Momentum and heat transport scalings in laminar vertical convection

We derive the dependence of the Reynolds number Re and the Nusselt number Nu on the Rayleigh number Ra and the Prandtl number Pr in laminar vertical convection ( VC), where a fluid is confined between two differently heated isothermal vertical walls. The boundary layer equations in laminar VC yield two limiting scaling regimes: Nu similar to Pr-1/4 Ra-1/4, Re similar to Pr-1/2 Ra-1/2 for Pr << 1 and Nu similar to Pr-0 Ra-1/4, Re similar to Pr-1 Ra-1/2 for Pr >> 1. These theoretical results are in excellent agreement with direct numerical simulations for Ra from 105 to 1010 and Pr from 10(-2) to 30. The transition between the regimes takes place for Pr around 10(-1).