Order and chaos in two-dimensional Rayleigh-B\'enard convection

TL;DR

This study investigates two-dimensional Rayleigh-Bénard convection with free-slip boundaries, revealing various convective states, energy cascades, and a traveling rolls instability, using high-resolution numerical simulations up to high Rayleigh numbers.

Contribution

It provides a comprehensive numerical analysis of convective states, energy transfer mechanisms, and instabilities in 2D Rayleigh-Bénard convection at high Rayleigh numbers.

Findings

Identification of stationary, oscillatory, chaotic, and soft-turbulent states.

Observation of traveling rolls instability in the chaotic regime.

Evidence of inverse kinetic energy cascade and forward entropy cascade.

Abstract

A detailed study of the Rayleigh-B\'enard convection in two-dimensions with free-slip boundaries is presented. Pseudo-spectral method has been used to numerically solve the system for Rayleigh number up to $3.3 \times 10^7$. The system exhibits various convective states: stationary, oscillatory, chaotic and soft-turbulent. The `travelling rolls' instability is observed in the chaotic regime. Scaling of Nusselt number shows an exponent close to 0.33. Studies on energy spectrum and flux show an inverse cascade of kinetic energy and a forward cascade of entropy. This is consistent with the shell-to-shell energy transfer in wave number space. The shell-to-shell energy transfer study also indicates a local energy transfer from one shell to the other.