*3.4. Scaling of Experiment*

According to Verhey et al. [7], the scaling effects caused by viscosity are negligible if the Reynolds number for flow (*Rflow*) and the propeller (*Rprop*) are greater than 3 × 10<sup>3</sup> and 7 × 104. The Reynolds numbers can be calculated using Equations (1) and (2).

$$R\_{flow} = \frac{V\_0 D\_P}{\nu} \tag{1}$$

$$R\_{prop} = \frac{nL\_mD\_p}{\nu} \tag{2}$$

where *Lm* is the length that depends on β as defined by Blaauw and van de Kaa [6] as Equation (3).

$$L\_m = \beta D\_p \pi \left( 2N \left( 1 - \frac{D\_h}{D\_p} \right) \right)^{-1} \tag{3}$$

where *V*0 is the efflux velocity (m/s); *Dp* is the diameter of the propeller (m); *Dh* is the diameter of the hub (m); ν is the kinematic viscosity of the fluid (8.54 × 10−<sup>7</sup> m<sup>2</sup>/s); *n* is the rotational speed (rps); β is the blade area ratio; and *N* is the number of blades.

**Figure 4.** 3D printing for propeller model. (**a**) Printing status; (**b**) print completed; (**c**) propeller-A; (**d**) propeller-B.

Typical propeller sizes and rotational speeds that might cause seabed scouring in British ports and harbours ranged between 1.5 m and 3 m for propeller diameters running at 200–400 rpm according to Qurrain [21]. A typical ship propeller with a diameter of 1.65 m that operated at 200 rpm with a coefficient of thrust (Ct) of 0.35 was used as the prototype for the twin-propeller. The propeller rotation speeds used for this investigation were set to 500 and 700 rpm for observing the resulting scour profiles. Reynolds numbers for the proposed speed ranges were 2.9 × 10<sup>4</sup> *Rflow* and 1.3 × 10<sup>4</sup> for *Rprop*. In the current study, the motor speed was limited. *Rprop* was slightly smaller than the specified value. However, Blaauw and van de Kaa [6] and Verhey [7] proposed that these scale effects were insignificant. The Reynolds numbers for the jets were all greater than 3 × 10<sup>3</sup> satisfying the criteria for Froudian scaling.

The dimensions of the scour profile induced by twin-propeller are shown in Figure 5. Hong et al. [10] stated that scour induced by a propeller can be divided into small scour hole (Zone A), primary scour hole (Zone B) and deposition dune (Zone C). The current research focused on the scour caused by the twin-propellers. Ten different twin-propeller experiments and one single-propeller experiment were set up as summarised in Table 3.

**Figure 5.** Dimensions of the scour profile induced by twin-propeller.



#### **4. Twin-Propeller Jet Induced Sandbed Scour**

## *4.1. Temporal Depth of Scour*

The experiments found the maximum scour position of a twin-propeller occurred symmetrically at the rotational axis of both two propellers. The scour holes were mirrored for two propellers in the twin system at the port and starboard sides with time, as shown in Figure 6. The depth of the scoured hole increased with time from the observation of various experiments. The motor speed reached 500 rpm in a very short time and the propeller reached a stable rotation speed in 3 s. The scour data were recorded at 60, 120, 300, 600, 1200 and 1800 s in the experiment. The increase of the scour depth was insignificant after 1800 s (30 min), which reached the equilibrium state. Hong et al. [10] stated the scour process includes initial stage, developing stage, stabilisation stage and asymptotic stage. The scour process for T-1 case in current experiments is shown in Figure 6. The evolution of a typical scouring profile along the longitudinal direction was measured.

**Figure 6.** Temporal scour processes for twin-propeller, (**a**) 60 s; (**b**) 120 s; (**c**) 300 s; (**d**) 600 s; (**e**) 1200 s; (**f**) 1800 s.

