**1. Introduction**

The hydrodynamics of the coastal environment usually correspond to the result of the interaction of several force, such as waves, tides, and winds, that act at different spatial and temporal scales, thereby modulating circulation. Meanwhile, rivers run mainly due to the gravitational action that moves the waters resulting from snowmelt or rain, which flow into the alluvial channel that transports to the ocean by runoff. The convergence of coastal and fluvial environments is known as an estuary zone, and the resulting currents correspond to a complex interaction between tides, waves, winds, and river flow.

The hydrodynamics of environments where waves and currents interact have been previously studied by various authors both co-directionally [1–3] and perpendicularly [4–6].

The research developed by Umeyama [1–3], sought to analyze the behavior of Reynolds stress and velocity vertical distributions [1], the changes induced by the combined wave currents over the turbulent flow structures [2], and the surface elevation and particle velocities [3].

In the case of waves orthogonal to currents, the experimental research developed by Feraci et al. [4], Lim and Madsen [5] and Feraci et al. [6] allows us to understand the effects of joint action on the behavior of the resulting velocity of the fluid. For example, Feraci et al. [4] experimentally demonstrated

the joint action of orthogonal waves and currents, which speeds up the evolution process of a sandpit. Lim and Madsen [5] analyzed, via an experimental study the e ffects of the roughness in an experimental study on the velocity distribution in a wave-current interaction. A complete statistical analysis of the near bed velocity behavior due to waves and a current acting perpendicularly was developed by Feraci et al. [6]. They concluded that the probability distribution of near-bed velocity follows a Gaussian distribution in a flow field generated by a current alone. In the presence of waves, the distribution changes and another peak over a Gaussian distribution appears.

However, all the studies in the previous paragraphs do not include any type of obstruction to the flow, which generates additional modifications to the hydrodynamic characteristics of the flow field.

It is well known that when placing a circular pile in an environment that has a specific current (that can be produced by waves/tides, river flow, or both), a hydrodynamic modification will be produced around it and, therefore, vortexes will be produced (a horseshoe vortex and vortex shedding), which are the main elements responsible for the scour around the pile.

Through time, di fferent authors have studied pile scour due to a uniform flow. Among these authors it is worth mentioning Hjorth [7], Melville [8], Ettema [9], Chiew and Melville [10], Melville and Chiew [11], Oliveto and Hager [12], Link et al. [13], Diab et al. [14], and Link et al. [15], who focused their interests mainly on the scour around bridge piles. When it comes to scour by waves, the number of studies is limited. On this subject, the authors of this paper consider the contributions of Sumer et al. [16], Sumer et al. [17] and Sumer and Fredsøe [18] to be fundamental to our understanding of multiple hydrodynamic processes responsible for the movement of sediments near the pile.

Experimental studies on the scour around piles under a flow associated with the combined action of waves and currents have been carried out by di fferent authors [19–26], who have contributed, through their laboratory tests, to our understanding of the scour phenomenon in this type of environments. The following is a brief bibliographic description.

Eadie and Hernich [19] studied a physical model with the purpose of evaluating the e ffects that the combined action of two co-directional forces, waves (random) and currents, have over the scour around cylindric piles. The main results of Eadie and Hernich [19] indicate that the scour process due to waves and currents together is faster and reaches higher equilibrium compared to currents acting alone. Similar results were determined by Kawata and Tsichiya [20], who characterized the scour process for clear-water and live-beds in a similar manner to Eadie and Hernich [19].

Raaijmakers and Rudolph [15] studied the temporal dependency of the scour around a pile due to the combined action of waves and currents with the purpose of analyzing the equilibrium scour, the temporal scales needed to reach such depths and the backfilling process. As part of their results, Raaijmakers and Rudolph [21] propose an equation to determine the scour as a function of time and additionally concluded that the equilibrium scour is of a higher magnitude in cases of currents acting alone compared to the conditions reached for the combined action of waves and currents. The equilibrium scour equation as a function of time, presented by Raaijmakers and Rudolph [21], was validated through the comparison of field data, as presented by Rudolph et al. [22].

Zanke et al. [23], through an analysis of data gathered by other authors, proposed a unified equation to determine the scour depth due to the actions of waves and currents, through the incorporation of a transition function (*xrel*) defined by the e ffective scour (*xeff*). Similarly, Ong et al. [24] developed a stochastic method by which the maximum equilibrium scour could be determined in piles exposed to long-crested and short-crested nonlinear random waves plus a current. They validated their approach by comparison with the experimental data provided by Sumer and Fredsøe [25].

The contribution carried out by Sumer and Fredsøe [25] to understand the process of scour is significant, since through its dimensional analysis, their model is able to represent the dimensionless scour (*S*) over the pile diameter ( *D*) as a function of relative flow velocity ( *Ucw*), as defined by Equation (1), where *Uc* corresponds to the undisturbed current velocity at the transverse distance *z* = *D*/2 and

*Um* is the maximum value of the undisturbed orbital velocity at sea bottom just above the wave's boundary layer:

$$\mathcal{U}\_{cw} = \frac{\mathcal{U}\_c}{\mathcal{U}\_c + \mathcal{U}\_m}.\tag{1}$$

Evidently, the relative flow velocity will have values close to zero when the environment is dominated by waves, but it will approach one if currents are the main flow mechanism.

The main conclusions presented by Sumer and Fredsøe [25] indicate that in a wave environment, the scour increases significantly in the presence of a current, even if the current is mild. This current, is mainly associated with a strong horseshoe vortex in front of the pile, even in the case of a mild vortex. In addition, the current apparently dominates the pile's scour when *Ucw* ≥ 0.7; the scour approaches this value due to the current acting alone.

Even though the articles mentioned above have studied the scour around cylindrical piles due to the combined action of waves and currents, they considered forcing to act co-directionally. Qi and Gao [26], in their experimental work, studied the scour around cylindrical piles under the combined action of co-directional and opposite waves and currents, for di fferent pile diameters, waves conditions and currents. The main conclusion they reached was that the scour in the combined flows of waves and currents is a nonlinear process, and the time required to reach scour equilibrium is much lower than that required for waves or currents acting independently. Additionally, Qi and Gao [26] mentioned that the maximum flow velocity in waves and co-directional currents is much higher than that in waves and currents from opposite directions, thereby a ffecting the maximum scour magnitude, which is lower in opposite flows.

While there is a number (albeit limited) of experimental articles related to the study of scour caused by combined waves and currents, investigations based on numerical models are even more scarce. It is only possible to find simulations of scour acting separately around piles due to currents or oscillatory flows. A literature review on this subject is available on Quezada et al. [27].

The application of Reynolds-averaged Navier–Stokes equations (RANS) in simulated environments, in which waves and currents coexist, has been demonstrated by several authors [28–30], who nonetheless fail to include the vertical pile in the flow. Ahmad et al. [31] recently developed a numerical study based on the REEF3D model in order to study scour on a horizontal pile (pipeline) caused by combined waves and currents. This study is relevant to the research presented in this article, since the same numerical model used by Ahmad et al. [31] was applied.

Based on the above, the main objective of this article is to study, through numerical models, scour's hydrodynamics around cylindrical piles where waves and currents coexist, both co-directional and opposite to the wave direction as well.
