Scaling of heat flux and energy spectrum for "very large" Prandtl number convection

TL;DR

This paper derives analytical expressions and performs simulations to understand heat flux and energy spectra in very large Prandtl number convection, revealing specific scaling laws and spectral behaviors.

Contribution

It provides new analytical scaling relations and numerical validation for heat flux and energy spectra in the infinite Prandtl number limit.

Findings

Nu scales as Ra^{0.30-0.32}

Pe scales as Ra^{0.57-0.61}

Energy spectrum E_u(k) ~ k^{-13/3}

Abstract

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., P\'{e}clet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations $\theta_\mathrm{rms}$. We complement the analytical work with direct numerical simulations, and show that $\mathrm{Nu} \sim \mathrm{Ra}^{\gamma}$ with $\gamma \approx (0.30-0.32)$, $\mathrm{Pe} \sim \mathrm{Ra}^{\eta}$ with $\eta \approx (0.57-0.61)$, and $\theta_\mathrm{rms} \sim \mathrm{const}$. The Nusselt number is observed to be an intricate function of $\mathrm{Pe}$, $\theta_\mathrm{rms}$, and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum $E_u(k)\sim k^{-13/3}$, which is in a very good agreement with our numerical results. The entropy spectrum $E_\theta(k)$ however exhibits dual branches consisting of $k^{-2}$ and $k^0$ spectra; the $k^{-2}$ branch corresponds to the Fourier modes $\hat{\theta}(0,0,2n)$, which are approximately $-1/(2n \pi)$. The scaling relations for Prandtl number beyond $10^2$ match with those for infinite Prandtl number.