Penetration height of transitional round fountains in a linearly stratified fluid

The transition to asymmetry of transitional fountains is further complicated by the stratification of the ambient fluid. In this study, a series of three-dimensional direct numerical simulations were carried out for transitional round fountains in linearly-stratified fluids over the ranges of 100 ≤ Re ≤ 400, 1 ≤ Fr ≤ 8 and 0.0 ≤ s ≤ 0.3, where Fr, Re and s are the Froude, Reynolds, and dimensionless temperature stratification parameters, respectively, to examine, both qualitatively and quantitatively, the effect of these parameters on their axisymmetric transition and their maximum penetration heights. The results show that when Fr or Re are small enough, a fountain remains axisymmetric for all time; however, when Fr or Re are increased sufficiently, the fountain will be axisymmetric initially, before becoming asymmetric and unsteady, ultimately reaching a fully developed quasi-steady stage when the maximum fountain penetration height fluctuates over a constant, time-average, value. The stratification is found to play a positive role to stabilize the flow and to reduce or even to eliminate the asymmetric behavior. The numerical results were also used to develop a series of scaling relations, in conjunction with the scalings obtained with dimensional analysis, for the initial and time-averaged fountain penetration heights, and the time for the fountain to attain the initial penetration height. It is further found that in general these characteristic parameters are strongly dependent on Fr and s, while only weakly dependent on Re.