Proper orthogonal decomposition investigation of turbulent Rayleigh-Bénard convection in a rectangular cavity

We consider the large-eddy simulation (LES) of turbulent Rayleigh-Bénard convection for air in a parallepipedic cavity of ratio (1:5:1) over the range Ra = 6 × 108 up to Ra = 1010 previously studied in Sergent and Le Quéré (Proceedings of the 13th European Turbulence Conference, 2011). Using proper orthogonal decomposition (POD) analysis, we confirm the existence of a large-scale circulation (LSC) consisting of quasi-stationary cross-stream rolls (y-rolls) which are aligned with the small direction of the box. Strong changes in the LSC are observed to take place over a few hundred convective time units, defined as [12pt]{minimal}{document}κ /(Lₓ² Ra1/2){document}κ/(Lx2Ra1/2), where κ is the fluid diffusivity, Lx is the height of the box and Ra is the Rayleigh number. We also show the existence of a secondary flow, which consists of horizontal rolls (z-rolls) surrounding the core of the cavity and orthogonal to the cross-stream rolls. The amplitude of these longitudinal rolls oscillates on a time scale of 50 convective units. The longitudinal rolls are associated with strong variations in the vertical momentum transfer, while cross-stream rolls are primarily responsible for more than half of the convective heat transfer and make little contribution to the convective momentum transfer. Integration of a simplified dynamical model with stationary y-rolls leads to an oscillation in the z-rolls with a characteristic period of 70 units, which supports the idea that the oscillation of the secondary rolls is determined by the primary flow structure. Using 2D linear stability analysis based on the contributions from POD modes, we show that the presence of longitudinal shear leads to the stabilization of the high wavenumber range, and we predict a roll size which is reasonably close to that observed in the LES. We conjecture that changes in the large-scale circulation of the flow are related to the fluctuating shear created by the secondary rolls and higher-order modes outside the boundary layer.