensor

heating/cooling plate

heat flux sensor (fcutoff=3Hz)qw (t)

Convection flow

20mm

1.5m

m

Sensor: The local wall heat fluxes were measured using special sensors called heat flux plates.

1. Motivation


3. Plume dynamics


5. Conclusion


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20

10

z[mm]T(z)

Where we are with our heat flux sensor? – Ra ≈ 5x1011, ≈ 1

‹v›(z)

bhp TTq

The heat flux sensor reflects the local convection coefficient depending on the local velocity v(t) and the local temperature difference T(t).

Thickness of the sensor

2. Heat flux sensor

1. Motivation

3. Plume dynamics


5. Conclusion


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1. Motivation

3. Plume dynamics

Thermal plumes at the cooling plate of a RB cell filled with water , Pr ≈ 5 and at Ra = 2.6x109 .

(Y. Du and P. Tong, 52nd APS meeting, Nov 21-23, 1999, New Orleans, Louisiana)



5. Conclusion


1. A thermal plume develops at the surface of the cooled top plate.

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1. Motivation

3. Plume dynamics

Thermal plumes in an aspect ratio one cell filled with Dipropylene glycol withPr = 596 and at Ra = 6.8x108 .(Shang et al., PRL 90, 074501)



5. Conclusion


1. Thermal plumes arise from the heated bottom plate and fall down from the cooled top plate.

2. They organize themselve to a large scale motion of one single role occupying the whole space of the cell.

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1. Motivation

3. Plume dynamics

Generalized sketch of the plume motion in turbulent convection.

Can we generalize this picture to other fluids, e.g. with Pr = 0.7.



5. Conclusion


Can we generalize it to aspect ratios > 1.

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1. Motivation

v

Hypothesis:

1. The plume has to be reflected as a positive burst in the time series of the local heat flux.

2. For a given velocity of the LSC the width can be calculated from the time of the burst.

vtww

t

q


3. Plume dynamics


5. Conclusion


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1. Motivation


Time resolved local heat flux at the surface of the cooling plate.position: centre; =1.13, Ra=4.27x1011, Pr=0.7

1. The local heat flux fluctuates over the time by ≈ 40%.

0 1000 2000 3000 4000 500070

80

90

100

110

120

130

time [s]

he

at

flu

x [W

/m²]

2. Positive eruptions occur as often as negative ones do.

3. Plume dynamics


5. Conclusion


1800 1850 1900 1950 200080

90

100

110

120

time [s]

he

at

flu

x [W

/m²]

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1. Motivation


1. The typical time of a strong eruption is ≈ 8s.

2. According to a typical mean velocity of the LSC the width of such an eruption is w = v*t = 0.34 ms-1

*8 s = 2.72 m!

7s

8s

3. Plume dynamics


5. Conclusion


75 80 85 90 950

0.02

0.04

0.06

0.08

0.1

0.12

heat flux [W/m²]

pro

ba

bili

ty

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1. Motivation


Probability density of the local heat flux at the surface of the heating plate.position: centre; =1.13, Ra=4.27x1011, Pr=0.7, meas. time: 53 hours

492.3kurt

307.0skew

Wm27.3std

Wm7.85

2

2

q

q

q

q

1. For aspect ratio = 1.13 the fluctuations of the local heatflux are distributed gaussian.

gaussian

3. Plume dynamics


5. Conclusion


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1. Motivation


1. The mean local heat flux and the fluctuations at the heatingand the cooling plate differ, but both are distributet gaussian.

Is the local heat flux symmetrical at the heating and the cooling plate?

Question:

67.3kurt

34.0skew

Wm00.4std

Wm8.98

2

2

q

q

q

q

heating plate cooling plate

49.3kurt

31.0skew

Wm27.3std

Wm7.85

2

2

q

q

q

q

3. Plume dynamics


5. Conclusion


70 80 90 100 110 1150

0.05

0.1

0.15

heat flux [W/m²]

pro

ba

bili

ty

Ra=4.27x1011, T=20K

22 24 26 28 30 32 34 360

0.05

0.1

0.15

0.2

0.25

heat flux [W/m²]

pro

ba

bili

ty

200 250 3000

0.01

0.02

0.03

=7

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1. Motivation


1. For aspect ratios ≈ 4 the fluctuations of the local heat flux are notdistributet gaussian, there are more positive than negative eruptions.

Question:

03.4kurt

76.0skew

Wm92.1std

Wm6.28

2

2

q

q

q

q

Does it depend on aspect ratio?

second measurementposition: centre; =4.00, Ra=4.00x109, Pr=0.7

2. There is a sharp cut towards lower heat fluxes.

3. Plume dynamics


5. Conclusion


94.2kurt

28.0skew

Wm25.15std

Wm9.245

2

2

q

q

q

q

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1. Motivation


1. For aspect ratios ≈ 1 and Pr ≈ 1 the LSC is stronger as the localadvection of plumes. In this case the interface between bulk and boundary layer fluctuates.(K.-Q. Xia, ETC 11, Marburg)

Question: How can we interpret these results?

≈ 1, Pr ≈ 1

≈ 4, Pr ≈ 1

fluctuating boundary layer

LSC LSC

2. For aspect ratios ≈ 4 and Pr ≈ 1 the LSC is weak, plumes

can emerge freely, the sharp cut towards low heat fluxes can be associated with maximum boundary layer thickness

3. Plume dynamics


5. Conclusion


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3. Local heat flux and the large-scale circul.

1. Motivation


Question: Are the boundary layers coupled by the large scale circu-lation and is the emission of plumes a periodic process?

