Scale effects on wave forces and flow structures around a perforated semicircular submerged breakwater

Perforated semicircular submerged breakwaters can attenuate incident waves while allowing water exchange, making them suitable for coastal protection and marine infrastructure. Their hydraulic performance is often assessed using physical models based on Froude similarity, for which Reynolds similarity cannot be preserved. Extrapolating wave loads from reduced models to prototype conditions may therefore introduce bias. To quantify this effect, a validated numerical wave flume was used to examine six geometric scale ratios (λ L = 1, 5, 10, 15, 25, and 35), four perforation rates (ϵ = 0–15%), and five wave conditions. Validation used measurements from physical model experiments and field pressures obtained at prototype scale. Comparisons considered horizontal force coefficients, free surface elevations, velocity fields, and vorticity fields. Smaller models increasingly overestimated the shoreward peak force. At λ L = 35 and ϵ = 10%, the mean model to prototype ratio of the shoreward peak force coefficient reached 1.23. The differences between model and prototype responses were mainly associated with changes in wave transmission, reverse flow, and vortical motion over and behind the structure as scale decreased. These findings improve the interpretation of reduced model results and support more reliable prototype extrapolation of wave loads on perforated submerged structures.

Coastal protection requires structures that reduce wave energy without unnecessarily restricting water exchange or altering nearshore processes. Recent coastal engineering research has increasingly considered multifunctional infrastructure that combines hazard mitigation with broader environmental and operational functions (Firth et al., 2024). Modern breakwater design considers basin tranquility, circulation, sedimentation, navigation, and habitat-related functions in addition to wave attenuation (Cox and Czlapinski, 2016). Submerged and perforated breakwaters are compatible with this approach because they can dissipate wave energy while retaining partial transmission and water exchange. A semicircular breakwater mainly consists of a curved concrete shell and a base slab. Its curved surface distributes wave pressure and can reduce horizontal force and overturning moment relative to a vertical wall (Tanimoto and Takahashi, 1994; Tanimoto et al., 1987). Under suitable conditions, submerged semicircular breakwaters exhibit favorable hydrodynamic performance (Young and Testik, 2011; Jiang et al., 2017; Al-Towayti and Teh, 2024; Al-Towayti et al., 2025). These characteristics make the perforated configuration a promising option for coastal protection and marine infrastructure, provided that its hydraulic performance and structural loads can be evaluated reliably.

The hydrodynamic response of semicircular breakwaters has been examined through experiments, analytical solutions, and numerical models. Yuan and Tao (2003) studied submerged, alternately submerged, and emerged configurations and proposed a simplified wave force formula. Dhinakaran et al. (2002) measured dynamic pressure and resultant force on impermeable and seaward side perforated structures. Liu and Li (2012) analyzed a submerged perforated semicircular breakwater using a multipole method, which was later extended to partially perforated structures by Lyu et al. (2020). Other studies addressed porous resistance under solitary waves (Zhao et al., 2021), CFD based pressure and force prediction (Gomes et al., 2020), and multiple semicircular breakwaters protecting floating facilities (Venkateswarlu et al., 2023). Previous studies have established how wave conditions and perforation affect the global structural response. Perforation, however, also introduces local processes that are important for both load reduction and water exchange. Oscillatory jets form at the openings, while cavity return flow changes the pressure difference between the seaward and leeward sides. These processes create strong velocity gradients, vorticity, and energy dissipation. Numerical simulations and particle image velocimetry show that curved boundaries affect flow separation, near wall velocity, and vortex development (Gomaa et al., 2023). Surface configuration can also modify wave breaking and turbulence (Romya et al., 2022). Studies of porous breakwaters indicate that the timing of local velocity, wave run down, and pressure gradients can control critical loading (Rahmani et al., 2023). Although hydrodynamic and seabed responses around marine structures have been widely investigated (Teh et al., 2023; Wang et al., 2025; Yao et al., 2026), corresponding studies of scale effects remain limited, particularly with respect to the similitude of local flow and structural loading. Quantifying these local processes is therefore important for evaluating the hydraulic response and structural loads of perforated submerged structures.

Physical models of wave interaction with marine structures usually follow Froude similarity because gravity and inertia dominate free surface motion (Heller, 2011). Wave impact pressure may nevertheless show scale dependent deviations during prototype extrapolation (Cuomo et al., 2010). Recent numerical and experimental coastal studies also illustrate the need to represent wave transformation and nearshore response accurately (Zhang et al., 2026; Yuan et al., 2026). When the same fluid is used in the model and prototype, Reynolds similarity cannot be satisfied simultaneously with Froude similarity. The Reynolds number decreases as the model becomes smaller, so viscous effects become relatively stronger (Jensen and Klinting, 1983). This issue is particularly important for perforated structures because resistance depends on opening size, local velocity, boundary layer development, and flow regime. Scale dependent friction has been reported in oscillatory porous flow (Pérez-Romero et al., 2009), while scale effects in wave reflection have been observed for low reflective quay walls (Altomare and Gironella, 2014). Major breakwater projects still rely on laboratory testing under severe wave conditions (Lauridsen et al., 2024). For perforated structures, scale distortion can affect both predicted loads and the interpretation of local flow through the openings. Prototype performance cannot be assessed confidently without quantifying the distortion introduced at reduced model scales.

