Depth-energy and depth-force relationships in open channel flows: analytical findings

In the present work the depth-specific energy relationship and the depth-total force relationship in open channel flows of wide rectangular cross-section are analytically inverted. The nondimensional expressions of the specific energy and of the total force, as functions of the nondimensional water depth, are considered. The inversion of such functions consists of finding the roots of third degree algebraic equations; simple analytical solutions are obtained. More specifically, for a given specific discharge and for each meaningful value of the specific energy, a subcritical and a supercritical depth are found analytically. Similarly, for a given specific discharge and for each meaningful value of the total force, a subcritical and a supercritical depth are found analytically. For both functions, it is also shown that the third root corresponds to a negative depth, which can be discarded on the basis of physics. Examples from classical open channel hydraulics and a numerical application show the consistency of these analytical solutions.