Turbulent Convection in Air

Transcription

1 SMR Conference and Euromech Colloquium #480 on High Rayleigh Number Convection 4-8 Sept., 2006, ICTP, Trieste, Italy Structure of the thermal boundary layers in turbulent Rayleigh-Benard convection A. Thess Ilmenau University of Technology Ilmenau Germany These are preliminary lecture notes, intended only for distribution to participants

2 Turbulent Convection in Air A. Thess, Ch. Resagk, R. du Puits Technische Universität Ilmenau Part I: Structure of the thermal boundary layer In collaboration with F. Busse, A. Tilgner Part II: Coherent oscillations In collaboration with F. Fontenele-Araujio, F. Dolzhansky, D. Lohse, S. Grossmann EUROMECH Colloquium 480, ICTP Trieste, 7 September 2006

3 Part I Temperature Profiles

4 Motivation J. Fluid Mech, vol 2 (1957) 473

5 Motivation Shear flows Convection global: λ(re) local: u(r) global: local: Nu(Ra) T(z) diffusive universal bulk

6 Background: shear flows Wall shear stress: Friction velocity: Viscous scale: Inner variables: τ u τ = τ / ρ δ = ν / uτ + u = u / u, y = y / + δ τ Universal logarithmic law (Prandtl 1933) Power law (Barenblatt 1993) u = ln y + κ B u + = C y +α

7 Background: shear flows

8 Background: convection Kinematic heat flux: Temperature scale: Inner variables: Diffusive scale: Θ + T q q = = q / c p ρ δ = κt q / q = ( T T ) / ( 3 q / αgκ ) 1/ 4 c T q, + y = y / δ Prandtl's law Logarithmic law Power law A y + Θ = /3 B Θ + + = C ln y + D Θ + = E y + α

9 Background: temperature measurements Thomas & Townsend, J. Fluid Mech., vol. 2 (1957) 473 Townsend, J. Fluid Mech., (1959) 209 Deardorff, Willis, J. Fluid Mech., vol. 28, (1967) 675 Fitzjarrald, J. Fluid Mech., vol. 73 (1977) 693 Tilgner, Belmonte, Libchaber, Phys. Rev. E vol 47 (1993) R2253 Belmonte, Tilgner, Libchaber, Phys. Rev. Lett, vol 70 (1993) 4067 Belmonte, Tilgner, Libchaber, Phys. Rev. E vol 50 (1994) 269 Lui, Xia, Phys. Rev. E, vol. 57 (1998) 5494 Fernandes, Adrian, Exp. Thermal Fluid Sci. vol. 26 (2002) 355

10 Challenge for experiments 1 - Attain high Rayleigh numbers Use low viscosity fluids Ra = αg TH νκ 3 Build large experiments 2 - Measure with high resolution Use very small sensors R = size of exp eriment size of sensor Build large experiments

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13 Technical Parameters R 50,000 5, Barrel of Ilmenau (Air) Hongkong Barrel (Water) Niemela et al 2000 (Helium) Glazier et al 1999 (Mercury) Castaing et al 1989 (Helium) Fitzjarrald 1976 (Air) log(ra)

14 Measurements at constant temperature difference

15 6a) 1.2 Profile at Γ=1.13, Ra=7.7x Θ 0.6 σ z/h

16 6a) 1.2 Profile at Γ=1.13, Ra=7.7x Θ 0.6 6b) σ Θ σ z/h δ z/h x 10 3

17 6c) Profile at Γ=1.13, Ra=7.7x Θ 1 σ z/h 10 2

18 Profile at Γ=1.13, Ra=7.7x10 11 Ξ Ξ = dθ d ln z Log-law diagnostic function Ψ Ψ = dln Θ dln z Power-law diagnostic function

19 Scaling exponents of temperature

20 Scaling of temperature fluctuations -1/3-1/3 Ra=7.7x10 11 Ra=7.7x /3-1/3 Ra=7.6x10 9 Ra=3.5x109

21 Scaling exponents of fluctuations

22 Summary of Part I Power law scaling of mean temperature Scaling exponent 1/2 weakly dependent on Ra Priestley scaling for temperature fluctuations qualitative changes for Ra near du Puits, Resagk, Tilgner, Busse, Thess Structure of thermal boundary layers in turbulent thermal convection J. Fluid Mech. (in press)

23 Outlook

24 Part II Coherent Oscillations

25 Observations measuring volume "cross flow" direction "mean flow" direction "wind"

26 Observations: : Time series magnitude/angle of velocity (x=0m, z=90mm); Ra=7.48e+011, Γ = v abs [m/s] time [sec] Φ [degree] time [sec]

27 Observations: Autocorrelation velocity magnitude polar angle autocorrelation function - total velocity, A=1.24, Ra=5.5e11 autocorrelation function - angle, A=1.24, Ra=5.5e sec R R time in sec time in sec

28 Motion in a triaxial ellipsoid a c b v(x, y,z, t) = u(t) f u (y,z) + v(t) f v (x,z) + w(t) f w (x, y) is a solution of the 3d Euler equations if

29 Motion in a triaxial ellipsoid cont'd J J J w u v u v w = (J = (J = (J w u v J J J w u v )vw )uw )uv where u 2 2 J = b + c, J = a + c, J = a + b v 2 2 w 2 2 Addition of a temperature gradient: T(x, y,z, t) x y = ϕ (t) χ(t) ψ (t) a b z c is a solution of the 3d nondissipative Boussinesq equations if

30 Motion in a triaxial ellipsoid cont'd J J J w u v u = (J v = (J w = (J w u v J J J w u v ϕ = vψ wχ χ = wϕ uψ ψ = uχ vϕ )vw -αgcχ )uw + αgcϕ )uv Let then

31 A model model for for thermal thermal convection convection ) ( αgc q 0 q q ω q L q k L ω L = + = µ λ viscous friction thermal diffusion forcing nondimensional form ) ( 1 R k q q ω q L q k L ω L = + = σ σ (Dolzhansky 1977,1982)

32 Calibration of the model steady state solution for sphere 1 Lω Re = σ R 2 ψ 1 R Nu = 2 Grossmann-Lohse (JFM 2000) phenomenological theory Re = f ( Ra,Pr) Nu = g(ra, Pr) --> calibration formula R = 2g( Ra,Pr) σ = 2g( Ra,Pr) 2 f ( Ra,Pr)

33 Results of simulations II

34 Results of simulations I

35 Summary of Part II Experiments show coherent oscillations with high level of regularity Large scale oscillations can be modelled using six-dimensional phenomenological model Coherent oscillations are the fluid-dynamical analog to precession of a heavy top Resagk, du Puits, Thess, Dolzhansky, Grossmann, Fontenele-Araujo, Lohse Oscillations of the large scale wind in turbulent thermal convection Phys. Fluids (in press)

36 Summary of the summary Θ α = Ez with α 0.5 Coherent oscillations heavy top precession

37 Acknowledgment Deutsche Forschungsgemeinschaft Thüringer Ministerium für Wissenschaft