2.1.2. Supercritical Flows

The study in [20] recently showed that the flow pattern around an obstacle in a supercritical flow varies as a function of the obstacle width ( *D*) to flow depth (*h*) ratio, distinguishing the following patterns:

The detached hydraulic jump pattern: at high ratios of obstacle width to flow depth, i.e., *D*/*h* > 0.5–2.0 a bow-wave like, detached, hydraulic jump takes place in front of the obstacle [21]. From up to downstream, the flow first slows down to pass from the supercritical to subcritical regime and eventually stops at the upstream face of the obstacle. This transition from the super- to sub-critical regime takes place through the detached hydraulic jump wrapping around the obstacle (Figure 2a). The foot of the hydraulic jump then follows a hyperbolic curve in the horizontal plane. Near the obstacle, the flow is in the subcritical regime, so a horseshoe vortex occurs in the near-bed region at the foot of the foundation. The foot of the horseshoe vortex then follows an elliptic curve [21]. Moreover, the separation distance between the obstacle and both the detached jump and the horseshoe vortex appears to increase with the non-dimensional flow depth (*h*/*D*) and decreases with the Froude number of the incoming flow [22].

The wall jet pattern: at small ratios of foundation width to flow depth (i.e., *D*/*h* < 0.5–2.0) a so-called wall-jet-like bow wave develops (Figure 2b; [20]). The flow remains in the supercritical regime (una ffected by the presence of the obstacle) until reaching the foot of the obstacle. There, it deviates and goes up along the upstream face of the obstacle and slightly towards its sides, where it is evacuated and falls down in the flow further downstream. Part of the up-going flow rolls backward and falls down at the foot of the obstacle in periodic reverse spillage.

**Figure 2.** Flow patterns upstream of a rectangular emerging foundation in supercritical flow: (**a**) detached hydraulic jump and (**b**) wall jet pattern (adapted from [20]). HSV, horseshoe vortex.

## *2.2. Foundations with Scour Hole*
