The VOWS ProjectThe VOWS Project

The VOWS project (Violent Overtopping by Waves at Seawalls) was a three year project funded by the EPSRC to study violent wave overtopping and provide guidance on the construction of seawalls. As part of the VOWS project, a series of experiments were carried out at the University of Edinburgh.

Using the data from these experiments we can use our numerical model to simulate the experiments and compare the results to get an idea of accuracy.

VIOLENT WAVE OVERTOPPINGVIOLENT WAVE OVERTOPPINGJon ShiachJon Shiach

Supervisors: Clive Mingham and David IngramSupervisors: Clive Mingham and David Ingram

What is Violent Wave Overtopping?What is Violent Wave Overtopping?

Violent wave overtopping events occur when waves break against seawalls throwing water up and over the top (see Figure 2). There are hundreds of kilometres of seawalls in Great Britain many of which have footpaths, roads and railway running alongside. Extreme overtopping events have been known to wash cars, trains and more worryingly people into the sea. It is for these reasons that we wish to make predictions about violent wave overtopping so that coastal engineers can design safer sea defences.

Figure 2. These photographs were taken by the author of the west harbour seawall at Whitby on the North Yorkshire coast (November 2002). They illustrate how violent wave overtopping can occur even in relatively calm wave conditions.

Mathematical Modelling of Water Mathematical Modelling of Water FlowFlow

One model of water flow is the Shallow Water Equations (SWE). The SWE use depth-averaged momentum and water depth to describe the behaviour of water and are commonly used by mathematicians and engineers to model problems such as dam breaks and tidal flow in estuaries. In one spatial dimension they are;

where

As the SWE are depth-average any velocity in the vertical direction is neglected. Therefore for modelling violent wave overtopping the SWE should be a poor choice of governing equations. However, compared to alternative equation sets such as the Navier-Stokes equations the SWE require orders of magnitude less effort to solve and before we discard them they must be tested.

Figure 1. A news story describing the dangers of violent wave overtopping (BBC Website - news.bbc.co.uk)

Corresponding authors Email: j.shiach@mmu.ac.ukProject Supervisors Email: c.mingham@mmu.ac.uk, d.m.ingram@mmu.ac.uk

Centre for Mathematical Modelling and Flow AnalysisManchester Metropolitan University, Chester Street, Manchester, M1 5GD

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Numerical Model of the VOWS ExperimentsNumerical Model of the VOWS Experiments

The SWE equations cannot be solved by using analytical methods so instead a numerical approach is needed. We have used a finite volume method with MUSCL reconstruction (Mingham and Causon, 1998) which uses a Surface Gradient Method to deal with the source terms (Zhou et al. 2001).

The numerical model of the VOWS experiment concentrated on an area 2 metres away from the wall as seen in Figure 4.

ReferencesReferencesN.W. Allsop, P. Besley and L. Madurini (1995) Overtopping performance of vertical walls and composite breakwaters, seawalls and low reflection alternatives. Technical report.

P.Besley (1999) Overtopping of Seawalls – design and assessment manual. R&D Technical report, Environmental Agency

C.G. Mingham and D.M. Causon (1998) High-resolution finite-volume method for shallow water flows J.Hyd.Eng. 124:605 – 614

J.G. Zhou, D.M. Causon, C.G. Mingham and D.M. Ingram (2001) The surface gradient method for the treatment of source terms in the shallow water equations J. Comp. Phys. 168: 1–25

Figure 3. Digital images of the VOWS experiments showing a wave breaking against the battered wall and the resulting violent wave interaction

(Bruce and Pearson, 2001).

Figure 5. A plot of the water surface elevation for a point 1 metre away from the battered wall comparing the experimental surface (red line) against the

numerical surface (blue line).

ResultsResults

The water surface elevation 1 metre from the battered wall for the numerical model have been compared against the experimental surface in Figure 5. It is clear that the numerical model over predicts the wave heights. To analyse the accuracy of the numerical model in predicting overtopping discharge we have used dimensionless parameters to compare the numerical results against the experimental data and empirical formulae. Figure 6 shows the calculated values of the dimensionless discharge and dimensionless freeboard for both the numerical and experimental data along with and empirical formula calculated by Besley (1999). For dimensionless freeboard values of 0.17 and greater the numerical model gives excellent agreement with both the experimental data and the Besley curve.

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Figure 6. Scatter plot of the dimension less discharges and the dimensionless freeboards for the experimental (red) and numerical (blue) data along with the Besley

curve.

In order to compare the numerical results for varying impacting wave conditions we have used quantitative measure, the h* parameter, derived by Allsop et al. (1995). The lower the value of h*, the more impacting the wave conditions therefore the less accurate the SWE model should be. Figure 7 shows a scatter plot containing the percentage error values of the significant wave height (the mean of the top 1/3 wave heights), the dimensionless discharge and the number of waves overtopping between the experimental readings and the numerical results. It clearly shows as the waves become more impacting (h* gets smaller) the overall accuracy decreases.

Figure 4. Diagram showing the numerical solution domain.

Figure 7. Scatter plot of the percentage errors between the experimental readings and numerical results for the significant wave height (red), dimensionless discharge

(blue) and number of waves overtopping (green) against h*

ConclusionsConclusionsThe numerical model based upon the shallow water equations performed well when compared to real life physical experiments. The numerical water surface over estimated the heights of the wave crests which can be explained by the lack of dispersion term within the shallow water equations. The discharges recorded by the numerical model compared well against both the experimental discharges and empirical formulae. Analysis of the percentage errors for the significant wave heights, the dimensionless discharge and the number of waves overtopping showed that for h* values of greater than 0.075 a SWE equation gives values to within 20% of the experimental readings.

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