Energy Spectrum of Buoyancy-Driven Turbulence

TL;DR

This paper uses high-resolution simulations to analyze the energy spectra in buoyancy-driven turbulence, confirming Bolgiano-Obukhov scaling in stratified flows and Kolmogorov scaling in convection, with detailed insights into energy flux behaviors.

Contribution

The study provides a comprehensive numerical validation of the Bolgiano-Obukhov and Kolmogorov scalings in buoyancy-driven turbulence, clarifying the role of buoyancy in energy transfer.

Findings

Kinetic energy spectrum follows $k^{-11/5}$ in stratified flows.

Entropy spectrum follows $k^{-7/5}$, consistent with Bolgiano-Obukhov scaling.

In convection, energy flux $\Pi_u(k)$ increases with $k$, leading to Kolmogorov scaling.

Abstract

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux $\Pi_u$, we demonstrate that for stably stratified flows, the kinetic energy spectrum $E_u(k) \sim k^{-11/5}$, the entropy spectrum $E_\theta(k) \sim k^{-7/5}$, and $\Pi_u(k) \sim k^{-4/5}$, consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence $E_u(k)$ follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh B\'{e}nard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing $\Pi_u(k)$ with $k$, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant $\Pi_u(k)$ and $E_u(k) \sim k^{-5/3}$ for a narrow band of wavenumbers.