Two-dimensional jets falling from funnels and nozzles

Abstract

Two-dimensional flow in a domain bounded on one side by an infinite rigid wall and on the other by a semi-infinite rigid wall and a free surface is considered. Gravity is included in the free surface condition and surface tension is neglected. The flows are characterized by the parameters H=(Q2/gw3)1/3 and the angles α and β between the walls and the horizontal. Here, Q is the total flux, g the acceleration of the gravity and w the distance between the separation point (the point of intersection of the free surface with the semi-infinite wall) and the infinite wall. The configurations include, as particular cases, flows from nozzles and funnels, the Kirchhoff jet and the flow under a gate. Numerical solutions are computed by series truncation. It is shown that there are three distinct configurations at the separation point corresponding to contact angles 2π/3, π−α (horizontal free surface at the separation point), and π (free surface tangent to the rigid wall at the separation point). For given values of β and α≳π/3, there is a critical value Hc of the parameter H such that H=Hc for a contact angle of 2π/3, H≳Hc for a contact angle of π and H<Hc for a contact angle of π−α. On the other hand, for given values of β and α≤π/3, there is only one configuration with a contact angle of π. Free surface profiles are presented for various values of α, β, and H.