*3.3. Relationship between Scour Dimensions*

The temporal development of the scour volumes for all the tests was extracted from the scour contour profiles. Although the secondary flows in the cross sections of the flume influence the shape of the scour hole in the transverse direction, the scour hole that forms downstream of submerged weirs can still be approximately regarded as two-dimensional. The mean longitudinal scour area, *As*, of the

scour hole downstream of the weir can be approximately considered to be a triangle. Therefore, the scour area *As* can be expressed as:

$$A\_s = \frac{V\_s}{b} = B \cdot d\_s \cdot l\_s \tag{6}$$

where *b* = weir width, *B* = coefficient to be determined.

Figure 6 plots the relationship between the scour area, *As*, and the product of *d*s and *ls*. It can be seen that the data trends are linear for all the tests, which may be used to infer that the assumed form of Equation (6) is valid and that the scour hole profiles exhibit geometric similarity.

**Figure 6.** Relationship between averaged longitudinal scour area and scour hole dimensions.

For all the tests, the flow regimes over the submerged weir were surface flow regime. Under this flow regime, the flow remains as a surface jet downstream of the weir; and the scour hole downstream of the submerged weir is caused by the increasing jet thickness and turbulence mixing with the tailwater [17,19]. As indicated in [11], the geometric similarity of the scour hole profiles can be affected by the jet thickness and drop ratio. Therefore, the coefficient *B* could be expressed as a function of the overtopping ratio:

$$B = f\left(\frac{z}{h}\right) \tag{7}$$

Substituting Equation (7) into Equation (6) yields:

$$A\_s = c\_1 \cdot \left(\frac{z}{h}\right)^{\varepsilon\_2} \cdot d\_s \cdot l\_s \tag{8}$$

where *e*1 and *e*2 are coefficients to be determined. A regression analysis is carried out by using the data in Figure 6, which yields *e*1 = 0.308 and *e*2 = −0.156 with a coefficient of determination of *R*<sup>2</sup> = 0.991. The final form of Equation (8) is as follows:

$$A\_s = 0.308 \cdot \left(\frac{z}{h}\right)^{-0.156} \cdot d\_s \cdot l\_s \tag{9}$$

A comparison of the measured and calculated averaged longitudinal scour area is shown in Figure 7, which implies that the fitted Equation (9) can provide a good estimate of the scour area downstream of a 2D submerged weir. The area of the computed final scour hole may be used as the area for rock placement as a type of armoring countermeasure against scour.

**Figure 7.** Comparison of measured and calculated averaged longitudinal scour area.
