Lipps, F. B., 1971: Two-dimensional numerical experiments in thermal convection with vertical shear. Journal of Atmospheric Sciences, 28 (1), 3-19.

Abstract: The dynamics of thermal convection through a shallow layer with vertical shear is examined using an idealized numerical model. The convection is assumed to take the form of two-dimensional rolls. The mean shear flow and the unstable temperature gradient are maintained by no-slip, conducting boundary conditions applied at the upper and lower boundaries of the model. When the convective rolls are transverse to the mean current, the flow approaches a steady state with time for the cases of primary interest. In agreement with previous numerical studies the shear has a stabilizing influence on the convection: the transformation of potential energy into disturbance kinetic energy is reduced, and disturbance kinetic energy is transformed into basic kinetic energy. A new result, in agreement with linear stability theory, is that shear can significantly increase the horizontal distance between disturbances over that expected with no shear.
Steady-state results were also obtained when the rolls are parallel to the mean current. In this case basic kinetic energy is transformed into disturbance kinetic energy. Results for momentum transfer and heat transfer obtained from the present numerical model are compared with the experimental results of Ingersoll. This comparison suggests that for low values of the Rayleigh number his convection is primarily in the form of rolls parallel to the shear flow. However, for Rayleigh numbers >20,000 the experimental and numerical results start to diverge, suggesting that three-dimensional effects are becoming important in this range.