Modulation of turbulent Rayleigh-Bénard convection under spatially harmonic heating
Chao-Ben Zhao, Yi-Zhao Zhang, Bo-Fu Wang, Jian-Zhao Wu, Kai Leong Chong, Quan Zhou
Abstract
We numerically study turbulent Rayleigh-B\'enard (RB) convection under spatial temperature modulation, where the bottom temperature varies sinusoidally around a mean value in space. Both two- and three-dimensional simulations are performed over the Rayleigh number range ${10}^{7}\ensuremath{\le}\text{Ra}\ensuremath{\le}{10}^{10}$ and the wave number range $1\ensuremath{\le}k\ensuremath{\le}120$ at fixed Prandtl number $\text{Pr}=0.7$. It is demonstrated that spatial temperature modulation with small wave numbers can enhance the global heat transfer (characterized by the Nusselt number $\text{Nu}$) in the turbulent regime, while $\text{Nu}$ is close to that in standard RB convection in the case of large wave numbers. Further, we propose two characteristic modulation length scales: one is the penetration depth ${\ensuremath{\delta}}_{k}$ above which spatial modulation is negligible, the other is the inversion depth ${\ensuremath{\delta}}_{k2}$ below which there exists a stable inverse temperature gradient. Based on the relative thickness of the thermal boundary layer (BL) ${\ensuremath{\delta}}_{\mathrm{th}}$ compared with these two length scales, the underlying modulation mechanism is physically explained and three regimes are identified: (1) an unperturbed BL regime (${\ensuremath{\delta}}_{k}<{\ensuremath{\delta}}_{\mathrm{th}}$), in which the modulation effect does not penetrate through the thermal BL and $\text{Nu}$ is nearly unchanged; (2) a partially modulated BL regime (${\ensuremath{\delta}}_{k2}<{\ensuremath{\delta}}_{\mathrm{th}}<{\ensuremath{\delta}}_{k}$), in which hot spots trigger more plume emissions from the thermal BL, resulting in $\text{Nu}$ enhancement; and (3) a fully modulated BL regime (${\ensuremath{\delta}}_{\mathrm{th}}<{\ensuremath{\delta}}_{k2}$), in which the stable temperature inversion over the cold phases begins to affect convective flows, which alters the trend of $\text{Nu}$ enhancement.