*2.2. Hydrodynamics Calibration Test*

Prior to executing a numerical simulation of scour due to the combined waves and currents around a circular pile, the process of hydrodynamically calibrating the model REEF3D was conducted by comparing the numerical results with the experimental results obtained by other authors.

The calibration process was carried out in two phases. The first phase to verified the numerical model's capacity to represent a flow of waves and currents combined, without the pile in the flume, while the second calibration phase included the presence of a vertical pile in the center of the flume.

In Table 3, cases executed in the hydrodynamic calibration process without considering the pile are described. The experimental information for each of the cases was obtained from Umeyama [3], and these data correspond to the unevenness data for the water surface and the vertical profiles of flow horizontal velocity for different times steps, both for the flow caused only by the actions of waves (case W1 to W3) and by waves and currents (cases WC1 to WC3).

The variables in Table 3 are defined as follows: h is water depth, H is wave height, T is the wave periodic time, and *UC* is the undisturbed current velocity defined according to Sumer and Fredsøe [25].

In order to compare the results obtained by the numerical model and those provided by Umeyama's [3] experimental work, a virtual sensor was established in the middle of the numerical domain, from which the vertical distribution (from the total velocity), and the surface elevation were extracted. The simulation time for all scenarios of the situation without pile (W1 to WC3) was five minutes (300 waves).

Simulated cases for the hydrodynamic calibration of the numerical model, including one pile in a flow due to waves and the combined action of currents, are summarized in Table 4, which considers simulated cases for codirectional (C01 to C03) and perpendicular (C04) waves and currents.

The numerical results obtained by the simulation for cases C01 to C03 (codirectional) contrasted with the experimental data provided by Qi and Gao [26], where the flow velocity was the comparison variable. For the case of the perpendicular waves and currents (C04), the experimental information provided by Miles et al. [51] was used to verify the numerical results, where the contrasted variable was the average vertical profile of the total flow velocity.

The comparison of the numerical and experimental data obtained by Qi and Gao [26] (C01 to C03) was carried out by obtaining the time series of the total velocity in a virtual monitoring station located at the horizontal 20D and 1D relative to the bed. The total time of the simulation was seven minutes (300 waves).

For case C04, the comparison of numerical and experimental data published by Miles et al. [51] was conducted via the vertical distribution of the longitudinal velocity averaged in eight points around the pile, which were denominated P1 to P8 and distributed as shown in Figure 1. The monitoring stations were located 0.75*D* from the center of the pile, while their angular separation was 45◦.

**Figure 1.** Identification of the comparison points of the average velocity profiles obtained from the numerical modeling and those published by Miles et al. [51].

The velocities were nondimensionalized according to the characteristic velocity (*C*) described in Equation (22) for all compared cases between the modeled and experimental data (Umeyama [3], Qi and Gao [20], and Miles et al. [51]).

$$C = \sqrt{gh} \tag{22}$$

#### *2.3. Hydrodynamics Behavior of the Flow around a Cylindrical Pile*

To numerically determine the hydrodynamic behavior around a cylindrical pile subjected to the combined action of waves and currents, a set of 8 simulations were developed, as described in Table 5. The cases were defined to cover a wide range of waves and current interactions according to the flow relative velocity (*Ucw*) proposed by Sumer and Fredsøe [25]. Thus, we formed scenarios dominated by currents (E01 and E02), waves and currents but with a tendency toward current domination (E03 and E04), waves and currents but with a tendency toward wave domination (E05 and E06), and environments dominated by waves (E07 and E08).

The estimation of the flow relative velocity (*Ucw*) was conducted by considering the maximum value of the undisturbed orbital velocity at the sea bottom just above the wave boundary layer (*Um*), according to Equation (23), while the undisturbed current velocity at the transverse distance *z* = *D*/2 (*Uc*) was defined as an edge condition for each of the modeling scenarios. Additionally, the Keulegan-Carpenter number was estimated as indicated in Equation (24), in order to identify the influence that waves have over maximum scour.

$$\mathcal{U}\_{\rm m} = \frac{\pi H}{T \sin h(Kh)}\tag{23}$$

$$KC = \frac{\mathcal{U}\_{\text{m}}T}{D}.\tag{24}$$

All simulations conducted (E01 to E08) were set to solve the hydrodynamics model within 30 minutes throughout the entire numerical domain, using the potential flow and a hydrostatic distribution of pressures as the initial condition (see Table 1).

To analyze the velocities and vortexes associated with the flow, two main vertical planes of the channel were designed. The first of these plains corresponds to the longitudinal axis (flow development) passing through the center of the pile from the beginning of the channel to its end. The second is associated with the axis perpendicular to the channel, as shown in Figure 2.

**Figure 2.** Analysis of the hydrodynamic planes around the cylindrical pile.

As the first stage of the analysis conducted on the results obtained from the numerical model, the velocity fields along the channel were inspected in order to identify patterns in spatiotemporal flow performance. Subsequently, for each of the planes traced around the cylindrical pile (see Figure 2), the vorticity's average performance was determined and the associated streamlines were traced to study and analyze the average performance of the horseshoe vortex. Additionally, eight monitoring stations were defined to obtain the vertical distribution of the flow velocity Figure 1 indicates the distribution of these stations, which is concordant with the methodological approach of Miles et al. [51].

