Measurements of the thermal dissipation field in turbulent Rayleigh-B\'enard convection

Measurements of the thermal dissipation field in turbulent Rayleigh-Bénard convection

Xiaozhou He and Penger Tong

Phys. Rev. E 79, 026306 – Published 6 February 2009

Abstract

A systematic study of the thermal dissipation field and its statistical properties is carried out in turbulent Rayleigh-Bénard convection. A local temperature gradient probe consisting of four identical thermistors is made to measure the normalized thermal dissipation rate $ϵ_{N} (r)$ in two convection cells filled with water. The measurements are conducted over varying Rayleigh numbers Ra $(8.9 \times 10^{8} ≲ Ra ≲ 9.3 \times 10^{9})$ and spatial positions $r$ across the entire cell. It is found that $ϵ_{N} (r)$ contains two contributions; one is generated by thermal plumes, present mainly in the plume-dominated bulk region, and decreases with increasing Ra. The other contribution comes from the mean temperature gradient, being concentrated in the thermal boundary layers, and increases with Ra. The experiment provides a complete physical picture about the thermal dissipation field and its statistical properties in turbulent convection.

Received 13 October 2008

DOI:https://doi.org/10.1103/PhysRevE.79.026306

Authors & Affiliations

Xiaozhou He and Penger Tong^*

Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

^*penger@ust.hk

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Vol. 79, Iss. 2 — February 2009

