One method for obtaining solutions for single- and multi-layer flow over bottom humps and through sidewall contractions is to consider the variation in mechanical energy with layer thickness at key cross-sections. However, previous models of gradually-varying multi-layer flow have been restricted to flows which remain subcritical with respect to all but the slowest internal wave speed. This paper shows how the energy extrema method can be used over the full range of flow conditions; from subcritical with respect to all internal wave speeds to supercritical with respect to the free surface wave.
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