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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
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
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 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
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
•
•
•
•
•
• ○ ○
○
-180-225
-248
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)
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.