The onset of nonpenetrative convection in a suddenly cooled layer of fluid

TL;DR

This paper investigates the conditions triggering nonpenetrative convection in a cooled fluid layer, revealing how critical parameters depend on Prandtl number and boundary conditions, and extending previous research with new theoretical insights.

Contribution

It extends existing theories by analyzing the onset of nonpenetrative convection across a wide Prandtl number range and different boundary conditions using propagation theory.

Findings

Critical conditions match unsteady Rayleigh-Bénard instability at high Rayleigh numbers.

Rigid-rigid and free-rigid critical curves tend to converge at low Prandtl numbers.

Good agreement with previous results for high Prandtl numbers; suggests a 10-fold ratio for onset times in stress-free systems.

Abstract

Conditions for the onset of nonpenetrative convection in a horizontal Boussinesq fluid layer subject to a step change in temperature are studied using propagation theory. A wide range of Prandtl numbers and two different kinematic boundary conditions are considered. It is shown that for high Rayleigh numbers, critical conditions for the onset of convective motion reproduce exactly those for the unsteady Rayleigh-B\'enard instability. Present results extend those of previous research and show a tendency of the rigid-rigid and free-rigid critical curves to converge for low Prandtl numbers. Comparison between present and previously reported results on critical conditions for the onset of instabilities and onset time using different methods yields good agreement on a middle to high Prandtl number range. A ratio of 10 between experimentally measured and theoretically predicted onset times is suggested for stress-free bounded systems.