Velocity and thermal boundary layer equations for turbulent Rayleigh-Bénard convection

In turbulent Rayleigh-Bénard convection, the boundary layers are nonsteady with fluctuations, the time-averaged large-scale circulating velocity vanishes far away from the top and bottom plates, and the motion arises from buoyancy. In this paper, we derive the full set of boundary layer equations for both the temperature and velocity fields from the Boussinesq equations for a quasi-two-dimensional flow above a heated plate, taking into account all the above effects. By solving these boundary layer equations, both the time-averaged temperature and velocity boundary layer profiles are obtained.