Plume dynamics in quasi-2D turbulent convection

We have studied turbulent convection in a vertical thin (Hele-Shaw) cell at very high Rayleigh numbers (up to ${\text{7×10}}^4$ times the value for convective onset) 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 the 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-free). At finite cell thickness $\delta$ , 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, $\mathrm{Nu)}$ and rms vertical velocity (the Péclet number, $\mathrm{Pe)}$ with the Rayleigh number ${\mathrm{(Ra}}^{*})$ for the simulations. Both no-slip and stress-free solutions exhibit the scaling ${\mathrm{NuRa}}^{*}{\mathrm{\sim Pe}}^2$ that we develop from simple arguments involving dynamics in the interior, away from cell boundaries. In addition, for stress-free solutions a second relation, $\mathrm{Nu\sim }\sqrt{\mathrm{nPe}}$ , 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 ${\mathrm{Nu\sim Ra}}^{\text{1/3}}$ .