Prediction of temperature distribution in turbulent Rayleigh-Benard convection

She, Zhen‐Su; Chen, Xi; Zou, Hong-Yue; Bao, Yun; Hussain, Fazle

doi:10.48550/arxiv.1401.2138

Cited by 4 publications

(7 citation statements)

References 18 publications

(31 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This result differs from the analogous universal inverse von Kármán constant of shear flow which is independent of the Reynolds number. Although it agrees quite well with a recent prediction by She et al (2014), we note that very recent as yet unpublished measurements for a much larger Pr = 12.3 by P. Wei and G. Ahlers yielded a Ra-independent A and A ′ . Thus, it seems that the analogy to a universal shear-flow von Kármán constant is perhaps recovered in the large-Pr limit.…”

Section: Discussionsupporting

confidence: 92%

“…In any case, it differs from the experimental value. A recent analysis based on a multi-layer model (She et al 2014) arrived at a log layer with an amplitude of the logarithm varying with Ra as Ra −0.162 (this model does to seem to distinguish between the classical and the ultimate state) which is slightly higher than the experimental value. For log profiles that extend all the way to the horizontal mid plane (Z 0 = Z ′ 0 = 1/2) we expect (see Eq.…”

Section: Dependence Of the Parameters Of The Logarithmic Term On Ramentioning

confidence: 87%

See 1 more Smart Citation

Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8

Ahlers

Bodenschatz

2014

J. Fluid Mech.

View full text Add to dashboard Cite

We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh-Bénard convection (RBC) for a cylindrical sample with an aspect ratio (diameter D over height L) equal to 0.50, both in the classical and in the ultimate state. The measurements are for the Ra range from 6 × 10 11 to 10 13 in the classical and for Ra = 7 × 10 14 to 1.1 × 10 15 (our maximum accessible Ra) in the ultimate state. The Prandtl number was close to 0.8. Although to lowest order the bulk is often presumed to be isothermal in the time average, we find a "logarithmic layer" (as already reported by Ahlers et al. (2012a)) in which the reduced temperature Θ = [ T (z) − T m ]/∆T (with T m the mean temperature and ∆T the applied temperature difference) varies as A * ln(z/L) + B or A ′ * ln(1 − z/L) + B ′ with the distance z from the bottom plate of the sample. In the classical state the amplitudes −A and A ′ are equal within our resolution, while in the ultimate state there is a small difference with −A/A ′ ≃ 0.95. For the classical state the width of the log layer is about 0.1L, the same near the top and the bottom plate as expected for a system with reflection symmetry about its horizontal mid plane. For the ultimate state the log-layer width is larger, extending through most of the sample, and slightly asymmetric about the mid plane. Both amplitudes A and A ′ vary with radial position r, and this variation can be described well by A = A 0 [(R − r)/R] −0.65 where R is the radius of the sample. In the classical state these results are in good agreement with direct numerical simulations (DNS) for Ra = 2 × 10 12 ; in the ultimate state there are as yet no DNS. The amplitudes −A and A ′ varied as Ra −η , with η ≃ 0.12 in the classical and η ≃ 0.18 in the ultimate state. A close analogy between the temperature field in the classical state and the "Law of the Wall" for the time averaged down-stream velocity in shear flow is discussed. Although there are strong similarities, the Ra dependence of A differs from the universal (Reynolds-number independent) von Kármán constant in shear flow. In the ultimate state the result η ≃ 0.18 differs from the prediction η ≃ 0.043 by .

show abstract

Section: Discussionsupporting

confidence: 92%

Section: Dependence Of the Parameters Of The Logarithmic Term On Ramentioning

confidence: 87%

Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8

Ahlers

Bodenschatz

2014

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…For the ultimate state of RBC a logarithmic dependence had been predicted for T (z) by Grossmann and Lohse [11,23]. For classical RBC its discovery came as a surprise, but one theoretical explanation was offered very recently [24].…”

mentioning

confidence: 98%

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

Gils

Bodenschatz

et al. 2014

Phys. Rev. Lett.

View full text Add to dashboard Cite

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...

show abstract

“…Moreover, the fluid near the wall being advected downstream undergoes the varying pressure gradient along with emission of thermal plumes. To obtain a function of boundary layer (BL) in CR, we apply the structure ensemble dynamics (SED) theory 16 , by introducing the stress length and the symmetry of the wall taking into account the constraint of solid wall, pressure gradient, and thermal effects 19 .…”

Section: (B)mentioning

confidence: 99%

“…power law, defect power law and generalized invariant relation by Lie group analysis 16 . Having been examined by canonical wall-bounded turbulence 17,18 , the SED also unifies the temperature profile and the Ra-scaling of coefficient of the log-law 19 . In applications, the SED has been extended to develop turbulent transition model and RANS model for the flow around foils considering the effects of pressure gradient and finite Re number 20 .…”

mentioning

confidence: 99%

Similarity model for corner roll in turbulent Rayleigh-Benard convection

Zhou,

Chen

2018

Preprint

Self Cite

View full text Add to dashboard Cite

The corner roll (CR) in the Rayleigh-Bénard (RB) convection accounts for the behaviors of heat transport and convection flow at the corner. Streamlines of the three-dimensional direct numerical simulations for 10 8 < Ra < 5 × 10 9 show that CR presents well-defined similarity and multi-layer structure. A stream function for CR is developed by homotopy and the structure ensemble dynamics. The model presents the scaling of Reynolds number of corner roll Re cr ∼ Ra 1/4 . Scaling of CR scale r = 0.77Ra −0.085 indicates strong near-wall shearing induced by wind and provides a probability of the 'ultimate regime' at high Ra.

show abstract

Prediction of temperature distribution in turbulent Rayleigh-Benard convection

Cited by 4 publications

References 18 publications

Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8

Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8

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

Similarity model for corner roll in turbulent Rayleigh-Benard convection

Contact Info

Product

Resources

About