Natural convection in a bottom heated horizontal cylinder

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 (Ra) varying between 102 and 106 at a fixed Prandtl number (Pr) taken as 0.71. The numerical experiments reveal that for Ra⩽8500, the flow always attains a steady state. In the steady regime, at very low Rayleigh numbers (Ra<300), 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 Ra<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 Ra⩾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 Ra≈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 (Ra>8.9×103), 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.