Analytical Solutions of Energy Equation for Rectangular Channels: Direct Approach

An analytical solution of nondimensional hydraulic energy equation is derived in cosine form for the flow in rectangular open channels considering nonhydrostatic pressure and nonuniform velocity distributions across flow depth and friction and turbulent losses. The new inverted energy equation is a single equation describing all the three roots of the nondimensional hydraulic energy equation. The energy losses due to turbulence and bed shear are also implicitly accounted. Only two roots out of the three roots are practically significant, which denote the two alternate depths and are distinguishable for subcritical and supercritical flow, respectively. These equations yield direct determination of alternate depths in a single step, avoiding an iterative procedure. The use of the new equations is illustrated through worked-out examples. The new single-term analytical inverted equations are computationally simple and useful for academicians, field engineers, and practitioners in directly solving in a single step the problems of energy-equation inversion often encountered when dealing with transitions in a channel section, and flows over a dam-spillway and under a sluice gate, with correction-factors and loss coefficients estimated or known.