1 Introduction

Breakwaters are coastal structures built for protection against sea wave action [1]. Typically, breakwaters are hundreds to thousands of meters long, and their construction costs may exceed tens of millions of dollars [2]. These costs can be reduced using prefabricated components and modern technology [3]. Hence, the high-tech semi-circular breakwater (SBW) [4] was introduced in Japan in the 1990s due to its strength, stability, and construction advantages [5].

Careful design of breakwaters is essential to avoid the large social and economic costs of their failure. The available design options will be briefly reviewed [6]. The first option is field measurements which are expensive and difficult. For instance, field measurements for SBWs are almost limited to Miyazaki port SBW [5], despite their construction in other locations [7]. The second option is to design based on theory and experience, which generally suffers limited validity [6]. For instance, the widely accepted Goda’s formula [8] was subject to several modifications to account for SBW’s characteristics [9,10,11]. The third and last tool is numerical models, which offer cost-effectiveness and detailed design data [6]. However, the numerical models must be validated using experimental measurements and both complement each other [12].

The wave-induced flow field (WIFF) due to breakwaters is complex due to the simultaneous occurrence of strong free surface deformation and viscous rotational flow [13]. Accurate modeling of this complex WIFF is a necessary design requirement to clarify scouring effects [14], and structural stability [15]. Numerical models were adopted to study WIFF due to SBW using inviscid flow assumptions in [10], and viscous multiphase assumptions in [16]. Validation via comparison with the same experimental pressure measurements set showed discrepancies in [10] and improvement in [16]. However, the numerically simulated WIFF adjacent to SBW in [16] suffered discrepancy with the experimentally visualized free surface, and experimental verification of the subsurface velocity was missing.

In numerical models, the flow domain is discretized by a set of points called the numerical grid [17]. Cartesian grids offer simplicity and fast convergence, but unstructured boundary-fitted grids are more accurate. Unstructured grid points are located directly on the curved SBW boundary, and their spacing can be refined near high gradients and vice versa [17]. The cartesian grid model of [16] approximated the SBW boundary by stair-case-like steps. The error introduced by this approximation may have been the main reason for the discrepancy mentioned in the WIFF adjacent to SBW. A more accurate presentation of the SBW boundary was provided using the immersed boundary method [18]. Still, the immersed boundary method is inefficient for problems that require very small grid spacing in the wall-normal direction [17]. Unstructured boundary-fitted grids are available in the widely used commercial package ANSYS-FLUENT [19] and were not used yet to model WIFF due to SBW.

Particle image velocimetry (PIV) is an effective experimental method, that provides complete velocity fields using particle displacements. Hence, PIV measurements are valuable because they are suitable to clarify WIFF and other complex flows [20]. For instance, the PIV dataset of WIFF due to solitary wave propagation over a rectangular breakwater [14], was used to validate several numerical models [21, 22]. However, the PIV measurements for WIFF due to SBW of [23] have not been used yet to validate numerical models.

The present study aims to address the aforementioned numerical and experimental gaps. Numerical simulations are performed using ANSYS-FLUENT, and an unstructured grid is adopted to accurately model the curved SBW boundary. In the next section, the model and the detailed design of the unstructured grid are provided, and the PIV experimental setup is explained. In Sect. 3, validation is performed using the PIV measurements of [23], and the resolution of the numerical results adjacent to SBW is illustrated. The ability to provide novel explanations of WIFF due to SBW is summarized in Sect. 4.

2 Research methodology

2.1 Numerical simulation

2.1.1 Governing equations

The wave-induced flow field (WIFF) is modeled by adopting the assumptions of two-dimensional, incompressible, turbulent, and multiphase flow. The Navier Stokes equations are given by

$$\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} = 0,$$
(1)
$$\rho \left( {\frac{\partial u}{{\partial t}} + u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}}} \right) = - \frac{\partial P}{{\partial x}} + \left( {\mu + \mu_{t} } \right)\left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right),$$
(2)
$$\rho \left( {\frac{\partial v}{{\partial t}} + u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}}} \right) = - \frac{\partial }{\partial y}\left( {P + \rho gy} \right) + \left( {\mu + \mu_{t} } \right)\left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right).$$
(3)

Equation (1) is the continuity equation. Equations (2) and (3) are the horizontal and vertical momentum equations, respectively. Here, \(x\) and \(y\) stand for horizontal and vertical space coordinates, \(u\) and \(v\) stand for the horizontal and vertical velocity components, \(t\) represents time, \(\rho\) and \(\mu\) are the fluid density and viscosity, respectively. The right-hand side of the momentum equations includes the effects of the pressure \(P\) gradients, body forces (\(- \rho gy\)), laminar viscosity \(\mu\) and turbulence eddy viscosity \(\mu_{t}\).

