*3.2. Supercritical Flows*

The study in [50] measured the maximum scour depths at piers in live-bed experiments for several flow configurations, including seven flows in the supercritical regime (Figure 4). For these flow conditions, no major increase in the maximum scour depth was reported as the Froude number exceeds 1.

**Figure 4.** Measured maximum scour depth for flows in the supercritical regime: available configurations in the literature, with *Fr* as the Froude number.

The authors in [27] reported on scour at an abutment (*h*/*D* = 0.095) in a supercritical flow with clear-water conditions. The flow pattern was of the detached hydraulic jump type. The scour hole extended far downstream from the abutment sides (Figure 5a). Two symmetric deposition zones were observed downstream, separated by a streamwise valley of almost zero deposition elevation, with a magnitude of maximum deposition elevation about half the maximum erosion measured in the scour hole. The maximum scour depth at equilibrium was observed along the nose and the upstream part of the lateral faces of the abutment (see Figure 5b). Interestingly, the temporal evolution of the maximum scour depth agreed well with the scour formula in [51] for subcritical flows, when considering the conjugate depth and velocity downstream the straight hydraulic jump (see Figure 5c) as input flow parameters. More experimental evidence for the extrapolation of the results in [27] is needed, as the researchers in [27] investigated only a single flow configuration, with a low Reynolds number (Figure 4).

The study in [6] investigated scour at a pier or abutment in a supercritical flow using a 1:50 physical model of the projected bridge on the Rivière des Galets, located in the CNR (Compagnie Nationale du Rhône) laboratory (Figure 5d) to assess the shape and depth of the scour hole for different foundation diameters and approaching flow conditions. The ratio between the flow depth and pile diameter ranged between 0.4 and 1.8 (Figure 4), resulting in a wall-jet or detached jump flow patterns. Incoming velocities ranged from 1 to 2 m/s under a steady flow regime. The model sediment was scaled geometrically and consisted of a non-uniform mixture of sand and gravel, with a mean diameter of 1.8 mm. Around 20 tons of sand were supplied during running experiments to keep the sediment bed in equilibrium. Unfortunately, experimental conditions (turbidity of water, bed load displacement, high flow velocity) did not allow the researchers to measure the hydrodynamics in the scour hole. The measurements of the scour depth during the test were carried out using three rows of metal rods in a comb arrangemen<sup>t</sup> (see Figure 5f), while the equilibrium scour (see Figure 5g) was measured using an automatic tacheometer after drying the bed (Figure 5h). Figure 5i shows the maximum scour depths for a range of pier diameters and approaching flow velocities, with a flow intensity *I* up to 4. The maximum scour depth after the experiments appears to vary between 0.9 and 2.5 times the pier diameter, demonstrating that the scour is not as high as expected from the extrapolation of the results from subcritical flows, even though the flow velocities are much higher.

**Figure 5.** Clear-water scour at bridge foundations in experiments with a supercritical flow. Photograph of the clear water scour from [27] (**a**), with the corresponding normalized topography (*zb* being the bed elevation) at equilibrium (**b**) and the measured time evolution of normalized maximum scour depth *zmax* (symbols) along with the prediction (plain line) obtained with the scour formula for subcritical flows [51] (**c**); live bed scour experiments from [6] (**d**), the flow pattern seen from downstream the pile (**e**), the metal rod comb used to measure the scour hole during the experiment (**f**), the scour shape at quasi-equilibrium after 2 hours at a pile of 6 cm in diameter in a 1.4 m/s flow condition (**g**, **h**), and the maximum normalized scour depth measured at the foot of the piles as a function of the flow intensity, *I* (**i**).

#### **4. Recommended Methods for the Study of Flow and Scour in Supercritical Flows**

Future research should be devoted to better understanding the flow processes responsible for scour at bridge foundations in watercourses with a supercritical regime along with the quantification of the maximum scour depths. Recommendations on methods to develop research on scour in supercritical flows are given below for laboratory and field measurements, as well as physical and numerical modeling.

#### *4.1. Laboratory Techniques for Flow and Scour Dynamics*
