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

Viscous fluid exiting a long horizontal circular pipe develops a complex structure comprising a primary jet above and a smaller secondary jet below with a thin fluid curtain connecting them. We present here a combined experimental, theoretical, and numerical study of this “Torricelli's curtain” phenomenon, focusing on the factors that control its morphology. The dimensional parameters that define the problem are the pipe radius a; the mean exit velocity U of the fluid; the gravitational acceleration g; and the fluid's density ρ, kinematic viscosity ν, and coefficient of surface tension γ. Rescaling of experimentally measured trajectories of the primary and secondary jets using a for the vertical coordinate and $LD=U(a/g)1/2$ for the horizontal coordinate x collapses the data onto universal curves for $x<10LD$⁠. We propose a theoretical model for the curtain in which particle trajectories result from the composition of two motions: a horizontal component corresponding to the evolving axial velocity profile of an axisymmetric viscous jet and a vertical component due to free fall under gravity. The model predicts well the trajectory of the primary jet, but somewhat less well that of the secondary jet. We suggest that the remaining discrepancy may be explained by surface tension-driven (Taylor–Culick) retraction of the secondary jet. Finally, direct numerical simulation reveals recirculating “Dean” vortices in vertical sections of the primary jet, placing Torricelli's curtain firmly within the context of flow in curved pipes.