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: (Ra)

G. Ahlers, SGn, D. LohsePhysik Journal 1 (2002) Nr2, 31-37Data: acetone, Pr = 4.0 (105

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 PrG. 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 > 1(2+1)-1 = 0.14 > () > 1() = (ra/rg)4 = [2-1(1+)/ ]4 = r1 / r2 = 0.83 M. Couette, 1890G.I. Taylor, 1923

Torque on inner cylinderKinematic torque independent of , of , if Experimentally:

Torque 2G versus R1 in TCLewis and Swinney, PRE 59, 5457 (1999)r1 = 16 cm, r2 = 22 cm, = 0.724 l = 69.4cm, = 11.4 8 vortex stateR1 = (r11d) / log-law2G ~ R1

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

u-profiles measured by Fritz Wendt 1933

r2 = 14.70cm, l = 40cm, = 8.5 / 18 / 42r1 = 10.00cm / 12.50cm / 13.75cm = 0.68 / 0.85 / 0.94R1=8.47 1044.95 104 2.35 104

N=M1/Mlam 52 37 21 Rw 2 670 1 580 820 /d 1 / 100 1 / 80 1 / 58 -180-225-248

Exact analogies between RB, TC, Pipe ,Ta=(ra1d/)2 =[2-1(1+)/ ]4

B.Eckhardt, SGn, D.Lohse, 2006Torque 2G versus R1Reduced torquedatavariable exponentlog-law

variable exponentlog-law

Rw ~ R1 1- 0.10-0.05 Wind = ur-amplitudeincreases less than control parameter R1~ufriction factor f or cf

dependence of N r1/r2 = = 0.500 0.6800.8500.935Data 0.935 0.850 0.680Fritz Wendt, Ingenieurs-Archiv 4, 577-595 (1933)(LS = 0.724)

Skin friction coefficient = 4 cffrom Hermann Schlichting, Grenzschichttheorie, Fig.20.1

Friction factor pipe flowB.Eckhardt,SGn,D.Lohse, 2006datavariable exponentlog-law

Summary1. Exact analogies between RB, TC, Pipe quantities Nu = Q/Qlam, M1/M1,lam, p/plam or cf Navier-Stokes based2. Variable exponent power laws M1 ~ R1 with (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/Nu4. N, N, Nu as well as w decomposed into BL and bulk contributions and modeled in terms of Rew 5. Exact relations between transport currents N and dissipation rates w = lam : w = Pr-2 Ra (Nu1), = -2 Ta (N-1), = 2Re2(Nu-1)6. Perspectives:Verify/improve by higher experimental precision.Explain very large Ra or R1 behaviour, very small .Measure/calculate profiles (r),ur,uz, etc and Ns, ws.Determine normalized correlations/fit parameters ci.Include non-Oberbeck-Boussinesque or compressibility.