Coherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-B\'{e}nard Convection

TL;DR

This paper links unstable steady solutions in 2D Rayleigh-Bénard convection to the transition to turbulence, showing how these solutions manifest in temperature fields and relate to heat transport scaling at different Prandtl numbers.

Contribution

It establishes a direct connection between unstable steady solutions and turbulence transition in 2D convection for specific Prandtl numbers, extending previous steady-state analyses.

Findings

Primary solutions appear in transitional turbulence at Ra near 1708.

Optimal heat transport solutions are more detectable at higher Prandtl numbers.

The prevalence of solutions aligns with Nu vs. Ra scaling laws.

Abstract

For two-dimensional Rayleigh-B\'{e}nard convection, classes of unstable, steady solutions were previously computed using numerical continuation (Waleffe, 2015; Sondak, 2015). The `primary' steady solution bifurcates from the conduction state at $Ra \approx 1708$, and has a characteristic aspect ratio (length/height) of approximately $2$. The primary solution corresponds to one pair of counterclockwise-clockwise convection rolls with a temperature updraft in between and an adjacent downdraft on the sides. By adjusting the horizontal length of the domain, (Waleffe, 2015; Sondak, 2015) also found steady, maximal heat transport solutions, with characteristic aspect ratio less than $2$ and decreasing with increasing $Ra$. Compared to the primary solutions, optimal heat transport solutions have modifications to boundary layer thickness, the horizontal length scale of the plume, and the structure of the downdrafts. The current study establishes a direct link between these (unstable) steady solutions and transition to turbulence for $Pr = 7$ and $Pr = 100$. For transitional values of $Ra$, the primary and optimal heat transport solutions both appear prominently in appropriately-sized sub-fields of the time-evolving temperature fields. For $Ra$ beyond transitional, our data analysis shows persistence of the primary solution for $Pr = 7$, while the optimal heat transport solutions are more easily detectable for $Pr = 100$. In both cases $Pr = 7$ and $Pr = 100$, the relative prevalence of primary and optimal solutions is consistent with the $Nu$ vs. $Ra$ scalings for the numerical data and the steady solutions.