On multiple states in single-layer flows

Abstract

For free surface flows over obstacles in a channel of constant width, there is a range of parameter values where two steady flow states are possible, with the state that is actually obtained being determined by the past history. Specifically, one of these flow states is wholly supercritical (i.e., no waves can propagate upstream against the flow) over the obstacle. The other has a hydraulic jump that travels to upstream infinity, and the flow undergoes a subcritical (i.e., waves can propagate in both directions) to supercritical transition at the obstacle crest. A new, third steady solution is described here, in which a hydraulic jump is stationary over the upstream face of a long obstacle. This new solution is contiguous with the other two, and in a sense, lies between them. It is shown that this new solution is unstable, in that if the stationary jump is displaced to a location with a slightly different bottom height, it will move further in the same direction. By this criterion, jumps are unstable on upslope flow, and stable on downslope flow. These properties, and the general character of hysteresis implied by these multiple hydraulic equilibria, have been tested with two series of experiments. The new solution is ordinarily not found because of its instability, but it can be viewed by manually balancing the unstable jump. Comparisons were also made between the observed abrupt transitions between flow states, and the predictions of hydraulic theory. Qualitatively the agreement is quite good, with differences attributable to experimental factors that are not contained in two-dimensional long wave hydraulics.