Flow Separation and Vortex Dynamics in Waves Propagating over A Submerged Quartercircular Breakwater

The interactions of cnoidal waves with a submerged quartercircular breakwater are investigated by a Reynolds-Averaged Navier-Stokes (RANS) flow solver with a Volume of Fluid (VOF) surface capturing scheme (RANS-VOF) model. The vertical variation of the instantaneous velocity indicates that flow separation occurs at the boundary layer near the breakwater. The temporal evolution of the velocity and vorticity fields demonstrates vortex generation and shedding around the submerged quartercircular breakwater due to the flow separation. An empirical relationship between the vortex intensity and a few hydrodynamic parameters is proposed based on parametric analysis. In addition, the instantaneous and time-averaged vorticity fields reveal a pair of vortices of opposite signs at the breakwater which are expected to have significant effect on sediment entrainment, suspension, and transportation, therefore, scour on the leeside of the breakwater.


Introduction
Offshore submerged breakwaters have recently become popular coastal defence to protect maritime infrastructure and retain sediments in the sheltered harbor through premature wave breaking (Zuo et al., 2015;Wang et al., 2016;Zheng et al., 2016;Ju et al., 2017). Better understanding of the flow field around submerged offshore breakwaters, especially small-scale hydrodynamic phenomena, is critical to improve the design of breakwaters.
Generation and shedding of vortices occur during the interaction of waves with submerged structures due to flow separations at the structure. The vortices may interact with sea bed at the submerged structures and therefore cause erosion and scour at the foundation and undermine the structure.
Optic and acoustic measurement techniques have been applied to investigate the vortex dynamics induced by wave-structure interaction in the laboratory recently. Ting and Kim (1994) used Laser Doppler Velocimetry (LDV) to measure regular waves travelling over a submerged rectangular obstacle. Lin et al. (2006) applied Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) to study a solitary wave interacting with a bottom-mounted rectangular dike. Poupardin et al. (2012) used PIV to investigate the evolution of vortices generated by waves before and after a submerged horizontal plate.
Various submerged breakwaters have been employed to protect the coast, such as vertical, rubble mound, and circular-shaped breakwaters. Previous studies paid more attention to the flow field around the former two structures, such as vortex generation over submerged trapezoidal and rectangular dikes by Huang & Dong (1999), and vertical breakwater by Hajivalie et al. (2015), and vortex shedding from a submerged rectangular obstacle subject to a solitary wave by Lin & Huang (2010) and Zhang et al. (2010), and vortex evolution in Bragg scattering by Hsu et al. (2014). In the past decades, circular-shaped breakwaters, including semi-and quarter-circular breakwaters, have attracted considerable attentions for their aesthetically pleasing view and economic feasibility, especially in deep water. A comprehensive review of semicircular breakwaters can be found in Dhinakaran & Sundar (2012). Xie et al. (2006) developed the concept of quartercircular breakwaters in China based on semicircular breakwaters. Quartercircular breakwaters are more economical than semicircular breakwaters because they consume much less concrete and rubble mound. However, there is a lack of studies on the hydrodynamic processes of wave propagation over a quartercircular breakwater. Previous studies related to the quartercircular breakwater have been focused on wave reflection and transmission (Jiang et al., 2008;Shi et al. 2011 andHafeeda et al. 2014), wave dynamic pressures (Liu et al., 2006;Qie et al., 2013), wave run-up and run-down (Binumol et al., 2015). The generation and shedding of vortices at a submerged coastal structure have significant impact on the hydrodynamics. Circular-shaped breakwaters reflect less and transmit more wave energy than vertical and rubble mound breakwaters, therefore, may produce greater local scouring (Young & Testik, 2009) and hydrodynamic loading behind the structure . In contrary to the recommendation of existing design criteria for the semi-and quarter-circular breakwaters,  experimental and numerical studies suggest that wave trough instead of wave crest plays a dominant role in the stability of circular-front breakwaters against seaward sliding. Under wave trough, although a stronger trailing vortex is formed on the leeside of the quartercircular breakwater than the semicircular breakwater, it is away from the rear wall and thus leads to a small impact on the dynamic pressures exerted on the structure.
The present study focuses on the characteristics of flow fields around a submerged quartercircular breakwater with special attentions to the vortex dynamics in the leeside of the breakwater. In the following sections, the setup of numerical flume is first described in section 2.
Next, the numerical results are presented in section 3, including the wave generation and validation, the temporal evolution of velocity and vortex fields within a wave period, the effects of hydrodynamic parameters on vortex intensity, and the time-averaged flow field in the vicinity of the submerged quartercircular breakwater. Conclusions are summarized in section 4.

