Turbulent spherical Rayleigh–Bénard convection: radius ratio dependence

Direct numerical simulation (DNS) is performed to explore turbulent Rayleigh–Bénard convection in spherical shells. Our simulations cover six distinct values of radius ratio, $\eta = r_i/r_o = 0.2$ , 0.3, 0.4, 0.5, 0.6 and 0.8, under the assumption of a centrally condensed mass with gravity profile $g \sim 1/r^{2}$ ; where $r_i$ , $r_o$ and $r$ denote the inner shell radius, the outer shell radius and the local radial coordinate, respectively. The Prandtl number is kept constant at unity while the Rayleigh number ( $Ra$ ) is varied from $3 \times 10^{3}$ to $5 \times 10^8$ . Our primary aim is to analyze how the radius ratio influences the global transport properties and flow physics. To gain insights into the scaling behaviour of the Nusselt number ( $Nu$ ) and the Reynolds number ( $Re$ ) with respect to $Ra$ and $\eta$ , we apply the Grossmann–Lohse (GL) theory (Grossmann & Lohse, J. Fluid Mech. , vol. 407, 2000, pp. 27–56) to the system. It is observed that the scaling exponents for $Nu$ and $Re$ in relation to $Ra$ are more significant for smaller $\eta$ values, suggesting that the simulations with smaller $\eta$ reach the classical $Nu\sim Ra^{1/3}$ regime at a relatively lower $Ra$ . This observation could also imply the systems with smaller $\eta$ might transition to the ultimate regime earlier at a smaller $Ra$ . Based on our extensive DNS data, we establish that the thickness of the inner thermal boundary, $\lambda _{\vartheta }^{i}$ , follows a scaling relationship of $\lambda _{\vartheta }^{i} \sim \eta ^{1/2}$ . This relationship, in turn, leads to a scaling law for $Nu$ in the form of $Nu \sim f(\eta ) Ra^{\gamma }$ , where the function $f(\eta )$ is defined as $f(\eta ) = {\eta ^{1/2}}/{(1+\eta ^{4/3})}$ , and the exponent $\gamma$ depends on both $Ra$ and $\eta$ . Additionally, we characterize and explain the asymmetry in the velocity field by introducing the separate Re