Intermittency in Boundary Layers

Abstract

Having in mind the recent experiments by Heslot, Libchaber and Castaing¹ on convection at very large Rayleigh number, I tried to understand why the temperature fluctuations become more and more intermittent at increasing distances from the horizontal plates. The explanation I have found is as follows. The thermohydrodynamics equations express the global conservation of quantities as the total momentum and energy. In the turbulent regime some amount of those conserved quantities is injected into the fluid through the boundaries. But this flux of conserved quantities is expected to be fluctuating in space and time on the boundary because of turbulence (the same remark was made by Lallemand and Zaleski⁴ for the Kuramoto-Sivashinsky equation). As indicated by the experiments, those wall fluctuations have no specific feature, in particular they are not intermittent. This may be explained by the fact that near the wall the fluctuation dynamics are given by a local Rayleigh number of order 1, as in the boundary layer theory of Howard². But as one goes away from the boundary, two phenomena act in opposite direction: the molecular heat conductivity tends to decrease the amplitude of those fluctuations whereas the nonlinearities increase the local Rayligh number measured with the distance to the wall as a length scale. A simple scaling analysis show that the nonlinearities increase more rapidly than the molecular damping. This makes inconsistent any estimate of the decay of fluctuations by molecular diffusion only since the corresponding term in the equations becomes formally smaller and smaller far from the wall.