**3. Results and Discussion**

#### *3.1. Temporal Evolution of Scour Depth and Scour Length*

The 3D bed geometry at three different stages of the scour hole development for Test CS2 is shown in Figure 2.

**Figure 2.** Temporal evolution of bed topography contours (Test CS2) in cm (flow from left to right).

The maximum scour depth downstream of the weir, which is located at longitudinal distance = 50 cm in the figure, occurs close to the weir at the beginning of each test before moving downstream as the scour hole develops. Although the tests were conducted in the straight flume with a 2D weir, the scour hole that forms downstream of the weir exhibits 3D characteristics. The maximum scour depth in the developing scour hole downstream of the weir was found to be close to the flume sidewalls rather than along the centerline of the flume. This is due to the secondary flows in the transverse sections of the flume. According to previous studies [3], the secondary flows are characterized by paired circular flow cells, which are quasi-symmetrically located on both sides of the centerline sand ridge.

The temporal evolution of scour depth and scour length for all the tests is shown in Figure 3. The results show that the maximum scour depth and scour length develop very quickly in the first 20 h, before progressing at a decreasing rate. This trend is consistent with local scour at other structures, such as monopile foundations, submarine pipelines, etc. [29–32].

**Figure 3.** Temporal evolution of scour depth (**a**) and scour length (**b**).

Due to the insu fficient scour time, none of the tests, arguably, has reached the final equilibrium stage (see Figure 3a). The equilibrium clear-water scour depth, which is reached asymptotically with time, may take a very long time to form, perhaps infinite time [24]. Therefore, the final equilibrium scour time and depth will not be discussed in this paper. As shown in Figure 3a, the development of scour depth, *ds*, is influenced by the flow intensity, *U*0/*Uc*, and overtopping ratio, *<sup>z</sup>*/*h*. The increase of *U*0/*Uc* or *z*/*h* accelerates the scour rate and increases the magnitude of scour depth. Figure 3b shows the temporal development of scour length *ls*. The trends are similar to those in Figure 3a. The development rate and magnitude of *ls* are a ffected by *U*0/*Uc* and *<sup>z</sup>*/*h*. However, the influence of the overtopping ratio *z*/*h* on *ls* appears to be less significant (see data trends of Tests CS4 and CS5). It should be noted that the values of *U*0/*Uc* for Tests CS3, CS4, and CS5 are not exactly the same in this study, thus more work is needed to determine clearer e ffects of overtopping ratios on the scour dimensions.

Over the past decades, extensive studies [24,30,31,33–35] have shown that exponential functions could be used to describe the temporal evolution of clear-water scour process around a variety of hydraulic structures, such as bridge piers, bed sills, abutments, etc. The typical form of the exponential function is as follows:

$$\frac{Y\_s}{Y\_{sc}} = 1 - \exp\left[-\mathcal{C}\left(\frac{t}{T}\right)^n\right] \tag{1}$$

where *Ys* = scour dimension (depth or length) at any time *t*; *Yse* = scour dimension (depth or length) at the equilibrium stage; *T* = time scale (represents the equilibrium scour time in most cases); and *C*, *n* = coe fficient and exponent to be determined experimentally. Considering the universality of Equation (1), it is reasonable to adopt the same form for describing the time evolution of clear-water scour at submerged weirs. However, unlike previous studies, the scour processes in this study may not have not reached equilibrium, i.e., the parameters *Yse* and *T* are not available to fit Equation (1). As mentioned in the previous paragraph, the temporal evolution of scour dimensions was found to be significantly influenced by the magnitude of weir height *z*, thus the length scale *z* may be an appropriate parameter to substitute for the parameter *Yse* in Equation (1). Therefore, in this study, Equation (1) is modified as follows:

$$\frac{\mathcal{Y}\_s}{\mathcal{Y}\_{sz}} = \mathcal{C}\_1 \Big\{ 1 - \exp \left[ -\mathcal{C}\_2 \left( \frac{t}{T\_z} \right)^n \right] \right\} \tag{2}$$

