Minimum Stream Power: Theory

Variational formulations of the equations of motion applicable to irrotational, nonaccelerating, as well as gradually varied open channel flows are presented. It is shown that there exists a suitable functional, J, from which the equation of motion can be obtained as the condition of minimization with respect to the velocity profile. Minimization of J is shown to be equivalent to the minimization of the total rate of energy loss, E, providing that the discharge, Q, and the total head, H, at upstream and downstream ends of the river is known. Furthermore, if the speed of the movement of the river bed is small, then the total rate of energy loss is shown to be equal to the total stream power. Thus, under certain restrictive conditions the hydraulic problem reduces to the problem of minimizing total stream power. The minimization theory presented here is somewhat similar to the second writer's theory of minimum unit stream power.