Random wave runup and overtopping a steep sea wall: Shallow-water and Boussinesq modelling with generalised breaking and wall impact algorithms validated against laboratory and field measurements

Coastal Engineering 74 (2013) 33–49

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Random wave runup and overtopping a steep sea wall: Shallow-water and Boussinesq modelling with generalised breaking and wall impact algorithms validated against laboratory and field measurements M.V. McCabe a, b,⁎, P.K. Stansby a, D.D. Apsley a a b

School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester, M13 9PL, UK Halcrow, 304 Bridgewater Place, Birchwood Business Park, Warrington, Cheshire, WA3 6XG, UK

a r t i c l e

i n f o

Article history: Received 18 November 2011 Received in revised form 28 November 2012 Accepted 29 November 2012 Available online 11 January 2013 Keywords: Wave overtopping Wave runup Random waves Boussinesq equations Nonlinear shallow water equations Sea walls Recurves Wave breaking

a b s t r a c t A semi-implicit shallow-water and Boussinesq model has been developed to account for random wave breaking, impact and overtopping of steep sea walls including recurves. At a given time breaking is said to occur if the wave height to water depth ratio for each individual wave exceeds a critical value of 0.6 and the Boussinesq terms are simply switched off. The example is presented of waves breaking over an offshore reef and then ceasing to break as they propagate inshore into deeper water and finally break as they run up a slope. This is not possible with the conventional criterion of a single onset of breaking based on rate of change of surface elevation which was also found to be less effective generally. The runup distribution on the slope inshore of the reef was well predicted. The model is tested against field data for overtopping available for Anchorsholme, Blackpool and corresponding 1:15 scale wave flume tests. Reflection of breaking waves impacting a steep sea wall is represented as a partial reversal of momentum flux with an empirically defined coefficient. Offshore to nearshore significant wave height variation was reasonably predicted although nearshore model spectra showed distinct differences from the experiments. The breaking wave shape described by a shape parameter was also not well represented as might be expected for such a simple model. Overtopping agreement between model, field and flume was generally good although repeatability of two nominally identical flume experiments was only within 25%. Different distributions of random phase between spectral components can cause overall overtopping rates to differ by up to a factor of two. Predictions of mean discharge by EurOtop methods were within a factor of two of experimental measurements. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Coastal flooding is caused by a combination of large waves and high water levels. Such conditions can cause sea water to overtop coastal defences and flood low lying areas; in severe cases, the overtopping can cause the defences to be breached, causing considerably more flood damage. The ability to accurately predict, or to accurately calculate the probability of wave overtopping is therefore of great importance. Firstly, the forecasting of overtopping due to an incoming storm enables people and emergency services to plan for the event; secondly, it allows coastal engineers to make good designs, reducing the need to “over-engineer” defences; and thirdly, by being able to make more accurate predictions of potential flood damage, infrastructure planning decisions can be more effectively optimised. There are three possible approaches for the prediction of wave overtopping. Physical modelling should give accurate results (although

⁎ Corresponding author at: Halcrow, 304 Bridgewater Place, Birchwood Business Park, Warrington, Cheshire, WA3 6XG, UK Tel.: +44 (0)7751899172. E-mail address: [email protected] (M.V. McCabe). 0378-3839/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coastaleng.2012.11.010

scale effects could be an issue), but it is expensive and timeconsuming. Alternatively, there exist a range of empirical tools and formulae, based on data from numerous physical model and field experiments; the EurOtop Manual (Pullen et al., 2007) gives guidance on many such tools. However, a wave by wave analysis is not available and some of the tools may be difficult to use with input parameters being open to interpretation. Also some tools and formulae are only suited to certain shapes of structure. The third approach is through numerical modelling. Ideally, a numerical model will provide the accuracy of a physical model test, but with increased flexibility and reduced expense. Some predictions of random wave overtopping have been carried out using the nonlinear shallow water (NLSW) equations, by Dodd (1998) for example. However, these equations are only valid for breaking waves close to the shore, and therefore require another modelling approach to transform the incident waves into the surf zone; McCabe et al. (2011), for example, demonstrated the coupling of a spectral energy model (SWAN, Booij et al., 1999) to a NLSW solver, giving good results for the prediction of random wave runup on a slope. Here we develop the more general Boussinesq modelling approach which is validated against field and laboratory measurements.

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Several Boussinesq-type models have been developed over the past few decades. The equations are an extension of the nonlinear shallow water equations, with extra terms to take account of the non-hydrostatic pressure effect associated with wave propagation, allowing frequency dispersion to occur. Various forms of these equations have been derived; those of Madsen and Sorensen (1992) have been well tested and give accurate representation of frequency dispersion for waves in intermediate water depths, with values of depth parameter kd of up to 3; k is wave number and d is depth. However, the extra terms in Boussinesq-type equations, with third (and higher) order derivatives, can cause numerical instability issues with breaking waves. To simulate breaking waves, a Boussinesq-type model requires some modification to take account of the associated change in structure and energy losses. Although, the initiation of breaking is generally considered, its cessation is generally not (for example when a wave propagates into deeper water after breaking over a reef). Most Boussinesq-type models retain the Boussinesq terms throughout the surf zone. However, this may not be necessary as the frequency dispersion becomes less important as waves propagate into shallow water, e.g. Borthwick et al. (2006). Therefore, there is some justification for using the NLSW equations for modelling depth-limited breaking waves, which become shallowwater bores as they approach the shore. Boussinesq and NLSW models are valid for mild slopes and are limited when interacting with steep-sided coastal structures, although the vertical sea wall is a special case which can in principle be handled. However, when a breaking wave hits a structure, the structure imposes a force on the flow, generally directing it upwards, as well as seawards, at a recurve wall for example. Fig. 1 shows a photograph of a wave impacting a recurve wall. This appears not to have been accounted for within Boussinesq and NLSW models to our knowledge. In this paper a particular shallow-water and Boussinesq (SWAB) model will be developed for these applications, making a comparison with overtopping and runup data from random wave laboratory experiments and field measurements. A generalised breaker model will be developed, appropriate for random waves, and a simple solution will be proposed for wave impact on near vertical sea walls, using time series of wave overtopping data for comparison. In addition, runup data from experiments with a submerged reef will be compared. The following section will describe the model, including the breaking and wall impact algorithms; Section 3 describes the experimental data used for this investigation; Section 4 describes results from random wave overtopping tests with a sea wall, demonstrating

how recurve walls can be modelled; Section 5 describes results from random wave runup experiments for waves propagating over a submerged reef; Section 6 discusses the results; and Section 7 presents the conclusions. 2. The shallow-water and Boussinesq (SWAB) model 2.1. Model equations and solution The SWAB model uses the equations of Madsen and Sorensen (1992), in one horizontal dimension. They consist of a continuity Eq. (1) and a momentum Eq. (2): ∂h ∂ðhuÞ þ ¼0 ∂t ∂x

