Spectral analysis of the correlation between temperature and wall-normal velocity in Rayleigh-Bénard convection at moderate Rayleigh numbers

Abstract

Turbulent Rayleigh-Bénard convection holds significant scientific and engineering implications, with one of the most debated topics in convection research being the ultimate scaling of the Nusselt number at extremely high Rayleigh numbers. A widely adopted method for investigating this issue is direct numerical simulation (DNS); however, achieving the ultimate Rayleigh number with DNS can be prohibitively expensive. To overcome this limitation, one approach is to employ a small simulation domain size for high Rayleigh number cases. This study explores the impact of simulation domain size by conducting DNS with a large geometrical aspect ratio. Specifically, we perform spectral analysis on the covariance of wall-normal velocity and temperature, along with its transport equations. The results reveal that the characteristic length scale in the intermediate length-scale region linearly increases with wall-normal distance. Moreover, the spectral analysis indicates that large-scale structures play a crucial role in the dynamics of the covariance of wall-normal velocity and temperature. However, the study suggests that these large-scale structures have a minimal impact on the scaling of the Nusselt number. This is attributed to the fact that large-scale structures remain in regions away from the wall, and no processes transfer them from large to small-scale structures. As a consequence, the dynamics of the covariance of wall-normal velocity and temperature in the close vicinity of the wall experience little influence from large-scale dynamics.