Scaling transition in horizontal convection near the density maximum

Horizontal convection (HC) serves as a canonical model for geophysical and industrial flows driven by differential heating along a surface. While the classical Oberbeck-Boussinesq (OB) approximation is well-established, the impact of a nonlinear equation of state, specifically the density maximum of water near $4^\circ\mathrm{C}$, remains underexplored. This study investigates Non-Oberbeck-Boussinesq (NOB) effects on HC via direct numerical simulations (DNS) over a Rayleigh number range of $10^6 \le Ra \le 5\times 10^{10}$. We examine two configurations: Classical HC (CHC) and Symmetric HC (SHC). Our results reveal that the NOB-SHC case undergoes a structural transition, evolving from a bicellular structure to a full-depth, single-roll circulation driven by central `mixing plumes'. This reorganization manifests as transitional anomalies in Reynolds number ($Re$) scaling, whereas the emergence of full-depth plumes fundamentally alters the heat transport mechanism. Consequently, unlike the classical Rossby scaling ($Nu \sim Ra^{1/5}$) observed in reference cases, the NOB-SHC regime exhibits an enhanced heat transport scaling ranging from $Nu \sim Ra^{1/4}$ to $Ra^{1/3}$. To rationalize this behavior, we extend the Shishkina-Grossmann-Lohse (SGL) theory by incorporating a generalized potential energy transfer term ($Φ_{i2}$). The theoretical framework demonstrates that the global scaling law is dictated by the characteristic plume height ($\hat{z}$). Specifically, when plumes penetrate the entire cavity depth ($\hat{z} \sim H$), as observed in the NOB-SHC case, the flow transcends classical bounds for OB HC, accessing a regime analogous to Rayleigh Bénard convection. The proposed theory successfully unifies the scaling laws for both OB and NOB fluids, showing excellent agreement with numerical data.

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