Non-linear Topographic Effects in Two-Layer Flows

We consider the nature of non-linear flow of a two-layer fluid with a rigid lid over a long obstacle, such that the flow may be assumed to be hydrostatic. Such flows can generate hydraulic jumps upstream, and the model uses a new model of internal hydraulic jumps, which results in corrections to flows that have been computed using earlier models of jumps that are now known to be incorrect. The model covers the whole range of ratios of the densities of the two fluids, and is not restricted to the Boussinesq limit. The results are presented in terms of flow types in various regions of a Froude number-obstacle height (F-0-H-m) diagram, in which the Froude number F-0 is based on the initial flow conditions. When compared with single-layer flow, and some previous results with two layers, some surprising and novel patterns emerge on these diagrams. Specifically, in parts of the diagram where the flow may be supercritical (F-0 > 1), there are regions where hysteresis may occur, implying that the flow may have two and sometimes three multiple flow states for the same conditions (i.e., values of F-0 and Hm).