(295e) Effect of Side and Bottom Wall Temperature Non-Uniformities On Multiplicity of Steady Solutions for Mixed Convection In A Square Cavity

Galerkin penalty finite element method with 28x28 bi-quadratic elements has been used to investigate the role of bottom and side wall temperature non-uniformities on the existence of multiple steady solutions for mixed convective flows within a square enclosure with thermally insulated moving top wall. Here, non-uniformities of bottom and side wall temperatures are varied such that side walls always remain colder than bottom wall with continuity of temperature maintained at the bottom corners. Variations of bottom and side wall temperature non-uniformities are parametrically represented by thermal aspect ratio (0 \leqslant \Theta \leqslant 1), which is the ratio of temperature gradient of side walls to the total temperature gradient of bottom and side walls. Side walls remain isothermally cold with entire temperature gradient confined at the bottom wall at \Theta = 0. On the other hand, side walls drive entire temperature gradient at \Theta = 1, where bottom wall remains uniformly hot. Any intermediate non-uniformity of bottom and side walls in proportion to \Theta and 1 - \Theta, respectively can be simulated by varying \Theta from 0 to 1. Present work traces possible steady solution branches in the parameter space of \Theta at constant Grashof and Prandtl numbers (Gr =105, Pr = 10) and for three different Reynolds numbers as Re = 1, 10 and 100 representing almost natural to truly mixed convective flows. Bifurcation diagrams of various steady branches at Re = 1, 10 and 100 are presented in terms of trajectories of average bottom wall Nusselt number along \Theta. It has been shown that there exists a critical thermal aspect ratio, which differentiates the region of unique and multiple solutions. Steady multiple solutions appear for \Theta > \Theta_{cr}, where \Theta_{cr} = 0.626, 0.6446 and 0.218 for Re = 1, 10 and 100, respectively. This shows that a certain non-uniformity of side wall temperature is essential for existence of multiple solutions. We have obtained six steady branches for Re = 1 resulting in eleven steady solutions at \Theta = 1. Number of steady branches decreases to three for Re = 10 and 100. Correspondingly, number of steady solutions at \Theta = 1 also decreases to five at Re = 10 and 100. Transition of flow and temperature profiles along each branches are also shown, which illustrate a sequence of formation, merging, splitting, disappearance an reappearance of secondary cells near bottom wall of the enclosure along each trajectory.