RB Nusselt number, TC torque, pipe friction Analogies between thermal convection and shear flows by...

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B Nusselt number, TC torque, pipe frict Analogies between thermal convection and shear flows by Bruno Eckhardt, Detlef Lohse, SGn ilipps-University Marburg and University of Tw

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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

  • 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.