The present investigation deals with the numerical computation of laminar natural convection in a gamma of right-angled triangular cavities filled with air. The vertical walls are heated and the inclined walls are cooled while the upper connecting walls are insulated from the ambient air. The defining apex angle α is located at the lower vertex formed between the vertical and inclined walls. This unique kind of cavity may find application in the miniaturization of electronic packaging severely constrained by space and/or weight. The finite volume method is used to perform the computational analysis encompassing a collection of apex angles α compressed in the interval that extends from 5° to 63°. The height-based Rayleigh number, being unaffected by the apex angle α, ranges from a low 103 to a high 106. Numerical results are reported for the velocity field, the temperature field and the mean convective coefficient along the heated vertical wall. Overall, the matching between the numerically predicted temperatures and the experimental measurements of air at different elevations inside a slim cavity is of ordinary quality. For purposes of engineering design, a Nu¯H correlation equation was constructed and also a figure-of-merit ratio between the Nu¯H and the cross sectional area A of the cavity was proposed.
The archival literature contains an abundance of correlations for the characterization of laminar-free convection in rectangular, square, and annular cavities, but unfortunately does not disclose a single correlation for triangular cavities. A prominent application of triangular cavities specialized to isosceles shapes arises in air-filled attic spaces of houses and buildings with sloped roofs and horizontal suspended ceilings. In this work, two contrasting situations are considered embodying three apex angles and several Grashof numbers: a hot base and symmetrically cold inclined sides for winter conditions, and a cold base and symmetrically hot inclined sides for summer conditions. The goal of this article is to construct two dependable correlations, one for the mean convective coefficient at the air/wall interface, and the other for the temperature variation of the ascending air along the center plane of attic spaces. The input data was taken from precise measurements available in the specialized literature.
Transient, laminar thermal convection of air confined to an isosceles triangular cavity heated from the base and symmetrically cooled from the upper inclined walls has been investigated numerically. The system of conservation equations, subject to the proper boundary conditions, along with the equation of state assuming the air behaves as a perfect gas are solved with the finite volume method. In the conservation equations, the second-order-accurate QUICK scheme was used for the discretization of the convective terms and the SIMPLE scheme for the pressure-velocity coupling. The maximum height-to-base aspect ratio A is fixed at 0.5, while the Grashof number extends from a low Gr=103 to a high Gr=106. The influence of Gr on the flow and temperature patterns is analyzed and discussed for two opposing scenarios, one corresponding to increasing Gr and the other corresponding to decreasing Gr. It is found that two steady-state solutions are possible, excluding their solution images through a vertical mirror plane. The symmetrical solution prevails for relatively low Grashof numbers. However, as the Gr is gradually increased, a transition occurs at a critical value of Gr. Above this critical value of Gr, an asymmetrical solution exhibiting a pitchfork bifurcation arises and eventually becomes steady. The existing ranges of these unsteady and steady solutions are reported for the two opposing scenarios. Also, issues related to the observed hysteresis phenomenon are discussed in detail.
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