Dimensional and Non-Dimensional Implementations for the Differentially Heated Square Cavity Benchmark: Accuracy and Computational Efficiency

This study presents a numerical comparison of dimensional and non-dimensional implementations of the classical benchmark problem of steady natural convection in a two-dimensional differentially heated square cavity over the Rayleigh-number range 103 ≤ Ra ≤ 1012. The novelty of this work lies in the systematic comparison of both formulations under identical numerical conditions, providing an implementation-oriented assessment of their benchmark accuracy, mesh sensitivity, continuation strategy, and computational efficiency. The governing equations of mass, momentum, and energy conservation were solved under the Boussinesq approximation using primitive variables and the finite-element method. Since both formulations are theoretically equivalent descriptions of the same physical problem, the purpose of this work is not to reassess their physical validity, but to examine their numerical behavior under identical benchmark conditions in terms of mesh sensitivity, continuation strategy, benchmark accuracy, and computational efficiency. In the dimensional implementation, temperature differences of 1, 10, 25, and 50 K were considered, and the corresponding cavity lengths were determined from the Rayleigh-number definition. The main calculations were performed with ΔT = 25 K, while ΔT = 50 K was retained only as an exploratory sensitivity case. Boundary-layer refinement was applied along the vertical walls over the range 103 ≤ Ra ≤ 1012. The results show that, once the near-wall gradients are properly resolved, both implementations predict essentially identical average Nusselt numbers, with a maximum relative difference of 0.022%. Temperature contours, stream-function distributions, and centerline profiles also exhibited the same structural behavior in both cases. For the reference mesh, the dimensional implementation exhibited a lower computational cost than the non-dimensional implementation, resulting in an approximately 31% reduction in computation time. These results demonstrate that solving the governing equations directly in dimensional variables provides a numerically efficient and physically interpretable alternative while preserving the benchmark accuracy of the classical non-dimensional formulation.