where *Tz* = characteristic time at *ds* = *z*; *Ysz* = scour dimension (depth or length) at *Tz* (estimated by interpolating the observed data in Figure 3a); *C*1, *C*2, and *n* = coe fficients and exponent to be determined experimentally. Based on the experimental data in Figure 3 and the form of Equation (2), the temporal evolution of scour depth and scour length can be derived as follows:

$$\frac{d\_s}{d\_{sz}} = 4.22 \left\{ 1 - \exp \left[ -0.27 \left( \frac{t}{T\_z} \right)^{0.40} \right] \right\} \tag{3a}$$

$$\frac{l\_s}{l\_{sz}} = 2.93 \left\{ 1 - \exp \left[ -0.50 \left( \frac{t}{T\_z} \right)^{0.38} \right] \right\} \tag{3b}$$

where *dsz* and *lsz* = scour depth and scour length at *Tz*, respectively. Figure 4 shows *ds*/*dsz* and *ls*/*lsz* versus *t*/*Tz* with an excellent fit. The regression analysis yielded coe fficients of determination of *R*<sup>2</sup> = 0.994 and 0.976 for Equations (3a) and (3b), respectively. This indicates that Equation (3) has a high adaptability for the prediction of the scour dimensions downstream of submerged weirs in clear-water scour conditions.

**Figure 4.** Temporal evolution of dimensionless scour depth (**a**) and scour length (**b**).

It may also be inferred from Equation (3) that the final equilibrium scour depth *dse* = 4.22*dsz* = 4.22*z* and scour length *lse* = 2.93*lsz* when the scour time *t* approaches infinity. However, this simply is an inference at this stage; its verification by more data with su fficiently longer experimental time is needed in future studies.

#### *3.2. Temporal Evolution of Scour Hole Profiles*

It is observed that when the scour process downstream of the submerged weir moves into the slow development stage (*t* > 20 h in this study, see Figure 3), the scour patterns exhibits a high similarity in spite of the di fferent experimental conditions. This is similar to the scour process downstream of bed sills [11,22]. To further explore this similarity, an empirical model is proposed in this section to predict the temporal evolution of the scour hole profile along the centerline of the flume. Based on this model, hydraulic engineers may design the scour countermeasures downstream of submerged weirs more economically. In this study, *ys* and *xs* are, respectively, defined as the vertical and longitudinal coordinates for describing the scour hole measured with respect to the origin *O*, as shown in Figure 1c. Based on the experimental data analysis, the temporal evolution of the dimensionless scour hole profile can be described as:

$$\frac{y\_s(t)}{y\_{s,m}(t)} - \frac{z}{h} = m\_1 \exp\left(m\_2 \frac{\mathbf{x}\_s(t)}{\mathbf{x}\_{s,m}(t)}\right) \left[1 - \exp\left(m\_3 \frac{\mathbf{x}\_s(t)}{\mathbf{x}\_{s,m}(t)}\right)\right] \tag{4}$$

where *ys* (*t*) and *xs* (*t*) = vertical and longitudinal coordinates for describing the scour hole measured with respect to the origin *O* at any time *t* (> 20 h); *ys,m* (*t*) and *xs,m* (*t*) = maximum vertical and longitudinal coordinates of the scour profiles at any time *t* (>20 h), respectively, i.e., *ys,m* (*t*) = −*ds* (*t*) and *xs,m* (*t*) = *ls* (*t*); *m*1, *m*2, *m*3 = coe fficients. By fitting the experimental data to Equation (4), the following prediction equation for describing the temporal evolution of the scour hole profile is obtained:

$$\frac{y\_s(t)}{y\_{s,m}(t)} - \frac{z}{h} = 181.98 \exp\left(-3.96 \frac{\mathbf{x}\_s(t)}{\mathbf{x}\_{s,m}(t)}\right) \left| 1 - \exp\left(0.06 \frac{\mathbf{x}\_s(t)}{\mathbf{x}\_{s,m}(t)}\right) \right| \tag{5}$$

The coe fficients of determination, *R*<sup>2</sup> = 0.903. Comparisons of the results obtained from the proposed Equation (5) with the experimental data in this study are shown in Figure 5. It can be seen from the figure that the proposed equation may be used to predict the temporal evolution of scour hole profiles well.

**Figure 5.** Temporal evolution of dimensionless scour profiles for *t* > 20 h.
