Thermo-fluid-dynamics of natural convection around a heated vertical plate with a critical assessment of the standard similarity theory
Authors: Guha Abhijit , Nayek Subhajit 
Published: October 2017

A compulsory element of all textbooks on natural convection has been a detailed similarity analysis for laminar natural convection on a heated semi-infinite vertical plate and a routinely used boundary condition for such analysis is u = 0 at x = 0. The same boundary condition continues to be assumed in related theoretical analyses, even in recent publications. The present work examines the consequence of this long-held assumption, which appears to have never been questioned in the literature, on the fluid dynamics and heat transfer characteristics. The assessment has been made here by solving the Navier-Stokes equations numerically with two boundary conditions—one with constrained velocity at x = 0 to mimic the similarity analysis and the other with no such constraints simulating the case of a heated vertical plate in an infinite expanse of the quiescent fluid medium. It is found that the fluid flow field given by the similarity theory is drastically different from that given by the computational fluid dynamics (CFD) simulations with unconstrained velocity. This also reflects on the Nusselt number, the prediction of the CFD simulations with unconstrained velocity being quite close to the experimentally measured values at all Grashof and Prandtl numbers (this is the first time theoretically computed values of the average Nusselt number Nu¯ are found to be so close to the experimental values). The difference of the Nusselt number (ΔNu¯) predicted by the similarity theory and that by the CFD simulations (as well as the measured values), both computed with a high degree of precision, can be very significant, particularly at low Grashof numbers and at Prandtl numbers far removed from unity. Computations show that within the range of investigations (104 ≤ GrL ≤ 108, 0.01 ≤ Pr ≤ 100), the maximum value of ΔNu¯ may be of the order 50%. Thus, for quantitative predictions, the available theory (i.e., similarity analysis) can be rather inadequate. With the help of the CFD simulations, the details of the fluid dynamics, particularly the physics of fluid entrainment, are thoroughly studied. It is shown that the relative proportions of the fluid entrainment from the bottom, top, and side of the vertical plate depend on the size of the region of interest (ROI). As the size of the ROI is made large, most of the entrained fluid comes from the bottom, a little bit from the top and almost no fluid enters from the side; the nature of entrainment is opposite in the similarity analysis for which all the fluid enters from the side and no fluid enters either from the bottom or the top. The two sets of CFD simulations establish, in particular, the conclusion that it is the inappropriateness of the age-old boundary condition u = 0 at x = 0, and not the boundary layer approximation, that is the principal cause for the vulnerability of the standard similarity analyses (and integral theories) for natural convection. The CFD solutions further demonstrate the effects of finite length and finite thickness of the plate on the flow field and the shape of the buoyant jet. The different boundary conditions on the two sides of the vertical plate and the presence of its finite thickness make the buoyant jet bend over the top edge of the plate and make the evolution of entrainment from the two sides of the free buoyant jet different. The entrainment velocity from the two sides, however, equilibrates at a certain distance above the plate. The asymmetry in the velocity and temperature fields above the plate decreases more rapidly when Pr is smaller and GrL is greater. It is shown that sufficiently above the plate, the distributions of axial velocity and temperature in the buoyant jet tend to be symmetric with respect to an axis that seems to pass through the vertical mid-plane of the plate, i.e., the jet tends to lose its history of origination.