Turbulent Rayleigh-Benard Convection A Progress Report

1. Turbulent Rayleigh-Benard Convection A Progress Report Work done in collaboration with Eric Brown, Denis Funfschilling And Alexey Nikolaenko, supported by the US Department of Energy Guenter Ahlers Department of Physics and iQUEST UCSB

2. Rayleigh-Benard Convection Cell Fluid: H2O Wall: Plexigl. G = D/L = 1.0 D Q Three samples: Small: D = L = 9 cm Medium: D = L = 25 cm Large: D = L = 50 cm DT = 20 oC : R = 1011 Q = 1500 W L DT R = a g L3 DT / k n Q Prandtl No. s = n/k = 4.4 ( H2O, 40 oC )

3. Why is it interesting? Important process in the Atmosphere: Weather Mantle: Continental Drift Outer core: Magnetic field Sun: Surface temperature etc.: ??? Interesting Physics

4. Models

5. A central prediction: Heat transport. Define Nusselt number N = leff / l ; leff= Q / ( DT / L ) Various models were proposed: Malkus (1954), Priestly (1959): N ~ R1/3 Kraichnan (1962): N ~ R1/2 Castaing et al. (1989) Shraiman + Siggia (1990): N ~ R2/7 Grossmann and Lohse (2000): No single power law; Crossover between two power laws

6. C.H.B. Priestley [Quarterly Journal of the Royal Meteorological Society 85, 415 (1959)] Define N = leff / l ; leff= Q / ( DT / L ) Q = N l DT / L = heat-current density Assume that there are power laws and that the R- and s-dependence of N separates: N = f( s ) Rg ; R = (a g DT / k n) L3 Q = f( s ) Rg l DT / L Q = f( s ) (a g DT / k n)gl DT L(3g - 1) Assume that the heat-current density Q is determined by the BLs and does not depend on the distance between them. Then 3 g - 1 = 0; g = 1/3; N ~ R1/3

7. R. Krishnamurty and L.N. Howard, Proc. Nat. Acad. Sci. 78, 1981(1981): Large Scale Circulation (“Wind of Turbulence”) R = 6.8x108 s = 596 • = 1 cylindrical slightly tilted in real time Movie from the group of K.-Q Xia, Chinese Univ., Hong Kong connects the BLs and invalidates the simple models.

8. Q Q T1 thermal boundary layer cold plumes schematic drawing of the flow structure “wind” or large scale circulation ( T1 + T2 ) / 2 thermal boundary layer hot plumes T2 > T1 See, e.g., X.-L. Qiu and P. Tong, Phys. Rev. E 66, 026308 (2002)

9. However, assuming no interaction between BLs is not needed to get 1/3 ! W.V.R. Malkus, Proc. Roy. Soc. (London) A 225, 196 (1954) Assume laminar BLs with conductivity l and DT/2 across each: Q = N l DT / L = l (DT/2) / l l = BL “thickness” l = L / 2N Assume laminar BLs are marginally stable: R = Rc = a g l3 DTc / k n = O(103); DTc =DT/2 l ~ (DT/2)-b ~ R-b; b = 1/3; N = L/2 l; N ~ R1/3 z / l Wind direction x Experiment: b not = 1/3 and depends on horizontal location ! S.-L. Lui and K.-Q. Xia, Phys. Rev. E 57, 5494 (1998). s = 7.

10. 1975: D.C. Threlfall, J. Fluid Mech. 67, 17 (1975). 1987: F.Heslot, B. Castaing, + A. Libchaber(Chicago), Phys. Rev. A 36, 5870 (1987). N ~ R0.282 Mixing layer model (bulk, BL, and plume region between them) ofthe Chicago group [Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thome, Wu, Zaleski, & Zanetti, J. Fluid Mech. 204, 1 (1989)] and of B.I. Shraiman and E. Siggia, Phys. Rev. A 42, 3650 (1990): N ~ s-1/7 R2/7; 2/7 = 0.2857… .

11. S. Grossmann and D. Lohse, J. Fluid Mech. 407, 27 (2000) (GL) start with the kinetic and thermal dissipation rates Their volume averages follow from the Boussinesq equations and are given by GL set each equal to a sum of a BL and a bulk contribution: They assume that the separate contributions can be modeled using approximations to the length, temperature, and time scales, e.g. (assumes laminar BLs, uniform in the x-y plane, with conductivity l) (based on Blasius BL model) etc.

12. log( s ) log( R ) No simple power laws, but rather cross-overs from a small-R to a large-R asymptotic region. Various regions in the R - s plane, depending on which dissipative term dominates, etc. For s > 1 and large R, IVu pertains. There eu and eq are both bulk dominated.

