Vertical temperature and velocity structure in horizontal convection: analytical profiles under varying velocity and thermal boundary conditions

We report results from experiments and direct numerical simulation (DNS) of the temperature and velocity fields in horizontal convection (HC) over a range of Rayleigh numbers 10 Superscript 8 Baseline less than or slanted equals upper R a less than or slanted equals 5 times 10 Superscript 12 10 8 ⩽ R a ⩽ 5 × 10 12 $10^{8} \leqslant Ra \leqslant 5\times 10^{12}$ and Prandtl numbers 4.3 less than or slanted equals upper P r less than or slanted equals 7 4.3 ⩽ P r ⩽ 7 $4.3 \leqslant Pr \leqslant 7$ in a rectangular cell of aspect ratio upper Gamma identical to upper L colon upper W colon upper H equals 10 colon 1 colon 1 Γ ≡ L : W : H = 10 : 1 : 1 $\varGamma \equiv L:W:H = 10:1:1$ ( upper L L $L$ , upper W W $W$ and upper H H $H$ denote the length, width and height of the cell, respectively). The DNS considers four combinations of velocity and thermal boundary conditions (BCs), spanning free-slip and no-slip velocity constraints together with isothermal and adiabatic thermal conditions. In the experiments, the upper boundary consists of an open water–air interface, which approximates a slippery surface. The DNS results reveal systematic differences in the thermal boundary-layer thickness, vertical temperature gradients and overall flow organisation across the four BC configurations. Building on these findings, we develop a theoretical model based on a prescribed streamfunction that satisfies the relevant boundary constraints. From this formulation, general analytical expressions for the vertical mean temperature profile are derived for each configuration, capturing the interplay among advection, diffusion and the imposed BCs. The model successfully reproduces self-similar features of the temperature profiles observed in the DNS and agrees with available experimental data. This framework thus provides a unified description of vertical temperature and velocity structure in HC and clarifies how velocity and thermal BCs jointly regulate boundary-layer formation and heat transport efficiency. The findings also offer insight into analogous geophysical processes, including urban near-surface heating and oceanic stratification.