Effects of Rayleigh Number and Inclination Angle on Natural Convection in a Differentially Heated Square Cavity

Abstract

This study presents a numerical investigation of natural convection in a two-dimensional inclined square cavity filled with air and subjected to differential heating. The effects of the Rayleigh number and cavity inclination on heat transfer and flow behavior were investigated for 103 ≤ Ra ≤ 109 and inclination angles of 0°, 15°, 30°, and 45°. The dimensionless Navier–Stokes and energy equations were solved using the finite-element method under the Boussinesq approximation with a steady laminar formulation. A structured quadrilateral mesh with boundary-layer refinement was employed near the differentially heated walls. Mesh-refinement tests and comparisons with benchmark data for the classical square cavity were used to assess the numerical accuracy of the model. The results show that the average Nusselt number increases with the Rayleigh number, reflecting the progressive intensification of buoyancy-driven heat transfer. The effect of inclination is non-monotonic and depends on the Rayleigh number. At Ra = 104, the highest average Nusselt number is obtained at 45°, whereas for Ra ≥ 105, the maximum is consistently observed at 15°. At Ra = 109, the average Nusselt number is 54.475, 55.617, 53.224, and 49.408 for inclination angles of 0°, 15°, 30°, and 45°, respectively. The results indicate that moderate cavity inclination can enhance heat transfer by favorably modifying the interaction between buoyancy and the imposed thermal gradient, whereas larger inclinations progressively reduce the heat-transfer rate. The present results provide a systematic characterization of the coupled effects of Rayleigh number and cavity inclination within the scope of the steady two-dimensional formulation considered.