1. 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

2. data: cryogenic helium approximate power law only exponent varies with Ra: β(Ra) β β β

3. G. Ahlers, SGn, D. Lohse Physik Journal 1 (2002) Nr2, 31-37 Data: acetone, Pr = 4.0 (10 5 <Ra<10 10 ) Γ = 2 Γ = 1 Γ = 1/2

4. Physical origin of variable exponent Varying weight of BL and bulk

5. Variable exponent scaling in convective transport

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

7. Very large Ra scaling

8. 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)

9. Conditions for „thin“ BLs: d > 1 (2π+1) -1 = 0.14 > σ(η) > 1 σ(η) = (r a /r g ) 4 = [2 -1 (1+η)/ √η ] 4 η = r 1 / r 2 = 0.83 M. Couette, 1890 G.I. Taylor, 1923

10. Torque on inner cylinder Kinematic torque independent of, of, if Experimentally:

11. Torque 2πG versus R 1 in TC Lewis and Swinney, PRE 59, 5457 (1999) r 1 = 16 cm, r 2 = 22 cm, η = 0.724 l = 69.4cm, Γ = 11.4 8 vortex state R 1 = (r 1 ω 1 d) / ν log-law 2π2π G ~ R 1 α

12. Reduced TC torque data, variable exponent power law, log-law B.Eckhardt, SGn, D.Lohse, 2006

13. u φ -profiles measured by Fritz Wendt 1933 r 2 = 14.70cm, l = 40cm, Γ = 8.5 / 18 / 42 r 1 = 10.00cm / 12.50cm / 13.75cm η = 0.68 / 0.85 / 0.94 R 1 =8.47 ·10 4 4.95 ·10 4 2.35 ·10 4 N ω =M 1 /M lam ≈ 52 ≈ 37 ≈ 21 R w ≈ 2 670 ≈ 1 580 ≈ 820 δ/d ≈ 1 / 100 ≈ 1 / 80 ≈ 1 / 58 ○ ○ ○ -180 -225 -248

14. Exact analogies between RB, TC, Pipe ► ► ►,Ta=σ(r a ω 1 d/ν) 2 σ=[2 -1 (1+η)/√η ] 4

15. B.Eckhardt, SGn, D.Lohse, 2006 Torque 2πG versus R 1 Reduced torque data variable exponent log-law

16. variable exponent log-law

17. R w ~ R 1 1-ξ ξ ≈ 0.10-0.05 Wind = u r -amplitude increases less than control parameter R 1 ~u φ friction factor f or c f

18. η dependence of N ω r 1 /r 2 = 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) (η LS = 0.724)

19. Skin friction coefficient λ = 4 c f from Hermann Schlichting, Grenzschichttheorie, Fig.20.1

20. Friction factor pipe flow B.Eckhardt,SGn,D.Lohse, 2006 data variable exponent log-law

21. Summary 1. Exact analogies between RB, TC, Pipe quantities Nu = Q/Q lam, M 1 /M 1,lam, Δp/Δp lam or c f Navier-Stokes based 2. Variable exponent power laws M 1 ~ R 1 α with α(R 1 ) etc due to varying weights of BLs relative to bulk 3. Wind BL of Prandtl type, width ~ 1/√Re w, time dependent but not turbulent, explicit scale L transport BL width ~ 1/Nu 4. N, N ω, N u as well as ε w decomposed into BL and bulk contributions and modeled in terms of Re w 5. Exact relations between transport currents N and dissipation rates ε w = ε – ε lam : ε w = Pr -2 Ra (Nu–1), = σ -2 Ta (N ω -1), = 2Re 2 (Nu-1) 6. Perspectives: Verify/improve by higher experimental precision. Explain very large Ra or R 1 behaviour, very small η. Measure/calculate profiles ω(r),u r,u z, etc and N‘s, ε w ‘s. Determine normalized correlations/fit parameters c i. Include non-Oberbeck-Boussinesque or compressibility.