13. yield and At large R else

14. D. Funfschilling, E. Brown, A. Nikolaenko, and G. A., J. Fluid Mech., submitted.

15. 1.) No power law 2.) 4 parameters of the GL model were determined from a fit to these data X. Xu, K.M.S. Bajaj, and G. A., Phys. Rev. Lett. 84, 4357 (2000); G. A. + X. Xu, Phys. Rev. Lett. 86, 3320 (2001)

16. A.) The important components have been identified: 1.) top and bottom boundary layers 2.) “plumes” 3.) large-scale circulation B.) The nature of the interactions between boundary layers, plumes, and large scale circulation is not so clear. C.) The GL model can be fitted to existing Nusselt data by adjusting its four undetermined coefficients D.) Adjustment of a fifth parameter gives reasonably good agreement with the measured Reynolds numbers of the LSC.

17. New Nusselt-Number Measurements

18. N / R1/4 Prandtl Number s R = 1.8x109 s -1/7 R = 1.8x107 X. Xu, K.M.S. Bajaj, and G. A., Phys. Rev. Lett. 84, 4357 (2000); G. A. + X. Xu, Phys. Rev. Lett. 86, 3320 (2001) K.-Q. Xia, S. Lam, and S.-Q. Zhou, Phys. Rev. Lett. 88, 064501 (2002). S. Grossmann and D. Lohse, Phys. Rev. Lett. 86, 3316 (2001).

19. Foam inside of here Water cooled Cu top plate adiabatic side shield Plexiglas side wall Joule heated Cu bottom plate Adiabatic bottom-plate shield Leveling and support plate Catch basis E. Brown, A. Nikolaenko, D. Dunfschiling, and G.A., Phys. Fluids, submitted.

22. D. Funfschilling, E. Brown, A. Nikolaenko, and G. A., J. Fluid Mech., submitted.

23. 2 % D. Funfschilling, E. Brown, A. Nikolaenko, and G. A., J. Fluid Mech., submitted.

24. The GL model can not reproduce the effective exponent • = 0.333 ~ 1/3 derived from the Nusselt number data for R > 1010.

25. Reynolds-Number Measurements

26. 0.63 cm

27. 3 4 2 5 1 5 6 7 0 1 q 0 6 7

30. C00 -C04

32. Re = (L / t1)(L / n) Medium Sample X.-L. Qiu and P. Tong, Phys. Rev. E 66, 026308 (2002). unpublished Large Sample

34. A.) The important components have been identified: 1.) top and bottom boundary layers 2.) “plumes” 3.) large-scale circulation B.) The nature of the interactions between boundary layers, plumes, and large scale circulation is not so clear. C.) Models yielding relationships between, the Nusselt number, Rayleigh number, Prandtl number, and Reynolds number (of the LSC) are at best good approximations, but for large R miss important physics.

35. LSC Reversals E. Brown, A. Nikolaenko, and G. A., Phys. Rev. Lett., submitted.

36. Ti = <T> + d cos( ip/4 + q )

37. Rotation

38. Cessation

40. probability distribution of |Dq| for reorientations with d/<d> < 0.25

42. A.) LSC “reversal” can occur via 1.) rotation of the vorticity vector (“rotation) 2.) shrinking of the vorticity vector, followed by re-development with a new orientation (“cessation”) B.) Cessation is followed by re-development of the LSC in a circulation plane with an arbitrary new orientation, i.e. P(Dq) ~ constant. C.) Rotation through an angle Dq has a powerlaw probability distribution P(Dq) ~ Dq-g with g ~ 4. D.) Reversals are Poisson distributed.

43. More LSC Dynamics D. Funfschilling and G. A., Phys. Rev. Lett. 92, 194502 (2004).

44. Shadowgraph lens Hot plumes/rolls near the bottom plate appear dark Cold plumes/rolls Near the top plate Appear bright pin hole and LED ligh source beam-splitter lens Rayleigh Bénard cell 2o tilt mirror

45. dt = 0.0s 34 Correlation Functions: 0.3s dt = 0.0s 0.6s 0.9s 37 1.5s 1.2s dt = 0.9s

46. The maximum of the correlation Function is located at DX, DY Relative to its origin (center). Viewed from Above: direction of plume movement and presumed direction of circulation of LSF Lowest point Speed Q Direction of 2 deg. tilt Highest point

47. The angle qof the plane of the large-scale-flow circulation, and its time correlation function R = 7.0 x 108

49. Assumption: Near the center of the top and bottom plate plumes/rolls follow the large-scale flow Conclusion Near the center of the top and bottom plate the large-scale flow direction oscillates about the vertical axis of the cell. This oscillation has the same frequency as the periodic signals seen by others in measurements at individual points. The frequency yields a Reynolds number consistent with measurements by other methods and the GL model.