**4. Discussion**

The experimental data with no scour provided by Umeyama [3] have been used to compare the numerical results of other authors, such as Zang et al. [30] and Ahmad et al. [31] who used the RANS approach to solve the hydrodynamics of waves and currents acting codirectionally over a grid featuring finite di fferences with a regular element ( Δ*x* = Δ*y* = <sup>Δ</sup>*z*), which correspond to the methodology applied in this investigation.

For the construction of the numerical domain, Zang et al. [30] utilized Δ*x* = 0.002 m to solve the vertical domain in 150 layers, while the configuration applied by the authors of this study considered Δ*x* = 0.01 m which determines the 30 layers in the vertical direction for the water flow adopted by Umeyama [3]. Despite of the coarser grid used in this research, the results are consistent with the experimental data for the vertical profile of velocities as well as for the instantaneous surface elevation of the water.

In order to model the scour around cylindrical piles, this study used the same element dimension as Ahmad et al. [31], who proved that the use of an element of 0.01 m is su fficient to estimate the scour under a pipeline [31]. This conclusion was reached by a grid analysis and time convergence study which analyzed the numerical behavior of REEF3D for element sizes of Δ*x* = 0.04, 0.03, 0.02, 0.01 and 0.005 m.

The results obtained from the eight simulations (scenarios E01 to E08), showed a low variability of the mean velocity profile around the pile (stations P1 to P8), as illustrated in Figure 10, although these simulations were constructed to represent both, mixed and current or waves dominated environments, according to the criteria of Sumer and Fredsøe [26]. The results associated with the expected bed shear stresses for each of the eight scenarios (see Table 6), show that the e ffect of the waves on the first six scenarios (E01 to E06) is not significant in the bed dynamics, since the dimensionless shear stress due to waves (τ<sup>∗</sup> *w*) is an order of magnitude less than the dimensionless shear stress due to currents (τ<sup>∗</sup> *c*).

Based on the above, if it is considered that the shear stress is the hydraulic boundary condition to build at vertical profile of flow velocities, and the e ffect of the current dominates over the waves, it is expected that the first six scenarios present a high similarity for both a co-directional and opposed flow. On the other hand, in the remaining scenarios (E07 and E08), where the dimensionless shear stresses associated to waves and currents are of the same order of magnitude, greater e ffects on the velocity profile around the pile could be noticed (Figure 10), which would indicate that both the co-directional and opposed flow develop di fferences in the velocity mean behavior.

This di fference between the shear stresses and the Sumer and Fredsøe criteria [26] seems to imply that the use of the dimensionless number called the relative velocity of the current (*UCW*), does not fully describe the domain of the forcing over the total hydrodynamics, a discussion that is presented in the following paragraphs of this investigation.

From the results obtained, it is possible to verify that the presence of waves in the hydrodynamic behavior for scenarios E01 to E06 was less significant than for scenarios E07 and E08. This can be clearly seen in the mean profile analysis, since when the current dominated, the direction resulting from the flow agreed with the streamwise direction, these results were previously described by Feraci et al. [4], who, while performing wave and current test acting orthogonally and without the presence of a pile, obtained results comparable to those obtained in this investigation.

The vertical distribution of the mean velocity illustrated in Figure 10 is consistent with that described by Lim and Madsen [5], who mentioned that although the flow field can be mixed (waves and currents acting together and orthogonally), the velocity distribution can be simply modeled by a uniform return current. The foregoing analysis is also consistent with the results presented by Feraci et al. [6].

The equation by Sheppard et al. [54] was applied to estimate the equilibrium scour based on an extrapolation of the numerical model. These results were similar to those obtained experimentally by other authors, which indicates that the methodology applied as well as the configuration adopted by the numerical model are appropriate to describe the phenomenon under study, not only from the perspective of element size but also for the sediment transport equations applied.

An important aspect to highlight is that this investigation has used the relaxation factor previously applied by Quezada et al. [27], which also allowed the authors to correctly represent the scour for unsteady current and oscillatory flow. According to the results obtained in the current study, this value will also allow us to estimate the scour for uniform and oscillatory flow, not only codirectionally but also opposite. The relaxation coe fficient permits in an auxiliary manner e ffects inherent to the structure of the fluid interaction produced around the pile, thereby improving the estimations of sediment transport and the resulting scour.

Sumer and Fredsøe [26] propose the relative velocity of the current (*UCW*) as a dimensionless number relevant for the description of the scour due to codirectional or perpendicular waves and currents. This is defined in Equation (1), where the current magnitude ( *UC*) is estimated at a height of *D*/2 from the bed, while the velocity of the wave is considered as the maximum value of the undisturbed orbital velocity at the bottom, just above the wave boundary layer ( *Um*). By this dimensionless definition, Sumer and Fredsøe [26] established that values of *UCW* higher than 0.7 indicate that the current dominates in the center, while waves have a significant e ffect when *UCW* is close to zero (cases waves alone) and less than 0.4.

This dimensionless number considers that *UC* and *Um* are added, independently of the direction of incidence of the currents and waves. In this respect, Sumer and Fredsøe [26] consider a single value of *UCW* for codirectional and perpendicular flow, if *UC* and *Um* are the same in magnitude but di fferent in direction.

The above, according the authors of this paper, would not be appropriate as a general indicator of wave and current interaction, nor would their e ffects on the scour for cases in which the forcings are not codirectional, since when both flows face in the opposite direction, the wave would propagate with greater di fficulty and modify the net velocity of the channel, such that the current present in the center would correspond to the residual value of both forcings.

