Generalised critical free-surface flows

Abstract

Nonlinear waves in a forced channel flow are considered. The forcing is due to a bottom obstruction. The study is restricted to steady flows. A weakly nonlinear analysis shows that for a given obstruction, there are two important values of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, F _C>1 and F _L<1 such that: (i) when F<F _L, there is a unique downstream cnoidal wave matched with the upstream (subcritical) uniform flow; (ii) when F=F _L, the period of the cnoidal wave extends to infinity and the solution becomes a hydraulic fall (conjugate flow solution) – the flow is subcritical upstream and supercritical downstream; (iii) when F>F _C, there are two symmetric solitary waves sustained over the site of forcing, and at F=F _C the two solitary waves merge into one; (iv) when F>F _C, there is also a one-parameter family of solutions matching the upstream (supercritical) uniform flow with a cnoidal wave downstream; (v) for a particular value of F>F _C, the downstream wave can be eliminated and the solution becomes a reversed hydraulic fall (it is the same as solution (ii), except that the flow is reversed!). Flows of type (iv), including the hydraulic fall (v) as a special case, are computed here using the full Euler equations. The problem is solved numerically by a boundary-integral-equation method due to Forbes and Schwartz. It is confirmed that there is a three-parameter family of solutions with a train of waves downstream. The three parameters can be chosen as the Froude number, the obstruction size and the wavelength of the downstream waves. This three-parameter family differs from the classical two-parameter family of subcritical flows (i) but includes as a particular case the hydraulic falls (ii) or equivalently (v) computed by Forbes.