Momentum transfer across an open-channel, turbulent flow

The distribution of stress generated by a turbulent flow matters for many natural phenomena, of which rivers are a prime example. Here, we use dimensional analysis to derive a linear, second-order ordinary differential equation for the distribution of stress across a straight, open channel, with an arbitrary cross-sectional shape. We show that this equation is a generic first-order correction to the shallow-water theory in a channel of large aspect ratio. It has two adjustable parameters – the dimensionless diffusion parameter, $\chi$ , and a local-shape parameter, $\alpha$ . By assuming that the momentum is carried across the stream primarily by eddies and recirculation cells with a size comparable to the flow depth, we estimate $\chi$ to be of the order of the inverse square root of the friction coefficient, $\chi \sim C_f^{-1/2}$ , and predict that $\alpha$ vanishes when the flow is highly turbulent. We examine the properties of this equation in detail and confirm its applicability by comparing it with flume experiments and field measurements from the literature. This theory can be a basis for finding the equilibrium shape of turbulent rivers that carry sediment.