Localized turbulent flows on scouring granular beds

In many applications a sustained, localized turbulent flow scours a cohesionless granular bed to form a pothole. Here we use similarity methods to derive a theoretical formula for the equilibrium depth of the pothole. Whereas the empirical formulas customarily used in applications contain numerous free exponents, the theoretical formula contains a single one, which we show can be determined via the phenomenological theory of turbulence. Our derivation affords insight into how a state of dynamic equilibrium is attained between a granular bed and a localized turbulent flow.