Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions

Penetrative turbulent Rayleigh–Bénard convection which depends on the density maximum of water near 4∘C is studied using two-dimensional and three-dimensional direct numerical simulations. The working fluid is water near 4\,∘C with Prandtl number $Pr=11.57$ . The considered Rayleigh numbers $Ra$ range from 10⁷ to 10¹⁰ . The density inversion parameter [STIX]x1D703ₘ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thicknesses ( F_[STIX]x1D706=[STIX]x1D706ₜ^[STIX]x1D703/[STIX]x1D706b^[STIX]x1D703 ) increases with increasing [STIX]x1D703ₘ , and the relationship between F_[STIX]x1D706 and [STIX]x1D703ₘ seems to be independent of $Ra$ . The centre temperature [STIX]x1D703c is enhanced compared to that of Oberbeck–Boussinesq cases, as [STIX]x1D703c is related to F_[STIX]x1D706 with 1/[STIX]x1D703c=1/F_[STIX]x1D706+1 , [STIX]x1D703c is also found to have a universal relationship with [STIX]x1D703ₘ which is independent of $Ra$ . Both the Nusselt number $Nu$ and the Reynolds number $Re$ decrease with increasing [STIX]x1D703ₘ , the normalized Nusselt number Nu([STIX]x1D703ₘ)/Nu(0) and Reynolds number Re([STIX]x1D703ₘ)/Re(0) also have universal relationships with [STIX]x1D703ₘ which seem to be independent of both $Ra$ and the aspect ratio [STIX]x1D6E4 . The scaling exponents of Nu~ Ra^[STIX]x1D6FC and Re~ Ra^[STIX]x1D6FD are found to be insensitive to [STIX]x1D703ₘ despite of the remarkable change of the flow organizations.