ð1Þ


 2 ∂ðhuÞ ∂ hu ∂h ∂z τ ¼ −gh −gh b − b þ ρ ∂x ∂x ∂t ∂x ! (  3 3 1 2 ∂ ðhuÞ 3∂ η þ Bgd þ Bþ d 3 ∂x2 ∂t ∂x3 ð2Þ !) 2 2 ∂d 1 ∂ ðhuÞ ∂ η þd þ 2Bgd 2 ∂x 3 ∂x∂t ∂x non−breaking    ∂ ∂u þ hðν þ ν e Þ ∂x ∂x breaking where u is depth-averaged velocity, h water depth, d still-water depth, zb the bed level, τb bed shear stress, ρ water density, ν kinematic viscosity of water and νe the wave breaking eddy viscosity. The Boussinesq terms are the non-breaking part on the second and third lines of Eq. (2), where B is a constant that controls the linear dispersion characteristics. The horizontal diffusion terms are the breaking part on the fourth line. Madsen and Sorensen (1992) found that B = 1/15 gives the best linear dispersion. The model calculates numerical solutions for h using a Crank–Nicolson semi-implicit method following Stansby (2003). This has the advantage of being more stable than the fully explicit methods, and without the damping effects of the fully implicit methods. A staggered mesh finite volume scheme is used. The continuity Eq. (1) becomes: nþ1

hi

n

¼ hi −θ

 Δt
Δt  nþ1 nþ1 n n ðhuÞiþ1 −ðhuÞi ðhuÞiþ1 −ðhuÞi −ð1−θÞ Δx Δx

ð3Þ

where θ represents the degree of “implicitness” (i.e. fully implicit where θ = 1 and fully explicit where θ = 0) and is set to ½ giving second-order time stepping. The subscripts n and i represent the temporal and spatial steps respectively. The momentum Eq. (2) can be manipulated to give (hu) n + 1 in terms of h n + 1. (hu) n + 1 is then substituted into Eq. (3) to give an expression for h n + 1 in the following form: nþ1

nþ1

A1 hi−1 þ A2 hi

Fig. 1. Photograph of wave impact and reflection at a recurve wall in a wave flume.

 n nþ1 n n þ A3 hiþ1 ¼ f hi ; ðhuÞi ; ðhuÞiþ1

ð4Þ

where A1, A2 and A3 are functions of h n and u n. Eq. (4) gives a tridiagonal matrix, from which solutions for hin−+11, hin + 1 and hin++11 can be obtained using a tridiagonal equation solver. Flux, (hu)n +1, and depth averaged velocity, u n+1, are calculated through substitution back into the discretised momentum equation. The advection terms, Boussinesq terms and horizontal diffusion terms are all part of the function on the right-hand side of Eq. (4). These terms are calculated at the beginning of a time-step; the advective and horizontal diffusion terms are calculated explicitly first, followed by the Boussinesq terms. The advection is calculated using a linear upwind differencing scheme (LUDS). First-order time stepping was generally used for both advection and diffusion giving reliable stability

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

with negligible loss of accuracy compared with second-order Adams– Bashforth time stepping, which was also available. Wave breaking locations are also calculated at the beginning of the time step. Methods used for calculating these will be described in Section 2.2. To calculate the Boussinesq terms, Eq. (2) can be rewritten by gathering all the ∂(hu)/ ∂t terms on the left-hand side:       ∂ðhuÞ 1 2 ∂2 ∂ðhuÞ d ∂d ∂ ∂ðhuÞ d − Bþ − 2 3 3 ∂x ∂x ∂t ∂t ∂t ∂x
 2   ∂ hu ∂η τ ∂ ∂u hðν þ νe Þ −gh − b þ ¼− ρ ∂x ∂x ∂x ∂x 3

þBgd

ð5Þ

with the Boussinesq terms being the last two terms on the left and right hand sides of the equation. From Eq. (5) a discretised equation of similar form to Eq. (4) is derived, giving a solution for ∂(hu)/∂t, for time step n + 1, using a tridiagonal solver. A4

      ∂ðhuÞ ∂ðhuÞ ∂ðhuÞ þ A5 þ A6 ¼ Fi ∂t ∂t ∂t i−1 i iþ1

ð6Þ

where A5 and A6 are functions of B, d and ∂d/∂x and Fi represents the right-hand side of Eq. (5). For convenience with a moving break point, the Boussinesq terms are calculated throughout the domain, with breaking values then set to, or phased towards zero. To prevent saw tooth numerical instability in non-breaking waves, a digital filter was applied to the water depth h, of the form effectively used by Longuet-Higgins and Cokelet (1976) in their boundary-integral computations, following Stansby (2003). The third term on the right hand side of Eq. (2), τb/ρ, represents the bed shear stress which can be expressed as: τb C f jujn nþ1 ¼ ðhuÞ ρ 2 hn

ð7Þ

where Cf is the friction coefficient. The SWAB model calculates this term implicitly, using values of hu from the present time step. Bed friction has little effect on surface gravity waves over the distances simulated by the SWAB model, although it can affect runup levels. The method of Larsen and Dancy (1983) is employed for wave input. The change in free surface level, Δη, required at each time step to generate the necessary waves is given as: I

Δη ¼ 2η cg

Δt Δx

2.2. Wave breaking The SWAB model has a similar breaking formulation as those of Zelt (1991) and Kennedy et al. (2000), who used an additional horizontal diffusion term. In the SWAB model, for breaking waves the Boussinesq terms are set to zero and the horizontal diffusion is computed. Because there is no definitive knowledge on how the Boussinesq terms should be switched off after the break point, both an instant switch-off and a phasing-out algorithm have been tested. The horizontal diffusion is represented by the last term in Eq. (2), with the eddy viscosity associated with the breaking waves, νe, calculated using a similar method to Kennedy et al. (2000):



2 ∂η

νe ¼ δ h



∂t

2 ∂3 η 2 ∂d ∂ η þ 2gd ∂x ∂x2 ∂x3

ð8Þ

where η I(t) is the incident wave train, and cg is the wave group, or energy, velocity corresponding to the Boussinesq equations of Madsen and Sorensen (1992) and is a function of wave frequency and water depth, h(t). To calculate a random wave input from a time series, a Fourier transform was used to separate η I(t) into its components of amplitude and phase, calculating Δη for each component based on its corresponding value of cg and recombining the components to give an input free surface level; this method was found to give rather more accurate incident waves in comparison with using a single representative value of cg (McCabe, 2011). The direct use of a model wave paddle is thus avoided and outgoing waves are absorbed by a sponge layer, using the method of Yoon and Choi (2001). When setting up a SWAB model domain, one must allow at least one wavelength offshore of the wave input location to accommodate the sponge layer. This wave input provides a radiation boundary condition with reflected waves allowed to propagate offshore.

35

ð9Þ

where δ is a mixing length coefficient. For the SWAB model δ = 0.5 was used throughout, and Kennedy et al. (2000) found results fairly insensitive to varying this coefficient. In addition, some criterion is needed to initiate the breaking process. A convenient choice is a limiting value of the rate change of free surface elevation, employed by Kennedy et al. (2000): pffiffiffiffiffiffi ∂η > C bt gh ∂t

ð10Þ

where Cbt is a breaking coefficient. Note that breaking is only initiated when ∂η/∂t is positive, as breaking starts on the front face of a wave. Eq. (10) has the advantage of being easily calculated during the running of the model. Reasonably accurate predictions of wave runup and overtopping a trapezoidal structure with a 1:2 slope have been made for solitary waves, regular waves and wave groups (Stansby, 2003; Stansby and Feng, 2004; Stansby et al., 2007) and good representation of wave reflection was simulated by Stansby et al. (2008). However these empirical representations for the onset of wave breaking and associated turbulence for these particular wave conditions do not necessarily apply to more general wave propagation situations with random waves or wave propagation over offshore features such as bars or reefs. The main drawback of this wave breaking model is the assumption that all waves inshore from the break point will be assumed to be shallow water waves which is not generally true. It is quite possible for a wave in a random wave train to start breaking in relatively deep water but for inshore waves of smaller height not to be breaking. Another possibility is to use the well-known “rule of thumb” of Miche (1944), which imposes a limiting wave height to water depth ratio: H ¼ C bh h

ð11Þ

where H is the local wave height and Cbh is a breaking coefficient. This criterion has not been used in Boussinesq-type models to our knowledge, possibly because it requires an algorithm to separate individual waves before calculating the wave height and mean depth for each wave, and there may be stability issues with less robust numerical Boussinesq schemes. To separate into individual waves, the following algorithm was used, as demonstrated in Fig. 2. At every cell in the domain, the cumulative time mean water level is calculated; this value is updated at every time step. At each time step, spatial zero up-crossing points are identified within the domain. By using spatial up-crossing points each individual wave consists of a crest with a preceding trough. Note that at the start of a run the cumulative time mean water level will be rather changeable; therefore during the first four periods of a model run the still water level is used instead to calculate the zero up-crossing points. The wave height of each individual wave, H, is the difference between the trough and crest level of each wave; the

