3.2.2. Maximum Scour Depth

After scour equilibrium, the maximum scour depths were 2.7*D*, 2*D*, and 2.2*D* for *R* = 37.4 mm, *R* = 45.9 mm, and *R* = 56.3 mm, respectively. The maximum scour depth decrease first, and then increases with the increase in rotor radius. The results can be explained that the change of energy extract efficiency of turbine is nonlinear against tip speed ratio. In the experiment, the rotational speed and inlet velocity is remained constant. Hence, the tip speed ratio increases linearly with rotor radius. Accordingly, the equilibrium scour depth shows a complex nonlinear relationship with rotor radius.

## 3.2.3. Position of Maximum Scour Depth

According to the model tests, the position of the maximum scour occurs at both back sides of the monopile regardless of the rotor radius. This result agrees with previous researches of scour at piers [28] and scour induced by turbine [18]. The specific location is x/*D* = −0.5, y/*<sup>D</sup>* = 0.5 in each case. These results are same as the cases with different tip clearance. The maximum scour depth position is still decided by horseshoe vortex system. In each case, the equilibrium deposition of sediment cannot be observed within 8D downstream. However the moving process of sand dunes behind the turbine's foundation can be seen at different scour time.

#### *3.3. Overall Temporal Evolution of Scour Depth*

The temporal evolution of scour depths at different rotor radius and tip clearance are plotted in Figure 9. In each case, the scour hole develops rapidly in the beginning process and then grows gradually until 150 min. At first, at approximately 10 min, the scour depth increases rapidly. The general appearance of scour around piles: turbine scour starts with an increase of flow speed around turbine supporting pile. The sediment is easily washed away by large bed shear stress on both sides of pile. The scour hole expands by the energy of downflow and horseshoe vortex. However, the less bed shear stress and weaker energy of horseshoe vortex exists inside the hole along with the development of scour hole until the equilibrium situation. The maximum final scour depth shows two scour types against different tip clearance when the turbine radius remains unchanged. When *C*/*H* > 0.5, the equilibrium scour hole depth *S*/*D* is less than 1.7. An equilibrium scour depth is reached after ~150 min of run time. However, when *C* ≤ 0.5, the equilibrium scour depth is ~2.2*D*, and the scour equilibrium is reached after ~110 min. The reason is illustrated below.

**Figure 9.** Evolution of maximum scour depth in time with different rotor radius and tip clearance.

For all model tests, even though each test has the same inlet velocity, the bed velocity is different due to different setup of turbine model. The scour depth around pile increases with the increase of flow velocity near sediment bed in clear water scour, but the depth shows little increase when the inlet velocity is faster than critical velocity [11]. In cases 1–4 with same rotor radius but different tip

clearance, flow moves faster at lower tip clearance generally. This is due to greater flow acceleration below the turbine at lower tip clearance, thus the velocity of downflow and near-bed flow is amplified. Hence the evolution of the scour hole can be sped up. In cases 3, 5, and 6 with the same tip clearance but di fferent rotor radius, there is also a big di fference in scour depth. When dimensionless tip clearance is maintained as *C*/*H* = 0.5, the equilibrium scour hole depth reaches a maximum value of 2.6 *D* when *R*/*D* = 3.74, and minimum value of 2.0 *D* when *R*/*D* = 4.59. As discussed in Section 3.2, this result reflects the impact of the turbine's hydrodynamic performance on the seabed scour.

To further illustrate the shape of scour hole impacted by turbine's set up, the temporal development of scour width is shown in Figure 10. Final scour hole width is located between 3.5 *D* to 4.8 *D* with di fferent rotor radius and tip clearance. It can be seen that final hole width shows same trends as scour depth, the max hole width occurs in case *C*/*H* = 0.50, *R*/*D* = 3.74. In each case, the final scour hole width is ~2 times that of the scour depth. This can be explained by the development mechanism of the scour hole. The horizontal expansion of the scour hole is a result of a sand slide. With the sustained evolution of scour depth, the slope of hole exceeds sediment repose angle, hence the sand at edge of hole slide into the scour hole. Hence the equilibrium scour hole slope is fixed, which is approximately the sediment repose angle. Thus the horizontal and vertical dimensions of the scour hole show a 2:1 ratio.

**Figure 10.** Evolution of scour hole width in time with di fferent rotor radius and tip clearance.

#### **4. Equations to Predict Temporal Evolution of Scour Depth**

Tip-bed clearance and turbine radius are used to propose novel empirical equations used to predict Darrieus turbine-induced scour. Temporal scour equations are compared to the established works from bridge pier scour equations, ship propeller scour equations, and relevant tidal turbine scour data.