Soulsby [52] and Van Rijn [57] indicates that current wave interactions must be treated in terms of the net current produced between the two forcing agents, which corresponds to an algebraic sum that is usually treated according to Equation (34), where is the angular frequency, *UB* is the bulk velocity of the flow due to the current, *K* is the wave number, ϑ is the angle between current and wave direction (ϑ = 0 for codirectional, and ϑ = 180◦ for opposing), *g* is the gravity and *h* is the water depth:

$$
\omega - \iota \mathcal{U}\_{\mathcal{B}} K \cos(\mathfrak{d}') = \sqrt{\mathcal{g}K \tan h(\mathcal{K}h)}.\tag{34}
$$

The left term of Equation (34) correspond to the net velocity (defined according the relative velocity between the current and waves). Meanwhile, the right term corresponds to the dispersion relationship of the waves.

From Equation (34) it can be seen that in cases of co-directional or opposite waves, the *cos*(<sup>ϑ</sup>) changes its sign and therefore, the system net velocity is the sum or subtraction of both forcings. Therefore, defining a dimensionless number that summarizes the wave and current interaction, must include a differentiation when the action is codirectional or when it is counter current.

An approximation to the description of scour due to opposite and codirectional currents and waves was conducted by Qi and Gao [58] who, using experimental data, obtained by the same authors in previous works (Qi and Gao [26]) and by third parties as well (Sumer and Fredsøe [18] and Sumer et al. [55]), propose the use of the Froude number (*Fra*) defined in Equation (35), as a function of absolute velocity (*Ua*, defined by Equation (36)) and the pile diameter.

$$F\_{\text{ff}} = \frac{\mathcal{U}\_a}{\sqrt{gD}}\tag{35}$$

$$
\mathcal{U}\mathcal{U}\_{\mathfrak{d}} = \mathcal{U}\_{\mathbb{C}} + \frac{2}{\pi} \mathcal{U}\_{\mathfrak{m}}.\tag{36}
$$

From this analysis Qi and Gao [52] proposed a formula fitted to the experimental data for a dimensionless scour (*S*/*D*) which is presented in Equation (37) and is valid for the range 0.1 < *Fra* < 1.1 and 0.4 < *KC* < 2.6.

$$
\log\left(\frac{S}{D}\right) = -0.8 \exp\left(\frac{0.14}{F\_{ra}}\right) + 1.11.\tag{37}
$$

A comparison of the numerical results gathered in this paper, the experimental data and the equation proposed by Qi and Gao [58] are shown in Figure 14.

**Figure 14.** Equilibrium scour distribution according to absolute Froude number proposed by Qi and Gao [58].

In Figure 14, it is observed that the information available in the literature (experimental) and that generated in this study (numerical) are adequately concordant with the equation proposed by Qi and Gao [58], and such a description may be enough to collect information on the equilibrium scour around a dimensionless number that represents its behavior.

Notwithstanding the above, Qi and Gao [58], as well as Sumer and Fredsøe [26], consider current and wave actions added equally if they act in a codirectional or opposite manner, which, in general, is not consistent with a residual flow estimation that would be generated by the interaction. The foregoing disagrees with the results by Soulsby [52] and Van Rijn [57], who indicate that the sum of the forcing agents must be algebraic, respecting the angle between current and wave direction. Although the

arguments are contradictory, both proposals (Qi and Gao [58], Sumer and Fredsøe [26]) compile reasonably well the scour information regardless of the direction. This should be analyzed in greater detail as proposed below.

Thus, the Froude number may be rewritten according to Equation (39) if the absolute velocity defined by Qi and Gao [58] is considered, albeit modified according to Equation (38), which is a proposal of the authors of this paper, and where *<sup>C</sup>*ϕ is a coefficient to describe the flow direction, and where *<sup>C</sup>*ϕ = 1 describes codirectional flows and *<sup>C</sup>*ϕ = −1 describe opposite flows.

$$
\mathcal{U}\mathcal{U}\_a = \mathcal{U}\_\mathbb{C} + \frac{2}{\pi} \mathcal{U}\_m \mathcal{C}\_\mathbb{P} \tag{38}
$$

$$F\_m' = \frac{\mathcal{U}\_a'}{\sqrt{gD}}.\tag{39}$$

*<sup>C</sup>*ϕ was included in order to incorporate the recommendations of Soulsby [52] and Van Rijn [57], in order to consider the effects of waves and currents directionality acting together on the pile.

Considering this proposal and collecting scour data (experimental and numerical from this paper), Figure 15 is obtained. The blue circles indicate codirectional cases and red circles indicate opposite flow cases. Two trends are found in two areas of the figure. The first trend corresponds to codirectional data, which are still represented by the equation proposed by Qi and Gao [58]. Nevertheless, the opposite flow cases are to the left of the codirectional data and apparently adjust to an equation different than that proposed by Qi and Gao [58], which, according to the available data set (experimental and numerical data from this paper) correspond to Equation (39).

**Figure 15.** Equilibrium scour distribution according to the absolute Froude number (*Fra*) proposed in this research.

$$\frac{S}{D} = 0.39LN(F\_m') + 1.51.\tag{40}$$

Equation (40) seems to agree with the solution proposed by Qi and Gao [58] for high values of the Froude number. However, such behavior may be verified by adding new experimental and/or numerical antecedents, which enable us to complement equilibrium scour data in a combined domain of currents and waves acting in opposite directions.