Sketch of the large-scale circulation shearing the boundary layers.Villermaux, PRL 75, 4619 (1995)

1. Boundary layer instabilities (plumes) are emerged at one plate.

The Model of Villermaux:

2. They travel within the large-scale circulation to the other plate and create a second instability (plume) there.

3. The process runs periodically and can be described by a model of two coupled oscillators.

3. Plume dynamics


5. Conclusion


-5000 -2500 0 2500 5000-0.2

0

0.2

0.4

0.6

0.8

1

[s]

Cx

y

-100 -50 0 50 100-0.2

0

0.2

0.4

0.6

0.8

1

[s]

Cxy

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1. Motivation


Local heat flux sensors

= 1.13, Ra = 4.27x1011, Pr = 0.7

11

0

myxmC n

mN

nmnxy

We compute the cross-correlation function Cxy

1. For ≈ 1 and Pr ≈ 1 the local heat fluxes at the centre of the heating and the cooling plate are not correlated.


3. Plume dynamics


5. Conclusion


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1. Motivation



= 2.00, Ra = 0.52x1011, Pr = 0.7

1. For ≈ 2 and Pr ≈ 1 the local heat fluxes at the centre of the heating and the cooling plate are correlated.

-5000 -2500 0 2500 5000-0.2

0

0.2

0.4

0.6

0.8

1

[s]

Cx

y

-250 -125 0 125 250-0.2

0

0.2

[s]

Cxy

100s

3. Both boundary layers are coupled by the large-scale circulation.

2. Thermal plumes emerging from the cooling plate arrive at the heating plate and vice versa.


3. Plume dynamics


5. Conclusion


-5000 -2500 0 2500 5000-0.2

0

0.2

0.4

0.6

0.8

1

[s]

Cxy

-1000 -500 0 500 1000-0.2

0

0.2

[s]C

xy

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1. Motivation



= 4.00, Ra = 4.00x109, Pr = 0.7


3. Plume dynamics


5. Conclusion



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1. Motivation


Question: Can we identify the recently found „sloshing mode“ of the large-scale circulation also in RB convection at Pr ≈ 1?


Model of the large-scale circulation with the „oscillatory“ mode (red) and the „sloshing mode“ (green) .

Xi et al., PRL 102, 044503 (2009)Brown et al., JFM 638, 383 (2009)

q

t

q

t

oscillatory mode

sloshing mode3. Plume dynamics


5. Conclusion

0 1000 2000 3000 4000 5000-0.2

0

0.2

0.4

0.6

0.8

1

1.2

[s]

Cxx

0 50 100 150 200-0.5

0

0.5

1

[s]

Cxx

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1. Motivation


1. For Pr ≈ 1 and ≈ 1 the sloshing mode could not be identified.


= 1.13, Ra = 4.27x1011, Pr = 0.7

11

0

mxxmC n

mN

nmnxx

We compute the auto-correlation function Cxx

3. Plume dynamics


5. Conclusion


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1. Motivation


0 2000 4000 6000-0.2

0

0.2

0.4

0.6

0.8

1

[s]

0 300 600 900-0.2

0

0.2

0 2000 4000 6000-0.2

0

0.2

0.4

0.6

0.8

1

[s]

0 300 600 900-0.2

0

0.2

Auto-correlation function of the local heat flux for larger aspect ratios .

1. For larger aspect ratios 2 and 4 (Pr ≈ 1) we do not have sufficient information to interprete these results.


= 4.00, Ra = 4.00x109, Pr = 0.7

= 2.00, Ra = 0.52x1011, Pr = 0.7

? ?

3. Plume dynamics


5. Conclusion


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• Local heat flux measurements have been carried out at the surface of the heating and the cooling plate.

• The advection of plumes is a stochastic process which depends on the geometry of the cell.

• For aspect ratio ≈ 1 and Pr ≈ 1 the boundary layers at the centre of the cell are not coupled by the large-scale circulation while for larger aspect ratios correlations have been found.

• The „sloshing mode“ of the large-scale circulation in cylindrical samples of ≈ 1 and Pr ≈ 5 could not be found in a Pr ≈ 1 cell.

Conclusion:

4. Conclusion and Outlook

1. Motivation

3. Plume dynamics


5. Conclusion


It would be nice to have a complete picture of the distribution of the heat flux at the plates!

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Les Houches, 24.-29.01.2010

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1. Motivation


3. Plume dynamics


5. Conclusion


-5000 -2500 0 2500 5000-0.2

0

0.2

0.4

0.6

0.8

1

[s]

Cxy

-1000 -500 0 500 1000-0.2

0

0.2

[s]

Cxy

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• In an ongoing national Research Group we are going to study particularly these boundary layer dynamics like local heat fluxes, plume emissions etc.

• The group runs from 2010 to 2015 and the following groups will participate:- Bruno Eckardt (Uni Marburg) - C. Egbers (Uni Cottbus) - A. Delgado (Uni Erlangen)- B. Hof (MPI Goettingen)- J. Schumacher (Uni Ilmenau)- R. du Puits (Uni Ilmenau)

• Aim: to find similarities in the boundary layer transport in turbulent RB convection, TC- and pipe flow.

Outlook:

4. Conclusion and Outlook

1. Motivation

3. Plume dynamics


5. Conclusion


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1. Motivation


D

HT ,qu

INPUT

HdΓ

3THgRa

Pr

• Rayleigh number (buoyancy):

• Prandtl number (fluid prop.):

• Aspect ratio (geometry):

RESPONSE

dqqNu

HuRe

• Nusselt number (heat transport):

• Reynolds number (velocity):

1. Motivation

3. Plume dynamics


5. Conclusion


Local wall heat flux in turbulent Rayleigh-Bénard convection

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