China Ocean Eng., 2018, Vol. 32, No. 5, P. 514–523 DOI: https://doi.org/10.1007/s13344-018-0054-5, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: [email protected] Flow Separation and Vortex Dynamics in Waves Propagating over A Submerged Quartercircular Breakwater JIANG Xue-liana, b, YANG Tiana, ZOU Qing-pingc, *, GU Han-bind aTianjin Key Laboratory of Soft Soil Characteristics & Engineering Environment, Tianjin Chengjian University, Tianjin 300384, China bState Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China cThe Lyell Centre for Earth and Marine Science and Technology, Institute for Infrastructure and Environment, Heriot-Watt University, Edinburgh, EH14 4AS, UK dSchool of Naval Architecture &Mechanical-Electrical Engineering, Zhejiang Ocean University, Zhoushan 316022, China Received October 22, 2017; revised June 7, 2018; accepted July 5, 2018 ©2018 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature Abstract 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. Key words: submerged quartercircular breakwater, cnoidal wave, flow separation, vortex dynamics, scour Citation: Jiang, X. L., Yang, T., Zou, Q. P., Gu, H. B., 2018. Flow separation and vortex dynamics in waves propagating over a submerged quartercircular breakwater. China Ocean Eng., 32(5): 514–523, doi: https://doi.org/10.1007/s13344-018-0054-5 1 Introduction Kim (1994) used Laser Doppler Velocimetry (LDV) to Offshore submerged breakwaters have recently become measure regular waves travelling over a submerged rectan- popular coastal defence to protect maritime infrastructure gular obstacle. Lin et al. (2006) applied Particle Image Ve- and retain sediments in the sheltered harbor through prema- locimetry (PIV) and Laser Induced Fluorescence (LIF) to ture wave breaking (Zuo et al., 2015; Wang et al., 2016; study a solitary wave interacting with a bottom-mounted Zheng et al., 2016; Ju et al., 2017). Better understanding of rectangular dike. Poupardin et al. (2012) used PIV to invest- the flow field around submerged offshore breakwaters, es- igate the evolution of vortices generated by waves before pecially small-scale hydrodynamic phenomena, is critical to and after a submerged horizontal plate. improve the design of breakwaters. Recent development of novel computational fluid dy- Generation and shedding of vortices occur during the in- namics (CFD) technique for free surface flow has been ap- teraction of waves with submerged structures due to flow plied to study hydrodynamics around submerged structures. separations at the structure. The vortices may interact with Examples include the Reynolds-Averaged Navier–Stokes the sea bed at the submerged structures and therefore cause (RANS) solver by Chang et al. (2001), Hsu et al. (2004), Ji- erosion and scour at the foundation and undermine the ang et al. (2010), Zarruk et al. (2015), the boundary ele- structure. ment method (BEM) by Ning et al. (2014), Wang and Optic and acoustic measurement techniques have been Zhang (2018), the Smoothed Particle Hydrodynamics (SPH) applied to investigate the vortex dynamics induced by wave- flow model by Shao (2010) and Vandamme et al. (2011), structure interaction in the laboratory recently. Ting and and the meshless Lagrangian vortex method by Lin and Foundation item: The study was financially supported by the National Natural Science Foundation of China (Grant Nos. 51509178 and 51509177), the Natural Science Foundation of Tianjin City (Grant No. 14JCYBJC22100), and the Natural Science Foundation of Tianjin Education Commission (Grant No. 2017KJ046). *Corresponding author. E-mail: [email protected] JIANG Xue-lian et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 514–523 515 Huang (2009), and Chang et al. (2015). vortex fields within a wave period, the effects of hydro- Various submerged breakwaters have been employed to dynamic parameters on the vortex intensity, and the time- protect the coast, such as vertical, rubble mound, and circu- averaged flow field in the vicinity of the submerged quarter- lar-shaped breakwaters. Previous studies paid more atten- circular breakwater. Conclusions are summarized in Sec- tion to the flow field around the former two structures, such tion 4. as vortex generation over submerged trapezoidal and rectan- gular dikes by Huang and Dong (1999), and vertical break- 2 Numerical model setup water by Hajivalie et al. (2015), and vortex shedding from a In this study, a numerical wave flume is developed to in- submerged rectangular obstacle subject to a solitary wave vestigate the dynamics of vortex around a