The \(k - \omega - SST\) turbulence model [24] was designed to yield improved simulations, and was adopted to simulate modern free surface flow applications [25, 26]. Hence, it is selected in the current work. The model equations ate given by

$$\frac{\partial }{\partial t}\left( {\rho k} \right) + \frac{\partial }{{\partial x_{i} }}\left( {\rho ku_{i} } \right) = \frac{\partial }{{\partial x_{j} }}\left( {\Gamma_{k} \frac{\partial k}{{\partial x_{j} }}} \right) + G_{k} - Y_{k} + S_{k} ,$$
(4)
$$\frac{\partial }{\partial t}\left( {\rho \omega } \right) + \frac{\partial }{{\partial x_{i} }}\left( {\rho \omega u_{i} } \right) = \frac{\partial }{{\partial x_{j} }}\left( {\Gamma_{\omega } \frac{\partial \omega }{{\partial x_{j} }}} \right) + G_{\omega } - Y_{\omega } + S_{\omega } .$$
(5)

Here, \(k\) and \(\omega\) stand for turbulence kinetic energy and specific dissipation, respectively. The symbols \(G_{k}\) and \(G_{\omega }\) represent generation terms, \(\Gamma_{k}\) and \(\Gamma_{\omega }\) represent effective diffusivity, and \(Y_{k}\) and \(Y_{\omega }\) represent the dissipation of \(k\) and \(\omega\), respectively. The eddy viscosity is denoted by \(\mu_{t}\) and calculated as \(\mu_{t} = \rho \frac{k}{\omega }.\)

The air–water interface is tracked using the volume fraction equation represented by:

$$\frac{\partial }{\partial t}\left( {\alpha \rho } \right) + \nabla .\left( {\alpha \rho \overline{v}} \right) = 0$$
(6)

Here, \(\alpha\) represents the volume fraction whose value is: \(\alpha = 0\) for air, \(0 < \alpha < 1\) for the interface cells, and \(\alpha = 1\) for water.

Equation (7) describes how material properties have been calculated, whereas \(\rho\) can be replaced by \(\mu\).

$$\rho = \alpha \rho_{2} + \left( {1 - \alpha } \right)\rho_{1}$$
(7)

Values of fluid properties were selected corresponding to water properties: \(\rho_{2} = 998.2 \,{\text{kg m}}^{ - 3}\), and \(\mu_{2} = 1.003 \times 10^{ - 3} {\text{kg m}}^{ - 1} {\text{ s}}^{ - 1}\), and air properties: \(\rho_{1} = 1.225 \,{\text{kg m}}^{ - 3}\), and \(\mu_{1} = 1.7894 \times 10^{ - 5} {\text{kg m}}^{ - 1} {\text{ s}}^{ - 1}\).

2.1.2 Boundary and initial conditions

The following types of boundary conditions available in ANSYS-FLUENT [25] are imposed:

  • Wave inlet condition is imposed on the left side of the domain, adopting Stokes's second-order wave theory formulas.

  • Pressure exit condition is imposed at the upper boundary.

  • Slip wall conditions are adopted at the bottom boundaries.

  • The no-slip wall condition is adopted at the SBW boundary.

The boundary conditions at the right sloped region are considered carefully to avoid wave reflection as follows:

  • Like a natural beach, a sloped bed of \(3{\text{m}}\) in length is defined. Hence, as waves propagate towards this region, breaking occurs and energy is dissipated.

  • A relatively coarse grid is adopted at this beach region to increase dissipation and almost eliminate wave reflection. Wave breaking at the beach is out of the current scope, and the reduced resolution due to this coarse grid is justified.

  • A numerical damping beach zone is defined at this sloped bed region using the option available in ANSYS-FLUENT [19]. The Linear and quadratic damping coefficients are assigned to \(100 {\text{s}}^{ - 1}\) and \(100 {\text{m}}^{ - 1}\), respectively.

