Temporal evolution of jet induced scour depth in cohesionless granular beds and the phenomenological theory of turbulence

In this work, we investigate the temporal evolution of the jet-driven scour depth in a pothole lying on a cohesionless granular bed by using diverse approaches. First, we present new experiments which encompass cases with jet angles ranging from 45° to 90° from the horizontal, several initial water depths, and different particle sizes, supplementing experiments developed recently by the last two authors. In particular, we address relatively large angles, mostly absent in previous analyses. Our results initially confirm the existence of two very different stages in the scour process, essentially overlooked in datasets used to obtain the traditional formulas—developing and developed phases; they then provide unprecedented evidence of the very distinct behavior at 90°, characterized by a step-wise behavior. Second, after revisiting the rationale of a theory for the equilibrium condition developed elsewhere by the first author and a collaborator, we employ the existing and new datasets to determine the multiplicative constants embedded in the equilibrium scour formulas. Third, we present a novel theory for the temporal evolution of the scour depth during the developed phase (but with good prediction capabilities in the developing phase as well). By invoking the conservation of mass of sediment in the pothole, in addition to the energy conservation within the pothole and the phenomenological theory of turbulence, we obtain ordinary differential equations which we solve by numerical means. We validate the theory using our new and other datasets. Finally, we provide interesting interpretations of the scour process by using the results of the theory.