36

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

Fig. 2. Wave identification methodology.

mean depth of each wave, h, is the mean depth of water between zero up-crossing points for that particular time step. 2.3. Sea walls The nonlinear shallow water and Boussinesq-type equations used in the SWAB model were derived for mild bed slopes. However, when modelling wave overtopping, one encounters steep slopes, near vertical walls, and recurve walls with slopes beyond the vertical. Shiach et al. (2004) tested a model based on the nonlinear shallow water equations to simulate violent wave overtopping of vertical sea walls with promising results. However, the nonlinear shallow water equations do not take account of the impact force imposed by a wall when a jet of water impacts against a wall in the x direction. If the x momentum is reduced to zero the wall will have imposed a force back on the flow, given by F wall ¼ ρAujet

2

ð12Þ

where, A is the cross-sectional area of the jet of water and ujet is its horizontal velocity. If a jet of water is reflected backwards at the same speed then: F wall ¼ 2ρAujet

2

ð13Þ

When a wave impacts a sea wall the situation is more complex. Firstly, the wall may not be as high as the depth of the flow, in which case it can only impose a force on a proportion of the depth. Reflection by a wall may enhance energy dissipation as the jet is redirected. Transmission will generally occur and reflected flow will interact with the incident flow. Here a wall force term is proposed, proportional to ρAujet 2 with a constant of proportionality defined empirically. Fig. 3 shows two adjacent cells in the SWAB model, i and i + 1. The SWAB momentum equation is in terms of force per unit bed area per unit density (i.e. F/(ρ Δx)); in all the following references to force and Fwall, it will be factored by 1/(ρ Δx). When the flow is directed towards the wall, the force is applied to cell i. hF represents the height of wall that imposes a force on the flow; if the depth of the flow is less than the height of the wall (i.e. hi b Δz) then the representative height hF is equal to hi; otherwise it is equal to Δz. Therefore: F wall;i ¼

kwall hF;i u2i Δx

ð14Þ

where hF = min(hi, Δz) and kwall is an empirical constant; different values for kwall will be considered in the overtopping analysis. When the flow is seaward Fwall is obviously zero. To specify where this wall force is calculated and applied in the model, a text file input is required specifying which cells are located adjacent to a wall, and what kwall should be at each wall location. The effectiveness of this algorithm will be discussed in the overtopping analysis. 3. Experimental data 3.1. Wave runup experiments Mase et al. (2004) performed tests for the runup of random waves on a sloping sea wall with a submerged breakwater offshore. Fig. 4 shows the experimental set-up. From these tests, time-series data are available for a significant wave period T1/3 of 1.1 s and 2.1 s with a water depth of 0.425 m. Table 1 contains the incident wave conditions for these tests; surf similarity parameters, ξ0, are calculated: tanβ ξ0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 0 =L0;p

ð15Þ

where tan(β) ≈ 0.05, which is the slope where most breaking is expected to occur, with the deep water wave heights and wavelengths, H0 and L0,p, being calculated from incident values of significant wave height, H1/3, and peak period, Tp. In the flume there were six wave gauges; the second, WG2, was placed at the base of the slope, with the first, WG1, being L/4 further offshore (with L based on the significant wave period, T1/3). A runup meter was placed 3 mm above the

Fig. 3. Application of wall force in the SWAB model.

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

Fig. 4. Experimental set-up of Mase et al. (2004).

Table 1 Incident wave data for selected Mase et al. (2004) tests. Run

H1/3 (m)

Hm0 (m)

T1/3 (s)

Tp (s)

Distance WG1–WG2 (m)

ξ0,p

T1 T2

0.0709 0.0671

0.0726 0.0739

1.05 2.11

1.14 2.25

0.43 1

0.253 0.519

sea wall face; the lower end of the runup meter was placed approximately 10 cm distance from the base of the wall. This means that the base of the runup meter was located 3.45 cm above the still water level. Mase (2008) provided the time-series data for all six wave gauges and the runup meter for the tests described in Table 1, as well as data on wave heights and runup parameters from all their model tests. However, the SWAB modelling that will be described in the next section will only make use of the time-series data. 3.2. Physical model tests of wave overtopping at the Anchorsholme sea wall Simultaneous wave and overtopping field data were collected at Anchorsholme, Blackpool by HR Wallingford for the Environment Agency (Bocquet et al., 2009). Two acoustic wave and current profilers (AWACs) were placed approximately 30 m and 450 m offshore from the base of the seawall, collecting time-series of wave conditions. The offshore AWAC was placed at a bed level of approximately − 3.4 mOD; the crest of the sea wall is at 7.8 mOD. Wave overtopping

37

was measured on 24th January 2008, using a collection tank placed behind the crest of the sea wall. By carrying out physical model tests using the conditions from the field data based on Froude scaling, this enables a three-way comparison between field data, physical model test results, and numerical modelling. The wave flume tests were undertaken at HR Wallingford at 1/15 scale, with the layout shown in Fig. 5. The sea wall itself is shown in Fig. 6, and is a scale model of the sea wall at Anchorsholme. Eight wave gauges were located in the flume, at the locations shown in Table 2. A tank was placed between the sea wall and the back wall to collect overtopping water, with a chute placed between the sea wall and the tank to channel the overtopping flow. The chute was always placed after the waves had started to avoid recording overtopping from possible abnormally high waves at the start of the wave train; the chute placement time was recorded for each test. Inside the collection tank was a wave gauge, placed inside a tube to protect it from high-frequency splashing. The overtopping tank was emptied after each test; if it filled up during a test, this time was also recorded. A photograph of the layout behind the sea wall is shown in Fig. 7. Each model test was also run without the sea wall in place (calibration runs) to check the wave conditions without reflections. Some input wave spectra were calculated from the spectra recorded at the offshore AWAC in the field tests, with corresponding water levels; other input spectra used a JONSWAP spectrum (see Table 3). Details on the spectra taken from the field data are given by McCabe (2011). A gravel “beach” was placed in front of the back wall of the flume to absorb the incident waves for these calibration runs. All wave probes were in place during these runs, and the calibration run output from wave gauge WG1 will be used for input to the SWAB model. The wave conditions at this wave gauge are given in Table 3. Surf similarity parameters were calculated in the same way as those in Table 1. All the surf similarity parameters are low, below 0.1, corresponding to spilling breakers. For the overtopping tests the wave paddle used an active absorption method to absorb reflected waves; for the calibration runs, active absorption was not employed. Calibration runs and overtopping runs all consisted of 1024 waves. The P0785 model run was repeated twice: firstly as an exact repeat, with exactly the same time-series; and secondly as a repeat with a slightly reduced wave height (the P0775 run). The overtopping field dataset was collected between 11:04 am and 12:08 pm on 24th January 2008. The S55 wave spectrum and

Back Wall Gravel to absorb reflections Wave Paddle Wave Gauges

WG1

WG2

WG3

WG4

Sea Wall

WG5

WG6

WG7

WG8

0.446 m 0.20 m

≈ 10.0 m

3.0 m

20.0 m

1.8 m

5.2 m

Sea Wall Toe Fig. 5. Wave flume layout. Not to scale.

38

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49 293

67

57

156

57

113

132 33.3

154

Sea wall toe

Bed level Fig. 6. Model sea wall dimensions (mm). Fig. 7. Layout behind sea wall in wave flume, showing the overtopping collection tank.

water level conditions correspond with the field conditions between 11:32 am and 11:52 am, so these datasets can be compared directly with each other.