The initial values of \(k\) and \(\omega\) are set carefully as follows. The values of \(k\) and \(\omega\) should be high enough to avoid instability, and the ratio \(\frac{k}{\omega }\) should not be too high to avoid extra damping. The initial values of the turbulence model parameters were set to \(k = 1 \times 10^{ - 3} {\text{m}}^{2} {\text{s}}^{ - 2}\) and \(\omega = 500\, {\text{s}}^{ - 1}\).

Turbulence wall boundary conditions are imposed using enhanced wall functions. Hence, the restriction of an extremely fine near-wall grid is avoided, and the computational requirements are reduced without sacrificing accuracy.

2.1.3 Numerical scheme

The numerical algorithms adopted to solve the flow equations are explained briefly, and the reader may refer to [19] for more details. The free surface is tracked using the volume of fluid (VOF) method with implicit formulation and implicit body force. The second-order upwind scheme was adopted to discretize the momentum convection terms. The pressure was interpolated using the PRESTO method. The least-squares cell-based method was adopted to evaluate gradients. The second-order implicit method was used for time integration to improve accuracy. A constant time step of \(1 \times 10^{ - 3}\) s was adopted.

2.1.4 Domain

The model domain, including a semi-circular breakwater (SBW), is shown in Fig. 1 with a flat-bed of length \(10.5 \;{\text{m}}\) followed by a sloped bed of length \(3 {\text{m}}\) and slope angle \(\theta = \tan^{ - 1} \left( {0.45/3} \right)\). This domain is large enough to minimize wave reflection effects for all simulated cases. The Particle Image Velocimetry (PIV) measurements [23] are provided in a square window whose location is indicated in Fig. 1. Validation is done using the velocity profiles on the vertical and horizontal centerlines of the measurement window.

Fig. 1
figure 1

Semi-circle domain

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2.1.5 Numerical grid

The numerical grid is shown in Fig. 2. The grid is designed to satisfy accuracy and computational efficiency. The following techniques ae adopted:

  • Subdividing the domain into separate zones.

  • Adopting a refined grid at regions of strong gradients and high influence.

  • Using a coarse grid in areas of low gradients or low influence.

  • Assuring suitable values of grid aspect ratio to avoid numerical difficulties [17].

Fig. 2
figure 2

Grid of the domain (not to scale)

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The domain is divided into the following four zones:

  • Zone A is located at the flume inlet above the horizontal bottom. A relatively simple structured grid type was adopted. However, stretching is adopted to provide vertical refinement near the free surface to catch the wave profile accurately. On the other hand, the bed boundary layer details are out of the current scope, and the influence of the upper boundary exit flow on the WIFF should be negligible. Hence, a coarse vertical grid is adopted near the lower and upper boundaries. The maximum grid aspect ratio was 20 to avoid numerical instability.

  • Zone B is located adjacent to the SBW. The zone width covers about \(7\) flume depths. This dimension is large enough to capture all the complex phenomena of the WIFF. The crucial role of the unstructured boundary-fitted grid becomes evident in this zone. The curved boundary of the SBW is modeled accurately. A very fine resolution is adopted adjacent to the SBW, with a minimum grid spacing of \(0.01 \times R\). The fine grid in the normal-wall direction is illustrated in Fig. 2. A coarse grid is adopted in the air region adjacent to the upper boundary, due to its negligible effect. The unstructured grid accommodates the mentioned coarsening and refining. In addition, the unstructured grid is easily matched with the structured grid at zone A.

  • Zone C is located to the right of zone B, at the downstream side of the SBW. The design of zone C follows the same details of zone A.

  • Zone D is located at the sloped beach region. A relatively coarse unstructured grid is adopted without refinement at the free surface. This grid contributes to damping the wave energy and reducing reflection, and reducing the computational requirements. Again, the unstructured grid is easily matched with the structured grid at zone C.

2.2 PIV experimental setup

Complete details of the Particle Image Velocimetry (PIV) experiments are available in [23]. However, a brief discussion is provided in this section for convenience. The experiments were performed using the wave flume available in Bergische Universität Wuppertal—the hydraulic engineering section. The flume's total length is 24 m, width 0.3 m, and depth 0.5 m. The piston wave maker generates sinusoidal waves for a fixed water depth of 0.3 m.

