Borthwick, Alistair

Fitzgerald, Colm

Orszaghova, Jana

Raby, Alison

Taylor, Paul

Whittaker, Colin

Accurate estimation of irregular wave runup at sloping beaches is important in assessments of coastal erosion and flood risk. We define runup as the vertical elevation of the shoreline position, driven by swash oscillations about the wave setup level. To date there have been rather few investigations into irregular wave runup statistics, and these have assumed linear, Gaussian behaviour. This presentation will describe use of a 1-D hybrid Boussinesq-NLSW model to produce statistical information on the runup of irregular waves, arising from a random sea state offshore of the beach. A validated technique will be presented for simulating long-duration irregular runup time series whereby the incident waves are generated using second-order accurate paddle motions with reflected waves effectively absorbed before they reach the paddle

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     Edinburgh Research Explorer                                      Random wave runup at a plane beachCitation for published version:Borthwick, A, Fitzgerald, C, Taylor, P, Orszaghova, J, Whittaker, C & Raby, A 2015, Random wave runup ata plane beach. in 4th DNVA-RSE Norway-Scotland Symposium, Edinburgh 12-13 October 2015.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:4th DNVA-RSE Norway-Scotland Symposium, Edinburgh 12-13 October 2015General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019  4th DNVA-RSE Norway-Scotland Symposium, Edinburgh, 12-13 October 2015  Random wave runup at a plane beach  Colm Fitzgerald1, Paul H. Taylor1, Jana Orszaghova2, Alistair G. L. Borthwick3, Colin Whittaker4, and Alison C. Raby4   1University of Oxford, 2HR Wallingford, 3University of Edinburgh, 4University of Plymouth  Introduction Accurate estimation of irregular wave  runup at sloping beaches is important in  assessments of coastal erosion and flood risk.  We define runup as the vertical elevation of the shoreline position, driven by swash oscillations about the wave setup level. To date there have been rather few investigations into irregular wave runup statistics, and these have assumed linear, Gaussian behaviour.  This presentation will describe use of a 1-D hybrid Boussinesq-NLSW model to produce statistical informat ion on the runup of irregular waves, arising from a random sea state offshore of the beach.  A validated technique will be presented for simulating long-duration irregular runup time series whereby the incident waves are generated using second-order accurate paddle motions with reflected waves effectively absorbed before they reach the paddle.    Numerical Model The numerical wave flume solves a weakly dispersive, weakly nonlinear version of the Boussinesq-type equations (see Madsen & Sørensen, 1992) using fourth-order fin ite differences for non-breaking waves, and the nonlinear shallow water (NLSW) equations using a second-order MUSCL-Hancock HLLC Godunov-type finite volume scheme.  A local mapping is used to facilitate the paddle-type wave generator.  The switch between  the Boussinesq and NLSW solvers is triggered by a wave breaking criterion  set according to the local water free surface slope, with dispersive terms turned off gradually.  Flux gradient and source term balancing of the NLSW  solver is achieved by recasting the equations in deviatoric form with wetting and drying modelled using an algorithm developed by Liang & Borthwick (2009). Full details o f the model are given  by Ors zaghova et al. (2012, 2014).  To generate the irregular waves, we use the second-order wave generation theory by Schäffer (1996) without first-order evanescent contributions to second-order error waves.  A generating-absorbing sponge layer proposed by Zhang et al. (2014) is used to achieve effective wave absorption over the range of the incident wave frequency spectrum.  Irregular wave runup is  modelled by running concurrently (1) a simulat ion of incident irregular waves over a flat bed, in the absence of a beach, which are absorbed in a sponge layer, with (2) a simulation of the irregular waves propagating across the generating -absorbing sponge layer (with the solution from (1) imposed on the sponge layer) and up the plane beach.  