Resonant water-waves in ducts with different geometries: Forced KdV solutions

In a remarkable paper, Cox and Mortell (1986) (A.A. Cox, M.P. Mortell 1986. J. Fluid Mech. 162, pp. 99-116) 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) (W. Chester and J.A. Bones 1968. Proc. Roy. Soc. A, 306, 23 (Part II)). We compare the fKdV solutions with a number of channel flows with different geometry that have been studied experimentally and numerically. When sweeping the selected wide parameter range, extreme cases of the fKdV equation are covered: single soliton solutions as well as multiple solitons with a rather short wavelength challenging the long-wave fKdV assumption. The transition of solutions with a different number of solitons is rather abrupt and we show that the parameter values for transitions from single soliton towards multi-soliton solutions can be predicted and follow a simple exponential relation. In particular, we compare the fKdV model with solutions from a fully nonlinear Navier–Stokes model. We further consider a case for which the 2D assumption of the fKdV equation is strictly speaking violated.