3.2.2. Dimensional Analysis

The scour process is mainly influenced by the flow and fluid characteristics, the geometry of the piers, and the bed material properties [15]. However, the lateral flow through the river boundaries also plays significant role in modifying the channel geometry. So, it is necessary to include the downward seepage parameters for the prediction of scour hole characteristics. In this study, the seepage Reynolds number (R*es* = *Vsd50*/υ) is used as a dimensionless seepage parameter. Hence, in the analysis of the scour hole characteristics, such as the length (*ls*) and width (*ws*) of the pier, the following independent parameters can be considered to affect the scour process:

$$d\_{\mathbb{S}\_{\prime}} \, w\_{\mathbb{S}\_{\prime}} = \mathbf{f} \, (d\_{\mathbb{S} \mathbb{O}\_{\prime}} \, \sigma\_{\mathbb{S}^{\prime}} \, \mathop{\mathbb{S}}\_{\prime} \, \mathbf{v}, h, \, \mathbf{U}, \, \mathop{\mathbb{S}}\_{\prime} \, \mathop{\mathbf{V}s}\_{\prime} \, \mathop{\mathbf{R}}) \tag{3}$$

where, *d*50 (mm) is the median sand diameter, <sup>σ</sup>*g <sup>d</sup>*84/*d*16!is the standard deviation, *g* (m/s2) is the gravitation acceleration, υ (m<sup>2</sup>/s) is the kinematic viscosity, *h* (m) is the flow depth, *U* (m/s) is the depth average velocity, *Vs* (m/s) is the seepage velocity, and *R* is the hydraulic radius (m). Applying and suitably modifying Equation (1), it can be rearranged as Equation (2).

$$L\_{\rm s\nu} \ W\_{\rm s} = f\left\{\frac{\mathcal{U}}{\sqrt{g\mathcal{R}}} \; ; \; \frac{h}{d\_{\rm 50}} \; ; \; \frac{V\_{\rm s}d\_{\rm 50}}{\mathfrak{g}} \; ; \sigma\_{\mathcal{S}}\right\} \tag{4}$$

where *Ls* (*ls*/*D*) is the dimensionless scour length and *Ws* (*ws*/*D*) is the dimensionless scour hole width. The dimensionless characteristics of the scour hole are a function of the following dimensionless parameters: the Froude number is *Fd* (*U*/ *gR*); *h*/*d50* is aspect ratio, *Vsd50*/υ is the Seepage Reynolds number (Res), and <sup>σ</sup>*g* is the standard deviation. Starting from Equation (2), the length and width of the scour hole can be expressed as follows:

$$L\_s = a(F\_d)^b \left( h/d\_{50} \right)^c \exp(\text{Re}\_s)^d \left( \sigma\_{\%} \right)^{\varepsilon} \tag{5}$$

$$\mathcal{W}\_{\mathfrak{s}} = a(F\_d)^b \left( h/d\_{50} \right)^c \exp(\mathcal{R} \mathfrak{e}\_{\mathfrak{s}})^d \left( \sigma\_{\mathfrak{s}} \right)^{\mathfrak{s}} \tag{6}$$

where *a*, *b*, *c*, *d* and *e*, are the empirical parameters evaluated through experimental measurements with both sands.

The seepage Reynolds number is stated in an exponential, form in order to fulfil the condition without seepage. The final forms of Equations (3) and (4) are the following ones:

$$L\_{\circ} = 0.06 \left( F\_d \right)^{2.69} \left( h/d\_{\circ 0} \right)^{1.54} \exp(Re\_{\circ})^{0.703} \left( \sigma\_{\circ} \right)^{-0.644} \tag{7}$$

$$\mathcal{W}\_s = 0.02 \left( \mathcal{F}\_d \right)^{4.23} \left( h / d\_{50} \right)^{2.05} \exp(\mathcal{R} e\_s)^{0.707} \left( \sigma\_\% \right)^{-0.705} \tag{8}$$

The area and the volume of the scour hole are the function of the scour depth. Hence, Equation (2) can be modified as follows:

$$A\_{s\prime}, V\_s = f\left\{\frac{\mathcal{U}}{\sqrt{gR}}, \frac{h}{d\_{50}}, \frac{V\_s d\_{50}}{8}, \frac{d\_s}{D}\right\} \tag{9}$$

where, *As* = *as*/*ap*, and *as* is the scour hole area and *ap* is the pier area, and *Vs* = *vs*/*vp*, where *vs* is the scour hole volume and *vp* is the pier volume.

From the dimensional analysis, the following expressions for the dimensionless area and volume of the scour hole result in the following:

$$A\_s = 0.49 (F\_d)^{4.007} \left( h/d\_{50} \right)^{1.18} \exp(\text{Re}\_s)^{0.324} \left( \frac{d\_s}{D} \right)^{1.69} \tag{10}$$

$$V\_s = 0.81 \left( F\_d \right)^{3.39} \left( h / d\_{50} \right)^{1.05} \exp(\text{Re}\_s)^{3.15} \left( \frac{d\_s}{D} \right)^{2.17} \tag{11}$$

Scour hole characteristics, such as the length, width, area, and volume, have been calculated from Equations (5), (6), (8) and (9), and the results have been compared with the measured values, as shown in Figure 8. The overall agreemen<sup>t</sup> between Equations (5), (6), (8) and (9), and the measured values is good, with R<sup>2</sup> ranging between 0.88 and 0.92.

**Figure 8.** Comparison between the predicted and observed values of the dimensionless (**a**) scour length (Equation (5)), (**b**) scour width (Equation (6)), (**c**) scour area (Equation (8)), and (**d**) scour volume (Equation (9)).