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Figure 1
(a) Upper figure shows the assembly of a homemade temperature gradient probe consisting of four identical thermistor beads. Lower figure shows a preassembled thermistor (AB6E3-B05KA202R) purchased from GE Thermometrics. (b) Space coordinates used for the presentation of the measurements. The long horizontal (HS) and vertical (VS) arrows indicate, respectively, the tracks of the horizontal and vertical scans of the local dissipation measurements. The three crosses marked with a number indicate the positions used for the statistical study of dissipation fluctuations. The dashed arrows near the sidewall indicate the direction of the large-scale circulation.Reuse & Permissions
Figure 2
Measured standard deviation $σ_{2} (ℓ)$ as a function of separation $ℓ$ . The solid line shows a linear fit to the data point with small values of $ℓ$ . The measurements are made at the cell center with $Ra = 3.9 \times 10^{9}$ .Reuse & Permissions
Figure 3
(a) Measured temperature histograms $H (δ T) ∕ H_{0}$ as a function of $δ T ∕ σ_{T}$ . Four histograms (circles, squares, triangles, and diamonds) are obtained using the four thermistors, of which the temperature gradient probe is made. The solid line shows the fitted function, $H (δ T) ∕ H_{0} = \exp [- α (δ T ∕ σ_{T})]$ , with $α = 1.4$ . (b) Frequency power spectra $P (f)$ of the temperature signals obtained from the four thermistors. All the measurements in (a) and (b) are made at the center of the $Γ ≃ 1$ cell with $Ra = 3.96 \times 10^{9}$ .Reuse & Permissions
Figure 4
(a) Comparison between $ϵ_{m} (x)$ (open squares) and $ϵ_{f} (x)$ (solid circles) obtained from the horizontal scan along a cell diameter at the mid-height of the cell. (b) Comparison between $ϵ_{m} (z)$ (open squares) and $ϵ_{f} (z)$ (solid circles) obtained from the vertical scan along the central axis of the cell. All the measurements in (a) and (b) are made in the $Γ ≃ 1$ cell with $Ra = 3.9 \times 10^{9}$ .Reuse & Permissions
Figure 5
Measured horizontal profile $ϵ_{f} (x)$ as a function of $x ∕ D$ for three different values of Ra: $6.6 \times 10^{9}$ (circles), $2.7 \times 10^{9}$ (triangles), and $9.2 \times 10^{8}$ (diamonds). The solid line shows the fitted function, $ϵ_{f} (x) = a + b {(x ∕ D - 0.5)}^{4}$ , to the circles with $a = 0.56$ and $b = 74.4$ .Reuse & Permissions
Figure 6
(a) Measured vertical profile $ϵ_{f} (z)$ as a function of $z ∕ δ$ for different values of Ra. (b) Measured vertical profile $ϵ_{N} (z)$ as a function of $z ∕ δ$ for different values of Ra. The measurements in (a) and (b) are made in the $Γ ≃ 1$ cell with $Ra = 9.2 \times 10^{8}$ (triangles), $2.7 \times 10^{9}$ (diamonds), $3.9 \times 10^{9}$ (squares), and $6.6 \times 10^{9}$ (circles).Reuse & Permissions
Figure 7
(a) Measured horizontal profiles $ϵ_{f}^{x} (x)$ (diamonds), $ϵ_{f}^{y} (x)$ (circles), and $ϵ_{f}^{z} (x)$ (triangles) as a function of $x ∕ D$ . (b) Measured vertical profiles $ϵ_{f}^{x} (z)$ (diamonds), $ϵ_{f}^{y} (z)$ (circles), and $ϵ_{f}^{z} (z)$ (triangles) as a function of $z ∕ δ$ . All the measurements in (a) and (b) are conducted in the $Γ ≃ 1$ cell at $Ra = 2.7 \times 10^{9}$ .Reuse & Permissions
Figure 8
(a) Measured horizontal profile $ϵ_{f} (x)$ as a function of $x ∕ D$ for two different values of Ra. (b) Measured vertical profile $ϵ_{f} (z)$ as a function of $z ∕ δ$ for two different values of Ra. The measurements in (a) and (b) are made in the $Γ ≃ 0.5$ cell with $Ra = 9.1 \times 10^{9}$ (squares) and $2.7 \times 10^{10}$ (circles).Reuse & Permissions
Figure 9
Measured $ϵ_{f} (r)$ as a function of Ra at the cell center (circles) and near the sidewall (triangles). The solid lines show the power-law fits, $ϵ_{f} = α {Ra}^{β}$ , with $β = 0.33$ and $α = 1.9 \times 10^{3}$ (lower curve) and $α = 1.05 \times 10^{4}$ (upper curve).Reuse & Permissions
Figure 10
(a) Measured $ϵ_{f} (r)$ as a function of Ra near the lower conducting plate at distance $\sim 1 mm$ above the bottom plate (diamonds) and $\sim 2 mm$ above the bottom plate (triangles). The solid lines indicate the power-law fits, $ϵ_{f} = α {Ra}^{β}$ , with $β = 0.33$ and $α = 2.05 \times 10^{5}$ (upper curve) and $α = 9.2 \times 10^{4}$ (lower curve). (b) Measured Ra dependence of $ϵ_{m} (r)$ inside the thermal boundary layer ( $\sim 0.2 mm$ above the bottom plate). The solid line is a power-law fit, $ϵ_{m} = 1.1 \times 10^{- 2} {Ra}^{0.63}$ . The dashed line shows ${(H ∕ 2 δ)}^{2} ≃ 5.54 \times 10^{- 2} {Ra}^{0.57}$ (see text).Reuse & Permissions
Figure 11
Time series measurements of the local temperature fluctuation $δ T (t)$ , $z$ component of the temperature gradient $(d T ∕ d z) (t)$ , two components of the thermal dissipation rate $ϵ_{f}^{x} (t)$ and $ϵ_{f}^{z} (t)$ , and the total dissipation date $ϵ_{f} (t)$ (top to bottom curves). All the measurements are made at the center of the $Γ ≃ 1$ cell with $Ra = 3.9 \times 10^{9}$ .Reuse & Permissions