Direct studies of scale effects in perforated coastal structures remain limited. Jin et al. (2024) showed that geometric scale affects velocity, vorticity, reflection, and force around perforated caissons. However, these findings were obtained for perforated caissons, whose flow configuration differs from that of a fully submerged perforated semicircular breakwater. Research on perforated semicircular breakwaters has investigated the interaction among waves, the structure, and the seabed (Gao et al., 2024), yet the hydrodynamic discrepancies between model and prototype scales have not been systematically examined. A fully submerged semicircular shell is distinct because curved surface flow, crest overflow, perforation jets, and cavity return flow occur simultaneously. The interaction of these processes may lead to scale dependent force responses that cannot be inferred directly from studies of conventional perforated caissons. Its horizontal force dependence on geometric scale has not been linked systematically to global wave motion and local flow structures. The combined effects of perforation rate, relative wave height, and relative water depth also remain unclear. Accordingly, the main gap addressed here is the link between model scale force bias and the scale dependent evolution of local flow through and around the perforated semicircular structure. The absence of such evidence limits the interpretation of results obtained from reduced models of perforated submerged structures. This study employs a numerical wave flume validated against physical model data to investigate hydrodynamic scale effects on a perforated semicircular submerged breakwater. The numerical model is validated against physical model data at two selected scales and field measurements obtained at prototype scale. Six numerical scale ratios ranging from λ L = 1 to 35 are then considered in the scale effect analysis. The analysis quantifies Reynolds number distortion and examines shoreward and seaward horizontal forces under different wave and perforation conditions. Free surface elevations, velocity fields, and vorticity fields are used to identify the hydrodynamic origin of the force deviations. By linking the force bias between model and prototype with local flow through the perforations, the study provides a basis for interpreting reduced model results and for more reliable prototype extrapolation of wave loads on perforated submerged structures.

Similarity criteria and dimensionless parameters

The geometric scale ratio is defined in Equation 1 as:

where the subscripts p and m denote the prototype and model, respectively. Accordingly, λ L = 15, 25, and 35 denote the corresponding scaled models. Because gravity and inertia dominate free surface wave motion, Froude similarity was adopted according to Equation 2 (Heller, 2011):

where g is gravitational acceleration, D is the breakwater diameter, and U m is the characteristic velocity amplitude calculated from linear wave theory at the geometric center of the semicircular breakwater. The same characteristic velocity U m was used to calculate Fr, Re, and KC. Here, U m is used only as a common reference velocity for the global dimensionless parameters, while the actual wave kinematics in the simulations are represented by cnoidal waves. Under Froude scaling, this choice preserves the prototype to model velocity scaling and therefore does not affect the scale dependent trends examined in this study. The principal scale relations are listed in Table 1.

Since the horizontal wave force is expressed per unit breakwater width, its scale ratio is λ L 2. When the same fluid is used in the model and prototype, Reynolds similarity cannot be satisfied simultaneously with Froude similarity. The Reynolds number and its prototype–to–model relationship are:

where ν is the kinematic viscosity of water. The theoretical Reynolds-number ratios are 58.1, 125.0, and 207.1 for the models with λ L = 15, 25, and 35, respectively. The Reynolds number in (Equations 3, 4) is based on the breakwater diameter and characterizes the overall flow around the structure. To account for the local flow through the perforations, an opening based Reynolds number, Re o = U o d o / ν, was also evaluated. Here, U o is the characteristic local velocity in the opening region on the seaward curved surface, and d o is the opening diameter. As model size decreases, the Reynolds number decreases and viscous effects become relatively more important. Such Reynolds–number distortion is a recognized source of hydraulic model scale effects (Heller, 2011; Jensen and Klinting, 1983), and analogous scale–dependent resistance has been reported for oscillatory porous flow (Pérez-Romero et al., 2009). Surface tension effects were also examined for the smallest model scale. For the T1 case at λ L = 35, the Weber number is approximately 64. Although reported critical Weber numbers for small hydraulic models vary widely (Peakall and Warburton, 1996), the minimum water depth of 0.163 m and wave period of 1.6 s are well above the practical limits reported for reducing appreciable surface tension effects on free surface wave propagation (Hughes, 1993; Heller, 2011). Surface tension is therefore expected to have only a minor influence on the global wave response, although minor local effects at the smallest openings cannot be completely excluded. The Keulegan–Carpenter number and perforation rate are defined in Equation 5 as:

where T is the wave period, A o is the total opening area, and A s is the gross area of the corresponding perforated shell. The dimensionless horizontal wave–force coefficient is defined as

where F x (t) is the horizontal wave force per unit width. The shoreward direction is defined as positive, while the seaward direction is negative. The positive and negative peak coefficients are denoted by C F x + = max t [ C F x (t) ] and C F x - = min t [ C F x (t) ], respectively. Therefore, C F x + > 0 and C F x - < 0. The magnitude | C F x - | is used when the maximum seaward force is discussed.

