Scaling transition in horizontal convection near the density maximum

TL;DR

The paper tackles horizontal convection under a nonlinear equation of state near water's density maximum, advancing beyond the OB approximation. It develops an extended Shishkina-Grossmann-Lohse framework by introducing a NOB potential-energy transfer term $\Phi_{i2}$ and linking plume height $\hat{z}$ to global transport scalings, predicting a transition when plumes penetrate the full depth ($\hat{z}\sim H$). Direct numerical simulations reveal a flow-topology shift in NOB-SHC from a bicellular structure to a full-depth single-roll circulation, with heat-transport scaling transitioning from $Nu \sim Ra^{1/4}$ to $Ra^{1/3}$ and Reynolds scaling from $Re \sim Ra^{1/2}$ to $Ra^{4/9}$. The results unify OB and NOB HC scalings within the GL/SGL framework and elucidate how density anomalies can drive enhanced mixing and transport in geophysical-like systems.

Abstract

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.

Figures (9)

• Figure 1: Schematic of the computational domain and boundary conditions. No-slip conditions are applied to all boundaries. A fixed buoyancy profile (Dirichlet condition) is imposed on the bottom boundary, while the remaining walls are adiabatic. The parameters $b_0$ and $\Delta_b$ denote the reference buoyancy and the maximum buoyancy difference, respectively.
• Figure 2: Instantaneous temperature fields (colour contours) and streamlines for the SHC cases at various Rayleigh numbers. Solid lines represent clockwise circulation, while dashed lines represent counter-clockwise circulation.
• Figure 3: Instantaneous temperature fields (colour contours) and streamlines for the CHC cases at various Rayleigh numbers. Solid lines represent clockwise circulation, while dashed lines represent counter-clockwise circulation.
• Figure 4: Horizontally averaged vertical profiles of kinetic energy under different $Ra$.
• Figure 5: Time-averaged kinetic energy contributed by the (1, 1), (1, 2), (2, 1) and (2, 2) modes at different Rayleigh numbers in the NOB-SHC case.