Analytical inversion of specific energy–depth relationship in channels with parabolic cross-sections
2011
Vatankhah, Ali R. | Valiani, A.
The specific energy–depth relationship in open channels with parabolic cross-sections is analytically inverted. Two nondimensional expressions of the specific energy are considered, depending on the prescribed quantity (specific energy or alternate depth). The inversion of such functions consists of finding the roots of cubic and quartic equations. By solving a quartic equation for a given discharge and for each value of the specific energy, a subcritical depth and a supercritical depth are found analytically. In this case, two acceptable roots are recognized and two other roots are discarded, on the basis of their physical meaning. Moreover, by solving a cubic equation for a given discharge and for each value of the depth, the other corresponding alternate depth is found analytically. Then, one acceptable root is recognized and two other complex conjugate roots are discarded. Finally, different examples are presented, to show the efficiency of the proposed solutions. Such analytical solutions can be easily used in natural rivers and parabolic channels.
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