Velocity Profiles and Minimum Stream Power

The velocity distribution of laminar and turbulent flows in a wide open channel was analyzed using the theory of minimum rate of energy expenditure or, equivalently, the theory of minimum stream power. The solution for the laminar flow is the classical parabolic velocity distribution. For turbulent flow it is necessary to assume the functional forms of the velocity distribution and the corresponding eddy viscosity distribution. A number of constants or parameters in the assumed functions were determined using the minimization theory and appropriate constraint equations. By assuming that the turbulent flow may consist of a logarithmic inner layer and a parabolic outer layer, the minimization theory determines that the thicknesses of the two layers must be equal. Two constants remain to be determined using experimental data. It is suggested that the two empirical constants are best determined by measured maximum and mean velocities. Theoretical results are verified with data available in the literature.