Free-Surface Flows over an Obstacle: Problem Revisited

Abstract

Two-dimensional steady free-surface flows over an obstacle are considered. The fluid is assumed to be inviscid and incompressible; and the flow is irrotational. Both gravity and surface tension are included in the dynamic boundary condition. Far upstream, the flow is assumed to be uniform. Triangular obstruction is located at the channel bottom as positive bump or negative bump (dip). This problem has been investigated by many researchers, such as Forbes [5], Shen [8], and Dias and Vanden-Broeck [2], to seek for new types of solutions. In this paper, the fully nonlinear problem is formulated by using a boundary integral equation technique. The resulting integrodifferential equations are solved iteratively by using Newton’s method. When surface tension is neglected, a new solution type of subcritical flow is proposed, the so-called drag-free solution. Furthermore, solutions of flows over a dip in the bottom are also presented. When surface tension is included, there is an additional parameter in the problem known as the Bond number B. In addition, the weakly nonlinear problem is investigated and compared with the fully nonlinear results. Finally, solution diagrams for all flow regimes are presented on the (F,hob)-plane for which F is the Froude number and hob is the dimensionless height of the obstacle.