Variational problem of water-level stabilization in open channels

A method is presented far minimization of water-level fluctuations in an open channel controlled by pumping stations at bath ends. The situation is posed as a variational problem to determine the optimum upstream flow control, given constraints at the opposite end. The method of indefinite Lagrangian multipliers is used. A global measure-integrated in space and time-of the water-level deviation away from some desired water level is defined. The problem involves minimizing this deviation while still satisfying the governing equations and the initial boundary conditions for the flow. The solution is obtained numerically. The equations governing gradually varied, unsteady open-channel flow are assumed to be valid. A horizontal, frictionless channel of constant trapezoidal cross section is assumed, although the method could readily be extended to more general conditions. The problem has many applications related to hydropower and water supply canals. An example is presented to illustrate the utility and effectiveness of the method.