The Reynolds and Prandtl number dependence of weak fountains

The effect of the Prandtl number (Pr) and the Reynolds number (Re) on the behaviour of weak laminar axisymmetric and plane fountains has been studied using dimensional and scaling analyses and direct numerical simulation. For Fr similar to 1.0 and assuming viscous effects are important, the analysis shows that for both the axisymmetric and plane fountains, y(m) similar to FrRe-1/2, where Fr is the Froude number defined at the fountain source and y(m) is the non-dimensionalized fountain height. These scalings are also valid for the non-dimensionalized fountain width. The analyses also shows tau(ms) similar to Fr-2, where tau(ms) is the non-dimensionalized time scale for the fountain flow in the fountain core to reach steady state, and using this time scale Deltay(tau) similar to Fr(RePr)(-1/2), where Deltay(tau) is the non-dimensionalized thickness of the temperature layer on the symmetry axis over which the fountain fluid temperature changes from the inlet value to that of the ambient fluid. All these scalings have been quantified by the direct numerical simulations, hence confirming in certain ranges the phenomenological scaling obtained in this paper.