Geometry-controlled heat transport pathways and optimal heat transfer in differentially heated cavities

We perform direct numerical simulations of natural convection in a differentially heated cavity over Rayleigh number $Ra=10^6$--$10^8$ at Prandtl number $Pr=0.7$, systematically varying the aspect ratio over $0.1 \leq \Gamma \leq 60$. Across this nearly three-decade range, the Nusselt number $Nu$ exhibits four distinct power-law regimes as a function of $\Gamma$, arising solely from geometric confinement. We show that these transport regimes are governed by qualitative changes in the anisotropy and structure of the large-scale circulation (LSC), quantified by the ratio of Reynolds numbers based on the root-mean-square horizontal and vertical velocities, $Re_u/Re_v$. For small $\Gamma$, vertical confinement promotes a horizontally dominant LSC and strong enhancement of heat transport. At intermediate aspect ratios, the circulation reorganizes into an efficient heat-carrying structure for which $Nu$ becomes nearly independent of $\Gamma$. At larger $\Gamma$, the LSC becomes increasingly vertically elongated and transitions to shear-driven dynamics associated with Kelvin--Helmholtz-type instability, leading to a progressive reduction in heat transport before approaching an asymptotic large-$\Gamma$ limit. A central result is that the heat flux is maximized when the circulation anisotropy satisfies $Re_u/Re_v \approx 0.45$, which remains robust across all Rayleigh numbers considered. The corresponding optimal aspect ratio follows the scaling $\Gamma_{\mathrm{opt}} \sim Ra^{-0.19}$. Resolvent analysis further reveals that optimal transport is associated with stationary, slender response modes, whereas larger $\Gamma$ results in oscillatory shear-layer amplification. These findings establish geometric confinement as the key control parameter governing transport pathways in differentially heated cavities and provide a predictive framework for geometry-driven heat-transfer optimization.