For all the input waves (with and without the sea wall), calibration run (i.e. without the sea wall) wave flume time-series from WG1 were used (see Table 3).

4. Nearshore waves and wave overtopping 4.2. Nearshore waves without the sea wall 4.1. SWAB model setup The results from the scale model tests were used for this investigation. For most tests, the SWAB model was set up with the same bathymetry as the flume. The wave input was located at wave gauge WG1, with 10 m of domain added offshore of this location to contain the sponge layer. The sea wall was represented as a change in level over one horizontal cell in the model with wall forces applied at the locations shown in Fig. 8 (Inshore Section). These represent momentum flux reversal by the recurve, determined through empirical calibration, as described in Section 2.3. Landward of the sea wall, a collection tank was included, with a zero flux boundary condition. The SWAB model was also set up to simulate the wave calibration tests in the flume, with the bathymetry shown in red in Fig. 8. In this case a sponge layer was included at the inshore boundary, to mimic the gravel absorption layer in the flume; therefore the model domain was made long enough to incorporate a sponge layer between the inshore boundary and Wave Gauge 8. Analysis of the time-series data from the wave flume occasionally showed noisy behaviour, possibly caused by damaged wave gauges. When this occurred as sporadic bursts of noise a low-pass filter (b5.0fp) was applied; all data shown in the figures has been filtered as required. Unfortunately some data, especially from wave gauge WG6, could not be used.

Table 2 Wave gauge locations.

Significant wave heights in the nearshore for three of the calibration runs are shown in Fig. 9. Two SWAB runs are shown: the first uses the limiting height to depth ratio (H/h) (Eq. (11)), with a coefficient Cbh of 0.60; the second uses the ∂η/∂t breaking criterion (Eq. (10)), with the Boussinesq terms phased out over a distance h at the break point, and a coefficient Cbt of 0.25. From Fig. 9, it is apparent that the ∂η/∂t model does not perform as well as the H/h breaking model. This is further demonstrated by the moving average of ηrms shown in Fig. 10 for the S55 wave conditions. The variance of the free surface level is equivalent to the integral under the wave energy spectrum, m0. Therefore 4ηrms, where ηrms ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðη−η Þ2

ð16Þ

is equivalent to the significant wave height, Hm0. A moving average value of ηrms represents a moving representative significant wave height over time. Long-term offshore variations are clearly seen in Fig. 10 and averaging over 20Tp effectively smoothes wave by wave variation in ηrms while being a much smaller time scale than long-term variations; averaging over 10Tp produced slightly noisier variations. In Fig. 10 values of ηrms have been normalised using the local still water depth to indicate when waves become depth limited. The time axis is given as (t−xprobe/cg)/Tp Table 3 Incident wave parameters from calibration run, at Wave Gauge 1 (model scale).

Wave gauge number

Distance from back wall (m)

Bed level (m)

WG1 WG2 WG3 WG4 WG5 WG6 WG6

36.94 32.45 26.99 20.00 13.61 7.00 8.07

0.000 0.000 0.139 0.316 0.379 0.446 0.437

WG7 WG7 WG7 WG8 WG8 WG8

5.85 5.46 6.35 4.54 4.46 5.85

0.446 0.446 0.446 0.446 0.446 0.446

Notes

During calibration and S57 test. For all other sea wall tests and post-calibration. For pre-calibration For post-calibration For all tests with sea wall For pre-calibration For post-calibration For all tests with sea wall

Test name

Hm0 (m)

Tp (s)

Still water depth (m)

ξ0

Spectrum

S23 S55 S57 S59 S131 S135 P01 P02 P03 P04 P06 P0785 P0775

0.117 0.223 0.233 0.219 0.196 0.172 0.290 0.238 0.228 0.261 0.268 0.229 0.204

1.495 1.945 2.041 2.04 1.815 1.992 2.235 2.220 1.963 2.123 2.235 2.220 2.220

0.593 0.644 0.655 0.635 0.666 0.593 0.655 0.654 0.644 0.593 0.644 0.666 0.666

0.052 0.050 0.051 0.053 0.049 0.058 0.051 0.055 0.049 0.051 0.053 0.056 0.060

Spectrum from field Spectrum from field Spectrum from field Spectrum from field Spectrum from field Spectrum from field JONSWAP γ = 3.3 JONSWAP γ = 3.3 JONSWAP γ = 3.3 JONSWAP γ = 3.3 JONSWAP γ = 3.3 JONSWAP γ = 3.3 Repeat of P0785 with reduced wave height

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

1

39

SWAB Model Bathymetry: Whole Domain

z (m)

0.8 Wave Input Location

0.6 0.4 0.2

With Seawall Without Seawall

0

0

5

10

15

20

25

30

35

40

45

50

x (m) 0.9

SWAB Model Bathymetry: Inshore Section

z (m)

0.8

F5

0.7 0.6

F1

F2

F3

F4

0.5 0.4 41.6

41.7

41.8

41.9

42

42.1

42.2

42.3

42.4

42.5

42.6

x (m) Fig. 8. SWAB model bathymetry for Anchorsholme sea wall, showing locations where wall force, F, (Eq. 14) has been applied.

so that the wave energy arrives at different wave probes at almost the same location on the graphs. For pre-breaking at wave gauge WG2, the SWAB model gives a good representation of ηrms. Depth-limited breaking starts to occur around wave gauge WG4 where H/h ≈ 0.6. Wave breaking can start some way offshore especially for the larger waves; in this case the run using the ∂η/∂t breaking condition will transfer to the NLSW equations at this offshore location, causing wave decay throughout most of the

0.26

0.3

Test S 55 Wave Heights

domain. Surprisingly at wave gauge WG4 predictions using the ∂η/∂t breaking are a little better than for the H/h breaking condition. However the initiation of breaking is a very complex phenomenon, e.g. Stansby and Feng (2005) and Ting (2001), to be represented by a simple depth-limited criterion and importantly once bore-like behaviour has become established by WG7 the predictions by the H/h breaking condition are considerably better than for the ∂η/∂t breaking condition. The long-term offshore

Test S 57 Wave Heights

0.22

0.24 0.22

0.2

0.25

0.18

0.2

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0.16

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Hm0 (m)

Test S 131 Wave Heights

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0.1

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0.1

0.08

0.08 0.06 0.06 10

5

0

x / Lp,in

0.05

10

5

x / Lp,in SWAB H/h Breaking

SWAB ∂η /∂t Breaking

0

10

5

0

x / Lp,in

Experimental Data

Fig. 9. Significant wave heights for model without sea wall – for first 410 s of waves (x is the distance from the inshore boundary of the SWAB model).

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M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

S55 Waves - No Seawall 0.8

SWAB – WG7

SWAB – H/h Breaking Condition

Normalised rms = 4 rms/ dprobe

0.7 Experiment – WG7

0.6 SWAB – WG4

0.5

Experiment – WG4

0.4

WG2 SWAB – Solid Line Experiment – Dotted Line

0.3 0.2 50

100

150

200

250

(t-xprobe/cg) / Tp 0.8

SWAB –

Breaking Condition

Experiment – WG7

Normalised rms = 4 rms/ dprobe

0.7

SWAB – WG4

0.6 SWAB – WG7

0.5

Experiment – WG4

0.4

WG2 SWAB – Solid Line Experiment – Dotted Line

0.3 0.2 50

100

150

200

250

(t-xprobe/cg) / Tp Fig. 10. Time-series of 20 wave moving average normalised ηrms, at various wave gauge locations, for S55 wave conditions without the sea wall.

