Energy spectrum of Buoyancy-driven Flows

TL;DR

This paper uses high-resolution simulations to analyze the energy spectra in buoyancy-driven flows, confirming Bolgiano-Obukhov scaling in stratified flows and contrasting it with Kolmogorov scaling in weaker buoyancy conditions.

Contribution

It demonstrates the applicability of Bolgiano-Obukhov scaling in stratified flows and clarifies the conditions under which different energy spectra emerge.

Findings

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

Entropy spectrum follows $k^{-7/5}$ in stratified flows.

Kolmogorov spectrum applies in weaker buoyancy regimes.

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 $\Pi_u(k) \sim k^{-4/5}$ (Bolgiano-Obukhov scaling). This scaling is due to the depletion of kinetic energy because of buoyancy. For weaker buoyancy in stratified flows, $E_u(k)$ follows Kolmgorov's spectrum with a constant energy flux. We also argue that for Rayleigh B\'{e}nard convection, the Bolgiano-Obukhov scaling will not hold for the bulk flow due to the positive energy supply by buoyancy and non-decreasing $\Pi_u(k)$.