Modulation of turbulent Rayleigh-Bénard convection under spatially harmonic heating

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 10⁷≤Ra≤10¹⁰ and the wave number range 1≤k≤120 at fixed Prandtl number 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 Nu) in the turbulent regime, while 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 δₖ above which spatial modulation is negligible, the other is the inversion depth δₖ₂ below which there exists a stable inverse temperature gradient. Based on the relative thickness of the thermal boundary layer (BL) δₜₕ compared with these two length scales, the underlying modulation mechanism is physically explained and three regimes are identified: (1) an unperturbed BL regime (δₖ<δₜₕ), in which the modulation effect does not penetrate through the thermal BL and Nu is nearly unchanged; (2) a partially modulated BL regime (δₖ₂<δₜₕ<δₖ), in which hot spots trigger more plume emissions from the thermal BL, resulting in Nu enhancement; and (3) a fully modulated BL regime (δₜₕ<δₖ₂), in which the stable temperature inversion over the cold phases begins to affect convective flows, which alters the trend of Nu enhancement.