Fitzgerald et al. (2015) provide further details of the methodology and its verification tests.   Results Irregular waves were simulated over a 1/10 reflect ive, 1/20 intermediate, and 1/40 dissipative beach slopes  with corresponding domain  lengths of 18, 24, and 32 m. The input spectrum was  Pierson-Moskowitz with peak frequency 0.464 Hz.  The still water depth in the flat bed region between the paddle and beach toe was 0.5 m.  The spectral zero-crossing period was Tz = 1.75 s, with 280 waves equivalent roughly to a return period.  The significant wave height was 0.1 m.  For each slope, sixty wave records were acquired, each of 491.52 s repeat period.  Figs. 1and 2 d isplay part of the free surface elevation (at beach toe) and shoreline  elevation t ime histories for the 1/20 beach slope (with Froude scaling for 20 m offshore depth), the red  dashed lines are for linear paddle signal motion and the black solid for linear + 2nd order.  Fig . 3 presents histograms of frequency of occurrence against maximum runup crests that were determined from the full repeat duration shoreline elevation records obtained for each beach slope, and include the wave setup effect.  A secondary peak (indicated by the red arrow) can be discerned in the upper tail of each histogram.  Its presence indicates that there may be an upper limit to the maximum runup crest elevation (as has been observed from the analysis of swash spectra on beaches of mild slope).  It can also be seen that the width of the probability distribution decreases progressively as the beach slope reduces (noting that the plots have different scales).   There  are more runup events / time on the steeper slopes, meaning that the distributions are not exactly equivalent statistically.          Fig. 1 Free surface elevation at beach toe at 1/20 beach slope (after Froude scaling for 20 m offshore depth).         Fig. 2 Shoreline elevation time history at 1/20 beach slope (after Froude scaling for 20 m offshore depth).            Fig. 3  Max. runup crest values on beach slopes of (a) 1/10, (b) 1/20, and (c) 1/40. Arrows = ensemble maxima.  Conclusions A validated methodology has been described for the simulation of irregular wave runup at beaches in a 1-D flume, with wave generation according to second-order wavemaker theory, and reflected waves properly absorbed by a sponge layer before reach ing the offshore boundary.  The method has been used to obtain random wave runup statistics at examples of reflective, intermediate, and dissipative beaches.  Acknowledgements:  This work was supported by EPSRC Grant EP/K024108/1.  References Fitzgerald C., Tay lor P.H., Orszaghova J., Borthwick A.G.L., Whittaker C., and Raby A.C. (2015) Irregular wave runup statistics on plane beaches: application of a Boussinesq-type model incorporating a generating-absorbing sponge layer and second-order wave generation.  Submitted to Coastal Engineering. Liang, Q., and Borthwick A.G.L. (2009)  Adaptive quadtree simulation of shallow flows with wet -dry fronts over complex topography.  Computers & Fluids, 38(2): 221-234. Madsen, P.A., and Sørensen, O.R. (1992) A new form of the Boussinesq equations with improved linear dispersion characteristics.  Part 2.  A slowly -varying bathymetry,” Coastal Engineering, 18(3), pp. 183-204, 1992. Orszaghova J., Borthwick A.G.L. and Taylor P.H. (2012) From the paddle to  the beach – a  Boussinesq shallow water numerical wave tank based on Madsen and Sørensen’s equations.  J. Comp. Phys, 231(2): 328-344. Orszaghova J., Tay lor P.H., Borthwick A.G.L., and Raby A.C. (2014)  Importance of second -order wave generation for focused wave group run-up and overtopping. Coastal Engrng, 94: 63-79. Schäffer, H.A. (1996) Second-order wavemaker theory for irregular waves.  Ocean Engineering, 23(1): 47-88. Zhang Y., Kennedy A.B., Panda N., Dawson C. and Westerink J.J. (2014) Generat ing -absorbing sponge layers for phase-resolving wave models.  Coastal Engineering, 84(0): 1-9.  

Random wave runup at a plane beach

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Random wave runup at a plane beach

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