Onset of entrainment in transitional round fountains

It is of fundamental interest to understand the behavior of transitional fountains with intermediate Froude and Reynolds numbers, together with the associated entrainment and turbulence. In this work, the transient behavior of axisymmetric fountains with 1 ⩽ Fr ⩽ 8 and 200 ⩽ Re ⩽ 800 is studied by direct numerical simulation. It is found that at Re ⩽ 200, there is little entrainment present at the upflow–downflow interface and at the downflow–ambient interface, even for a value of Fr as high as 8; however, at Re > 200, entrainment is present at these interfaces and the extent increases with Re, which clearly demonstrates that entrainment is strongly dependent on Re whereas the contribution from the Fr effect is relatively much smaller. The DNS results also show that zm, which is the maximum fountain penetration height, fluctuates, even when the flow reaches full development, due to the entrainment at the upflow–downflow and the downflow–ambient interfaces, and the averaged zm scales with Fr32Re14 for 1 < Fr ⩽ 8 and 100 ⩽ Re ⩽ 800.