4.3. Nearshore waves with the sea wall The H/h wave breaking model (with Cbh = 0.6) was used for these tests, with a wall force applied at the locations shown in Fig. 6. For the SWAB model of the sea wall, it is to be expected that the factor kwall (Eq. (14)) should be greater for the recurve wall with jets directed upwards or backwards than for the four steps. However, the recurve wall does not reflect all the flow back in the opposite direction; kwall will therefore be considerably less than 2. It is arguable whether this force should be imposed at the steps at all. To examine the effect of the wall forces, the three tests in Table 4 were carried out, and compared with the experimental data. Fig. 11 shows the free surface output for the S57 conditions for all three reflection tests, alongside that for the run without the sea wall, using the H/h breaking condition. Even without the wall force applied the wave train has changed significantly; therefore, the reflection caused by the runup and run-down of the incident waves clearly has a much greater effect than the wall force. At this time, the three runs with the sea wall in place are very similar, and this was found to be generally the case. The wall force itself therefore has little effect on reflected waves although, as Section 4.4 will show, it does have a considerable effect on wave overtopping. Fig. 12 shows spectral energy curves (for the run with kwall = 1.0, applied at the recurve wall only), for wave gauges WG2, WG4 and Table 4 SWAB model tests with sea wall.

WG8, which are 7.5 m from the paddle, 14.8 m from the sea wall toe and 0.67 m from the sea wall toe respectively. Note that wave gauge WG8 for these tests with the sea wall is at the same location as WG7 in the tests without the sea wall described in Section 4.2. At WG2, and also at WG4 where some waves are breaking, the SWAB model runs match very well with the experimental data. However, at wave gauge WG8, located 0.67 m from the sea wall toe and 1.30 m from the sea wall crest, the low frequency peak is somewhat overestimated by SWAB, and there is another peak, around 1.2fp, that is not present in the experimental data. Using linear wave theory, 1.2fp corresponds to a wavelength of 2.14 m, which would suggest that this peak is caused by reflections from the sea wall (i.e. an antinode at 0.5 wavelengths from the wall). The experimental data features increased spectral energy between f = 0.25 and f = 0.9fp, with a trough at about 1.25fp, and a smaller peak at 2.0fp. The troughs and peaks in the

0.9

Free surface level near seawall at t = 409 s, S57 Conditions

0.85 0.8

0.7 0.65 0.6

0.45 kwall at Steps

kwall at Recurve wall

No wall force Wall force at steps and recurve wall position Wall force at recurve wall position only

0 0.5 0

0 1.0 1.0

Without Seawall

0.55 0.5

Test description

WG8 (WG7 for Calib.)

With Seawall k wall = 1.0 at recurve only

0.75

(m)

variations in ηrms have effectively become uniform inshore due to depth-limited breaking.

0.4 35

With Seawall kwall = 1.0at recurve kwall = 0.5at steps

36

37

With Seawall No wall force

38

39

40

With Seawall Bed Level

41

42

43

x (m) from Offshore Boundary Fig. 11. SWAB free surface levels – with different wall force coefficients.

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M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

S55 Waves- With Seawall - Normalised Spectral Energy SWAB Test:

5

k wall = 1.0 at recurve k wall = 0 at steps

SWAB – WG8 4

S(f) / m0 (s)

41

Experiment – WG8

WG2 SWAB – Solid Line Experiment – Dotted Line

3

WG4 SWAB – Solid Line Experiment – Dotted Line

2

1

0

0

0.5

1

1.5

2

2.5

(f/ fp) Fig. 12. Normalised spectral energy density, at various wave gauge locations, for S55 wave conditions. SWAB uses the H/h breaking condition.

experimental data will also be caused by reflections, though it is less clear how they correspond with the nodes and antinodes calculated using linear wave theory. This suggests that the SWAB model is unable to fully capture the complex processes involving the incident and reflective waves near the sea wall. Fig. 13 shows the effect of the sea wall on the spectral energy at wave gauge WG8 (WG7 for calibration run), for the S55 test conditions. Note that the figure shows actual spectral energy density; it has not been normalised in any way. The presence of the wall causes a very large increase in the low frequency region: the surf beat. The magnitude of this increase is approximately four-fold, both in the SWAB model and in the wave flume. If one considers the perfect reflection of a wave from a vertical wall, the wave height is doubled at the antinodes. Spectral energy density is proportional to the square of the amplitude; perfect reflection would therefore quadruple the energy at an antinode. These low frequency waves can therefore be assumed to be almost perfectly reflected by the sea wall. The post-breaking differences between model and experiment almost certainly originate from the limitations of what is a simple breaker model for a very complex phenomenon, e.g. Stansby and Feng (2005) and Ting (2001). The greater spread of energy in the experiments suggests that strong turbulent mixing associated with breaking waves near and at the wall causes reflection to be less coherent than in the model.

-3

Fig. 14 shows moving averaged ηrms for the S55 test. There are some differences in timing between SWAB predictions and the experimental data; variations in ηrms over time at wave gauges WG4 and WG8 occur with both, but the two datasets are rarely in phase. This is probably due to the spectral differences described above. However, despite the differences between SWAB and the experiments in terms of spectral energy density, these figures show that the overall wave energy in all parts of the flume is predicted reasonably. 4.4. Wave overtopping Due to the nature of the method for measuring overtopping in the flume, it is not possible to produce SWAB time-series that are simultaneous with the overtopping data. Firstly, the overtopping water ran down a chute into the collection tank; the time taken for this to occur is dependent on the discharge down the chute. Secondly, the water fell into the overtopping tank, which had a gauge at the far end; the falling water caused seiching to occur in the tank, eventually settling at a higher water level. A typical extract from a time-series is shown in Fig. 15. Small overtopping events caused a slow trickle of water to run down the chute; in Fig. 15, three main overtopping waves can be detected from the extract at approximately 525 s, 560 s, 580 s and 695 s. However, between the two major events at 580 s and 695 s there is a

Test S55 Spectral Energy

x 10 7

SWAB, With Seawall k wall = 1.0 at recurve k wall = 0 at steps

6 5

Expt. Data , With Seawall at wave gauge WG8

4 3

SWAB, Without Seawall H/h breaking condition

Expt. Data , Without Seawall at wave gauge WG7

2 1

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(f/ fp) Fig. 13. Spectral energy density at the wave gauge closest to sea wall: comparison between run with sea wall (with wall force at recurve only) and test without sea wall (with H/h breaking condition), S55 wave condition.

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M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

1

SWAB Test: k wall = 1.0 at recurve k wall = 0 at steps

S55 Waves - With Seawall

0.9

SWAB – WG7

Normalised rms = 4 rms / dprobe

0.8

Experiment – WG7 0.7 0.6

SWAB – WG4

0.5

Experiment – WG4 0.4

WG2 SWAB – Solid Line Experiment – Dotted Line

0.3 0.2 50

100

150

200

250

(t-xprobe /cg) / Tp Fig. 14. Time-series of 20 wave moving average normalised ηrms, at various wave gauge locations, for S55 wave conditions. SWAB uses the H/h breaking condition.

noticeable increase in the overtopping volume of about 0.001 m 3/m due to the small overtopping events. Two particular features were noticed from the raw overtopping data. Firstly, an overtopping event caused an increase in oscillations in the tank, which slowly dampen as the water level settles; each new overtopping event is characterised by this sudden increase in oscillations. Secondly, an overtopping event is also characterised by a jump in the water level; these new oscillations occur at a higher level than the previous oscillations. So, each overtopping event causes a jump in the variance of the water level, as well as a jump in the skewness defined by: Eðx−x Þ3 Skewness ¼  3=2 Eðx−x Þ2

ð17Þ

Therefore, a moving variance or moving skewness filter is likely to be more effective than a moving average filter. A moving skewness filter was found to contain spikes of shorter duration than a moving variance filter, with less variation in the magnitude of spikes from test to test. For these reasons, the moving skewness algorithm was preferred. It is important to remember that a perfect detection algorithm for overtopping events is not possible, with some small events remaining undetected and some false events being obtained. The

0.029

Filtered Overtopping Time Series, S55 Conditions

Overtopping Volume, V(m3/m)

0.028 0.027 0.026 0.025 0.024 0.023 0.022

Raw data

0.021

Filtered data, moving skewness Detected overtopping events

0.02 520

540

560

580

600

620

640

660

680

t (s) Fig. 15. Extract from experimental overtopping time-series: S55 conditions.

