Transient almost-invariant sets reveal convective heat transfer patterns in plane-layer Rayleigh-Bénard convection

TL;DR

This work introduces an inflated generator framework to identify transient quasi-stationary families of almost-invariant sets in three-dimensional Rayleigh–Bénard convection and links these transport structures to convective heat transfer. By time-extending the transfer operator and applying SEBA to the leading spatial eigenvectors, the authors extract evolving convection-cell patterns that contribute least to heat transfer, effectively tracking the birth, evolution, and decay of mesoscale structures. The method is demonstrated on DNS data at Ra = 10^5 and Pr = 0.7, revealing multiple coexisting metastable plumes and establishing modest but meaningful correlations between SEBA-derived structures and CH transfer, especially within the xy midplane. These results showcase a powerful dynamical-systems approach to dissect the spatiotemporal organization of turbulent heat transfer and point to extensions for more complex rotating or curved geometries.

Abstract

Horizontally extended plane-layer convection flows are characterized by characteristic patterns of turbulent heat transfer due to the convective fluid motion consisting of a nearly-regular ridge network where hot fluid rises and cold fluid sinks. Here, we analyse this transport behavior by the so-called inflated generator framework, which identifies quasi-stationary families of almost-invariant sets, derived from leading inflated generator eigenvectors. We demonstrate the effectiveness of this data-driven analysis framework in three-dimensional turbulent flow, by extracting transient characteristic heat transfer patterns as families of almost-invariant sets subject to a transient evolution, which contribute least to the convective heat transfer.

Figures (12)

• Figure 1: Snapshots of the temperature fluctuation $\theta$, in panels (a,b), the vertical velocity $u_z$, in panels (c,d), and convective heat transfer field in panels (e,f) for the time instant at $t=2052 \,T_f$. The left column displays horizontal cuts at $z=1/2$, the right column vertical cuts at $y=0$.
• Figure 2: Mean values of convective heat transfer (CHT) across the horizontal plane at different $z$-levels plotted against these $z$ levels, at $t=2052 T_f$, and averaged over time windows of length 5, 9, 13, 17 and 21 $T_f$, centred at time $t=2052 T_f$. Similar results are obtained at other times.
• Figure 3: Sketch of the spatial domain $M$ divided into cubic cells for the construction of the spatial generators $G_t$ on each time slice with side length $\ell$. We use periodic boundary conditions on the side walls of the domain (which are highlighted in cyan) and Dirichlet boundary conditions for the temperature field at the top and bottom planes (which are highlighted in red).
• Figure 4: The spectrum of the inflated generator for the three-dimensional RBC flow with $a = a_{\mathrm{init}} \approx 4.95$, showing the leading thirty real-valued spatial eigenvalues (indicated by blue circles), the first three temporal eigenvalues (indicated by red crosses), and an extensive collection of complex valued eigenvalues (indicated by black dots).
• Figure 5: The evolution of a quasi-stationary family of almost-invariant sets (a transient plume of fluid forming a metastable parcel) within the RBC flow over six discrete time steps as identified through the SEBA vector $\mathbf{S}_{21}$. A cutoff of 0.4 has been applied to the SEBA vector.