Resonant water-waves in a circular channel: forced KdV solutions

Tidal bores are natural phenomena observed in at least 450 river estuaries all around the world from Europe to America and Asia. Tidal bores manifest as a series of waves propagating over long distances upstream in the estuarine zone of a river. Bores can be studied experimentally using sloshing water tanks where sloshing itself is a process with many applications, not only relevant for environmental flows. In a remarkable paper, Cox and Mortell (1986) showed that for an oscillating water tank, the evolution of small-amplitude, long-wavelength, resonantly forced waves follow a forced Korteweg-de Vries (fKdV) equation. The solutions of this model agree well with experimental results by Chester and Bones (1968). At first glance this is surprising since their experimental setup is in conflict with a number of assumptions made for deriving the fKdV equation. It is hence worth to repeat the experiment by Chester and Bones but using a long narrow channel setup.

We use a long circular channel and repeat the experiments by Chester and Bones. We compare the results with solutions from the fKdV equation but also with the one from a full nonlinear model solving the Navier-Stokes equations. Under resonance conditions, depending on the parameters, we find a range of nonlinear localized wave types from single and multiple solitons to undular bores. As shown by Cox and Mortell, when the fluid is considered to be inviscid a kind of Fermi-Pasta-Ulam recurrence is observed for the fKdV model. Stationarity is reached by including a weak damping to the fKdV equation. 

References
A.A. Cox, M.P. Mortell 1986. J. Fluid Mech. 162, pp. 99-116.
W. Chester and J.A. Bones 1968. Proc. Roy. Soc. A, 306, 23 (Part II).