Shallow enclosure horizontal convection: insights from asymptotic expansion analysis

Horizontal convection is a distinctive convection heat transfer process where heating and cooling occurs along the same horizontal boundary. This study employs the asymptotic expansion method to elucidate Nusselt and Rayleigh number relationships toward the limits of small aspect ratios. Expansion solutions are derived and compared with the high-order spectral element simulations obtained for a rectangular enclosure of length L, height, H (aspect ratio, A = H/L), at a Prandtl number of Pr = 6.14 (consistent with water) across a wide span of Rayleigh numbers, (10 ≤ Ra ≤ 10¹⁶) and aspect ratios (10^-3 ≤ A ≤ 0.16). The Boussinesq flow is driven by imposing a linear temperature variation from colder to warmer across the bottom boundary of the enclosure, and insulating temperature conditions on the remaining boundaries. The accuracy of the model is demonstrated by comparison with the numerical solutions, and second-order expansion solutions captured the Nusselt-Rayleigh number behaviour.