SMR.1771-33 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
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
Part I Temperature Profiles
Motivation J. Fluid Mech, vol 2 (1957) 473
Motivation Shear flows Convection global: λ(re) local: u(r) global: local: Nu(Ra) T(z) diffusive universal bulk
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) + 1 + u = ln y + κ B u + = C y +α
Background: shear flows
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 = + + 1 /3 B Θ + + = C ln y + D Θ + = E y + α
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
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
Technical Parameters R 50,000 5,000 500 50 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) 7 8 9 10 11 12 13 14 15 16 17 log(ra)
Measurements at constant temperature difference
6a) 1.2 Profile at Γ=1.13, Ra=7.7x10 11 0.15 1 0.8 0.1 Θ 0.6 σ 0.4 0.05 0.2 0 0.005 0.01 0.015 0.02 0.025 z/h
6a) 1.2 Profile at Γ=1.13, Ra=7.7x10 11 0.15 1 0.8 0.1 Θ 0.6 6b) 1.2 0.15 σ 1 0.4 0.8 0.05 0.1 0.2 Θ 0.6 0.4 0.05 σ 0.2 0 0.005 0.01 0.015 0.02 0.025 z/h 0 0.5 1 1.5 2 δ z/h x 10 3
6c) 101 0 Profile at Γ=1.13, Ra=7.7x10 11 0.5265 0.4248 10 1 Θ 1 σ 101 1 0.5 0 10 5 10 4 10 3 10 2 10 5 10 4 10 3 10 2 z/h 10 2
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
Scaling exponents of temperature
Scaling of temperature fluctuations -1/3-1/3 Ra=7.7x10 11 Ra=7.7x10 10-1/3-1/3 Ra=7.6x10 9 Ra=3.5x109
Scaling exponents of fluctuations
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 10 12 du Puits, Resagk, Tilgner, Busse, Thess Structure of thermal boundary layers in turbulent thermal convection J. Fluid Mech. (in press)
Outlook
Part II Coherent Oscillations
Observations measuring volume "cross flow" direction "mean flow" direction "wind"
Observations: : Time series magnitude/angle of velocity (x=0m, z=90mm); Ra=7.48e+011, Γ =1.13 0.8 v abs [m/s] 0.6 0.4 0.2 0 100 200 300 400 500 600 700 800 900 1000 time [sec] 200 100 Φ [degree] 0 100 200 0 100 200 300 400 500 600 700 800 900 1000 time [sec]
Observations: Autocorrelation velocity magnitude polar angle autocorrelation function - total velocity, A=1.24, Ra=5.5e11 autocorrelation function - angle, A=1.24, Ra=5.5e11 0.8 0.4 0.8 0.8 0.3 0.6 39.5sec 0.6 0.2 0.6 0.4 0.1 0.2 R 0.4 0 R 0.4 0 0.2-0.1 0 200 400 600 800 1000 0.2-0.2 0 100 200 300 400 500 0 0-0.2 0 5000 10000 15000 time in sec -0.2 0 5000 10000 15000 time in sec
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
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
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
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)
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)
Results of simulations II
Results of simulations I
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)
Summary of the summary Θ α = Ez with α 0.5 Coherent oscillations heavy top precession
Acknowledgment Deutsche Forschungsgemeinschaft Thüringer Ministerium für Wissenschaft