Depth–energy and depth–force relationships in open channel flows: analytical findings A. Valiani a,∗ , V. Caleffi a a Universit` a degli Studi di Ferrara, Dipartimento di Ingegneria, Via G. Saragat, 1 44100 Ferrara, Italy Abstract 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 non dimensional expressions of the specific energy and of the total force, as functions of the non dimensional 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 analitycal solutions. Key words: Open channel flows, Total force, Specific energy, Analytical results. 1 Introduction A short look to any handbook of hydraulic engineering or to classical textbooks on free surface flows [1–3] is sufficient to guess the importance of the concepts of specific energy and total force in the crucial aspects of open-channel flows. For example, the complex and rich physics of fluid motion around obstacles, at ∗ Corresponding author. Email addresses: alessandro.valiani@unife.it (A. Valiani), valerio.caleffi@unife.it (V. Caleffi). Preprint submitted to Advances in Water Resources 26 September 2007