Spectral dynamics of turbulent heat flux in plane Rayleigh–Bénard convection

The spectral budget of the vertical heat flux

is investigated in turbulent Rayleigh–Bénard convection by direct numerical simulation. Eight cases spanning three Rayleigh numbers (

italic Ra equals 1 times 10 Superscript 5 Ra = 1 × 10 5 $\textit{Ra} = 1\times 10^5$

,

6 times 10 Superscript 6 6 × 10 6 $6\times 10^6$

and

4 times 10 Superscript 8 4 × 10 8 $4\times 10^8$

) and up to three geometric aspect ratios each are computed at

italic probability equals 1 Pr = 1 $\textit{Pr} = 1$

. The premultiplied spectra of

left angle bracket w theta right angle bracket ⟨ w θ ⟩ $\langle w\theta \rangle$

reveal a two-peak structure when both

italic Ra Ra $\textit{Ra}$

and

upper Gamma Γ $\varGamma$

are sufficiently large. Production of

left angle bracket w theta right angle bracket ⟨ w θ ⟩ $\langle w\theta \rangle$

is dominated by temperature fluctuations; production by velocity fluctuations is confined to the near-wall region and the large-scale production that develops at sufficient

upper Gamma Γ $\varGamma$

is restricted to the bulk. At large

upper Gamma Γ $\varGamma$

an inverse inter-scale cascade transfers

left angle bracket w theta right angle bracket ⟨ w θ ⟩ $\langle w\theta \rangle$

from intermediate to superstructure scales, the transfer towards the superstructure scale taking place away from the boundary layer. This upscale transfer is specific to the covariance, as the vertical kinetic energy and the temperature variance cascade towards small scales, so that the turbulence selectively transfers the correlation between vertical velocity and temperature towards the superstructure scale, consistent with the strong large-scale coherence of the superstructures. Pressure transport is the only mechanism that delivers large-scale

left angle bracket w theta right angle bracket ⟨ w θ ⟩ $\langle w\theta \rangle$

towards the wall with appreciable magnitude, yet the pressure–strain term destroys the transported quantity so that the net pressure–temperature interaction is purely negative at all scales. No mechanism in the budget delivers large-scale spectral content into the boundary layer, providing a scale-by-scale explanation for the insensitivity of the Nusselt number to the aspect ratio beyond a moderate threshold.