The <i>universal two-thirds law</i> of pipeline scour

The estimation of scour depth under pipelines, aligned transversely to river flow, has attracted much attention because of its importance in hydraulic engineering. Pipeline scour results from the interaction between the wake vortices induced downstream of the pipeline and the sediment bed beneath it. The core insight of this study is to seek a universal law that governs sediment-bed scour beneath pipelines in steady flow, particularly under the clear-water condition. The analysis begins with an approach that incorporates the similarity concept, with a specific focus on the incomplete similarity in the asymptotic function of the ratio of particle size to scour depth ratio as a power law. The power law exponent can be determined exclusively through Kolmogorov's energy cascade theory. Stemming from Kolmogorov's energy cascade theory (also called the phenomenological theory of turbulence), a universal law emerges, revealing that the equilibrium scour depth to pipe diameter ratio follows a two-thirds scaling law with a newly introduced “pipeline-scour number,” which encapsulates all key factors influencing pipeline scour. This number includes variables such as the approach mean flow velocity and depth, the threshold shear velocity for sediment motion, and the sediment particle size. It also captures essential hydrodynamic forces, including inertia, drag, shear, and threshold shear forces. Moreover, the scaling law incorporates an additional term that involves the drag coefficient with an exponent of 2/3, thereby accounting for the influence of the pipe shape on the equilibrium scour depth. The derived universal law is validated using the experimental data.