Numerical model setup
In this study, a numerical wave flume is developed to investigate the dynamics of vortex around a submerged quartercircular breakwater subject to cnoidal waves.  (1) where , i j (=1, 2) refer to the horizontal ( x ) and vertical ( z ) direction, respectively, t is time, stress computed by the nonlinear k ε − turbulent model (Lin and Liu, 1998).
In this flume, a porous beach is used to dissipate the incoming wave energy before it reaches the outflow boundary. The fluid domain within the porous media is modeled by the spatially averaged Navier-Stokes equations by including additional frictional forces in RANS (Lin and Karunarathna, 2007), where the overbar denotes the spatially averaged quantities in porous media, n the porosity, is the characteristic velocity, D C the drag force coefficient, and 50 d the median particle size of porous material. The drag force coefficient can be computed using The diameter ( d ) and porosity ( n ) of the porous material for the beach are determined by performing a series of preliminary tests. The test parameters are listed in table 2. Table 1 lists the reflection coefficients obtained from the predicted surface elevation using the three-probe method by Mansard and Funke (1980). It can be seen that over 94% incident wave energy has been eliminated before reaching the outflow boundary, which is consistent with the experiment of Chang et al. (2005). As a result, d =0.1mm and n =0.4 are used in this study.  (1) -(2) whereas that in the porous media is calculated by the spatial-averaged Navier-Stokes equations (3) -(4) and by applying the continuity of velocity and pressure as the boundary condition across the interface of porous media and outside flow. The analytical solution of third-order cnoidal wave theory by Chappelear (1962) is used to specify the velocity components and the free surface displacement at the inlet of the flume on the left-hand side. As suggested by Le Méhauté (1976), cnoidal wave theory is most adequate to describe a shallow water wave. At the outlet of the flume, a Sommerfeld radiation condition is imposed as the boundary condition. At the air-water interface, zero atmospheric pressure and zero shear stress are applied. The turbulent kinetic energy ( k ) and the dissipation rate ( ε ) have a zero normal gradient at the free surface, i.e., 0 k n ∂ ∂ = and 0 n ε ∂ ∂ = , assuming there is no turbulent exchange between air and water. On the fluid-solid interfaces, no-slip velocity and zero normal pressure gradient ( =0 p n ∂ ∂ ) are imposed.
A non-uniform rectangular grid system is used in this study with cell size varying from locally refined 0.004 m near the obstacle to coarser 0.01 m away from the obstacle in both x-and zdirections. Finite difference method is used to discretize equations (1)-(4) on the non-uniform mesh system. The movement of the free surface is captured using VOF scheme by tracking the change of the volume fraction of fluid in each cell (Hirt & Nichols, 1981). The solid boundaries of the internal obstacle are represented by the partial cell technique (Lin, 2008;Peng et al., 2018).
The RANS-VOF model has been applied to investigate wave interactions with a composite porous structure by Liu et al. (1999), wave overtopping rubble mound breakwaters by Losada et al. (2008), and a seaward sloping dike by Peng and Zou (2011), wave transformation over a low-crested structure by Zou et al. (2013) and wave loading on submerged circular breakwaters by Jiang et al. (2015.

Model validation
To validate the numerical wave flume, Chang et al's (2005) experiment for cnoidal waves propagating over a submerged rectangular obstacle is reproduced here. Fig. 2 shows good agreement between the predicted and measured surface displacements.   Wave phase A Wave phase C PIV measurements (Chang et al., 2005) COBRAS model results (Chang et al., 2005) The present model

