Analytical inversion of specific energy–depth relationship in channels\ud with parabolic cross-sections

摘要

The specific energy–depth relationship in open channels with parabolic cross-sections is analytically\udinverted. Two nondimensional expressions of the specific energy are considered, depending on the prescribed\udquantity (specific energy or alternate depth). The inversion of such functions consists of finding the roots of cubic\udand quartic equations. By solving a quartic equation for a given discharge and for each value of the specific energy,\uda subcritical depth and a supercritical depth are found analytically. In this case, two acceptable roots are recognized\udand two other roots are discarded, on the basis of their physical meaning. Moreover, by solving a cubic equation\udfor a given discharge and for each value of the depth, the other corresponding alternate depth is found analytically.\udThen, one acceptable root is recognized and two other complex conjugate roots are discarded. Finally, different\udexamples are presented, to show the efficiency of the proposed solutions. Such analytical solutions can be easily\udused in natural rivers and parabolic channels.