Numerical model

A three–dimensional numerical wave flume was established using a finite–difference framework. The flow model followed the RANS–FAVOR–VOF framework previously applied to perforated semicircular breakwaters (Gao et al., 2024). Water was treated as an incompressible Newtonian fluid. In compact form, the continuity and Reynolds–averaged Navier–Stokes equations are given by Equations 7 and 8:

Here, u i ¯ and u i ' are the mean and fluctuating velocity components, respectively; p ¯ is the mean pressure; ν is the kinematic viscosity; and g i is gravitational acceleration. The Reynolds stresses were closed using the RNG k – ϵ turbulence model (Yakhot and Orszag, 1986; Yakhot and Smith, 1992). The curved shell and perforations were represented using the Fractional Area/Volume Obstacle Representation method on a structured Cartesian mesh. The free surface was captured using the volume–of–fluid method described by Equation 9 (Hirt and Nichols, 1981):

The fluid fraction F equals 1 in a water–filled cell and 0 in a cell containing no water, while 0< F < 1 identifies a free surface cell.

The prototype semicircular submerged breakwater consisted of a curved shell and a base slab, as illustrated in Figure 1A for the ϵ = 5% configuration. The outer and inner radii of the curved shell were 3.0 and 2.5 m, respectively, giving a shell thickness of 0.5 m. The base slab was 7.5 m long and 0.75 m thick, with a 0.75 m extension beyond the shell on both the seaward and leeward sides. Four perforation rates, ϵ = 0%, 5%, 10%, and 15%, were considered. Circular through-openings were uniformly arranged over the curved shell, with a longitudinal spacing of 1.0 m and an angular spacing of 22.5°. The perforation rate was varied by changing the opening diameter while maintaining the same opening arrangement. At prototype scale, the corresponding opening diameters were approximately 0.29, 0.41, and 0.51 m for ϵ = 5%, 10%, and 15%, respectively. Thus, perforation rate and opening diameter are coupled in the present design. All structural dimensions and opening layouts were geometrically scaled for the different model scales.

The computational domain and boundary conditions are shown in Figure 1B. At the prototype scale, the numerical flume was 275 m long, 10 m wide, and 12.5 m high. The seaward edge of the breakwater was located 125 m from the wave–generation boundary, and the downstream sponge layer was 100 m long. For a geometric scale ratio of λ L, all flume and structural dimensions were divided by λ L. Scaling all flume and structural dimensions by λ L preserved the relative positions of the wave generation boundary, breakwater, sponge layer, and outlet at every numerical scale.

A multi–block structured mesh with local refinement was used throughout the computational domain. The same meshing strategy was adopted at all scales, while the grid dimensions were reduced according to the corresponding geometric scale ratio. Taking the prototype–scale model as an example, the grid spacing in the x –direction was refined to 0.005 m within the breakwater region (x = 112.5–145 m), while an average spacing of approximately 0.015 m was used elsewhere, with gradual transitions between different mesh resolutions. A uniform spacing of 0.06 m was adopted in the y –direction. In the z –direction, the grid spacing was 0.005 m within the main free surface variation region (z = 3.875–7.5 m) and 0.010 m elsewhere. A locally refined mesh block was placed around the semicircular breakwater and coupled with the background mesh through an overlap length of 0.005 m. The breakwater surface and flume bottom were treated as smooth no-slip walls. The local mesh refinement around the breakwater reduced the grid spacing to 0.005 m near the solid boundary, which provided sufficient resolution for capturing the near-wall velocity gradients associated with viscous effects. For the scaled models, both the computational domain and grid dimensions were reduced geometrically, thereby maintaining a consistent relative mesh resolution among different scales. The local structural mesh around the breakwater was scaled in the same manner as the perforation geometry, so the relative mesh resolution of the openings was maintained among the reduced scale models. A grid-independence study was conducted for the representative T1 case with ϵ = 10% using coarse, medium, and fine meshes. The horizontal wave-force coefficient and free surface elevation were selected as the main comparison quantities. The grid configurations are summarized in Appendix A. The differences between the medium and fine meshes were less than 5%, indicating that further mesh refinement had only a minor effect on the numerical results. An additional mesh sensitivity analysis was also conducted for the T1–5 case. The differences between the medium and fine meshes remained below 6.3%, providing further support for the mesh resolution used at the smallest investigated scale. The medium mesh was retained for subsequent simulations because it provided a suitable balance between numerical accuracy and computational cost.

Cnoidal waves were generated at the upstream boundary using a Fourier–series solution (Fenton, 1988). The upper boundary was maintained at atmospheric pressure, whereas the flume bottom and breakwater surfaces were treated as wall boundaries. The lateral boundaries were specified as symmetry boundaries because a breakwater cross–section spanning the flume width was considered. A constant–pressure outlet and the sponge layer were used downstream to reduce reflected–wave interference. The horizontal wave force was expressed per unit breakwater width and normalized using (Equation 6), with positive and negative values denoting shoreward and seaward forces, respectively.

