Scale-resolved turbulent Prandtl number for Rayleigh-Bénard convection at $\boldsymbol{Pr =10^{-3}}$

TL;DR

This study addresses scale-dependent turbulent heat and momentum transport in Rayleigh-Bénard convection at extremely low Prandtl number ($Pr=10^{-3}$). It applies Kolmogorov's refined similarity hypothesis, extended for passive scalars, to define scale-resolved eddy viscosity $\nu_e(r)$ and diffusivity $\kappa_e(r)$ from DNS data, avoiding mean-gradient closures in the bulk. The results show the scale-resolved turbulent Prandtl number $Pr_t(r)$ can reach $\mathcal{O}(10)$, with $\nu_e(r) \sim r^{4/3}$ and $\kappa_e(r)$ increasing more steeply (roughly $r^{3.4}$–$3.9$), implying very low-$Pr$ convection behaves like a high-$Pr$ flow in the bulk. These insights have implications for subgrid modeling of low-$Pr$ convection in solar and stellar contexts and motivate extending the analysis across wider $Ra$ and $Pr$ ranges.

Abstract

We present a framework to calculate the scale-resolved turbulent Prandtl number $Pr_t$ for the well-mixed and highly inertial bulk of a turbulent Rayleigh-Bénard mesoscale convection layer at a molecular Prandtl number $Pr=10^{-3}$. It builds on Kolmogorov's refined similarity hypothesis of homogeneous isotropic fluid and passive scalar turbulence, based on log-normally distributed amplitudes of kinetic energy and scalar dissipation rates that are coarse-grained over variable scales $r$ in the inertial subrange. Our definition of turbulent (or eddy) viscosity and diffusivity does not rely on mean gradient-based Boussinesq closures of Reynolds stresses and convective heat fluxes. Such gradients are practically absent or indefinite in the bulk. The present study is based on direct numerical simulation of plane-layer convection at an aspect ratio of $Γ=25$ for Rayleigh numbers $10^5\leq Ra \leq 10^7$. We find that the turbulent Prandtl number is effectively up to 4 orders of magnitude larger than the molecular one, $\Pr_t\sim 10$. This holds particularly for the upper end of the inertial subrange, where the eddy diffusivity exceeds the molecular value, $κ_e(r)>κ$. Highly inertial low-Prandtl-number convection behaves effectively as a high-Prandtl number flow, which also supports previous models for the prominent application case of solar convection.

Figures (8)

• Figure 1: Contour plots of the dissipation rates in the horizontal midplane for $\Ray=10^7$. (a) Contours of the logarithm of the thermal dissipation rate $\chi$ in the full cross section plane. (b) Magnified image of the region enclosed by the green square in panel (a). (c) Contours of the logarithm of the viscous dissipation rate $\epsilon$ in the full cross section plane. (d) Magnified image of the region enclosed by the blue square in panel (c). The plots show that the characteristic length scale of $\chi$ is much larger than that of $\epsilon$, consistent with the fact that the Corrsin scale $\eta_C$ is much larger than the Kolmogorov length scale $\eta$.
• Figure 2: Probability density functions (PDFs) of normalized viscous and thermal dissipation rates. The top row exhibits the PDFs of $\mathrm{log}~\epsilon_r$, centered with respect to its mean $\mu_\epsilon$ and normalized by its standard deviation $\sigma_\epsilon$, with $3 \eta \leq r \leq 100 \eta$. Panel (a) is for $\Ray=10^5$ and (b) for $\Ray=10^7$. The bottom row exhibits the PDFs of $\mathrm{log}~\chi_r$, normalized with respect to its mean $\mu_\chi$ and its standard deviation $\sigma_\chi$, with $\eta_C \leq r \leq 4 \eta_C$. Panel (c) is for $\Ray=10^5$ and (d) for $\Ray=10^7$. For $\epsilon_r$, the PDFs collapse reasonably well with a Gaussian curve (solid line) with mean $\mu_G=0$ and standard deviation $\sigma_G=1$, with deviations only in the tails. Thus, $\epsilon_r$ closely follows a log-normal distribution over the accessible range of Rayleigh numbers. On the other hand, although the PDFs of $\chi_r$ are close to log-normal for $\Ray=10^7$, they are skewed for $\Ray=10^5$ because of the diffusion-dominated dynamics of temperature. These PDFs get closer to log-normal as $r$ is increased.
• Figure 3: Comparison of the PDFs of normalized viscous dissipation rates of thermal convection (blue) for $\Ray=10^7$ ($R_\lambda=709$) and homogeneous isotropic turbulence (red) for $R_\lambda=359$. Panel ($a$) shows the PDFs of $\epsilon_r$ for $r=3 \eta$, and ($b$) shows the PDFs for $r = 6 \eta$. The PDFs of $\epsilon_r$ for thermal convection and homogeneous isotropic turbulence (HIT) are similar, closely following the log-normal distribution (black solid curves).
• Figure 4: Scatter plots for Rayleigh number $\Ray=10^7$: (a) Turbulent momentum flux $\langle u_x' u_z' \rangle_{x,t}$ as a function of the mean velocity gradient $\partial \langle u_x \rangle_{x,t}/\partial z$. (b) Turbulent convective heat flux $\langle u_z' T' \rangle_{x,t}$ as a function of the mean temperature gradient $\partial \langle T \rangle_{x,t}/\partial z$. The velocity field exhibits strong fluctuations, resulting in a nearly homogeneous distribution of the phase points and thus making it challenging to estimate the eddy viscosity using the flux gradient method. Data are obtained in the midplane by a combined average with respect to $x$ direction and time $t$. A total of 5120 data points taken from 10 data snapshots are plotted.
• Figure 5: Plots of eddy viscosity, eddy diffusivity, and scale-resolved turbulent Prandtl number versus the normalized length scale $r/\eta$ for $\Ray=10^5$ (red lines), $\Ray=10^6$ (green lines), and $\Ray=10^7$ (blue lines) in decreasing order of thickness. (a) Normalized eddy viscosity $\nu_e/\nu$ and eddy diffusivity $\kappa_e/\kappa$. (b) Magnified plot of $\nu_e/\nu$ which fits closely with $r^{4/3}$ curve. (c) Magnified plot of $\kappa_e/\kappa$ along with the corresponding best-fit curves. (d) Scale-dependent turbulent Prandtl number $\Pran_t$. In panels (a) and (b), the vertical dotted line represents $r/\eta=40$, which marks the lower end of the inertial subrange. In panel (d), the dotted lines correspond to the regime where $\kappa_e<\kappa$, and the solid lines correspond to $\kappa_e \geq \kappa$. The inset in (d) exhibits the plots of $\Pran_t$ versus the length scale $r$ for $0.25 \leq r \leq 1$.