700

filtered overtopping volume, using the moving skewness algorithm, is shown in Fig. 15 alongside the raw data. Fig. 16 shows time-series of overtopping volumes for the S55 and S59 wave conditions, for the SWAB tests described in Table 4, in comparison with the experimental data. Note that time in Fig. 16 is zero at the chute placement time; the cumulative overtopping volume is also zero at the chute placement time. Clearly the wall force has a significant effect on the overtopping volumes. The test without the wall force consistently leads to overestimates of overtopping volumes, whereas the test with the force at both the steps and the recurve wall consistently leads to underestimates. For the S55 wave conditions, the best results are achieved when the force is applied only at the recurve wall; for the S59 conditions, applying the force at the recurve wall leads to a 50% overestimate in the overtopping volume. In this case, results could probably be improved by increasing the friction factor at the steps. As a note of caution, it can be seen that the SWAB overtopping time-series do not match the experimental data event by event; this is to be expected as the nearshore waves, described in the previous section, did not necessarily match over time. All the other tests of Table 3 were run applying the force at the recurve wall only. Fig. 17 shows results for all the SWAB tests, compared with the experimental data; the graph on the left shows mean overtopping rates and the graph on the right shows the volume of the maximum overtopping wave, Vmax, from each of the tests. The mean rates appear to overpredict within a factor of 2, except at the lower end of the scale. There is slightly more uncertainty with maximum overtopping volumes; there appears to be a trend towards SWAB underpredicting overtopping for low values and overpredicting for high values, although with the number of runs and the narrow range of the magnitudes (ten of the thirteen tests have a value for Vmax between 0.002 m 3/m and 0.02 m3/m) it is not possible to make a definite conclusion on this. The P0785 test, which was repeated in the flume with exactly the same wave input and repeated again as the P0775 test with a reduced wave height, is shown in Fig. 18 alongside the SWAB model runs. It is apparent that the SWAB model does overestimate overtopping rates for both wave heights. However, there is also considerable variation in the results from the experiments; repeating exactly the same test caused a 25% reduction in the cumulative overtopping volume; this reduction appeared to be consistent over the whole time-series, emphasising the sensitivity of wave overtopping to very small changes in conditions. The reduced wave height repeat run caused a slight further loss in overtopping. Therefore, whenever overtopping calculations are carried out, one must be aware that the uncertainties are significant. If one were to repeat the same wave conditions with a

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

Cumulative Volume (m3/m)

0.1

43

S55 Conditions: Overtopping Volumes

0.08

SWAB kwall = 1.0 at recurve kwall = 0 at steps

0.06

SWAB No wall force Experimental Data

SWAB kwall = 1.0 at recurve kwall = 0.5 at steps

0.04

0.02

0

0

100

200

300

400

500

600

700

Time after chute placement (s)

Cumulative Volume (m3/m)

0.1

S59 Conditions: Overtopping Volumes

0.08

SWAB No wall force

SWAB kwall = 1.0 at recurve k wall = 0 at steps

0.06 SWAB k wall = 1.0 at recurve k wall = 0.5 at steps

0.04

Experimental Data

0.02

0

0

100

200

300

400

500

600

700

Time after chute placement (s) Fig. 16. Time-series of overtopping volumes: SWAB tests with H/h condition compared with experimental data.

different wave train (i.e. the same spectrum with different random phases and same water level), rather different results are possible; this will be examined further below (see Fig. 19). Ten additional SWAB runs were set up, using the S55 energy spectrum, but assigning a different set of random phases to each set of components; these additional runs each consisted of 250 waves. Fig. 19 shows overtopping volume time-series for these runs, showing to what extent overtopping can vary from one wave train to another. Of the ten additional runs, the mean overtopping rate ranged from 3.50× 10−5 m 3/s/m to 7.61× 10−5 m 3/s/m. Also shown in Fig. 19 are extracts from the experimental data and the original SWAB run of the

10-3

experimental data. For the S55 conditions, mean overtopping rates in the flume and the original SWAB run were 4.24 × 10 −5 m 3/s/m and 3.67× 10−5 m3/s/m respectively, which is at the lower end of the range of overtopping rates for these conditions. Fig. 20 shows the field data with the experimental data and SWAB runs given at full scale (i.e. experimental time multiplied by 15 1/2 and volume multiplied by 15 2 according to Froude scaling). It can be seen that the field overtopping rate is approximately equal to the laboratory overtopping rate. Over the period shown in Fig. 20, the laboratory data overestimates overtopping by 6% and the SWAB model underestimates it by 21%, relative to the field data. However, as Figs. 18

SWAB vs Experiment: Mean Overtopping Rate

10-1

10-2

10-5 SWAB = Expt.

10-6

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10-7

10-8 -8 10

Vmax (SWAB) (m3/m)

qm (SWAB) (m3/m/s)

10-4

SWAB vs Experiment: Maximum Overtopping Wave

10-3

SWAB = Expt.

10-4 SWAB = 2 Expt. SWAB = Expt. / 2

10-5

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10-6

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qm (Experiment) (m3/m/s)

10-4

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10-6 -6 10

10-5

10-4

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10-1

Vmax (Experiment) (m3/m)

Fig. 17. Mean overtopping rates and volumes of maximum overtopping wave for all tests: SWAB values, with wall force at recurve wall only, versus experimental values.

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M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

0.2

Extract from Overtopping Time Series

Overtopping Volume (m3/m)

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

100

200

300

400

500

600

t (s) after chute placement Fig. 18. P07 overtopping time-series: SWAB, with wall force at recurve wall only versus experimental data, showing repeated runs.

and 19 both demonstrate, the close match between the laboratory and field data is somewhat fortuitous, as a wave train with the same spectrum but different phase components or even an exact repeat of the experiment could cause a change in the overtopping rates. Also, the wind will affect overtopping to some degree in the field. Data from Hilbre Island, 56 km to the south of Anchorsholme, supplied by the National Oceanography Centre (Liverpool Bay Coastal Observatory), shows that the wind direction was likely to be onshore, with a speed of 10 to 20 ms −1; this would have the effect of increasing overtopping rates. To relate to existing practice, the mean overtopping discharges are compared with methods recommended by the EurOtop Manual (Pullen et al., 2007). Calculations for wave overtopping for the tests of Table 3 were performed using the empirical equations for embankment seawalls (EurOtop, Eq. 5.8) and the overtopping neural network tool. The wave period, Tm − 1,0 and significant wave height, Hm0, are required at the toe of the structure. These are spectral parameters and low frequency energy in the surf zone causes, Tm − 1,0 to be larger than, Tp/1.1, the usual approximation, and Hm0, to be larger than the probabilistic significant wave height H1/3. Overtopping discharges

calculated using Tp/1.1 and H1/3 were closer to the experimental results, effectively avoiding low frequency oscillations. Fig. 21 shows mean overtopping discharges calculated using these tools, compared with the SWAB model and laboratory measurements. Except for cases with very small discharges, the SWAB and EurOtop values can be seen to be within a factor of two of the experiments. For larger discharges, the EurOtop equations and neural network tool tend to underestimate slightly, while the SWAB model as calibrated tends to overestimate slightly.