The experimental setup is presented in Fig. 3. The basic idea of PIV measurements is seeding the flow with small particles that move with the same fluid speed. The motion of these particles is recorded by a suitable camera and the fluid motion is detected. Once the PIV images are recorded, they are processed using a suitable program to obtain the velocity vectors. The high-speed video camera MotionScope M3 was used. The capturing speed was adjusted to 120 frames per second, with a resolution of 1280 × 1024 pixels. The images were processes using the open-source program MatPIV [27]

Fig. 3
figure 3

The PIV experimental setup. The far left picture is an image of the system. The top right figure is the description of the main system elements, lower left figure is an image of the particles

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Dry particles dropped on the water's free surface would stay floating due to the surface tension. In addition, once the particles penetrated the surface, they sank slowly due to their relatively heavy weight. Hence, the seeding device shown in Fig. 4, was developed to solve these difficulties. The device keeps the particles wet to avoid the surface tension effects. Small aquarium pumps were used to provide a low water supply and drip the particles continuously at the measurement section during the wave propagation.

Fig. 4
figure 4

The particle seeding device

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The turbulence effects introduced considerable noise to the measurements. To minimize this noise the phase average was adopted as follows. Measurements were recorded for nine consecutive waves, and the velocity was obtained as the median value of these nine values. As illustrated in Fig. 1, measurements were recorded in a window of size \(0.1\,{\text{m}} \times 0.1\,{\text{m}}\).

3 Results and discussion

In this section, the WIFF due to the SBW is simulated for several cases. The wave parameters of each case are provided in terms of dimensionless quantities. The following starred dimensionless variables are defined: \(H^{*} = H/h\), \(L^{*} = L/h\), \(R^{*} = R/h\). Here, \(h, H\), \(L\), and \(R\) stand for the water depth, wave height, wavelength, and SBW radius, respectively. The dimensionless period \(T^{*}\) is defined as \(T^{*} = T\sqrt {g/h}\). The wave period \(T\) is related to \(L\) through the dispersion relation given by [28]

$$\frac{{4\pi^{2} }}{{T^{2} }} = g\frac{2\pi }{L}{\text{tanh}}\left( {2\pi \frac{h}{L}} \right)$$
(8)

The normalized horizontal and vertical velocity components are defined as \(u^{*} = u/C\) and \(v^{*} = v/C\). Here, \(C = L/T\) is the wave celerity.

The water depth \(h\) is fixed for all cases to \(0.3 m\). Simulations are performed for two SBW sizes \(R^{*} = 2/3, 1\). For each size, \(16\) cases are simulated, with a total of \(32\) cases. The computational time for each case was about 10 h on a PC with an Intel Core i5-6500 processor and 8 GB RAM.

Simulations are performed for 32 cases. The wave parameters are selected to match the conditions of the PIV data provided in [23]. In addition, the values of \(T^{*}\), \(L^{*}\) and \(R^{*}\) belong to the range adopted in the large-scale flume data set [10]. However, the values of \(H^{*}\) are relatively smaller. Hence, an extra case will be added to the next section. Tables 1 and 2 show the parameters of all cases. The underlined cases indicate the cases used in the validation phase.

Table 1 The small semi-circle cases’ parameters (R* = 2/3)-(the validated cases are bolded)
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Table 2 The large semi-circle cases’ parameters (R* = 1)-(the validated cases are bolded)
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3.1 Flat bottom

Simulations are provided for a flat bottom to assure that periodic results are obtained. Results are shown for case 16. The free surface profile is provided in Fig. 5. The contours of velocity magnitude, \(P\), \(k\), and \(\omega\) are provided in Fig. 6. The spatial periodicity of all quantities is illustrated. The strong dissipation at the right beach can be observed in \(k\) and \(\omega\) contours.

Fig. 5
figure 5

Free surface profile, case 16 with a flat bottom

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Fig. 6
figure 6

The contours of velocity magnitude, pressure, turbulence kinetic energy \({\text{k}}\) and specific dissipation \({\upomega }\) case 16 with a flat bottom

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3.2 Semi-circular breakwater

3.2.1 Velocity profiles

Experimental and numerical profiles for \(u^{*}\) and \(v^{*}\) of the WIFF due to SBW are plotted in Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 for several cases. The profiles are provided at the PIV measurement window center lines. The origin of the plots is located at the lower-left corner of the measurement window. The profiles are plotted at six instants for each case to verify temporal resolution within one period.