Figure 7 shows the experimental results obtained from six scour times up to 1800 s (30 min). The external rotating twin-propeller produced the two largest scour holes at the propeller axis and the scour profile was approximately symmetrical. The maximum scour depth in the axial direction of the right single propeller (starboard) was recorded. The x-axis shows the axis distance, while the y-axis indicates scour depth. The main scour hole and deposition appeared downstream of the jet. A shallow hole formed immediately below the propeller. The sediments located on the surface of the sand bed were washed downstream with time due to the jet impingement. The maximum scour depth increased gradually with time. The location of the maximum scour depth did not change significantly with time. The deposition height increased gradually with time, but the position of dune peak moves downstream with time. For the non-dimensionalised scour profile, the x-axis shows the axis distance (*X*/*Xm*), where *Xm* is the length of the maximum scour hole from the propeller. The y-axis indicates scour depth (<sup>ε</sup>m/εmax). The non-dimensionalised scour profiles show the scour profile at different times had high similarity.

**Figure 7.** Dimensionless scour depth for a twin-propeller.

#### *4.2. 2D Scour Section for Single-propeller*

Previous researchers proposed an empirical formula to predict the maximum scour depth based on the experimental results. The scour structure of the entire scour section has not been studied. The current research studied the 2D scour profile of single-propeller and twin-propeller induced scour. The maximum scour occurred on the rotational axis of single-propeller. Hamill et al. [8] stated that the maximum scour depth location (*Xm*) can be calculated by Equation (4). The location of the maximum scour depth was proportional to the distance from the tip of the propeller blade to the sandbed.

$$X\_{\rm II} = F\_0^{0.94} \text{C} \tag{4}$$

For current propeller diameter of 55 mm, speed of 500 rpm, and *C* of 0.0275 m, *F*0 was calculated as 8.135. The maximum scour depth and position was measured, as presented in Figure 8.

The dimensionless scour profiles show that the current research has the same scour profile compared with Hamill et al. [8] and Hong et al. [10]. The scour profile is divided into three zones: small scour hole, primary scour hole and deposition dune in order to describe the whole scour section for single-propeller. The length of each zone was necessary to describe the scour profile. Table 4 summarises the suggested dimensionless length of each zone. The current work suggests the application of a Gaussian probability distribution to represent each scour section, which the scour depth is distributed in the holes. A Gaussian distribution curve was used to fit the three zones, as shown in Figure 9.

**Table 4.** Dimensionless length of each zone for single-propeller induced scour.


**Figure 8.** Dimensionless scour profiles for a single-propeller.

**Figure 9.** *Cont*.

**Figure 9.** Dimensionless scour section of different zones. (**a**) Small scour hole (−0.5 < *X*/*Xm* < 0); (**b**) primary scour hole (0 < *X*/*Xm* < 1.8); (**c**) deposition dune (1.8 < *X*/*Xm* < 3).

The entire scour profile can be predicted by using Equations (5)–(7).

$$-0.5 < \mathcal{X}/\mathcal{X}\_m < 0: \frac{\varepsilon\_m}{\varepsilon\_{\text{max}}} = -0.02 + \left( -0.138 \ast \exp\left( -0.5 \ast \left( \frac{\frac{\mathcal{X}}{\mathcal{X}\_S} + 0.23}{0.12} \right)^2 \right) \right) \tag{5}$$

$$0.0 < X/X\_m < 1.8: \frac{\varepsilon\_m}{\varepsilon\_{\text{max}}} = -0.038 + \left( -0.9 \* \exp\left( -0.5 \* \left( \frac{\frac{x}{X\_S} - 1.09}{0.449} \right)^2 \right) \right) \tag{6}$$

$$1.8 < \text{X/X}\_{\text{m}} < 3: \frac{\varepsilon\_{\text{m}}}{\varepsilon\_{\text{max}}} = 0.21 + \left(0.796 \ast \exp\left(-0.5 \ast \left(\frac{\frac{\chi}{\chi\_{\text{S}}} - 2.5}{0.22}\right)^2\right)\right). \tag{7}$$

The proposed calculation was compared to the experimental results by Hong et al. [10] in order to validate the proposed scour equation to form 2D scour profile. The maximum scour depth was 0.107 m. The location of the maximum scour depth was 0.75 m. The comparison between the experimental results and the predicted equation is shown in Figure 10.

**Figure 10.** Comparison between the experimental values and the predicted equation.

The results of the predicted scour profile are consistent with the experimental results. The predicted scour depth has an error of less than 20%. The current research suggests that the scour profile of a single propeller can be predicted by using Equations (5)–(7) after the maximum scour depth and position was calculated using the existing equations.

