RB Nusselt number, TC torque, pipe friction

Analogies between

thermal convection and shear flows

by

Bruno Eckhardt, Detlef Lohse, SGn

Philipps-University Marburg and University of Twente

data: cryogenic helium approximate power law only

exponent varies with Ra: β(Ra)

β

β

β

G. Ahlers, SGn, D. LohsePhysik Journal 1 (2002) Nr2, 31-37

Data: acetone, Pr = 4.0 (105<Ra<1010)

Γ = 2

Γ = 1

Γ = 1/2

Physical origin of variable exponent Varying weight of BL and bulk

Variable exponent scaling in convective transport

Funfschilling, Ahlers, et al.JFM 536, 145 (2005)

Very large Ra scaling

Nu versus Pr

G. Ahlers et al., PRL 84, 4357 (2000) ; PRL 86, 3320 (2001)K.-Q. Xia, S. Lam, S.-Q. Zhou, PRL 88, 064501 (2002)

Conditions for „thin“ BLs: d << 2 π ri , Γ >> 1

(2π+1)-1 = 0.14 << η < 1 thus 5.4 >> σ(η) > 1

σ(η) = (ra/rg)4 = [2-1(1+η)/√η ]4

η = r1 / r2 = 0.83

M. Couette, 1890G.I. Taylor, 1923

Torque on inner cylinder

Kinematic torque independent of , of , if

Experimentally:

Torque 2πG versus R1 in TCLewis and Swinney, PRE 59, 5457 (1999)

r1 = 16 cm, r2 = 22 cm, η = 0.724 l = 69.4cm, Γ = 11.4 8 vortex state

R1 = (r1ω1d) / ν

log-law

2πG ~ R1

α

Reduced TC torque

data, variable exponent power law, log-law

B.Eckhardt, SGn, D.Lohse, 2006

uφ-profiles measured by Fritz Wendt 1933

r2 = 14.70cm, l = 40cm, Γ = 8.5 / 18 / 42r1 = 10.00cm / 12.50cm / 13.75cmη = 0.68 / 0.85 / 0.94

R1=8.47 ·104 4.95 ·104 2.35 ·104 Nω=M1/Mlam≈ 52 ≈ 37 ≈ 21

Rw ≈ 2 670 ≈ 1 580 ≈ 820

δ/d ≈ 1 / 100 ≈ 1 / 80 ≈ 1 / 58

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-180-225

-248

Exact analogies between RB, TC, Pipe

►

►

►

,Ta=σ(raω1d/ν)2

σ=[2-1(1+η)/√η ]4

B.Eckhardt, SGn, D.Lohse, 2006

Torque 2πG versus R1

Reduced torque

datavariable exponentlog-law

variable exponent

log-law

Rw ~ R1 1-ξ

ξ ≈ 0.10-0.05

Wind = ur-amplitudeincreases less than control parameter R1~uφ

friction factor f or cf

η dependence of Nω

r1/r2 = η = 0.500 0.680

0.850

0.935

Data Δ 0.935 ○ 0.850 □ 0.680

Fritz Wendt, Ingenieurs-Archiv 4, 577-595 (1933)

(ηLS = 0.724)

Skin friction coefficient λ = 4 cf

from Hermann Schlichting, Grenzschichttheorie, Fig.20.1

Friction factor pipe flow

B.Eckhardt,SGn,D.Lohse, 2006

datavariable exponentlog-law

Summary

1. Exact analogies between RB, TC, Pipe quantities Nu = Q/Qlam, M1/M1,lam, Δp/Δplam or cf

Navier-Stokes based

2. Variable exponent power laws M1 ~ R1α with α(R1) etc

due to varying weights of BLs relative to bulk

3. Wind BL of Prandtl type, width ~ 1/√Rew , time dependent but not turbulent, explicit scale L transport BL width ~ 1/Nu

4. N, Nω, Nu as well as εw decomposed into BL and bulk contributions and modeled in terms of Rew

5. Exact relations between transport currents N and dissipation rates εw = ε – εlam : εw = Pr-2 Ra (Nu–1), = σ -2 Ta (Nω-1), = 2Re2(Nu-1)

6. Perspectives:Verify/improve by higher experimental precision.Explain very large Ra or R1 behaviour, very small η.Measure/calculate profiles ω(r),ur,uz, etc and N‘s, εw‘s.Determine normalized correlations/fit parameters ci.Include non-Oberbeck-Boussinesque or compressibility.