#### *4.1. Derivation of the Temporal Scour Equations*

Previous researchers proposed the importance of the scour and predict the temporal variation of scour depth [31–36]. The temporal scour depth *Sd* was predicted by a selected percentage of equilibrium scour depth *Se* with consideration of the e ffects of current velocity, pier diameter, and water height. The selected scour equations to compare the proposed temporal scour depth model are listed below. Several numerical equations were found to predict the scour depth induced by propeller jet [37,38]. The flow velocity, propeller diameter, tip clearance, and sediment diameter are the main factors that influence the evolution of the scour hole. During the scour process induced by turbine, the temporal scour hole is deeper and wider compared to the pier-induced scour due to the flow contraction caused

by rotating turbine, the increase ratio is ~10–75% with various turbine rotor radius and tip clearance in the model tests. However the scour depth is smaller than the propeller jet scour. Therefore, the pier scour equations underpredict the turbine-induced scour, whereas the propeller scour equations overpredict the temporal scour depth induced by the turbine. Engineers do not have the appropriate empirical equations to predict the temporal development of scour hole around turbine foundation. Accordingly, this paper proposed more applicable empirical equations to predict temporal evolution of scour depth induced by tidal current turbine.

For empirical equations to predict temporal scour depth at piers, Sumer et al. [31] proposed the Equation (3), where *Se* is maximum scour depth at equilibrium stage, *te* is the equilibrium time of scour:

$$\frac{S\_d}{S\_\ell} = 1 - e^{(-\frac{L}{t\_\ell})} \tag{3}$$

Melville and Chiew [32] proposed the Equation (4), where *Ucr* is critical velocity of sediment, *U* is approach velocity of flow:

$$\frac{S\_d}{S\_{\ell}} = \varepsilon^{\left\{-0.03 \left| \frac{\hbar L}{\hbar \ell\_c} \ln \left( \frac{\hbar}{\ell\_c} \right) \right|^{1.6} \right\}} \tag{4}$$

Oliveto and Hager [33] proposed the Equation (5), where *T* is dimensionless time of scour:

$$\frac{S\_d}{L\_R} = 0.068 \sigma^{-1/2} F\_d^{1.5} \log T \tag{5}$$

In Equation (5),

$$L\_R = D^{2/3} h^{1/3}, \; F\_d = \frac{\mathcal{U}\_c}{\left(g'd\_{50}\right)^{1/2}}, \; \sigma = \left(d\_{84}/d\_{16}\right)^{0.5}, \; T = \left(\frac{\left(g'd\_{50}\right)^{0.5}}{L\_R}\right)t, \; g' = \left[\frac{\rho\_s - \rho}{\rho}\right]g.$$

Lança et al. [34] proposed the Equation (6), where *D* is the diameter of pile:

$$\frac{S\_d}{S\_\ell} = 1 - \varepsilon^{\left[ -\mu\_1 \left( \frac{lL\_\ell}{D} \right)^{d/2} \right]} \tag{6}$$

In Equation (6),

$$a\_1 = 1.22 \left(\frac{D}{d\mathfrak{s}0}\right)^{-0.764}, \ a\_2 = 0.99 \left(\frac{D}{d\mathfrak{s}0}\right)^{0.244}$$

Choi and Choi [35] proposed the Equation (7), where *h* is flow depth:

$$\frac{S\_d}{S\_{\ell}} = \varepsilon^{\left\{ 0.065 \left( \frac{l\_{\ell}}{l\_{\ell \rm cr}} \right)^{0.35} \left( \frac{\mu}{D} \right)^{0.19} \ln \left( \frac{\star}{l\_{\ell}} \right) \right\}} \tag{7}$$

Harris et al. [36] proposed Equations (8) and (9), in these models, Breuser's [34] equation has been used to predict the maximum scour depth:

$$\frac{S\_d}{S\_\varepsilon} = 1 - e^{\left(-\frac{\Delta}{t\_\varepsilon}\right)^n} \tag{8}$$

$$\frac{S\_\varepsilon}{D} = 1.5 K\_s K\_0 K\_b K\_d \tanh(\frac{h}{D}) \tag{9}$$

In our equations, the parameters that influence the temporal scour depth *St* are related in Equation (10) and when dimensionless lead to Equation (11).

$$S\_t = f\_1(\mathcal{U}\_t, \mathcal{D}, \mathcal{C}, d\_{50\prime} \mathcal{R}, \mathcal{g}, \mathcal{g}, \mathcal{g}\_\prime \mathcal{g}\_\prime, h\_\prime t) \tag{10}$$

$$S\_t = f\_2^2 \left( \frac{\mathcal{U}\_c t}{D}, \frac{\mathcal{C}}{H'}, \frac{d\_{50}}{D}, \frac{R}{D'}, \frac{\mathcal{U}\_c}{(gh)^{0.5}}, \frac{\rho\_s - \rho}{\rho} \right) \tag{11}$$

