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Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions

  • Qi Wang (a1), Quan Zhou (a2), Zhen-Hua Wan (a1) and De-Jun Sun (a1)


Penetrative turbulent Rayleigh–Bénard convection which depends on the density maximum of water near $4^{\circ }\text{C}$ is studied using two-dimensional and three-dimensional direct numerical simulations. The working fluid is water near $4\,^{\circ }\text{C}$ with Prandtl number $Pr=11.57$ . The considered Rayleigh numbers $Ra$ range from $10^{7}$ to $10^{10}$ . The density inversion parameter $\unicode[STIX]{x1D703}_{m}$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thicknesses ( $F_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}_{t}^{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D706}_{b}^{\unicode[STIX]{x1D703}}$ ) increases with increasing $\unicode[STIX]{x1D703}_{m}$ , and the relationship between $F_{\unicode[STIX]{x1D706}}$ and $\unicode[STIX]{x1D703}_{m}$ seems to be independent of $Ra$ . The centre temperature $\unicode[STIX]{x1D703}_{c}$ is enhanced compared to that of Oberbeck–Boussinesq cases, as $\unicode[STIX]{x1D703}_{c}$ is related to $F_{\unicode[STIX]{x1D706}}$ with $1/\unicode[STIX]{x1D703}_{c}=1/F_{\unicode[STIX]{x1D706}}+1$ , $\unicode[STIX]{x1D703}_{c}$ is also found to have a universal relationship with $\unicode[STIX]{x1D703}_{m}$ which is independent of $Ra$ . Both the Nusselt number $Nu$ and the Reynolds number $Re$ decrease with increasing $\unicode[STIX]{x1D703}_{m}$ , the normalized Nusselt number $Nu(\unicode[STIX]{x1D703}_{m})/Nu(0)$ and Reynolds number $Re(\unicode[STIX]{x1D703}_{m})/Re(0)$ also have universal relationships with $\unicode[STIX]{x1D703}_{m}$ which seem to be independent of both $Ra$ and the aspect ratio $\unicode[STIX]{x1D6E4}$ . The scaling exponents of $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ and $Re\sim Ra^{\unicode[STIX]{x1D6FD}}$ are found to be insensitive to $\unicode[STIX]{x1D703}_{m}$ despite of the remarkable change of the flow organizations.


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