Numerical study of natural convection in an enclosure with localized heating from below—creeping flow to the onset of laminar instability

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Abstract
An analytical study was made of the natural convection induced in an enclosure by a small hot spot centrally located on the floor. The enclosure was a circular cylinder, vertically oriented, with height equal to radius. A Prandtl number of 0.7 (air) was assumed; the Grashof number (Gr) was based on cylinder height and hot spot temperature. The equations of fluid flow in axisymmetric cylindrical co-ordinates were simplified with the Boussinesq approximation. The equations were solved numerically with a computationally stable, explicit method. The computation, starting from quiescent conditions, proceeded through the initial transient to the fully developed flow. Solutions were obtained for Gr from 4 × 104 to 4 × 1010. The theoretical flows are in excellent agreement with experimentally observed laminar flows (Gr [lsim ] 1.2 × 109) which are discussed in a companion paper, Torrance, Orloff & Rockett (1969). Turbulence was observed experimentally for Gr [gsim ] 1.2 × 109. When the theoretical calculations were extended to Gr = 4 × 1010, a periodic vortex shedding developed, suggestive of the onset of laminar instability. The theoretical results reveal a √Gr scaling for the initial flow transients and, at large Gr, the velocities and heat transfer rates.

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