Numerical model validation

The numerical model was validated at both model and prototype scales. Physical-model measurements at λ L = 25 and 35 were used to assess the predicted free surface elevations and structural pressures. Prototype–scale validation was conducted using field pressure measurements from the Miyazaki Port semicircular caisson breakwater (Aburatani et al., 1996). The validation conditions are summarized in Table 2.

Validation against scaled physical–model data

The numerical wave gauges and pressure–monitoring points were placed at the same locations as those in the physical experiments, as shown in Figure 2A. The physical experiments were conducted in a wave flume at Hohai University with dimensions of 50.0 m in length, 1.0 m in width, and 1.5 m in height. Surface pressures were measured using a DJ800 pressure measurement system, while free surface elevations were measured using a DS30 multi point wave gauge system. Both systems operated at a sampling frequency of 50 Hz, and each record had a duration of 300 s. The measurement errors of both the pressure and wave height systems were within 5%, and the period error was less than 0.1%. Figures 2B–E compares the measured and simulated free surface elevations and surface pressures at λ L = 25 and 35. At both scales, the first incident wave produced a relatively large transient response, after which the signals became periodic. The numerical results reproduced the measured wave periods and the main changes in free surface elevation. Small differences occurred near the wave troughs at the gauge close to the breakwater crest, but the calculated minimum elevations remained close to the measured values. The calculated pressures also followed the periodic trends observed in the experiments. At λ L = 25, the mean percentage errors for the free surface elevation and surface pressure are 8.6% and 6.1%, respectively, while the corresponding RMSE values are 0.96 cm and 0.11 kPa. At λ L = 35, the mean percentage errors are 7.3% and 4.8%, and the corresponding RMSE values are 0.71 cm and 0.09 kPa. Agreement in wave period, free surface elevation, and pressure, together with these quantitative error measures, indicates that the model reproduces the principal responses within the tested scale range.

Prototype–scale field validation

Prototype–scale validation used field measurements from the semicircular breakwater at Miyazaki Port, Japan (Aburatani et al., 1996). Five pressure sensors, WPS–4 to WPS–8, were located on the seaward curved surface, as shown in Figure 3A. Numerical pressures were extracted at the corresponding locations and compared with the field measurements in Figure 3B. The simulations reproduced the main temporal variations and relative pressure levels at the five monitoring points. The calculated pressures were slightly lower than the measurements during the first wave crest, while closer agreement was obtained near the still–water level. Agreement at the five pressure sensors indicates that the model reproduces the main temporal variation and relative pressure response at prototype scale with acceptable accuracy. The agreement was considered sufficient for applying the same numerical settings in the subsequent scale effect analysis. However, the Miyazaki Port caisson differs from the present fully submerged perforated semicircular breakwater in overall geometry, submergence condition, and perforation arrangement. Therefore, this comparison is used to assess the ability of the numerical framework to reproduce prototype scale pressure responses on a curved caisson under wave loading. It should not be interpreted as a direct prototype validation of the exact structure or the local flow through the perforations considered in this study. This limitation is taken into account when interpreting the subsequent scale effect results.

Scale effect case design

Numerical cases were designed to examine the effects of geometric scale, wave height, wave period, and perforation rate. The prototype-scale case (λ L = 1) was used as the reference, together with model scale ratios of λ L = 5, 10, 15, 25, and 35. Within each group, the model dimensions and wave conditions were derived from the prototype values using the similarity relations given in Section 2.1. As summarized in Table 3, Group T1 considered all six scales and four perforation rates of 0%, 5%, 10%, and 15% under the reference wave condition. Groups T2 and T3 varied the prototype wave height, while Groups T4 and T5 varied the prototype wave period. Groups T2–T5 included λ L = 1, 15, 25, and 35 and considered perforation rates of 0% and 10%. Cnoidal waves were used in all cases.

Scale effects on horizontal wave forces

Under Froude scaling, KC remained nearly constant within each wave group, whereas Re decreased markedly as the model size decreased. For the models with λ L = 15, 25, and 35, the corresponding Reynolds numbers were approximately 1/58.1, 1/125.0, and 1/207.1 of the prototype values, respectively. This systematic reduction in Re indicates increasing dynamic dissimilarity associated with viscous effects as the model size decreases. The opening based Reynolds number shows the same scale dependent tendency at the local flow scale. For the T1 case with ϵ = 10%, Re o was approximately 2140 at prototype scale and decreased to approximately 33.3, 14.9, and 8.8 at λ L = 15, 25, and 35, respectively. The strong reduction in Re o indicates an increasing relative contribution of viscous effects to the flow through the perforations as the model size decreases.

Figure 4 presents the horizontal wave–force histories for the reference T1 condition with ϵ = 10%. At all six scales, the force varied periodically, with a period close to that of the corresponding incident wave. The maximum shoreward force at the prototype scale was approximately 50 kN/m, and the dimensional force amplitude decreased progressively as the model size decreased. The decrease in dimensional force amplitude reflects geometric scaling and is not, by itself, evidence of a scale effect. At each scale ratio, the shoreward peak was greater than the magnitude of the seaward peak. The subsequent scale effect analysis was therefore based on the dimensionless peak coefficients C F x + and C F x -.