Figure 12
Measured histograms $H (d T ∕ d x) ∕ H_{0}$ (diamonds), $H (d T ∕ d y) ∕ H_{0}$ (circles), and $H (d T ∕ d z) ∕ H_{0}$ (triangles) for the three components of the temperature gradient at the cell center with $Ra = 3.96 \times 10^{9}$ . The solid curve shows the fitted function, $H (d T ∕ d x_{i}) ∕ H_{0} = \exp {- c {[(d T ∕ d x_{i}) ∕ σ_{g}]}^{d}}$ $(x_{i} = x, y)$ , to the diamonds and circles with $c = 2.25$ and $d = 0.64$ .Reuse & Permissions
Figure 13
(a) Measured histograms $H (ϵ_{f}) ∕ H_{0}$ as a function of $ϵ_{f} ∕ σ_{ϵ}$ . The measurements are made at the cell center with three values of Ra: $1.7 \times 10^{9}$ (diamonds), $3.9 \times 10^{9}$ (circles), and $8.2 \times 10^{9}$ (triangles). The solid line shows the fitted function, $H (ϵ_{f}) ∕ H_{0} = \exp [- c {(ϵ_{f} ∕ σ_{ϵ})}^{d}]$ , with $c = 3.9$ and $d = 0.35$ . (b) Histograms $H [\ln (ϵ_{f})] ∕ H_{0}$ as a function of the reduced variable, $k \equiv [\ln (ϵ_{f}) - μ] ∕ σ_{l}$ , using the same data in (a). The solid line shows a Gaussian fit, $H (k) ∕ H_{0} = \exp [- {(k - μ^{'})}^{2} ∕ 2 σ_{l}^{'} 2]$ , with $μ^{'} = 0.1$ and $σ_{l}^{'} = 0.92$ .Reuse & Permissions
Figure 14
Time series measurements of the local temperature fluctuation $δ T (t)$ , temperature derivative $(d T ∕ d z) (t)$ , two components of the thermal dissipation rate $ϵ_{f}^{x} (t)$ and $ϵ_{f}^{z} (t)$ , and the total dissipation date $ϵ_{f} (t)$ (top to bottom curves). All the measurements are made near the sidewall of the $Γ ≃ 1$ cell at $Ra = 3.9 \times 10^{9}$ .Reuse & Permissions
Figure 15
Measured histograms $H (d T ∕ d x) ∕ H_{0}$ (diamonds), $H (d T ∕ d y) ∕ H_{0}$ (circles), and $H (d T ∕ d z) ∕ H_{0}$ (triangles) for the three components of the temperature gradient vector near the sidewall at $Ra = 3.96 \times 10^{9}$ .Reuse & Permissions
Figure 16
(a) Measured histograms $H (ϵ_{f}) ∕ H_{0}$ as a function of $ϵ_{f} ∕ σ_{ϵ}$ . The measurements are made near the sidewall with three values of Ra: $1.7 \times 10^{9}$ (diamonds), $3.9 \times 10^{9}$ (circles), and $8.2 \times 10^{9}$ (triangles). The solid line shows the fitted function, $H (ϵ_{f}) ∕ H_{0} = \exp [- c {(ϵ_{f} ∕ σ_{ϵ})}^{d}]$ , with $c = 3.32$ and $d = 0.44$ . (b) Histograms $H [\ln (ϵ_{f})] ∕ H_{0}$ as a function of the reduced variable, $k \equiv [\ln (ϵ_{f}) - μ] ∕ σ_{l}$ , using the same data in (a).Reuse & Permissions
Figure 17
(Color online) Measured temperature histograms $H (δ T) ∕ H_{0}$ as a function of $δ T ∕ σ_{T}$ along the central axis (the $z$ axis) of the $Γ ≃ 1$ cell. The measurements are made at $Ra = 1.7 \times 10^{9}$ with four different values of $z$ : $0.2 δ$ (solid line), $1.0 δ$ (dashed line), $10 δ$ (dashed-dotted-dotted line), and $10^{2} δ$ (dashed-dotted line). Here $δ$ $(≃ 1.0 mm)$ is the boundary layer thickness at $Ra = 1.7 \times 10^{9}$ .Reuse & Permissions
Figure 18
(Color online) (a) Measured histograms $H (d T ∕ d y) ∕ H_{0}$ as a function of $(d T ∕ d y) ∕ σ_{y}$ along the central axis (the $z$ axis) of the $Γ ≃ 1$ cell. (b) Measured histograms $H (d T ∕ d z) ∕ H_{0}$ as a function of $(d T ∕ d z) ∕ σ_{z}$ along the $z$ axis. The measurements in (a) and (b) are made at $Ra = 1.7 \times 10^{9}$ with four different values of $z$ : $0.2 δ$ (solid line), $1.0 δ$ (dashed line), $10 δ$ (dashed-dotted-dotted line), and $10^{2} δ$ (dashed-dotted line). Here $δ$ $(≃ 1.0 mm)$ is the boundary layer thickness at $Ra = 1.7 \times 10^{9}$ .Reuse & Permissions
Figure 19
(Color online) (a) Measured histograms $H (ϵ_{f}) ∕ H_{0}$ as a function of $ϵ_{f} ∕ σ_{ϵ}$ along the $z$ axis of the $Γ ≃ 1$ cell. The measurements are made at $Ra = 1.7 \times 10^{9}$ with four different values of $z$ : $0.2 δ$ (solid line), $1.0 δ$ (dashed line), $10 δ$ (dashed-dotted-dotted line), and $10^{2} δ$ (dashed-dotted line). (b) Histograms $H [\ln (ϵ_{f})] ∕ H_{0}$ as a function of the reduced variable, $\ln (ϵ_{f}) ∕ σ_{l}$ , using the same data in (a).Reuse & Permissions
Figure 20
(Color online) Measured skewness of the local temperature fluctuation $δ T (t)$ (squares), horizontal temperature derivative $(d T ∕ d y) (t)$ (circles), vertical temperature derivative $(d T ∕ d z) (t)$ (triangles), and logarithmic dissipation rate $\ln (ϵ_{f}) (t)$ (diamonds) as a function of the normalized distance $z ∕ δ$ along the central axis of the $Γ ≃ 1$ cell. All the measurements are made at $Ra ≃ 6.0 \times 10^{9}$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Measurements of the thermal dissipation field in turbulent Rayleigh-Bénard convection

Xiaozhou He and Penger Tong

Phys. Rev. E 79, 026306 – Published 6 February 2009

Abstract

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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Measurements of the thermal dissipation field in turbulent Rayleigh-Bénard convection

Xiaozhou He and Penger Tong

Phys. Rev. E 79, 026306 – Published 6 February 2009

Abstract

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