5. Wave runup analysis with offshore reef 5.1. Setting up the SWAB model The SWAB model was set up with the same geometry as the experiment. The SWAB wave input requires incident waves only. The method of Frigaard and Brorsen (1995) was used to separate the incident and reflected waves from the experimental data from the offshore wave gauges WG1 and WG2, shown in Fig. 4. The location of WG1 was

Physical Model and Randomly Phased SWAB Model Overtopping Volumes: S55 Conditions 0.025

Cumulative Volume (m3/m)

Different randomly phased SWAB runs SWAB run of experimental time-series Experimental data

0.02

0.015

0.01

0.005

0

0

50

100

150

200

250

300

Time after chute placement (s) Fig. 19. Effect of random phase on overtopping volumes, showing 10 different randomly phased SWAB runs (with H/h condition) alongside the original SWAB run of experimental time-series and experimental data. Note that overtopping volumes have been set to zero at chute placement time.

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

Field, Physical Model and SWAB Model Overtopping Volumes S55, Full Scale

3.5

Field Data

3

Cumulative Volume (m3/m)

45

Experimental Data 2.5

SWAB Original Model Run

2 1.5 1 0.5 0

0

200

400

600

800

1000

1200

Time (s) (after chute placement for SWAB and expt.) Fig. 20. Overtopping volume time-series at full scale: comparison between SWAB run (with H/h condition), experimental data and field data.

used as the location of the wave input for the model, with 10.0 m added offshore to include the sponge layer. To take account of the reef the ∂η/∂t breaking criterion needed to be modified. Firstly, all waves were made to break on the reef as observed experimentally (i.e. Boussinesq terms switched off and horizontal diffusion switched on). When the waves moved inshore of the reef, the breaking process was stopped, with the ∂η/∂t breaking criterion applied (with Cbt = 0.30) for the waves approaching the shore. The SWAB runs using the H/h criterion need no such modification; as before, a coefficient Cbh = 0.6 was used. For the longer period wave condition, with the H/h breaking criterion, the SWAB model would became unstable; it was thought that the abrupt decrease in water depth over a small proportion of a wavelength

at the seaward slope of the reef was responsible. Therefore, for this particular run, the gradient of the seaward slope of reef was reduced by doubling its horizontal length from 0.32 m to 0.64 m. This change avoided instability.

5.2. Nearshore waves Figs. 22 and 23 show time-series of ηrms, normalised using the local still water depth, again taking the moving average over 20Tp. Only runs using the H/h breaking criterion are shown; the ∂η/∂t criterion gives larger errors, especially for the longer period wave conditions. In addition the H/h breaking algorithm allows, if required, the

Fig. 21. Mean overtopping rates from SWAB (●), EurOtop empirical formulae (○) and overtopping neural network (◊), in comparison with measured mean discharges from experiments.

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M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

1

SWAB - H/h Breaking Condition

SWAB – WG6

0.9

4

rms

/ dprobe

0.8

Experiment – WG6

0.7 0.6

SWAB – WG4

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Experiment – WG4

0.4

WG3 SWAB – Solid Line Experiment – Dotted Line

0.3 0.2 50

60

70

80

90

100

110

120

130

140

150

Time (s) Fig. 22. Time-series of moving average normalised ηrms, for T1 wave condition (Tp = 1.14 s) at wave gauges WG3 (before reef), WG4 (on reef) and WG6 (between reef and shore).

automatic cessation of breaking landward of the reef, and it does not require a manual specification of breaking on the reef. Goda (2000) gives two useful parameters for the study of nonlinear and breaking waves. Firstly, the skewness, γ1, of the free surface level is given as: γ1 ¼

N  3 1 1X η −η N i¼1 i

ð18Þ

η3rms

of γ3, which would increase again as these waves move closer to the shore. However, it was found that the simple breaking model of SWAB is inadequate to make good predictions of γ3. Because the atiltness parameter is a function of ∂η/∂t it is rather sensitive to errors in the shape of a modelled wave. It may require more complete Navier–Stokes equation solvers to accurately replicate the shape of the waves travelling across the reef towards the shore.

5.3. Wave runup Nonlinear waves are characterised by short steep crests with long shallow troughs; the skewness parameter will equal zero for linear waves but will become positive when the crests steepen and the troughs become flatter. If, as described by Beji and Battjes (1993), the nonlinear bound components are released when the waves move back into deeper water, the positive value of γ1 will move closer to zero. Fig. 24 shows part of the time-series for the moving average skewness parameter for the SWAB run using the H/h breaking condition, with the longer period (T2) wave conditions. For clarity, only approximately 25 periods of data have been shown in the figure, and only the wave gauge on the reef (WG4) and the two wave gauges landward of the reef (WG5 and WG6) are shown. It appears that the skewness parameter increases as waves propagate from the reef (from WG4 to WG5), maintaining a similar level as the waves move closer to the shore. The SWAB run does not produce a close match with the experimental data for both tests for the wave gauge closest to the shore (WG6). For the shorter period (T1) wave conditions, similar results were achieved, with the SWAB model underestimating the skewness parameter at both wave gauges WG5 and WG6. Goda (2000) also describes an “atiltness” parameter, γ3, which is effectively the skewness parameter applied to ∂η/∂t instead of η. Near-breaking and breaking waves are characterised by steepening of the front face of the wave; therefore γ3 will increase. Breaking waves returning to deeper water would give a decrease in the value

1.6

Fig. 25 shows extracts from the runup time-series for both wave conditions, showing the SWAB runs (only for the H/h breaking condition) alongside the experimental data. Note that the experimental data does not go below 0.0345 m; this is the level of the base of the runup meter. The SWAB model gives a reasonable simulation of the runup time-series. To generate runup distributions, a zero-crossing method was used to separate individual runups from the time-series. Therefore, a runup is said to begin when it exceeds a certain level; 0.036 m was selected as this level to avoid including noise in the smaller values. The runup level is said to be the peak level before the next runup begins. Some runup crests will be neglected where the run-down does not go below 0.036 m; however, this is not important for comparison as long as the SWAB data and the experimental data are analysed in the same way. Fig. 26 shows the wave runup distributions. Firstly, the SWAB run that uses the H/h breaking algorithm gives very good results for both test conditions. This is not unexpected as the nearshore ηrms was very well predicted (see Figs. 22 and 23). The run using the ∂η/∂t breaking condition tends to overpredict runup for both tests, but especially for the longer period wave conditions, where the nearshore ηrms (Fig. 23) is also overpredicted. It is worth expressing a note of caution with the SWAB runup distribution; the distributions shown in Fig. 26 show runup exceedance

SWAB - H/h Breaking Condition

4

rms

/ dprobe

1.4

WG6 SWAB – Solid Line Experiment – Dotted Line

1.2 1

SWAB – WG4

0.8

Experiment – WG4

0.6

WG3 SWAB – Solid Line Experiment – Dotted Line

0.4 0.2 100

120

140

160

180

200

220

240

260

280

300

Time (s) Fig. 23. Time-series of moving average normalised ηrms, for T2 wave condition (Tp = 2.25 s) at wave gauges WG3 (before reef), WG4 (on reef) and WG6 (between reef and shore).

