On the origin and structure of a stationary circular hydraulic jump

To elucidate the role played by surface tension on the formation and on the structure of a circular hydraulic jump, the results from three different approaches are compared: the shallow-water (SW) equations without considering surface tension effects, the depth-averaged model (DAM) of the SW equations for a flow with a parabolic velocity profile, and the numerical solutions of the full Navier-Stokes (NS) equations, both considering the effect of surface tension and neglecting it. From the SW equations, the jump can be interpreted as a transition region between two solutions of the DAM, with the jump’s location virtually coinciding with a singularity of the DAM’s solution, associated with the inner edge of a recirculation region near the bottom. The jump’s radius and the flow structure upstream of the jump obtained from the NS simulations practically coincide with the results from the SW equations for any flow rate, liquid properties, and downstream boundary conditions, being practically independent of surface tension. However, the structure of the flow downstream of the jump predicted by the SW equations is quite different from the stationary flow resulting from the NS simulations, which strongly depends on surface tension and on the downstream boundary conditions (radius of the disk). One of the main findings of the present work is that no stationary and axisymmetric circular hydraulic jump is found from the NS simulations above a critical value of the surface tension, which depends on the flow conditions, fluid properties, and downstream conditions.To elucidate the role played by surface tension on the formation and on the structure of a circular hydraulic jump, the results from three different approaches are compared: the shallow-water (SW) equations without considering surface tension effects, the depth-averaged model (DAM) of the SW equations for a flow with a parabolic velocity profile, and the numerical solutions of the full Navier-Stokes (NS) equations, both considering the effect of surface tension and neglecting it. From the SW equations, the jump can be interpreted as a transition region between two solutions of the DAM, with the jump’s location virtually coinciding with a singularity of the DAM’s solution, associated with the inner edge of a recirculation region near the bottom. The jump’s radius and the flow structure upstream of the jump obtained from the NS simulations practically coincide with the results from the SW equations for any flow rate, liquid properties, and downstream boundary conditions, being practically independent of sur...