The flip-through of a plane inviscid jet with a free surface

Abstract

The growth of a two-dimensional liquid jet is modelled, in which an inviscid incompressible fluid is in irrotational flow. On the moving free surface, the pressure is constant. The flow is symmetric with respect to the y-axis. The free-surface fluid particle that lies on the y-axis is Q and it has position y = Y(t). A velocity potential is presented that describes the local features of the flow near the centreline, and which contains essentially two unknowns: the velocity V(t) = d Y/d t of Q, and a length L(t). Near Q the free surface is a time-dependent parabola, whose curvature is directly proportional to a third unknown, F(t). The kinematic and dynamic free-surface boundary conditions constrain V, L and F to satisfy three ordinary differential equations. The solutions are a one-parameter family that separate into five types. For one type the free-surface curvature changes sign during the motion, so that the free surface changes from concave to convex—it executes flip-through. Soon after flip-through, there is a local maximum in the pressure, with respect to time and space. A short time later there is a maximum in the acceleration of Q. As t tends to infinity, V tends to a constant, V _∞. An estimate is made of the time scale for flip-through, based on pressure changes, and it is found to decrease as V _∞ increases. When V _∞ is chosen very large, the values of the large maxima in pressure and acceleration become sensitive to small changes in the initial conditions. The results focus on the highly transient pressure field. The findings may help to describe the flip-through of a steep fronted water wave meeting a plane impermeable wall.