A phenomenological interpretation of laminar natural convection correlations

Natural convection heat transfer is commonly described using dimensionless correlations relating the Nusselt number to the Grashof and Prandtl numbers. For gases and non-metallic liquids, these parameters combine into the Rayleigh number, (Formula presented.), while for liquid metals they form the Boussinesq number, (Formula presented.), whose physical roles are often underrated. Focusing on the classical configuration of laminar flow over an isothermal vertical plate, the present work adopts a phenomenological perspective to examine how the governing variables influence the thickness of the thermal boundary layer, and, consequently, the heat transfer rate. Within this framework, the Rayleigh and Boussinesq numbers emerge naturally as the controlling dimensionless parameters. Once a power law functional dependence between the Nusselt number and the Rayleigh and Boussinesq numbers is assumed, a physically consistent range for the associated exponents is identified, together with the necessity of introducing a weak corrective function of the Prandtl number, which plays just a refining role. An overall order-of-magnitude analysis is further developed to recover the classical one-fourth power law structure of natural convection correlations and to clarify the origin of the Prandtl number corrective function. The primary contribution of this study is to demonstrate that the Rayleigh and Boussinesq numbers are not merely the outcome of solving the governing equations, but are the dimensionless groups that encapsulate the underlying physics of natural convection, in contrast to the Grashof and Prandtl numbers considered separately. This perspective provides a comprehensive physical interpretation of the existing heat transfer correlations, also offering guidance for identifying the appropriate dimensionless parameters when developing new correlations, useful both to researchers and educators.