Dissipation-based proper orthogonal decomposition of turbulent Rayleigh-B\'enard convection flow

Abstract

We present a formulation of proper orthogonal decomposition (POD) producing a velocity-temperature basis optimized with respect to an $H^1$ dissipation norm. This decomposition is applied, along with a conventional POD optimized with respect to an $L^2$ energy norm, to a data set generated from a direct numerical simulation of Rayleigh-B\'enard convection in a cubic cell ($\mathrm{Ra}=10^7$, $\mathrm{Pr}=0.707$). The data set is enriched using symmetries of the cell, and we formally link symmetrization to degeneracies and to the separation of the POD bases into subspaces with distinct symmetries. We compare the two decompositions, demonstrating that each of the 20 lowest dissipation modes is analogous to one of the 20 lowest energy modes. Reordering of modes between the decompositions is limited, although a corner mode known to be crucial for reorientations of the large-scale circulation is promoted in the dissipation decomposition, indicating suitability of the dissipation decomposition for capturing dynamically important structures. Dissipation modes are shown to exhibit enhanced activity in boundary layers. Reconstructing kinetic and thermal energy, viscous and thermal dissipation, and convective heat flux, we show that the dissipation decomposition improves overall convergence of each quantity in the boundary layer. Asymptotic convergence rates are nearly constant among the quantities reconstructed globally using the dissipation decomposition, indicating that a range of dynamically relevant scales are efficiently captured. We discuss the implications of the findings for using the dissipation decomposition in modeling, and argue that the $H^1$ norm allows for a better modal representation of the flow dynamics.