A mixing layer theory for flow resistance in shallow streams

Abstract

A variety of surface roughness characterizations have emerged from nineteenth and twentieth century studies of channel hydraulics. When the water depth h is much larger than the characteristic roughness height ks, roughness formulations such as Manning's n and the friction factor f can be explicitly related to the momentum roughness height zo in the log‐law formulation for turbulent boundary layers, thereby unifying roughness definitions for a given surface. However, when h is comparable to (or even smaller than) ks, the log‐law need not be valid. Using a newly proposed mixing layer analogy for the inflectional velocity profile within and just above the roughness layer, a model for the flow resistance in shallow flows is developed. The key model parameter is the characteristic length scale describing the depth of the Kelvin‐Helmholtz wave instability. It is shown that the new theory, originally developed for canopy turbulence, recovers much of the earlier roughness results for flume experiments and shallow gravel streams. This study is the first to provide such a unifying framework between canopy atmospheric turbulence and shallow gravel stream roughness characterization. The broader implication of this study is to support the merger of a wealth of surface roughness characterizations independently developed in nineteenth and twentieth century hydraulics and atmospheric sciences and to establish a connection between roughness formulations across traditionally distinct boundary layer types.