A Three Dimensional Dam Break Flow: Small Time Behavior

Abstract

Small time behavior of gravity driven free surface flows resulting from the collapse of a cavity is studied. Initially there is a rigid vertical cylinder of circular cross section starting from the free surface of a liquid and ending at the rigid bottom. The cylinder disappears suddenly and gravity driven flow of the fluid starts. The flow in early stage is described by the potential theory. Attention is paid to the singular behavior of the velocity field at the intersection line between the bottom and the free surface of the cavity. The leading order linear problem is solved by the Fourier series method. The flow velocity is log-singular at the intersection line. In the limiting case where the radius and the center of the cavity approach infinity, the problem is reduced to the classical two dimensional dam break problem where the fluid is initially on one side of a vertical wall (dry bed case). The flow resulting from cavity collapse is a three dimensional dam break flow. It is concluded that the three dimensional effects are important when the radius of the cavity is small compared with its depth and that the local flow near the intersection line of the cavity is governed only by the hydrostatic pressure.