Velocity field
To elucidate the process of flow separation at the structure, the velocity fields at four representative wave phases over a wave cycle indicated in Fig.4 are extracted from the numerical model results (see Fig. 5 and Fig. 6). downwave (same as the wave direction) towards upwave (opposite to the wave direction). As shown in Fig. 5(a), a counterclockwise vortex is created above the obstacle due to the flow separation at the crown of the breakwater. Fig. 5(b) and Fig. 5(c) illustrate the evolution of the vertical profile of horizontal and vertical velocity components in the vicinity of the leeside wall, respectively. The horizontal velocity at x=-6.3 cm in Fig. 5(b) changes from negative at z=-10 cm to positive at z=-14 cm, indicating the flow separation and hence the existence of a vortex. A spatial evolution of the vertical velocity is shown in Fig. 5(c).  The center of the vortex locates at about half of the wave particle trajectory (S=6.9 cm for test 2a in table 2) from the leeside wall of the structure. The center position of the vortex is determined with the 2 λ factor criteria developed by Jeong and Hussain (1995). As shown in Fig. 6(b), the horizontal velocity above the structure increases with increasing x and changes from positive above the vortex center to negative below it at x=1.0 cm. Fig. 6(c) shows the value of the vertical velocity below the crown of the structure is negative at x=3.6 cm. These results indicate the formation of a clockwise vortex.

Vortex intensity
In this section, the relationships between the vortex intensity and a few nondimensional hydrodynamic and wave parameters (local Reynolds number, Keulegan-Carpenter number and Ursell number) are explored based on the numerical results. We use a nondimensional parameter to represent the intensity of the vortex in order to include the effect of wave period explicitly, where Ω is the maximum vorticity magnitude, U the local characteristic velocity, T the wave period, and d the water depth. Table 2 lists the values of these parameters. We examined the instantaneous vorticity field within the area of (-12 cm<x<8 cm, -24 cm<z<-6 cm) at each time step to obtain the maximum magnitude of counterclockwise (positive) and clockwise (negative) vorticities for a certain time interval. In comparison, the maximum magnitude of vorticity is used.
In order to minimize the effect of the fluctuation of predicted vorticities with time, the maximum vorticity magnitude is derived by averaging over twenty wave periods.
As shown in Fig. 8, the nondimensional vortex intensity increases linearly with the local Reynolds number for both clockwise and counterclockwise vortices. The PIV results of Chang et al. (2005) for a submerged rectangular obstacle are also included in Fig. 8. It can be seen that the quartercircular breakwater has stronger counterclockwise vortex but weaker clockwise vortex than the rectangular obstacle for the same Reynolds numbers owing to the different shape of these structures. This behavior is worth further explorations in the future.   The predictions of the above formula are compared with the numerical results in Fig. 11. The coefficient of determination, 2 R , is 0.9917 for the clockwise vortex and 0.9173 for the counterclockwise vortex.

Time-averaged flow field
The numerical results by RANS-VOF flow model and a partial cell morphological model by Peng et al (2018) indicate that vortex and jet formation play an important role in generating scour in front of a seawall. In order to investigate the accumulated effect of the wave-induced vortices around the quartercircular breakwater on the leeside sediment movement, the time-averaged velocity field was obtained by averaging the instantaneous velocity fields over multiple wave periods (Yeganeh-Bakhtiary, et al., 2010) ( ) where ( )

Conclusion
A Reynolds-Averaged Navier-Stokes (RANS) flow solver with a Volume of Fluid (VOF) surface capturing scheme (RANS-VOF) model is used to investigate the vortex dynamics in the interaction of cnoidal waves with a submerged quartercircular breakwater.
The vertical variation of velocity indicates that the change in flow direction due to negative pressure gradient and viscous effect in the immediate vicinity of the boundary layer causes the flow separation, and hence the vortex generation around the leeside wall of the quartercircular breakwater.
The temporal evolution of the vorticity fields reveals that the generation and shedding of the vortices at the structure depend on the variation of the wave-induced oscillatory flow and the shape of the breakwater. The direction of rotation of the vortices alters between counterclockwise and clockwise when the flow changes direction. The center of the vortex is found to be confined within a wave orbital particle trajectory from the crest of the structure.
The nondimensional vortex intensity is almost linearly proportional to local Reynolds number, Keulegan-Carpenter number and Ursell number. An empirical relationship between the nondimensional vortex intensity and these hydrodynamic parameters is proposed and compares well with the numerical results.
The time-averaged velocity and vorticity fields demonstrate that the vortex pairs before and after the crest of the structure persist over many wave periods, which are expected to play an important role in sediment transport and scour around the quartercircular breakwater.