Universal properties in penetrative turbulent Rayleigh-B\'enard convection

Penetrative convection refers to the phenomenon whenever convection in thermally unstable fluid layers penetrate into adjacent stable layers. In this work, we study penetrative turbulent Rayleigh-B\'enard convection which depends on the density maximum of water near 4°C using two-dimensional and three-dimensional direct numerical simulations. The working fluid is water near 4°C with reference Prandtl number Pr=11.57. The considered Rayleigh numbers Ra range from 10⁷ to 10¹⁰. The density inversion parameter T_m varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thickness (F_λ=λ_t^Θ/λ_b^Θ) increases with increasing T_m, and the relationship between F_λ and T_m seems to be independent of Ra. The center temperature T_c is enhanced compared to that of the Oberbeck-Boussinesq (OB) cases, and T_c is also found to have a universal relationship with T_m which is independent of Ra. T_c is related to F_λ with 1/T_c=1/F_λ+1. Moreover, the normalized Nu(T_m)/Nu(0) and Re(T_m)/Re(0) also have universal relationships with T_m which seem to be independent of both Ra and the aspect ratio Γ. The scaling exponents of Nu~Ra^α and Re~Ra^β are found to be insensitive to T_m.