Dimensionless tip clearance and turbine radius are important for scour process. ρ*s* is the sediment density, which is considered as constant in nature. By use of Vaschy–Buckingham theorem, Equation (12) is obtained. 

$$\frac{S\_t}{D} = f\_3 \left( \frac{\mathcal{U}\_t t}{D}, F\_{r\prime} \frac{d \mathfrak{s}\_0}{D}, \frac{\mathcal{C}}{H}, \frac{R}{D} \right) \tag{12}$$

$$F\_r = \mathcal{U}\_c / (\mathcal{g}\hbar)^{0.5} \tag{13}$$

The Froude number (*Fr*) is constant when considering constant flow depth and uniform flow velocity, as calculated by Equation (13). *Fr* = 0.13 in the current works. The sediment diameter is remained as 1.1mm at seabed. Equation (12) can be simplified as Equation (14):

$$\frac{S\_t}{D} = f\_4\left(\frac{\mathcal{U}\_c t}{D}, \frac{\mathcal{C}}{H}, \frac{R}{D}\right) \tag{14}$$

Live bed scour and clear water scour can be found in the experiment against different tip clearance. Clear water scour occurs when *C*/*H* > 0.5; the temporal scour depth is closely related to current velocity, foundation diameter, rotor radius, and tip clearance. When *C*/*H* ≤ 0.5, live bed scour takes place around monopile foundation, and temporal scour depth has no relevance to tip clearance or rotor radius. Dimensionless analysis produces the empirical equation to predict the temporal evolution of scour hole depth around monopile foundation of Darrieus tidal current turbine in Equations (15)–(19):

a. For clear water scour (*C*/*H* > 0.5):

$$\frac{S\_t}{D} = k\_1 \left[ \log\_{10} \left( \frac{lI\_c t}{D} \right) - k\_2 \right]^{k\_3} \tag{15}$$

where

$$k\_1 = 0.269 \left(\frac{C}{H}\right)^{1.1} \left|\frac{R}{D} - 5\right|^{-0.09} \tag{16}$$

$$k\_2 = -1.088 \left( \frac{C}{H} \right)^{0.171} \left| \frac{R}{D} - 5 \right|^{-0.3} \tag{17}$$

$$k\_3 = 1.233 \left(\frac{\text{C}}{H}\right)^{-0.7} \left| \frac{R}{D} - 5 \right|^{0.15} \tag{18}$$

b. For live bed scour (*C*/*H* ≤ 0.5):

$$\frac{S\_t}{D} = 0.131 \Big[ \log\_{10} \left( \frac{lI\_c t}{D} \right) + 1.11 \Big]^{1.869} \tag{19}$$

Figure 11 shows the comparison between experimental data and data calculated by proposed temporal scour depth equation; *R*<sup>2</sup> is 0.93. The proposed model well agrees with the experimental data.

**Figure 11.** Comparison between observed and computed time-dependent scour depth.

#### *4.2. Comparison of the Proposed Model with the Previous Works*

The proposed empirical model was compared to the bridge pier scour model, propeller scour model, and the scour data of tidal turbine.

#### 4.2.1. Comparison with Bridge Pier Scour Equations

Figure 12 shows the comparison between temporal evolution of scour depth around turbine foundation calculated by our proposed model and scour depth at bridge piers predicted by existing equations. In present study, six previously developed time-dependent scour depth equations [31–36] are chosen for checking the accuracy. According to Gaudio et al. (2010) [39] and Gaudio et al. (2013) [40], due to different mathematical structures in scour prediction formulas, the available bridge pier scour equations may give significantly different predictions.

**Figure 12.** Comparison of time-dependent scour depth around turbine and scour at piers.

For comparison, the equilibrium scour hole depth *Se* in Equations (3)–(9) is set to be 1.45*D*. This is the equilibrium scour depth around single pile for the same test conditions measured in the current experiment. Two cases (*C*/*H* = 0.5 or 1) in the current experiment have been chosen to be references. These two cases are typical turbine scour process as live bed scour or clear water scour. Equations

(4), (5), and (7) show grea<sup>t</sup> agreemen<sup>t</sup> with the temporal scour depth at *C*/*H* = 1.0. It shows a grea<sup>t</sup> increase in scour depth in the first 10 min and low growth in the later scouring time. When *C*/*H* = 1.0, the seabed scour around turbine is more like scour at piers. The scour process is not much affected by turbine rotor. When the tip clearance lower than 0.5*H*, the evolution of scour hole is greatly affected by turbine rotor. The equilibrium scour hole depth is much more than other cases in Figure 12. The increasing extent is ~50%.