Differences between the model and prototype were evaluated using the dimensionless peak force coefficients. When averaged across Groups T1–T5, the model to prototype ratios of C F x + for the unperforated structure were 1.04, 1.07, and 1.09 at λ L = 15, 25, and 35, respectively. For the 10% perforated structure, the corresponding mean ratios were 1.08, 1.15, and 1.23. Direct extrapolation based on Froude similarity therefore tends to overestimate the prototype shoreward peak force, particularly for smaller perforated models.

To examine the direct pressure response associated with the force differences, Figure 5 compares the dimensionless surface pressure distributions for the T1 cases with ϵ = 10% at all six geometric scales. The pressure field varied markedly over the wave cycle. At the wave crest, p * max generally increased towards the smaller models and reached its highest value at λ L = 25, while the pressure gradients near the seaward and leeward portions of the curved shell became more pronounced at λ L = 25 and 35 than at the prototype scale. The pressure level decreased during the upward zero crossing and became predominantly negative at the wave trough, with a generally stronger negative response at the smaller scales. During the downward zero crossing, the pressure returned to positive values, and the largest local maximum occurred at λ L = 35. Although the local pressure extrema did not vary monotonically with scale, the pressure distribution over the curved surface changed clearly, indicating that the scale dependent horizontal force is governed by the spatial redistribution of pressure rather than by a single local pressure maximum.

Figure 6 shows the influence of perforation rate under the reference wave condition. At all scales, C F x + decreased as ϵ increased from 0% to 15%. Meanwhile, C F x - moved towards zero, indicating a reduction in its magnitude, | C F x - |. Increasing the perforation rate reduced both the maximum shoreward and seaward loads. In contrast, the scale effect deviation in C F x + varied non–monotonically with perforation rate. At ϵ = 5%, the deviations for the selected models with λ L = 5, 15, 25, and 35 were approximately 3.9%, 9.1%, 12.3%, and 16.1%, respectively. The deviation generally increased as the model size decreased, with the λ L = 35 model showing the largest values. The seaward-peak deviation showed a less regular variation and reached approximately 10% at its maximum. Thus, perforation reduced the shoreward and seaward peak loads but generally increased the scale sensitivity of the force coefficients, particularly for the smaller models. Because ϵ was varied by changing the opening diameter, these trends reflect the combined effects of open area, opening size, and the associated local flow conditions. The present cases do not independently separate the effects of opening diameter and perforation rate.

Figure 7 compares the effects of relative wave height and relative water depth. As H / d increased from 0.263 to 0.439, C F x + decreased at all scales for both structural configurations. At the prototype scale, C F x + for the unperforated structure decreased from 0.3853 to 0.3513, corresponding to a reduction of 8.8%. For the 10%–perforated structure, C F x + decreased from 0.3221 to 0.2690, giving a reduction of 16.5%. The model–scale values of C F x + remained above the prototype values throughout the tested range. The difference between the model and prototype results was generally greater for the perforated structure and was most pronounced near H / d = 0.386. The magnitude | C F x - | decreased with increasing H / d, while the separation between the model and prototype results generally narrowed at larger relative wave heights.

Relative water depth was varied by changing the wave period while keeping H = 2.2 m and d = 5.7 m constant. As d / L increased from 0.065 to 0.101, C F x + decreased slightly for both structural configurations. At the prototype scale, the reduction was 4.4% for the unperforated structure and 5.7% for the 10%–perforated structure. At a given relative water depth, the model–scale values of C F x + remained above the prototype values, and the difference generally increased as the model size decreased. Unlike C F x +, the magnitude | C F x - | increased with d / L at all scales. Relative water depth therefore affected the shoreward and seaward peak forces in different ways. Across the tested cases, the difference between model and prototype increased as model size decreased, while perforation rate and wave conditions mainly altered the magnitude of this difference. To provide a compact quantitative measure, the shoreward force deviation was defined as Δ F = C F x, m + / C F x, p + − 1. Under Froude scaling, Re p / Re m = λ L 3 / 2, so Reynolds number distortion and scale ratio were not treated as independent variables. Regression of the 44 scaled cases gave the relationship in Equation 10:

with R 2 = 0.874 and RMSE = 0.024. This empirical relationship applies to λ L = 5 to 35 and ϵ = 0 to 0.15 for the present opening arrangement and tested wave conditions. Section 3.2 next examines where the corresponding scale-dependent changes emerge in the free surface response.

Scale dependence of the free surface response

Section 3.1 showed that the difference between model and prototype in the shoreward peak-force coefficient increased as the model size decreased and was more pronounced for the perforated structure. To identify where the associated hydrodynamic distortion develops, Figure 8 compares the normalized free surface elevations at the five wave gauges WG1–WG5 shown in Figure 2A for the T1 cases with ϵ = 10%. For direct comparison among scales, the time histories in Figure 8 were converted to prototype equivalent time according to Froude similarity. These gauges span the seaward region, the breakwater crest, and the leeward region, allowing the scale-dependent wave transformation to be traced as the wave passes across the structure. Phase differences were calculated from the mean lag between corresponding wave crests in the steady records after expressing time as t / T, with ΔΦ = 360° Δt *, where Δt * is the dimensionless time lag relative to the prototype.