Fig. 7
figure 7

Horizontal velocity u* profile, Case 10—Experimental × and Numerical —

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Fig. 8
figure 8

Vertical velocity v* profile, Case 10—Experimental × and Numerical —

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Fig. 9
figure 9

Horizontal velocity u* profile, Case 14—Experimental × and Numerical —

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Fig. 10
figure 10

Vertical velocity v* profile, Case 14—Experimental × and Numerical —

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Fig. 11
figure 11

Horizontal velocity u* profile, Case 15—Experimental × and Numerical —

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Fig. 12
figure 12

Vertical velocity v* profile, Case 15—Experimental × and Numerical —

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Fig. 13
figure 13

Horizontal velocity u* profile, Case 30. Experimental × and Numerical —

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Fig. 14
figure 14

Vertical velocity v* profile, Case 30—Experimental × and Numerical —

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Fig. 15
figure 15

Horizontal velocity u* profile, Case 31—Experimental × and Numerical —

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Fig. 16
figure 16

Vertical velocity v* profile, Case 31—Experimental × and Numerical —

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The flow reversal in the horizontal and vertical directions is well-predicted. For instance, the instants of horizontal reversal are identified at phases \(330^{o}\) and \(300^{o}\) in Figs. 7 and 9, respectively. The vertical reversal instant is identified at phase 180 in Fig. 12. The trends of the numerical and experimental profiles are generally consistent. For instance, both the numerical and experimental profiles provided in Fig. 7 are proportional to the vertical coordinate. Hence, spatial resolution is clarified.

The discrepancy between numerical and experimental results may be attributed to the following deficiencies:

  • The PIV experimental setup is based on volume illumination using Halogen lights instead of using a laser light sheet. Hence, the particles are tracked from several planes which may contribute to the measurement noise.

  • Numerical simulations are performed assuming two-dimensional flow and neglecting three-dimensional effects. However, three-dimensional velocity fluctuations should occur due to the flow turbulence [17].

  • Despite the important advantages offered by the \(k - \omega - SST\) turbulence model [24], it may yield artificial wave damping due to the large production of \(k\) at the air–water interface [29]. This feature, which is clarified in Fig. 6, may deteriorate the accuracy of the numerical simulations.

3.2.2 Turnover—trough breaking

For a large SBW, violent free surface deformation on the upstream side may occur. To investigate this phenomenon, an extra case is added adopting the following parameters: \(H^{*} = 0.4\), \(R^{*} = 1.15\) and \(T^{*} = 7.754\). The value of \(H^{*} = 0.4\) is high enough to match the data set used in [10]. The results are provided in Fig. 17 at five equally spaced instants during one period. The experimentally visualized free surface is provided in column (ii). The current numerical model results are provided for the free surface and the velocity vectors imposed over \(k\) contours, in columns (iii) and (iv), respectively. The numerically simulated free surface of [16] is provided in column (i).

Fig. 17
figure 17

Experimental and numerical simulation of WIFF adjacent to SBW at five successive time steps with \(\Delta t = T/5\). Detailed description of the results are provided in text

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The five instants illustrate the periodic evolution of the WIFF. As the wave crest approaches the SBW directed from left to right (Fig. 17a), the water climbs the SBW driven by inertia (Fig. 17b). As the trough approaches the flow reversal starts (Fig. 17c), and the reversal is intensified by the fast water descending from the SBW due to gravity (Fig. 17d). A skinny, thin water layer adjacent to the SBW can be observed, coexisting with a turnover of the free surface. This phenomenon is experimentally visualized in column (ii), and numerically reproduced in column (iii). The turnover induces localized intensive turbulence as shown by the contours of \(k\) in column (iv), and a bubble of rotational flow coexisting with free surface turnover is detected. Finally, the bubble is washed out by the approaching crest at the last instant (Fig. 17e). The numerical results of [16] suffer low accuracy at the last two instants (Fig. 17d and e).

To the authors’ best knowledge, the phenomenon that is illustrated in Fig. 17d and e is numerically reproduced for the first time. The authors denote this phenomenon as “trough-breaking”. This phenomenon is driven by three factors: wave trough, gravity force, and the curved SBW boundary, and contributes to strong dissipation of the incoming wave.

