Dynamics of line plumes on horizontal surfaces in trubulent convection

We study the dynamics of line plumes on the bottom horizontal plate in turbulent convection over six decades of Rayleigh number. 10(5) < Ra-w < 10(11) ) and two decades of Prandtl number or Schmidt number (0 : 7 < Pr < 5 : 3, Sc = 602). From the visualisations of these plumes in a horizontal plane close to the plate, we identify the main dynamics as (i) motion along the plumes, (ii) lateral merging of the plumes and (iii) initiation of the plumes; various other minor types of motion also occur along with these main dynamics. In quantifying the three main motions, we first find that the spatiotemporal mean velocity along the length of the plumes ((V) over bar (sh))) scales as the large-scale flow velocity (V-LS), with the fraction of the length of the plumes affected by shear increasing with Raw as L-ps/L-p similar to Ra-w(0.054) Pr-0.12. The mean time of initiation of the plumes (t*) over bar, scales as the diffusive time scale near the plate, Z(w)(2)/alpha where Z(w) is the appropriate length scale near the plate, in agreement with Howard (Proc. 11th Int. Congress Applied Mechanics, Munich, 1964, pp. 1109-1115). Merging occurs in a large fraction of the area of the plate, with similar to 70% of the length of the plumes undergoing merging at Ra-w approximate to 10(11) and Sc = 602. The fraction of the length of the plumes that undergoes merging decreases with increase in Raw as, L-pm /L-p similar to Ra-w(-0.054) Pr-0.12; the exponents of Raw and Pr being of the same magnitude but of opposite sign as that in the relation for L-ps/L-p. Measurements of the locational means of the velocities of merging of the plumes. (V-m) show that Vm is a constant during each merging cycle at any location. However, the values of these constant velocities depend on the location and the time of measurement, since the merging velocities are affected by the local shear, which is a function of space and time at any Raw Pr combination. The merging velocities at all Raw and Pr have a common lognormal distribution, but their mean and variance increased with increasing Raw and decreasing Pr. Using mass and momentum balance of the region between two merging plumes, we show that the spatiotemporal mean merging velocities (Vm), which are an order lower than (V) over bar (sh), scale as the entrainment velocity at the sides of the plumes, averaged over the height of the diffusive layer near the plate. This implies that Vm scales as the diffusive velocity scale near the plate v/Z(w). The Reynolds number in terms of Vm and the layer height H(Re-H) scales as Ra-w(1/3) w, in the same way as the Nusselt number (Nu) scales approximately; therefore Re-H similar to Nu. These relations imply that Re-w = (V) over bar (m)Z(w)/v a Reynolds number near the plate, is an invariant for a given fluid in turbulent convection.