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In this work a numerical investigation of two-dimensional steady and unsteady natural convection in a circular enclosure whose lower half is nonuniformly heated and upper half is maintained at a constant lower temperature has been carried out. An explicit finite difference method on a nonstaggered rectangular grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a range of Rayleigh number ͑Ra͒ varying between 10 2 and 10 6 at a fixed Prandtl number ͑Pr͒ taken as 0.71. The numerical experiments reveal that for Raഛ 8500, the flow always attains a steady state. In the steady regime, at very low Rayleigh numbers ͑RaϽ 300͒, it is shown that the velocity field is very weak and the heat transfer is predominantly by conduction. A series solution for the temperature field obtained by neglecting the fluid velocities is shown to agree well with the computed data for RaϽ 300. The convection takes place in the form of two cells with their interface aligned along the vertical diameter. As Ra is increased further, the isotherms distort to form a plume-like structure of hot fluid rising from the hottest point on the lower half of the cylinder wall. Local Nusselt number distribution over the wall shows that only a portion of the nonuniformly heated bottom half of the cylinder wall is responsible for heating the fluid. For Raജ 8900, the numerical simulations show that the steady flow looses its stability and the flow undergoes bifurcations to periodic and quasiperiodic states. On the basis of the data on the amplitude of the periodic flows obtained for a set of Ra slightly greater than 8900, it is shown that the steady flow undergoes a supercritical Hopf bifurcation at RaϷ 8830. An analysis using the proper orthogonal decomposition shows that the instability is in the form of a standing wave. The structure of the unstable mode is examined via empirical eigenfunctions obtained by the method of snapshots. In the unsteady regime ͑RaϾ 8.9ϫ 10 3 ͒, the cells start to swing their interface in an oscillatory manner with time. As Ra is increased further, the character of flow changes from periodic to quasiperiodic.

A novel extension to SMAC scheme is proposed for variable density flows under low Mach number approximation. The algorithm is based on a predictor—corrector time integration scheme that employs a projection method for the momentum equation. A constant-coefficient Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each time step. The proposed algorithm has second order centrally differenced convective fluxes with upwinding based on Cell Peclet number while diffusive flux are viscous fourth order accurate. Spatial discretization is performed on a collocated grid system that offers computational simplicity and straight forward extension to curvilinear coordinate systems. The algorithm is kinetic energy preserving. Further in this paper robustness and accuracy are demonstrated by performing test on channel flow with non-Boussinesq condition on different temperature ratios

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