Trough breaking is illustrated for various cases in Fig. 18. The origin is located below the free surface by \(0.167\) units (\(5\;{\text{cm}}\)) at the center line of the SBW. All large SBW cases are included except case 17 since trough breaking did not occur. For each case, three successive instants separated by the short time interval \(\Delta t = T/36\) are plotted. To resolve trough-breaking, the results are provided in a zoomed window adjacent to the SBW, and the thin fluid layer descending from the SBW is clarified. The model capability to model this feature is mainly attributed to the fine grid adopted adjacent to the SBW.

Fig. 18
figure 18

Turnover cases for three consecutive instants for four different wave periods: (i) \({\text{T}}^{*} = 4.575\), (ii) \({\text{T}}^{*} = 5.147\), (iii) \({\text{T}}^{*} = 5.718\), (iv) \({\text{T}}^{*} = 7.623\).. The origin is located below the free surface by \(0.167\) units (\(5{\text{ cm}}\)) at the center line of the SBW

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It should be noted that the main dynamics identified leading to trough breaking is water climbing the SBW and then descending back simultaneously with the wave trough. Hence, the trough-breaking intensity is proportional to wave energy which is, in turn, proportional to both \(H^{*}\) and \(T^{*}\). As a result, trough-breaking intensifies as both \(H^{*}\) and \(T^{*}\) increase, and the rotational flow bubble is driven lower. The effect of increasing \(H^{*}\) while fixing \(T^{*}\) is illustrated in each row Fig. 18. For instance, row (ii) displays the free surface for \(T^{*} = 5.147\) and four values of \(H^{*}\), and trough-breaking occurs in the lowest point for the highest \(H^{*}\). The effect of increasing \(T^{*}\) is clarified by considering any single column. For instance, in column (c) \(H^{*} = 0.167\) is fixed for cases 20, 24, 28, and 32, and \(T^{*} = 4.575, 5.147, 5.718\), and \(6.753\), respectively. The trough-breaking occurs at the lowest point for case 32 where \(T^{*}\) is maximum.

Trough-breaking provides novel explanations for the following two features of SBW:

  1. 1.

    Low wave reflectivity is observed at the upstream side [5] of an SBW and the disturbance to moving vessels is avoided [4]. One factor that contributes to this advantage is the strong trough-breaking dissipation that is localized adjacent to the SBW.

  2. 2.

    Generally, the wave drag in the left direction (seaward) is higher than that in the right direction (shorewardward) [30]. This asymmetry should be attributed to the strong decrease in the water level at the upstream side due to the trough-breaking.

Finally, it should be noted that the intensive rotational flow of the trough-breaking may have a strong effect on scouring and stability at the upstream side. This feature should be well considered in future projects of construction and upgrade of SBWs.

4 Conclusions

Numerical simulations of the wave-induced flow field near a semi-circular breakwater (SBW) are performed. The curved boundary of the SBW is described accurately using an unstructured boundary-fitted grid. To complement the numerical simulations, a PIV dataset is introduced and used for validation for the first time.

Extensive simulations are provided for several cases, illustrating agreement with the velocity field and the free surface experimental data. The phenomenon of strong free surface deformation adjacent to the SBW, denoted by trough-breaking, is reproduced numerically for the first time. The trough-breaking mechanism is explained, and the influence of the wave parameters is clarified. The impact of trough-breaking dynamics on SBW characteristics is explained. The study outcomes can be summarized in the following items:

  • The numerical simulations accuracy is considerably improved upon adopting unstructured boundary-fitted grids. Achieving the same accuracy using cartesian grids can be very challenging.

  • Fair agreement between the PIV dataset and the numerical simulations is illustrated. Hence, this PIV dataset should be used to validate future numerical models.

  • The localized trough-breaking results in simultaneous strong dissipation and rotational flow. The dissipation accounts for several advantages of SBW. However, the effects of the rotational flow should be carefully considered during the design of the SBW upstream armoring.

  • The numerical model complemented with the experimental data set can be used to perform accurate SBW designs. Hence, the deficiencies of other design tools are avoided.

  • Numerical simulations can be performed in the future to analyze more complex designs, including perforated breakwaters, flexible structures, and wave energy converters.