#### 4.2.2. Comparison with the Equations of Ship Propeller Induced Scour

The temporal scour depth induced by turbine and propeller jet has been compared in Figure 13. Two empirical equations for temporal propeller jet induced scour depth prediction proposed by Qurrain [37] and Hong [38] are chosen to compare the scour induced by turbine. These equations are listed below. Temporal scour depth induced by turbine and propeller jet has approximate similar variation trend, as shown in Figure 13. The scour depth increases greatly in the first 10 min. The scour depth increases slowly until equilibrium. The scour mechanisms of these two types of scour are different. In the ship propeller jet-induced scour, the propeller jet causes high shear force which can activate sediment. The maximum scour depth occurs at the centerline of the propeller in a streamwise direction downstream. However, the dominant feature of scour around foundation of turbine is horseshoe vortex. The evidence of turbine rotor disturbs the surrounding flow and accelerates flow under the turbine. Hence the scour hole can be deeper than the scour hole around piers. The position of maximum scour depth is around supporting pile of turbine. Generally, the scour depth induced by propeller jet is much deeper than scour around tidal current turbine at same installation height. In Figure 13, the equilibrium scour hole depth predicted by Hong's empirical equation reaches about 3.5D when C/H = 0.5, which is 58% deeper than equilibrium depth around turbine.

**Figure 13.** Comparison of time-dependent scour depth around the turbine with scour induced by propeller jet.

For empirical equations to predict temporal scour depth induced by propellers, Qurrain et al. [37] proposed Equation (20), where *Sp* is maximum scour depth in mm at any time *t*, in seconds.

$$S\_p = \Omega \lbrack \ln(t) \rbrack^\Gamma \tag{20}$$

In Equation (20),

$$
\Omega = \left[\frac{\mathbb{C}}{d\_{50}}\right]^{-4.758} \left[\frac{D\_p}{d\_{50}}\right]^{2.657} [F\_0]^{3.517}, \\
\Gamma = \left[\frac{\mathbb{C}}{d\_{50}}\right]^{0.758} \left[\frac{D\_p}{d\_{50}}\right]^{-0.339} [F\_0]^{-0.479}.
$$

Hong et al. [38] proposed Equation (21), where *Dp* is propeller diameter, *U*0 is e fflux velocity.

$$\frac{S\_p}{D\_p} = k\_1 \left[ \log\_{10} \left( \frac{Ll\_0 t}{D\_p} \right) - k\_2 \right]^{k\_3} \tag{21}$$

In Equation (21),

$$\begin{aligned} k\_{\mathfrak{a}} &= 0.014 F\_{\mathfrak{0}}^{1.120} \left( \frac{\mathbb{C}}{D\_{\mathbb{P}}} \right)^{-1.740} \left( \frac{\mathbb{C}}{d\_{\mathbb{S}0}} \right)^{-0.170} \\\ k\_{\mathfrak{b}} &= 1.882 F\_{\mathfrak{0}}^{-0.009} \left( \frac{\mathbb{C}}{D\_{\mathbb{P}}} \right)^{2.302} \left( \frac{\mathbb{C}}{d\_{\mathbb{S}0}} \right)^{-0.441} \\\ k\_{\mathfrak{c}} &= 2.477 F\_{\mathfrak{0}}^{-0.073} \left( \frac{\mathbb{C}}{D\_{\mathbb{P}}} \right)^{0.53} \left( \frac{\mathbb{C}}{d\_{\mathbb{S}0}} \right)^{-0.045} \end{aligned}$$

4.2.3. Comparison with Published Data of Turbine Scour

In addition, a comparison between the proposed equations in the current study for scour around Darrieus tidal current turbine and experimental data studied by Hill et al. [19] for horizontal axis turbine is presented in Figure 14. All the experiments had approached scour equilibrium after about 180 min. In Hill's experiment, the scour hole develops deeper and faster with upstream installed rotor. For turbine rotors installed upstream, the maximum scour depth was ~2.4 *D*. The temporal scour depth indicated similar pattern with live bed scour type in our proposed equation. For downstream installed turbine rotor, the maximum scour depth was about 1.5 *D*. This data was approximately same as *C*/*H* = 1 in our experiment. This was clear water type in our proposed equations. However, the temporal evolution of scour depth before scour equilibrium in downstream rotor condition in Hill's experiment was not match well with the current experiment. This may be caused by the di fferent type of tidal current turbine.

**Figure 14.** Comparison of time-dependent scour depth around two types of tidal current turbine.
