4.4.2. Multi-Phase Flow Approach

Over the last decades, a new generation of sediment transport models has emerged: the Eulerian–Eulerian two-phase flow approach [94]. Unlike classical sediment transport models, the two-phase flow approach is based on the resolution of momentum balance for both the fluid and the sediment phases, the latter being seen as a continuum with a peculiar rheology. Very recently, using an open-source multi-dimensional two-phase flow model for sediment transport applications [95], the study in [93] presented the first two-phase flow RANS simulation for scour around a vertical cylinder. In Figure 8b, the scour mark induced by the HSV is clearly visible, while the downstream erosion induced by the lee–wake vortices was also observed. The two-phase flow approach is able to reproduce the scour dynamics induced by the HSV without empirical parametrization for the sediment fluxes and the avalanching process.

Further, the authors in [96] reported the first two-phase flow turbulence-resolving simulations. This opens new possibilities for simulating the complex interactions between sediment transport and HSV dynamics. Another important research possibility has been recently developed by [97] and concerns the development of a two-phase flow sediment transport model, including a free surface resolving capability. This model will allow one to reproduce the free surface features observed in the supercritical flow regime by [20,21] while solving sediment dynamics based on mechanical principles, as in [93].

Advances on scour modeling also consider computational techniques that efficiently calculate the dynamic coupling between the flow and the bed. RANS–Exner models coded in GPU (Graphics Processing Unit) have already been developed to improve the computation of these problems [98]. Future models that incorporate LES and Exner in the vicinity of bridge piers using these strategies show grea<sup>t</sup> promise in tackling large computational domains or fine resolutions in scour problems.

#### *4.5. Moving from the Local Scale Phenomena Up to Long Term Dynamics*

As detailed above, the scour process has been investigated mainly with laboratory and numerical approaches under several simplifying assumptions, neglecting the stochastic nature of floods. Indeed, the scour process should be considered a stochastic process controlled by the dynamics of floods, sediment transport, and riverbed evolution over time. All these processes act over a bridge foundation throughout the bridge's lifespan, and it is likely that the entire history of these events is responsible of the failure risk of a bridge rather than a specific flood event. An example of the potential time evolution of flood characteristics (here: discharge and Froude number) and scour depth over time is given in Figure 9.

The limitation of the current knowledge of this phenomenon is highlighted in [99], which documented the magnitude of flood events causing bridge collapses in the US. Most of these bridges were designed for a flood event with a return period of about 100 years, but they collapsed under a variety of events with a return period ranging from 1 to 1000 years. This is clear evidence of the limitations of the current methodologies and understanding of bridge design. Lately, a number of authors have explored alternative strategies for bridge failure predictions in order to account for the non-stationarity of the flow and the stochastic nature of floods. These studies highlighted that scour evolution over time is strongly controlled by factors like the shape of flood hydrographs [83], and also the time-sequence of floods occurring at a given location [100]. Moreover, hydrological variability has been amplified even more significantly by the e ffects of global climate change [101].

Now, the interactions between infrastructure and river morphodynamics are characterized by many factors that span a wide range of temporal and spatial scales. Besides the hydrographs, additional factors at larger scales include variations in sediment availability, vegetation, land use in the watershed, channel managemen<sup>t</sup> and the e ffects of anthropic interventions (e.g., gravel mining), or other infrastructures built in the channel, such as reservoirs or intakes. The sum of these phenomena should be considered when designing a bridge foundation, but actual modeling schemes, as well as available information, are not able to fully describe such a complex dynamic. Therefore, the challenge in the coming years is to identify unifying principles able to interpret scour phenomena, both under subcritical and supercritical conditions, with the right compromise between model complexity and data availability [102–104]. Moreover, it is highly desired that forthcoming modeling approaches explicitly account for the stochastic nature of the flood process.

**Figure 9.** The synthetic time-series over a temporal window of approximately 4 years: (**a**) supercritical flow discharge, (**b**) the corresponding Froude number Fr, and (**c**) the evolution of the scour depth at the foot of a bridge pile, assuming it as an additive process. The graph displays the random nature of the flood events within a river system and its consequences in terms of the random evolution of scour depth over time, computed using the BRISENT model, as in [105].
