LONG-WAVELENGTH THERMAL-CONVECTION IN A WEAK SHEAR-FLOW

The problem of thermal convection in an imposed shear flow is examined for a horizontal layer of fluid between poorly conducting boundaries. The horizontal scale H of the convective motion near its onset is muc h greater than the depth h of the fluid layer, with h/H being proporti onal to the one-fourth power of a Blot number appearing in the conditi on applied to the temperature at the horizontal boundaries. It is know n that an asymptotic expansion in powers of h/H yields a nonlinear lon g-wavelength evolution equation for the depth-averaged temperature fie ld that is spatially isotropic in the absence of an imposed shear flow , but is strongly anisotropic for 'strong' shear. We derive in this pa per a nonlocal long-wavelength equation that bridges these two cases, and that contains each case in the zero-shear and large-shear limits. Using this evolution equation, we show how the shear flow stabilizes t he longitudinal rolls to the zigzag instability, and how a preference for a square planform on a periodic square lattice gives way to a pref erence for longitudinal rolls near onset. The longitudinal rolls may t hen become unstable as the Rayleigh number is increased. The analytica l work is illustrated by some numerical simulations of the full three- dimensional Boussinesq Navier-Stokes equations. The problem of pattern selection on a hexagonal lattice is also discussed, and some new resu lts are presented.