Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side

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Abstract
The stability of laminar natural convection in a shallow cavity has been studied theoretically. The flow is driven by a horizontal temperature gradient between isothermal vertical sidewalls of the cavity, the top and bottom of which are insulated. It was found that for a Prandtl number (Pr) less than 0.033, shear instability causes stationary transverse cells to be formed in the flow. For larger values of Prandtl number the instability sets in as oscillating longitudinal rolls in the range 0.033<Pr<0.2; and as stationary longitudinal rolls for larger values of Pr. The importance of three‐dimensional disturbances was investigated for Pr=0.02 and they were found to be less critical than two‐dimensional ones. Finite amplitude motions of the stationary transverse cells were simulated by solving the nonlinear equations numerically with the use of pseudospectral methods. This simulation supports the calculations of the onset of the instability by linear theory, and explains why the Nusselt number decreases when secondary flows are present.

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