Cellular convection in a chamber with a warm surface raft

We calculate velocity and temperature fields for Rayleigh-Benard convection in a chamber with a warm raft that floats along the top surface for Rayleigh number up to Ra = 20 000. Two-dimensional, infinite Prandtl number, Boussinesq approximation equations are numerically advanced in time from a motionless state in a chamber of length L′ and depth D′. We consider cases with an insulated raft and a raft of fixed temperature. Either oscillatory or stationary flow exists. In the case with an insulated raft over a fluid, there are only three parameters that govern the system: Rayleigh number (Ra), scaled chamber length (L = L′/D′), and scaled raft width (W). For W = 0 and L = 1, linear theory shows that the marginal state without a raft is at a Rayleigh number of $23π4=779.3$⁠, but we find that for the smallest W (determined by numerical grid size) the raft approaches the center monotonically in time for $Ra<790$⁠. For $790<Ra<811$⁠, the raft has a decaying oscillation consisting of raft movement back and forth accompanied by convection cell reversal. For $811<Ra<871$⁠, the oscillation amplitude is constant in time and it increases with larger Ra. Finally, there is no raft motion for $Ra>871$⁠. For larger raft widths, there is a range of W that produces raft oscillation at each Ra up to 20 000. Rafts in longer cavities (L = 2 and 4) have almost no oscillatory behavior. With a raft of temperature set to different values of $Tr$ rather than insulating, a fixed Rayleigh number $Ra=20000$⁠, a square chamber (L = 1), fixed raft width, and with internal heat generation, there are two ranges of oscillating flow.