At WG1 and WG2 on the seaward side, the waveforms at different scale ratios remained relatively close. The main periodic features were preserved, and only small differences occurred in the crest elevations and trough depths. The smallest model showed a slightly lower crest and a smoother waveform, but the overall upstream response remained close to the prototype result. The close agreement at WG1 and WG2 suggests that the incident wave field was reproduced consistently across the investigated scales. The main differences among scales developed after the waves reached the structure rather than during upstream propagation. The scale differences became clearer when the wave reached WG3 above the breakwater. More pronounced differences were observed at WG4 and WG5 on the leeward side. The mean phase differences among the scale cases at WG3, WG4, and WG5 were approximately 6.9°, 5.4°, and 4.4°, respectively. Compared with the prototype response, the smaller models generally produced lower transmitted crests, higher troughs, and smoother waveforms. As model size decreased, the difference between crest and trough amplitudes became smaller. This behavior is consistent with reduced wave nonlinearity and stronger relative attenuation during wave passage over and through the structure.

The contrast between the similar upstream waveforms and the increasingly different responses over and behind the breakwater shows that the main free surface distortion develops within the wave–structure interaction region. It is associated with the combined processes of wave passage over the crest, water exchange through the openings, and energy dissipation around the structure. Previous studies have also shown that structural permeability modifies wave transformation and transmitted-wave characteristics through internal water exchange and energy dissipation (Wang et al., 2023). In the present scale series, the geometry and Froude similarity were preserved, but Re decreased markedly as the model size decreased. Smoother and more attenuated transmitted waves at smaller scales are consistent with the increasing relative influence of viscous dissipation.

The free surface results alone do not determine the horizontal wave force. They nevertheless show that the hydrodynamic states on the seaward and leeward sides do not evolve in exact similarity across the investigated scales. In particular, the lower and phase-shifted leeward response may alter the timing of the pressure difference acting across the curved shell. Previous work on submerged circular-front breakwaters has shown that changes in wave transmission and flow separation can modify both the surface-pressure distribution and the total wave load (Jiang et al., 2017). The changes in surface elevation suggest that the pressure response may also deviate from prototype similarity at smaller scales. The local velocity field is examined next to determine how this wave-response mismatch is related to crest acceleration, through-flow, and reverse water exchange around the perforated structure.

Scale dependence of the velocity field

The free surface analysis in Section 3.2 showed that the scale-dependent differences developed mainly over and behind the breakwater. To examine how these differences are related to local water exchange and horizontal loading, the phase-resolved velocity fields of the T1 cases with ϵ = 10% are analyzed below. For a unit-width section, the pressure-induced contribution to the horizontal wave force can be expressed as

where S w is the wetted structural surface, p (x, t) is the instantaneous pressure, and n x is the horizontal component of the outward unit normal. (Equation 11) represents the pressure contribution rather than a direct force contribution from the local velocity. The velocity field affects the horizontal load by modifying the pressure distribution over the curved shell and the rate of water exchange through the openings. Here, the velocity fields are interpreted as hydrodynamic evidence for the force difference across scales rather than as a direct proxy for force.

To compare the local flow among different geometric scale ratios, the velocity magnitude was nondimensionalized as v *= vT / L, where v is the local velocity magnitude, and T and L are the corresponding wave period and wavelength. Figure 9 presents the dimensionless velocity fields at the wave crest, upward zero-crossing, wave trough, and downward zero-crossing phases. These phases describe the evolution of the local flow over a wave cycle and are not assumed to coincide exactly with the instant of the maximum shoreward force.

At the wave crest, the high- v * region extended over the breakwater crest and towards the leeward side, while local acceleration occurred around the openings. As the model size decreased, this region was redistributed towards the crest, openings, and leeward part of the structure. The flow pattern was not a geometrically scaled replica of the prototype field. The maximum v * at λ L = 25 was approximately 28.3% higher than the prototype value. More importantly, the region through which shoreward momentum was transferred over and through the perforated shell also changed in position and spatial extent. The wave-crest result therefore reflects both local velocity amplification and a scale-dependent redistribution of the shoreward flow.

During the upward zero-crossing phase, a distinct high-velocity band developed on the leeward side. This feature resulted from the interaction between the residual return flow and the developing shoreward flow, which generated a strong local velocity gradient. The maximum v * showed a non-monotonic scale response. It was approximately 31.4% higher than the prototype value at λ L = 15, but only about 15.7% higher at λ L = 35. This behavior differs from the shoreward peak-force coefficient, whose ratio between model and prototype continued to increase as the model became smaller. A single local velocity maximum cannot explain the observed difference in force across scales. Instead, the result suggests competition between local acceleration and the increasing relative influence of viscosity at lower Re, which changes both the intensity and the spatial organization of the flow. Scale-dependent friction has also been reported for oscillatory flow through porous structures (Pérez-Romero et al., 2009).

