Author(s): F. Bombardelli; M. Palermo; S. Pagliara
Linked Author(s): Michele Palermo, Simone Pagliara, Fabian Bombardelli
Keywords: Jet scour; Phenomenological theory of turbulence; Universal scour formula; Turbulent kinetic energy; Shields criterion
Abstract: Scour induced by jets constitutes a classic problem of fluid mechanics, for which no unique solution has been achieved for all combinations of sediment sizes and flow intensities. The safety of hydraulic structures might be compromised by the level of scour nearby. For almost one hundred years now, a number of widely-used, experimental formulas have been proposed around the world for the equilibrium scour depth (Schoklitsch, 1932; Mason and Arumugam, 1985; Bormann and Julien, 1991; and Hoffmans and Verheij, 1997). Most formulas have been developed on dimensional analysis and heuristic arguments; further, they possess experimentally-determined free exponents. Also, laboratory tests have been devoted to investigate the time evolution of the scour depth (Laursen, 1952; Blaisdell et al., 1981), and diverse expressions have been obtained. One key feature of those very useful formulas is that they are not easy to extrapolate to field situations where the flow and sediment conditions significantly differ from those at the laboratory. As an alternative to the purely empirical formulas, some semi-theoretical expressions have been put forward, such as those of Bormann and Julien (1991), and Stein et al. (1993), with the ability to represent not only the equilibrium state but also the transient scour phenomenon. Recently, a radically-different approach was pioneered by using the Phenomenological Theory of Turbulence (PTT) applied to the eddies of the scour process (Bombardelli and Gioia, 2005, 2006; Gioia and Bombardelli, 2005; Bombardelli et al., 2018; Palermo et al, 2021; Di Nardi et al., 2022 and 2023). In this presentation, we introduce the theory and walk through its different levels of prediction, which can be obtained to represent equilibrium as well as the time-dependent scour depths. The PTT applies to fully-developed flow turbulence, and addresses the steady production of turbulent kinetic energy (TKE). It follows two principles: (i) the TKE per unit mass is determined by scales associated with the largest (energy-containing) flow scales or eddies, being independent of viscosity; (ii) such turbulent energy, which is introduced at a rate İ, is transferred (it “cascades”) from large to small scales at that rate İ, until its dissipation into internal energy at sufficiently-small scales. In the case of jets, the large scales are dictated initially by the thickness of the water layer where the jet impacts. Such layer enlarges as the scour proceeds, and the energy-containing eddies increase their size, concomitantly. By assuming that eddies harbored in the coves in between the roughness elements are within the inertial sub-range, we can use the Taylor-Kolmogorov scaling to relate the velocity scales of the two sizes of eddies. Then, by adopting a physically-based expression for bed shear stress, an equation is derived as a function of the velocity scale of the largest eddies. Finally, by using Shields criterion for incipient sediment motion, a formula for the scour depth can be derived. This methodology has been checked to offer accurate predictions of the scour depth for laboratory tests, at the time of bringing more physical insight into the interpretation of the problem. The theory was also validated at large scales in Palermo et al. (2021), and it was successfully applied to conditions of variable jet discharges (see, for instance, Di Nardi et al., 2022 and 2023). In addition to discussing previous developments and validations, we present herein a description of current efforts to generalize the theory to all sizes and flow conditions, hoping to uncover a “universal” theory for scour.
Year: 2023