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

1.6

47

Skewness Parameter

SWAB – WG5

1.4 Experiment – WG6

γ1

1.2

Experiment – WG5

1 0.8

SWAB – WG6

0.6

WG4 SWAB – Solid Line Experiment – Dotted Line

0.4 0.2 200

210

220

230

240

250

260

270

280

290

300

Time (s) Fig. 24. Time-series of 20Tp moving average skewness and atiltness parameters, at various wave gauge locations, for the longer period (T2) wave condition at wave gauges WG4 (on reef), WG5 (landward of reef) and WG6 (between reef and shore).

probabilities relative to the number of runups. Tables 5 and 6 show the number of recorded runups for the two test conditions. It is apparent that both SWAB runs underestimate the number of runups. However, the number of recorded runups is affected by the 0.036 m level that was used to separate individual runups. This level is quite high and it is apparent from examination of Fig. 25 that a small underestimation of runup levels leads to many runup crests falling below the 0.036 m limit. In this paper, the effect of the friction coefficient (related to bed shear stress in Eq. 7), Cf, has not been presented. A value of Cf = 0.01 was used here; other testing with the SWAB model has shown that increasing Cf from 0.005 to 0.015 decreases runup levels by approximately 10% (McCabe, 2011). 6. Discussion

Filtered Runup Level - SWL (m)

A shallow-water and Boussinesq (SWAB) model with random wave input has been developed to simulate nearshore waves, runup and overtopping. The simple addition of a wall force has been incorporated

0.1

to take account of partial momentum flux reversal due to breaking wave impact on near vertical and/or recurve walls. Two different algorithms for the initiation of breaking have been tested: a conventional ∂η/∂t criterion where all waves inshore are said to be broken has the advantage of being calculated directly from the SWAB model solution, whereas the new H/h criterion for individual waves requires an algorithm to identify each wave. However, the latter criterion gives much better results in the nearshore. The stability of this semi-implicit numerical scheme is suited to the switching on and off of Boussinesq terms. The model is validated against field data for sea wall overtopping and laboratory experiments based on Froude scaling which gave similar overtopping rates (after appropriately scaling). We believe this is the first time such comprehensive comparisons have been undertaken. A moving average of significant wave height allows short-term wave-by-wave randomness to be smoothed and separated from the longer term variation and the moving average is generally well predicted from offshore to nearshore. However the nearshore wave

Test T1: Extract from Runup Time Series

0.08

0.06

0.04

0.02 120

125

130

135

140

145

150

155

160

290

300

310

320

Filtered Runup Level - SWL (m)

Time (s) 0.14 Test T2: Extract from Runup Time Series 0.12 0.1 0.08 0.06 0.04 0.02 240

250

260

270

280

Time (s) SWAB Run (H/h Breaking)

Mase Data

Fig. 25. Extracts from runup time-series: SWAB runs compared with experimental data.

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M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

Runup Distributions 1

1

T1 Waves (Tp = 1.14s) SWAB ∂η/∂t Breaking

0.7

SWAB H/h Breaking

0.6

0.8

Exceedance Probability

Exceedance Probability

0.8

0.5

Mase Data 0.4 0.3

0.7

0.5

0.3

0.1

0.1 0.05

0.06

0.07

0.08

0.09

0.1

SWAB H/h Breaking

0.4

0.2

0.04

SWAB ∂η/∂t Breaking

0.6

0.2

0

T2 Waves (Tp = 2.25s)

0.9

0.9

0

Mase Data

0.05

Runup Level (m)

0.1

0.15

Runup Level (m)

Fig. 26. Runup distributions: SWAB runs compared with experimental data.

spectra show marked differences from the experiments and the breaking wave shape, defined by two skewness parameters, is also not represented well. Switching off Boussinesq terms causes wave speed to reduce and the wave profile to steepen in order to represent breaking while conserving momentum flux; however this clearly does not represent the complex physics of the breaking process, particularly at its initiation. More complete models directly solving the Navier–Stokes equations may be expected to give a closer representation of these processes but at a computational cost several orders of magnitude greater. A SWAB model run of 500 waves was found to take about 20 min on a conventional PC. This makes wave by wave modelling of a complete storm event quite practical. For the case of waves passing over a submerged reef where breaking occurs before ceasing in deeper inshore water, good results for significant wave height landward of the reef are produced. Runup levels and distributions on the landward slope are well predicted. With sea wall overtopping, the additional wall force is made proportional to momentum flux over the height of the recurve wall with constant of proportionality kwall = 1.0, which is equivalent to the flow being directed vertically upwards, giving general agreement with flume experiments. In the wave flume it was observed that there was considerable splashing at the sea wall, with water being directed offshore, vertically upwards, as well as landwards, during overtopping events. kwall should

therefore be regarded as an overall calibration factor for a given geometry. It should be noted that the overtopping tests all had low surf similarity parameters, corresponding to spilling breakers. For plunging breakers, which may start to break closer to the sea wall, kwall may well be different for the same geometry. It should also be emphasised that there are uncertainties in overtopping rates in the experiments. Rates for nominally identical conditions differed by 25%. Also different random phases between spectral components can cause maximum overtopping rates to be double the minimum rates in the SWAB model. Given these uncertainties the overestimation of field overtopping rate by about 6% in the laboratory tests, and the underestimation of 21% by the SWAB model are both well within the bounds of accuracy which may be expected. Overtopping rates for the laboratory experiments were also calculated using both the empirical equations and neural network tool recommended by the EurOtop Manual. Estimates from the SWAB model and the EurOtop tools were generally within a factor of two of the experiments with the SWAB model tending to overestimate and the EurOtop tools underestimate mean discharges. However, the use of the EurOtop tools requires careful estimation and interpretation of significant parameters, and they cannot provide the wave by wave information on overtopping volumes given by the SWAB model. 7. Conclusions

Table 5 Numbers of runups and incident waves: Test T1 (Tp = 1.14 s). Breaking condition

Number of runups (NR)

Number of incident waves (NW)

Ratio of runups to waves (NR/NW)

∂η/∂t H/h Experimental data

46 35 54

214 214 214

0.215 0.164 0.252

Table 6 Numbers of runups and incident waves: Test T2 (Tp = 2.25 s). Breaking condition

Number of runups (NR)

Number of incident waves (NW)

Ratio of runups to waves (NR/NW)

∂η/∂t H/h Experimental data

172 115 180

220 220 220

0.782 0.523 0.818

A robust shallow-water and Boussinesq (SWAB) model has been extended to allow intermittent breaking based on a wave-by-wave height-to-depth ratio: a local Miche criterion. Boussinesq terms are simply switched off during breaking and turbulent diffusion switched on. This allows large waves to break while small waves do not and it allows breaking waves to cease breaking as they move into deeper water. Tests against experimental data showed a wave height to depth ratio of 0.6 proved to be effective. While the model is theoretically for mild slopes it has been extended to allow steep walls or recurves by adding a momentum flux reversal term or wall force to account for breaking wave jets being deflected upwards and/or seawards in an area local to the sea wall. This requires the empirical calibration of a given sea wall geometry. This has been achieved through comparison with field and corresponding laboratory experiments; for the recurve wall tested, a momentum flux reversal corresponding to vertical jet ejection was generally effective. Interestingly the wave

M.V. McCabe et al. / Coastal Engineering 74 (2013) 33–49

profile seaward of a sea wall was little affected by the wall force while the overtopping volume was markedly affected. While there was good direct comparison with the experiments, there could be differences in experimental overtopping volume by 25% for nominally the same wave conditions. For the range of tests considered the SWAB model could overestimate overtopping by up to a factor of 2 with a constant wall force factor. Mean overtopping discharges calculated using tools recommended by the EurOtop Manual were also generally within a factor of two of the experimental data. The SWAB model would give better agreement through improved calibration of the wall force factor, at the expense of further complication. The model was used to test sensitivity of overtopping to different random phase distributions for the amplitude components of a spectrum; it was found that ratio of minimum to maximum volume was about 2. As phase distribution cannot be predicted, this increases the uncertainty of any method used to make overtopping predictions.

Acknowledgements The authors would like to thank Professor Hajime Mase for his help with clarification and explanation of aspects of Mase et al. (2004) and for providing detailed data from their experiments for waves breaking over submerged reefs. The authors would also like to thank the referees for their helpful comments. This work contributes to the Coastal Inundation Super Work Package of the Flood Risk Management Research Consortium. The FRMRC is supported by grant EP/F020511/1 from the Engineering and Physical Sciences Research Council, in partnership with the DEFRA/EA Joint Research Programme on Flood and Coastal Defence, UKWIR, OPW (Ireland) and the Rivers Agency (Northern Ireland). The first author was funded on an EPSRC DTA studentship. The assistance of Dr Tim Pullen and Mr Dan Carter in setting up the wave flume tests at HR Wallingford is gratefully acknowledged. The field data were provided by the UK Environment Agency through HR Wallingford.

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