A different scale response occurred at the wave trough. The main high-velocity region was located inside the cavity, where the dominant motion was reverse flow through the structure. Unlike the shoreward acceleration observed at the wave crest, the maximum dimensionless return flow velocity decreased markedly as the model size decreased. At λ L =35, it was approximately 35.0% lower than the prototype value. Reverse flow through the structure weakened as model size decreased during this phase. This reduction indicates that reverse water exchange between the cavity and the surrounding water is more strongly distorted than the local shoreward acceleration. Weaker reverse exchange may also reduce the rate at which water levels and pressures are redistributed between the seaward side, cavity, and leeward side. The weakened return flow links the velocity distortion to the altered pressure response inferred from the free surface results in Section 3.2.

During the downward zero-crossing phase, the high-velocity region was mainly distributed on the seaward side and around the openings. The incoming flow interacted with the residual return flow and produced pronounced local shear. As the model size decreased, the secondary free surface feature associated with this phase became weaker, and the waveform became smoother. This change is consistent with the reduced reverse flow response identified at the wave trough. The downward zero-crossing field therefore records the interaction between the newly developing shoreward flow and a scale-dependent residual return flow.

Across the four wave phases, local flow intensity did not increase uniformly as model size decreased. Shoreward acceleration remained concentrated around the crest and openings, whereas reverse through-flow and cavity exchange were weakened at the smaller scales. The resulting distortion is therefore asymmetric between the shoreward and reverse flow phases. This asymmetry is particularly relevant to a perforated structure because flow through the openings involves local contraction, acceleration, and pressure loss. The resistance of perforated walls depends on the oscillatory-flow conditions, including KC, while pressure-loss formulations relate the pressure difference across a perforated plate to the normal through-flow velocity (Liu and Li, 2016; Li et al., 2024). Geometric similarity of the openings does not ensure dynamic similarity of water exchange when Re changes substantially. The weaker reverse exchange, together with the redistributed shoreward flow, may make pressure equalization across the perforated shell less synchronized with the prototype. This asymmetric velocity response provides a plausible hydrodynamic link to the systematic overestimation of the prototype shoreward peak force by the smaller models. Section 3.4 further examines how the associated velocity gradients, shear layers, and vortex transport are reorganized with scale.

Scale dependence of the vorticity field

The velocity fields in Section 3.3 showed that the local flow was reorganized rather than uniformly strengthened as the model size decreased. Vorticity is examined here to identify how the associated velocity gradients, shear layers, and vortical structures vary with scale. For the two-dimensional x–z plane, the out-of-plane vorticity was calculated as ω = ∂u / ∂z − ∂w / ∂x, where u and w are the horizontal and vertical velocity components, respectively. The dimensionless vorticity was defined as ω *= ωT. Figure 10 presents the ω * fields for the T1 cases with ϵ = 10% at the same four characteristic phases used in Figure 9. The sign of ω * indicates the direction of local rotation, while its magnitude reflects the local velocity gradient. Vorticity is treated here as evidence of shear, flow separation, and vortex development rather than as a direct measure of wave force or energy dissipation.

At all scale ratios, adjacent regions of positive and negative vorticity developed around the openings. These paired structures indicate strong local shear as water passes through the perforated shell. The through-flow accelerates near the opening and separates from the opening edges, producing shear layers with opposite rotational directions. Experiments on submerged slotted barriers have similarly observed flow separation at individual openings, followed by vortex shedding and interaction among the generated vortices (Wu and Hsiao, 2020). Although the opening geometry was scaled consistently in the present models, the strength and spatial organization of these shear layers did not remain identical across scales.

At the wave crest, regions of high | ω * | were located inside the structure and within the leeward cavity. Their overall spatial extent changed only moderately with scale, although the high-vorticity region extended slightly farther towards the leeward side in the smaller models. The positive vorticity extreme exhibited a clear non-monotonic response. Relative to the prototype, its largest increase was approximately 148.2% at λ L = 15, whereas the increase at λ L = 35 was approximately 54.9%. This trend differs from the shoreward peak-force coefficient, whose ratio between model and prototype increased towards the smallest model. A single vorticity extreme cannot account for the systematic force overestimation. The more relevant feature is the scale-dependent redistribution of shear and vortical motion around the openings and inside the cavity.

The scale differences became more pronounced during the upward zero-crossing phase. The high- | ω * | regions expanded, and distinct concentrations developed on the leeward side for λ L = 15, 25, and 35. The strongest response occurred at an intermediate scale and then weakened towards λ L = 35. This non-monotonic behavior is consistent with the velocity response identified in Section 3.3 during the same phase. The interaction between the residual return flow and the developing shoreward flow alters both the local shear intensity and the position of the shear layer. As Re decreases, the balance between inertia and viscosity also changes, so the shear-layer development cannot remain dynamically similar to the prototype (Heller, 2011; Jensen and Klinting, 1983).

