Torricelli's curtain: morphology of horizontal laminar jets under gravity

It has been ``known'' since the seventeenth century that a jet of water issuing horizontally from a hole in the side of a bucket describes a parabolic trajectory. However, this bit of canonical fluid mechanical lore is wrong in many cases. Our recent experiments performed on laminar jets issuing from a horizontal tube show that the initially round jet typically evolves into a thin vertical curtain bounded by bulbous rims at its upper and lower extremities. Moreover, injected dye reveals the presence of a recirculating flow with helical streamlines around the jet's axis. To understand this behavior, we formulate an analytical model for the near-orifice structure of the jet in the limit of large Froude number $Fr\equiv \epsilon^{-1}\gg 1$. We find that a recirculating flow is generated by the sinusoidal variation of the nonhydrostatic pressure around cross-sections of the jet at order $\epsilon$, and that deformation of the cross-section occurs at order $\epsilon^2$. We also use the volume-of-fluid code Gerris to study numerically the evolution of the jet's morphology as a function of the Reynolds, Froude and Ohnesorge numbers, and compare the results with our analytical theory and with laboratory experiments.