River restoration projects often make use of grade-control structures in order to prevent channel degradation, improve their ecological resources, and regulate floods. The time evolution of localized scour that can occur downstream of grade-control structures represents a crucial component in the design of those projects. This is because erosion processes may undermine the stability of hydraulic structures and compromise their effectiveness. Although localized scour has been widely investigated (mostly via laboratory models), there is significant scatter in results produced by the large number of obtained empirical equations. Consequently, there is a need for more general tools based on theoretical approaches that might overcome the limits of ad-hoc, experimental methods. In particular, a model based on the application of the Phenomenological Theory of Turbulence (PTT) has been recently developed by the last three authors to predict scour-depth evolution under steady flows. Nevertheless, in practical applications, scour phenomena usually occur during floods characterized by variable flow discharges. In this paper, for the first time to the authors’ knowledge, a detailed analysis of the basic assumptions of the PTT-evolution model allowed us to assess their validity for scour processes at different grade-control structures under unsteady (time-dependent) flows. By re-analyzing the dynamics of the scour evolution under unsteady flow conditions, we corroborate its consistency with the steady counterpart and with jet-driven scour problems, evincing that a homothetical evolution of the scour hole occurs during the developed phase. Finally, we provide a confirmation of the general validity of the theoretical approach presented herein. We show that it can be successfully applied to a large range of structure configurations and hydraulic conditions, without any significant modification. This last result is particularly important, since it paves the way to a unique, first-principles-based tool that can be helpful to design hydraulic structures.

河流修复工程通常利用坡度控制结构来防止河道退化、改善生态资源和调节洪水。可能发生在品位控制结构下游的局部冲刷的时间演变是这些项目设计中的一个关键组成部分。这是因为侵蚀过程可能会破坏水工结构的稳定性并影响其有效性。虽然局部冲刷已被广泛研究（主要通过实验室模型），但大量获得的经验方程产生的结果存在显着分散。因此，需要基于理论方法的更通用的工具，这些工具可能会克服临时实验方法的局限性。尤其，源源不断。然而，在实际应用中，冲刷现象通常发生在变流量特征的洪水期间。在这篇论文中，据作者所知，对 PTT 演化模型的基本假设的详细分析使我们能够评估它们在不稳定（时间相关）下不同等级控制结构下冲刷过程的有效性流动。通过重新分析非定常流条件下冲刷演变的动力学，我们证实了它与稳定对应物和射流驱动冲刷问题的一致性，证明冲刷孔的同位演化发生在发展阶段。最后，我们证实了本文提出的理论方法的普遍有效性。我们证明它可以成功应用于大范围的结构配置和水力条件，而无需任何重大修改。最后一个结果特别重要，因为它为一种独特的、基于第一性原理的工具铺平了道路，这种工具可以帮助设计水工结构。