Two-Dimensional Free-Surface Flows due to a Point Singularity

Two dimensional free-surface flows produced by a submerged source in a channel of finite or infinite depth are considered. It is assumed that there is a stagnation point on the free surface just above the source. Both supercritical and subcritical flows are computed using collocation or boundary integral equation techniques. It is shown that the family of subcritical solutions depends continuously on two parameters: the Froude number F and the dimensionless elevation b of the source above the bottom wall, and the family of supercritical solutions depends only on the Froude number F. In addition, a family of solutions with two stagnant regions, outside two shear layers, and a uniform velocity far below the source is considered. A limiting configuration for F = infty is also calculated. In the last part of the work, a boundary integral equation technique is used to study the disturbances caused by vortex in an open channel. Supercritical and subcritical families of solutions are calculated. Both families are shown to depend, continuously, on three parameters: the Froude number F, the strength of the vortex epsilon, and the dimensionless elevation b of the vortex above the bottom wall.