## *4.3. 2D Scour Section for Twin-Propeller*

Scour induced by twin-propeller produced a sand deposition downstream after the primary scour hole. The deposition is in the "M" distribution with two deposition peaks at the rotational axis of both propellers. The deposition decreased towards the two propellers laterally from the M-shaped lateral scour distribution. Previous researchers did not propose a corresponding formula to predict the deposition profile.

For propeller-B, the scour results were recorded at a speed of 700 rpm in T-6 case. The measured twin-propeller scour results in the rotational axis and the symmetrical plane are shown in Figure 11. The maximum scour depth on the rotational axis was significantly larger than the maximum scour depth on the symmetrical plane. The maximum deposition height on the symmetrical plane was higher than the deposition height on the rotational axis. The position of deposition peak on the symmetrical axis was 310 mm from the propeller, but the position of deposition peak on the rotational axis was 350 mm from the propeller. The experimental results show the ridge-like deposition profile with three peaks. The first deposition peak is bigger compared to the two other peaks downstream. The dimensionless scour profile on the rotational axis of twin-propeller (T-1 case) and single-propeller (T-11 case) induced scour are shown in Figure 12.

**Figure 11.** Scour on rotational axis and symmetrical planes for twin-propeller.

**Figure 12.** Scour induced by a twin-propeller and single-propeller.

The maximum scour position for twin-propeller can be calculated by Equation (8).

$$X\_{m,t\text{win}} = 2.1D\_p\tag{8}$$

The scour profile of twin-propeller was also divided into three zones including small scour hole, primary scour hole and deposition dune. Table 5 gives the position of the maximum scour depth (*Xm,twin*) and the length of primary scour hole (*Xs,twin*). The dimensionless length of each zone was also suggested. The three zones were predicted by using the Gaussian distribution curve, as shown in Figure 13.


**Table 5.** Dimensionless length of each zone for twin-propeller induced scour.

**Figure 13.** *Cont*.

**Figure 13.** Dimensionless scour section of different zones for twin-propeller. (**a**) Small scour hole (−0.5 < *X*/*Xm,twin* < 0); (**b**) primary scour hole (0 < *X*/*Xm,twin* < 1.6); (**c**) deposit mound (1.6 < *X*/*Xm,twin* < 2.6).

The entire scour profile can be predicted using Equations (9)–(11).

$$-0.5 < X/X\_{m,twin} < 0: \frac{\varepsilon\_{\text{ff}}}{\varepsilon\_{\text{max}}} = -0.23 \* \exp\left(-0.5 \* \left(\frac{\frac{X}{X\_{\odot}} + 0.14}{0.09}\right)^2\right) \tag{9}$$

$$0 < X/X\_{m, \text{train}} < 1.6: \frac{\varepsilon\_{\text{ff}}}{\varepsilon\_{\text{max}}} = 0.04 + \left( -1.03 \ast \exp\left( -0.5 \ast \left( \frac{\frac{X}{X\_{\text{S}}} - 0.87}{0.36} \right)^{2} \right) \right) \tag{10}$$

$$1.6 < \mathcal{X}/\mathcal{X}\_{m, \text{train}} < 2.6: \frac{\varepsilon\_m}{\varepsilon\_{\text{max}}} = -0.19 + \left(0.91\* \exp\left(-0.5\* \left(\frac{\frac{\mathcal{X}}{\mathcal{X}\_S} - 2.08}{0.22}\right)^2\right)\right). \tag{11}$$

The prediction of the scour profile by twin-propeller was compared with the experimental results. The measured maximum scour depth was 28 mm. The location of the maximum scour depth was 140 mm from propeller. The comparison between the predicted equation and previous research is shown in Figure 14.

The predicted scour profile for q twin-propeller proposed by the current research agreed with the scour profile of a single-propeller by Hamill et al. [8] and Hong et al. [10]. The predicted primary scour profile was different from the research provided by Yew [16]. The current research suggests that the scour profile of a twin-propeller can be predicted by Equations (9), (10) and (11). The maximum scour depth and position should be calculated first.

**Figure 14.** Comparison between the predicted equation and previous research.
