The transient behavior of the natural convection boundary-layer flow adjacent to a vertical plate heated with a uniform flux in a quiescent homogeneous ambient fluid with Prandtl number $\mathrm{Pr}<1$ is investigated by scaling analysis and direct numerical simulation (DNS). The flow is characterized by a startup stage, a short transitional stage and a steady state. The flow is parametrized by the thermal and velocity boundary-layer thickness scales, the vertical velocity scale, the time scale for the boundary layer to reach the steady state and the plate temperature scale. Scaling analysis is used to obtain laws relating these quantities to the flow governing parameters, the Rayleigh number Ra, the Prandtl number, and the Boussinesq number $\mathrm{Bo}=\mathrm{RaPr}$ which is a much more important control parameter than Ra for small Pr fluids. A series of DNS with selected values of Ra and Pr in the ranges of ${10}^6⩽\mathrm{Ra}⩽{10}^{10}$ and $0.01⩽\mathrm{Pr}⩽0.5$ are used to validate the scaling laws and obtain scaling constants.

• Received 23 May 2005

DOI:https://doi.org/10.1103/PhysRevE.72.066309