Free surface flows over submerged obstructions

Steady and unsteady two-dimensional free surface flows subjected to one or multiple disturbances are considered. Flow configurations involving either a single fluid or two layers of fluid of different but constant densities, are examined. Both the effects of gravity and surface tension are included. Fully nonlinear boundary integral equation techniques based on Cauchy’s integral formula are used to derive integro-differential equations to model the problem. The integro-differential equations are discretised and solved iteratively using Newton’s method. Both forced solitary waves and critical flow solutions, where the flow upstream is subcritical and downstream is supercritical, are obtained. The behaviour of the forced wave is determined by the Froude and Bond numbers and the orientation of the disturbance. When a second disturbance is placed upstream in the pure gravity critical case, trapped waves have been found between the disturbances. However, when surface tension is included, trapped waves are shown only to exist for very small values of the Bond number. Instead, it is shown that the disturbance must be placed downstream in the gravity-capillary case to see trapped waves. The stability of these critical hydraulic fall solutions is examined. It is shown that the hydraulic fall is stable, but the trapped wave solutions are only stable in the pure gravity case. Critical, flexural-gravity flows, where a thin sheet of ice rests on top of the fluid are then considered. The flows in the flexural-gravity and gravity-capillary cases are shown to be similar. These similarities are investigated, and the physical significance of both configurations, examined. When two fluids are considered, the situation is more complex. The rigid lid approximation is assumed, and four types of critical flow are investigated. Trapped wave solutions are found to exist in some cases, depending on the Froude number in the lower layer.