Constant Froude number in a circular hydraulic jump and its implication on the jump radius selection

The properties of a standard hydraulic jump depend critically on a Froude number
Fr defined as the ratio of the flow velocity to the gravity waves speed. In the case of a horizontal
circular jump, the question of the Froude number is not well documented. Our experiments
show that Fr measured just after the jump is locked on a constant value that does not depend
on the flow rate Q, the kinematic viscosity ν and the surface tension γ. Combining this result
with a lubrication description of the outer flow leads, under appropriate conditions, to a new and
simple law ruling the jump radius RJ: RJ(ln(R∞))3/8 ∼ Q5/8ν−3/8g−1/8, in excellent agreement RJ
with our experimental data. This unexpected result raises an unsolved question to all available models.