Moreover, the amplification of the shear stress (ατ) around the pile was determined, associated with the average flow conditions and (on a Cartesian plane) based on the numerical domain bed, prior to scour. For such purposes, the proportion of the bed shear stress (τ0) and the undisturbed bed shear stress (<sup>τ</sup>∞) according to Equation (25) were considered. To determine τ0 and τ<sup>∞</sup>, the shear velocity was computed by the numerical model in the bed's nearest cell for two locations: around the <sup>p</sup>ile *<sup>u</sup>*∗*T*,*pile*! for the calculation of τ0 and 2 meters downstream of the inlet (*<sup>u</sup>*∗*T*,*inlet*) for τ<sup>∞</sup>.

The relation employed for the bed shear stress estimations corresponds to the conventional hydraulic definition described in Equation (26) for τ0 and Equation (27) for τ<sup>∞</sup>, where *<sup>u</sup>*∗*T*,*inlet* or *<sup>u</sup>*∗*T*,*pile* corresponds to the total bed velocity, determined from the longitudinal and transverse velocity vector magnitude.

$$a\_{\tau} = \frac{|\tau\_0|}{\tau\_{\infty}} \tag{25}$$

$$
\pi\_0 = \rho u\_{\ast\_{T,\text{ pile}}}^2 \tag{26}
$$

$$
\pi\_{\circledast} = \rho u\_{\ast T, \text{ inlet}}^2 \tag{27}
$$

## *2.4. Scour around a Cylindrycal Pile*

Using a simulated scenario for the study of hydrodynamics, the scour around the pile was estimated by considering a bed composed of spherical sediments with diameter of 0.38 mm (*d*), 2650 Kg/m<sup>3</sup> in density, a 30◦ angle of repose, and a dimensionless critical shear stress (τ<sup>∗</sup>*cr*) equal to 0.036, in order to make the results obtained in this investigation for E01 and E02 comparable to those previously presented by Qi and Gao [26].

The scour estimation using the numerical model was activated during the last 25 minutes of the hydrodynamic modelling (*S*1), to obtain enough information at the beginning of the simulation for the immobile bed condition and, subsequently, the associated condition for the mobile bed.

The features of the numerical tests conducted on the scour around the modeled cylindrical pile are summarized in Table 6. The parameters associated with the incipient transport of sediments for the combined regimen of waves and currents have been estimated according to the methodology extensively described in Soulsby [52]. The fundamental equations for estimating shear stress are described below in summary.


**Table 6.** General characteristics of numerical tests of scour modeling.

Bed shear stress due to the combined action of waves and currents was estimated according to Equation (28), where τ*c* is shear stress considering only the action of currents, while τ*w* corresponds to the shear stress for waves alone. The shear velocity (*<sup>u</sup>*∗) was determined based on a resistance law according to Equation (31), where *UB* is the bulk velocity of the flow due to the current. The wave boundary layer velocity ( *Uf m*) is defined in Equation (32), where *fw* is the friction factor, which was determined according to Fredsøe and Deigaard [53] (page 25).

$$
\tau\_{\rm uvc} = \tau\_c \left[ 1 + 1.2 \left( \frac{\tau\_{\rm uv}}{\tau\_c + \tau\_{\rm uv}} \right)^{3.2} \right] \tag{28}
$$

$$
\pi\_c = \rho u\_\*^2 \tag{29}
$$

$$
\pi\_w = \frac{1}{2} \rho f\_w \mathcal{U}\_{fm}^2 \tag{30}
$$

$$\frac{\mu\_\ast}{\Pi\_B} = \frac{1}{7} \left( \frac{d}{h} \right)^\dagger \tag{31}$$

$$
\mathcal{U}\_{fm} = \sqrt{\frac{f\_w}{2}} \mathcal{U}\_{\text{m}}.\tag{32}
$$

The shear stress values associated with currents, waves and the combined action of both were nondimensionalized according to Equation (14). The results are presented in Table 6.

Considering that the numerical modeling duration of the scour was 25 minutes and that in this time scale the equilibrium condition was not reached, a projection was made via the equation proposed by Sheppard et al. [54], which corresponds to a four-parameter exponential function for the extrapolation of the equilibrium scour depth (*St*), which is presented in Equation (33) where *ai* corresponds to the adjustment coefficient *i* of the equation by Sheppard et al. [54]. This approach was also used by Qi and Gao [20], with experimental data.

$$S\_t = a\_1[1 - \exp(-a\_2t)] + a\_3[1 - \exp(-a\_4t)].\tag{33}$$

The equilibrium scour was compared with the results presented by Qi and Gao [26], Sumer and Fredsøe [25], Raaijmakers and Rudolph [15], Sumer et al. [55], and Mostafa and Agamy [56] to check whether the numerical results obtained are concordant with the experimental data developed by other authors.
