Plume dynamics in quasi-2D turbulent convection
We have studied turbulent convection in a vertical thin ͑Hele-Shaw͒ cell at very high Rayleigh numbers ͑up to 7ϫ10 4 times the value for convective onset͒ through experiment, simulation, and analysis. Experimentally, convection is driven by an imposed concentration gradient in an isothermal cell. Model equations treat the fields in two dimensions, with the reduced dimension exerting its influence through a linear wall friction. Linear stability analysis of these equations demonstrates that as
... e thickness of the cell tends to zero, the critical Rayleigh number and wave number for convective onset do not depend on the velocity conditions at the top and bottom boundaries ͑i.e., no-slip or stress-free͒. At finite cell thickness ␦, however, solutions with different boundary conditions behave differently. We simulate the model equations numerically for both types of boundary conditions. Time sequences of the full concentration fields from experiment and simulation display a large number of solutal plumes that are born in thin concentration boundary layers, merge to form vertical channels, and sometimes split at their tips via a Rayleigh-Taylor instability. Power spectra of the concentration field reveal scaling regions with slopes that depend on the Rayleigh number. We examine the scaling of nondimensional heat flux ͑the Nusselt number, Nu) and rms vertical velocity ͑the Péclet number, Pe) with the Rayleigh number (Ra*) for the simulations. Both no-slip and stress-free solutions exhibit the scaling NuRa*ϳPe 2 that we develop from simple arguments involving dynamics in the interior, away from cell boundaries. In addition, for stress-free solutions a second relation, NuϳͱnPe, is dictated by stagnation-point flows occurring at the horizontal boundaries; n is the number of plumes per unit length. No-slip solutions exhibit no such organization of the boundary flow and the results appear to agree with Priestley's prediction of NuϳRa 1/3 . Although three-dimensional "3D... turbulent convection has been the subject of a large amount of research, 1-3 the presumably simpler problem of two-dimensional "2D... turbulent convection has not been similarly graced. We have conducted laboratory experiments and numerical simulations on a quasi-two-dimensional version of a canonical problem in nonlinear dynamics: buoyancy driven convection, the best known example of which is Rayleigh-Bénard convection. We consider a very thin convection cell that effectively constrains the fluid motion to a vertical plane. However, the walls that define this plane not only confine the fluid, they also influence its motion through a frictional drag force. This drag force distinguishes quasi-2D convection from strictly 2D convection and leads to striking differences between the two. For instance, we find the drag force from the two large walls can be so important that the onset of convection can be effectively independent of the boundary conditions on the remaining four walls. The drag force stabilizes plumes, which dominate the flow pattern through their birth, coalescence, collision, and death. The existence of large plumes and the effect of drag on the plumes alters the transport properties from those of a strictly 2D flow. We construct scaling arguments for the average momentum and buoyant scalar fluxes based on an analysis of the plume motion and of the boundary layers at the top and bottom surfaces of the convection cell.