Logarithmic Spatial Variations and Universalf−1Power Spectra of Temperature Fluctuations in Turbulent Rayleigh-Bénard Convection

Abstract

We report measurements of the temperature variance σ 2 (z, r) and frequency power spectrum P (f, z, r) (z is the distance from the sample bottom and r the radial coordinate) in turbulent Rayleigh-Bénard convection (RBC) for Rayleigh numbers Ra = 1.6 × 10 13 and 1.1 × 10 15 and for a Prandtl number Pr ≃ 0.8 for a sample with a height L = 224 cm and aspect ratio D/L = 0.50 (D is the diameter). For z/L < ∼ 0.1 σ 2 (z, r) was consistent with a logarithmic dependence on z, and there was a universal (independent of Ra, r, and z) normalized spectrum which, for 0.02 < ∼ f τ0 < ∼ 0.2, had the form P (f τ0) = P0(f τ0)−1 with P0 = 0.208 ± 0.008 a universal constant. Here τ0 = √ 2R where R is the radius of curvature of the temperature autocorrelation function C(τ ) at τ = 0. For z/L ≃ 0.5 the measurements yielded P (f τ0) ∼ (f τ0)−α with α in the range from 3/2 to 5/3. All the results are similar to those for velocity fluctuations in shear flows at sufficiently large Reynolds numbers, suggesting the possibility of an analogy between the flows that is yet to be determined in detail.Turbulent thermal convection is an important phenomenon in many natural processes, for instance in climatology [1], oceanography [2], geophysics [3], astrophysics [4], and industry. In experiments, it can be generated for instance in a confined system between two horizontal plates separated by a distance L and heated from below in the presence of gravity. This system is known as Rayleigh-Bénard convection (RBC) [5][6][7][8].RBC is frequently studied in a cylindrical sample of height L and diameter D. Its properties are determined by the Rayleigh number Ra ≡ αg∆T L 3 /(νκ), the Prandtl number Pr ≡ ν/κ, and the aspect ratio Γ ≡ D/L. Here g is the gravitational acceleration, ∆T = T b − T t is the temperature difference between the bottom (T b ) and the top (T t ) plate, and α, ν, and κ are, respectively, the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity of the fluid.When Ra is not too large (say Ra < ∼ 10 14 ), there are thin laminar boundary layers (BLs) adjacent to the top and bottom plates with most of the temperature difference sustained by them, while the "bulk" of the fluid between these BLs is turbulent. The conventional view was that the bulk is nearly isothermal. This state is known as "classical" RBC. As Ra increases and exceeds a critical value Ra * which for Pr ≃ 1 is O(10 14 ), the shear stress from the turbulent bulk will become sufficiently large to force the BLs into a turbulent state as well and the system enters the "ultimate" state which is expected to be asymptotic as Ra tends toward infinity [9][10][11][12].Recently, it was found that the time-averaged temperature T (t, z, r) t (z is the vertical and r the radial coordinate), both in the classical and the ultimate state but outside the BLs, varies logarithmically with the distance z/L from the plates when this distance is not too large (say z/L < ∼ 0.1 or so) [13,14]. Similar logarithmic behavior is well known from mean velocity profiles of n...

Space-Time Correlations and Dynamic Coupling in Turbulent Flows

“…At the Max Planck Institute for Dynamics and Self-Organization and the University of California, Santa Barbara, He et al (2012bHe et al ( , 2014He et al ( , 2015 extensively tested the validity of the EA model in turbulent RBC, where Taylor's frozen-flow hypothesis is not valid because of large fluctuations (Lohse & Xia 2010). Their experimental results verify that the temperature correlation contours present similar elliptic shapes and the EA model leads to the collapse of all space-time correlations on the normalized space and time separations.…”

Vapor bubble nucleation, dynamics, and heat flux in turbulent Rayleigh-Bénard convection

“…[82] had a Rayleigh number Ra ƒ 10 15 , nearly five orders of magnitude larger than our Ra = 2.0 ◊ 10 10 . Further, their Prandtl number was about 0.8, while we have P r = 8.2.…”

“…For both reasons the turbulence in the flow of Ref. [82] is expected to be more fully developed, with larger Reynolds numbers, than is the case in our experiments. Nonetheless, the spectra suggest that the characteristic time scales in physical units of the two experiments turn out to be similar.…”

“…[82] showed the same frequency responses up to a critical value f c1 ƒ 0.05 Hz (see dotted vertical line) beyond which the spectrum of the 1.13 mm diameter thermistor (dotted curve) fell below the other two. Assuming attenuation was due to a di usive process, the critical frequency of the smaller thermistors should be larger than f c1 by the square of the corresponding thermistor size.…”

“…To check whether this is the case for the 0.36 mm thermistors used by us, we compared the normalized power spectrum P (f ) measured for 1-phase flow with the power spectra obtained by Ref. [82] for the same type of thermistor, and for a bigger and a smaller one with diameters of 1.13 mm and 0.18 mm. In figure 2.B.1 P (f ) is plotted for the three thermistors used in their work and for the thermistor we used to acquire data at z/L = 0.50 (our results at z/L = 0.28 are very similar).…”