submerged by Lin and Huang (2010) and Zhang et al. (2010), and vor- quartercircular breakwater subject to cnoidal waves. Fig. 1 tex evolution in Bragg scattering by Hsu et al. (2014). In the illustrates the configuration of the numerical flume follow- past decades, circular-shaped breakwaters, including semi- ing the PIV experiment by Chang et al. (2005) for cnoidal and quarter- circular breakwaters, have attracted consider- waves propagating over a submerged rectangular obstacle. able attention for their aesthetically pleasing view and eco- The flume is 36 m long, 0.6 m wide and 0.9 m deep. x co- nomic feasibility, especially in deep water. A comprehens- ordinate is positive in the wave propagation direction with ive review of semicircular breakwaters can be found in Dh- x=0 at the back wall of the structure and z coordinate is pos- inakaran et al. (2012). Xie et al. (2006) developed the itive upward with z=0 at the still water level. An imper- concept of quartercircular breakwaters in China based on meable quartercircular breakwater with a size of 0.12 m in semicircular breakwaters. Quartercircular breakwaters are radius (D), 0.175 m in width (B) is placed in the middle of more economical than semicircular breakwaters because the flume. A porous beach with a 1/10 seaward slope is placed they consume much less concrete and rubble mound. 6 m upwave towards the wavemaker from the outflow However, there is a lack of studies on the hydrodynamic processes of wave propagation over a quartercircular break- water. Previous studies related to the quartercircular break- water have been focused on the wave reflection and trans- mission (Jiang et al., 2008; Shi et al., 2011; Hafeeda et al., 2014), wave dynamic pressures (Liu et al., 2006; Qie et al., 2013), wave run-up and run-down (Binumol et al., 2015). Fig. 1. Sketch of the configuration of the numerical wave flume with a submerged quartercircular breakwater and the coordinate system. The generation and shedding of vortices at a submerged coastal structure have significant impact on the hydro- boundary following Chang et al.’s (2005) experiment. dynamics. Circular-shaped breakwaters reflect less and The motion of incompressible fluid in the numerical transmit more wave energy than vertical and rubble mound flume is described by the Reynolds-Averaged Navier– breakwaters, therefore, may produce greater local scouring Stokes equations (RANS) (Young and Testik, 2009) and hydrodynamic loading be- ∂u hind the structure (Jiang et al., 2017b). In contrary to the re- i = 0; (1) commendation of existing design criteria for the semi- and ∂xi ! quarter-circular breakwaters, Jiang et al.’s (2017a) experi- ∂ui + ∂ui = −1 ∂p + + 1 ∂ ∂ui − 0 0 ; u j gi μ ρui u j (2) mental and numerical studies suggest that wave trough in- ∂t ∂x j ρ ∂xi ρ ∂x j ∂x j stead of wave crest plays a dominant role in the stability of where i; j (=1, 2) refer to the horizontal (x) and vertical (z) circular-front breakwaters against seaward sliding. Under direction, respectively; t is time; u is the mean velocity in wave trough, although the trailing vortex formed on the i the i-th direction; ρ is the fluid density; p is the fluid pres- leeside of the quartercircular breakwater is stronger than sure; gi is the gravitational acceleration in the i-th direction; that formed on the leeside of the semicircular breakwater, it 0 0 μ is the dynamic viscosity; and −ρu u is the Reynolds is away from the rear wall and thus leads to a small impact i j − on the dynamic pressures exerted on the structure. stress computed by the nonlinear k ε turbulent model (Lin The present study focuses on the characteristics of flow and Liu, 1998). fields around a submerged quartercircular breakwater with In this flume, a porous beach is used to dissipate the in- special attention to the vortex dynamics in the leeside of the coming wave energy before it reaches the outflow bound- breakwater. In the following sections, the setup of numeric- ary. The fluid domain within the porous media is modeled al flume is first described in Section 2. Next, the numerical by the spatially averaged Navier–Stokes equations by in- results are presented in Section 3, including the wave gener- cluding additional frictional forces in RANS (Lin and Kar- ation and validation, the temporal evolution of velocity and unarathna, 2007), 516 JIANG Xue-lian et al. China Ocean Eng., 2018, Vol. 32, No. 5, P. 514–523 ∂u lists the reflection coefficients obtained from the predicted i = 0; (3) ∂xi surface elevation using the three-probe method by Mansard ! and Funke (1980).