At the wave trough, the leeward high-vorticity region became more extensive as the model size decreased. Its spatial pattern also changed from a relatively continuous band to more localized patches. This reorganization occurred at the same phase in which Figure 9 showed a marked reduction in the reverse flow velocity. Despite weaker bulk return flow, the vortical response did not weaken uniformly. Instead, the reverse flow process became weaker in magnitude but more localized in its velocity gradients and vortical structures. The simultaneous reduction in bulk return flow and localization of velocity gradients show that the return flow is distorted in both intensity and spatial organization. Previous numerical work on the same type of perforated semicircular breakwater has shown that inverse-head-driven return flow promotes mixing and vortex formation inside the cavity (Gao et al., 2024). The present results show that this return flow mechanism is not reproduced as a simple scaled version of the prototype.

During the downward zero-crossing phase, the vorticity distributions around the openings differed less among the scale ratios than during the other phases. Distinct regions of positive vorticity still appeared on the leeward side of the models with λ L = 15, 25, and 35. These regions correspond to the interaction between the incoming shoreward flow and the residual return flow identified in Figure 9. The result does not indicate a uniform increase in vorticity throughout the flow domain. Instead, it shows that the location of local shear shifts towards the leeward side as the relative contribution of the residual return flow changes.

Comparison of the free surface, velocity, and vorticity fields indicates that geometric similarity does not ensure complete dynamic similarity of the flow through the perforated structure. Froude scaling reproduced the main upstream wave motion, but Re decreased rapidly as the model size decreased. This change is particularly important for the perforated shell because its response involves local contraction, pressure loss, jet-like through-flow, separation, and repeated water exchange with the cavity. Scale-dependent friction has been reported in oscillatory porous flow (Pérez-Romero et al., 2009), while porous-media concepts have been used to describe the hydrodynamic resistance of both perforated walls and aquaculture net cages (Liu and Li, 2016; Miranda et al., 2025). In the present results, these effects appear as lower and phase-shifted transmitted waves, redistributed shoreward flow, weaker reverse exchange, and phase-dependent changes in the shear and vorticity fields. These quantities do not vary monotonically with scale, and no single velocity or vorticity extreme explains the force bias. Their combined evolution suggests that water exchange and pressure redistribution among the seaward side, cavity, and leeward side become less synchronized with the prototype. A larger instantaneous pressure difference may persist during part of the shoreward loading process. Delayed pressure redistribution offers a plausible explanation for the systematic overestimation of the prototype shoreward peak force by the smaller models. The same mechanism also helps explain why perforation reduces the absolute wave load while increasing its sensitivity to geometric scale. Perforation provides a path for pressure relief, but it also introduces additional Reynolds-sensitive flow processes that are absent from the unperforated shell.

Several limitations of the present study should be noted. The analysis is limited to the periodic cnoidal wave conditions considered here, and its applicability to irregular waves and to other nonbreaking or breaking conditions remains to be examined. The prototype scale comparison supports the numerical framework but does not directly validate the present perforated submerged breakwater. The empirical force relationship is limited to the investigated scale ratios, perforation rates, opening arrangement, and wave conditions. Other factors, such as surface roughness, alternative perforation layouts, and local surface tension effects at the smallest openings, were not examined independently.

This study investigated the scale effects on horizontal wave forces and local flow characteristics around a perforated semicircular submerged breakwater using a validated numerical wave flume. The main conclusions are as follows.

1. Froude similarity preserved the main periodic characteristics of the horizontal wave force, but Reynolds similarity was not maintained. As the model size decreased, the shoreward peak-force coefficient C F x + increasingly exceeded the prototype value. At λ L = 15, 25, and 35, the mean model to prototype ratios of C F x + were 1.04, 1.07, and 1.09 for the unperforated structure and 1.08, 1.15, and 1.23 for the 10% perforated structure, respectively. Direct extrapolation from a small model based on Froude similarity tends to overestimate the prototype shoreward peak force, particularly for a perforated structure. Within the investigated parameter range, the empirical relationship developed provides a quantitative estimate of this shoreward force deviation.

2. Perforation reduced both the shoreward and seaward peak loads but increased the sensitivity of C F x + to geometric scale. The difference between model and prototype in C F x + generally decreased as H / d increased, whereas the influence of d / L was weaker and differed between the two loading directions. The scale response of C F x + was stronger and more systematic than that of | C F x - |. The observed force scale effect depended jointly on model scale, perforation, and wave conditions.

3. Free surface differences were small upstream but became more evident over and behind the breakwater. Smaller models produced lower transmitted crests, higher troughs, and phase shifts. Shoreward acceleration remained concentrated around the crest and openings, whereas reverse flow through the structure and cavity exchange weakened as the model size decreased. This asymmetric response is consistent with the stronger scale effect in C F x + and the weaker, less regular variation in | C F x - |.

4. The spatial distribution and phase evolution of the vorticity field changed as the model size decreased. Local vorticity extremes varied nonmonotonically and could not alone explain the systematic overestimation of C F x +. The lower Re altered wave transmission, flow through the openings, reverse water exchange, and vortex development, making pressure redistribution across the perforated shell less synchronized with the prototype. A larger instantaneous pressure difference may therefore persist during shoreward loading, while the flow processes introduced by perforation become more sensitive to Reynolds number and further increase the force sensitivity to scale.