Natural Convection in Shallow Enclosures With Differentially Heated Endwalls

We consider a low-aspect-ratio two-dimensional rectangular cavity, differentially heated with an arbitrarily large horizontal temperature difference. Steady-state results obtained from numerical solutions of the transient Navier-Stokes equations are given for air using the ideal gas law and Sutherland law transport properties. We clarify the different flow regimes possible by using numerical results for aspect ratios 0.025 ≤ A ≤ 1 and for Rayleigh numbers (based on height) 10² ≤ Ra ≤ 10⁹. We present Nusselt numbers, and temperature and velocity distributions, and compare them with existing data. At high Ra in the Boussinesq limit we show the existence of weak secondary and tertiary flows in the core of the cavity. In addition we present novel solutions in the non-Boussinesq regime. We find that the classical parallel flow solution, accurate in the core of the cavity in the Boussinesq limit, does not exist when variable properties are introduced. For higher Rayleigh numbers, we generalize the well-known analytical boundary layer solution of Gill. The non-Boussinesq solutions show greatly reduced static pressure levels and lower critical Rayleigh numbers.