Transient Natural Convection Flow on an Isothermal Vertical Wall at High Prandtl Numbers: Second-Order Approximation

The method of matched asymptotic expansions, which has been used in previous studies of steady natural convection flow, is extended here to transient natural convection flow at high Prandtl number (Pr). Second-order expansion solutions, valid for large Prandtl numbers, are presented for the transient natural convection flow near a vertical surface which undergoes a step change in temperature. Throughout the transient, the flow is found to have the same dual-layer structure which is characteristic of the steady flow at high Prandtl number. For large Prandtl number, the time to steady state is shown to increase proportional to square root of Pr. The temperature and velocity overshoot, which occurs during the transient at moderate Prandtl number, is shown to disappear as Pr . Uniformly valid expansions for the velocity and temperature profiles near the surface are found to be in good agreement with the numerical solution of the full governing equations for as low as Pr=16. By increase of Prandtl number, the error because of instability in numerical solution of the full governing equations increases and the necessity of using singular perturbation techniques become more obvious.