On a nearly constant Froude number observed in circular hydraulic jumps

Circular hydraulic jumps are reminiscent of a shock for surface waves, but the flow is viscous, and analogous to boundary layer detachment. This yields [1] a scaling R$_{\mathrm{J}} \propto $ Q$^{5/8}\nu ^{3/8}$g$^{-1/8}$ that links the jump radius R$_{\mathrm{J}}$ to flow rate Q, viscosity $\nu $ and gravity g. In a recent experiment [2], with a jet of radius $\varphi $ impacting a horizontal disk of radius R, we observed that the Froude number Fr at the jump exit was constant, which yields a modified scaling R$_{\mathrm{J}}$(Log(R/R$_{\mathrm{J}})^{3/8} \approx $ (2$^{-11/8}$3$^{-3/8}\pi^{-5/8}$/Fr) Q$^{5/8}\nu^{-3/8}$g$^{-1/8}$ in good agreement with experiment. We show that this behavior is universal but Fr depends on phi/R. We also investigate the behavior of Fr (and more generally of the structure of the hydraulic jump) in the case of confinement walls. Theoretically, these results cannot be recovered by connecting two domains of negligible interface slope with a localized shock. Instead, a generalized inertial lubrication theory [3] seems able to explain these behaviors, that we relate to finite slope effects at the free surface. [1] T. Bohr et al., JFM 254, 635 (1993). [2] A. Duchesne et al., EPL 107, 54002 (2014). [3] N Rojas et al., PRL 104, 187801 (2010).