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Eddy thermal diffusivity model and mean temperature profiles in turbulent vertical convection

Ho Yin Ng, Emily S. C. Ching

TL;DR

This work addresses turbulent natural convection between two vertical walls at different temperatures by introducing a space-dependent eddy thermal diffusivity modeled with a three-layer structure. The authors derive closed-form inner and outer mean-temperature profiles expressed through universal scaling functions $F_i$ and $F_o$, with temperature and length scales $T_i$, $l_i$, $T_o$, and $l_o$ determined by $Nu$, $Pr$, $A$, and $C_m$, and assume $Pr\,C_m \gg 1$ for the outer region. Validation against DNS data for $1 \le \mathrm{Pr} \le 100$ across a broad $Ra$ range shows excellent agreement for the Nusselt number and the mean temperature, and a high-$Ra$ limit of $Nu \sim Ra^{1/3}$ consistent with prior theory. The approach provides a predictive, analytic framework for turbulent vertical convection with minimal adjustable parameters and highlights universal inner and outer temperature scaling across Prandtl numbers.

Abstract

In this paper, we propose a space-dependent eddy thermal diffusivity model for turbulent vertical natural convection in a fluid between two infinite vertical walls at different temperatures. Using this model, we derive analytical results for the mean temperature profile, which reveal two universal scaling functions in the inner region next to the walls and the outer region near the centerline between the two walls. These results are in good agreement with direct numerical simulation data for different Prandtl numbers.

Eddy thermal diffusivity model and mean temperature profiles in turbulent vertical convection

TL;DR

This work addresses turbulent natural convection between two vertical walls at different temperatures by introducing a space-dependent eddy thermal diffusivity modeled with a three-layer structure. The authors derive closed-form inner and outer mean-temperature profiles expressed through universal scaling functions and , with temperature and length scales , , , and determined by , , , and , and assume for the outer region. Validation against DNS data for across a broad range shows excellent agreement for the Nusselt number and the mean temperature, and a high- limit of consistent with prior theory. The approach provides a predictive, analytic framework for turbulent vertical convection with minimal adjustable parameters and highlights universal inner and outer temperature scaling across Prandtl numbers.

Abstract

In this paper, we propose a space-dependent eddy thermal diffusivity model for turbulent vertical natural convection in a fluid between two infinite vertical walls at different temperatures. Using this model, we derive analytical results for the mean temperature profile, which reveal two universal scaling functions in the inner region next to the walls and the outer region near the centerline between the two walls. These results are in good agreement with direct numerical simulation data for different Prandtl numbers.
Paper Structure (8 sections, 28 equations, 3 figures, 1 table)

This paper contains 8 sections, 28 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The problem
  3. Eddy thermal diffusivity and mean temperature profiles
  4. Eddy thermal diffusivity model
  5. Mean temperature profiles
  6. Validation and discussions
  7. Validation
  8. Conclusions

Figures (3)

  • Figure 1: (a) $(T_h-\overline{T})/T_i$ vs $x/l_i$ and (b) $(\overline{T} - T_m)/T_o$ vs $x/H$ at $Ra_{min}$ (circles) and $Ra_{max}$ (squares) for $\hbox{Pr}=1$ (black), $\hbox{Pr}=2$ (red), $\hbox{Pr}=5$ (orange), $\hbox{Pr}=10$ (blue) and $\hbox{Pr}=100$ (green). Here, $Ra_{min}$ and $Ra_{max}$ are the minimum and maximum values of $Ra$ for the corresponding $\hbox{Pr}$ (see table \ref{['tab:tab1']}). The solid lines in (a) and (b) are $F_i$ and $F_o$ given by (\ref{['scalinginner']}) and (\ref{['scalingouter']}), respectively. In the inset of (b), $(\overline{T} - T_m)/T_{o,GC}$ is plotted vs $x/H$.
  • Figure 2: Plots of $[T_h - \overline{T}(x)]/T_{i,GC}$ vs $x/l_{i,GC}$ for $\hbox{Pr}=1$ (left panel) and $\hbox{Pr}=100$ (right panel) at $Ra = 10^6$ (plusses), $2\times 10^6$ (crosses), $5\times10^6$ (stars), $10^7$ (circles), $2\times 10^7$ (squares), $5\times 10^7$ (diamonds), $10^8$ (triangles), $2\times 10^8$ (left triangles), $5\times 10^8$ (inverted triangles) and $10^9$ (right triangles). The red solid lines are (\ref{['Tinner2']}) and the blue dashed lines are (\ref{['inverseCubicIn']}) with $K_1=4.2$, $\phi_1(1)=5.20$ and $\phi_1(100)=5.15$.
  • Figure 3: Plot of $(Nu^{-3} Ra \hbox{Pr})^{1/4}$ vs $\log(Nu Ra \hbox{Pr})$ for DNS data of HNVL2022.The dashed lines are best fits of (\ref{['NuHH2005']}) with $K_2=0$.