1. Numerical Simulation of Wave-Seawall Interaction Clive Mingham, Derek Causon, David Ingram and Stephen Richardson Centre for Mathematical Modelling and Flow Analysis, Manchester Metropolitan University, UK

2. VOWS Violent Overtopping of Waves at Seawalls. Outline Background Experimental set up Numerical simulation Results Conclusions

3. VOWS Violent Overtopping of Waves at Seawalls. The VOWS Project (Violent Overtopping of Waves at Seawalls) http://www.vows.ac.uk Aim: To investigate the violent overtopping of seawalls and help engineers design better sea defences. Photo by G. Motyker, HR Wallingford

4. VOWS Violent Overtopping of Waves at Seawalls. Experimental Edinburgh, and Sheffield Universities 2D wave flume tests In Edinburgh. 3D wave basin tests at HR Wallingford. Numerical Manchester Metropolitan University AMAZON-CC to help experimental design AMAZON-SC to simulate overtopping

5. VOWS Violent Overtopping of Waves at Seawalls. VOWS Experimental Team: William Allsop (Sheffield). Tom Bruce, Jonathan Pearson and Nicolas Napp (Edinburgh) Funding: EPSRC - Grant M/42428

6. VOWS Violent Overtopping of Waves at Seawalls. VOWS: Numerical approach Use 1-D Shallow Water Equations to simulate wave flume and compare with experiments Use 2-D Shallow Water Equations to provide advice for wave basin experiments Simulate violent wave overtopping using more sophisticated numerics (see later)

7. VOWS Violent Overtopping of Waves at Seawalls. Edinburgh wave flume cross section bed seawall Wave maker Collection system Sloping beach Shallow water simulations were reasonable … so go to wave basin

8. VOWS Violent Overtopping of Waves at Seawalls. Experimental Investigation Schematic of HR Wallingford wave basin Wave guide Seawall Wave maker  8m 21m 19m 10m Water collection system

9. VOWS Violent Overtopping of Waves at Seawalls. Experimental Investigation Wave maker: 2 blocks, 8, 0.5m units in each SWL: 0.425 - 0.525m Elbow angle   Vertical or 1:10 battered wall Wave Climate: Regular waves and JONSWAP: period 1.5s, wave height 0.1m Variable wave guide length 5 – 10m

10. VOWS Violent Overtopping of Waves at Seawalls. Advice to Experimentalists Effect of gap between wave maker and wave guides - leakage Wave guide length to balance - Diffraction (around corners) - Reflection (from wall and sides) Wave heights at seawall Likely overtopping places

11. VOWS Violent Overtopping of Waves at Seawalls. Numerical Simulation of Wave Basin: AMAZON-CC Shallow Water Equations – provide a cheap 2D (plan) model of the wave basin which gives qualitative features (but not correct!) Cartesian cut cell Method –Automatic boundary fitting mesh generation –Moving boundary to simulate wave maker Surface Gradient Method (SGM) is used for bed topography

12. VOWS Violent Overtopping of Waves at Seawalls. Shallow Water Equations (SWE) U conserved quantities, H inviscid fluxes, Q source terms g gravity, h depth,  = g h, q = u i + v j velocity, b x, b y bed slopes,

13. VOWS Violent Overtopping of Waves at Seawalls. Semi-discrete approximation A ij : area of cell ij U ij, Q ij : averages of U, Q over cell ij defined at cell centre m : number of sides of cell ij n k : outward pointing normal vector to side k whose magnitude is the length of side k H k : interface fluxes

14. VOWS Violent Overtopping of Waves at Seawalls. 2-step Numerical Scheme Predictor step: grid cell ij showing interface fluxes and side vectors

15. VOWS Violent Overtopping of Waves at Seawalls. Corrector step: : solution to Riemann problem at cell interface H = H(U), find U at interface by MUSCL interpolation

16. VOWS Violent Overtopping of Waves at Seawalls. MUSCL interpolation U i R = U i + 0.5  x i  U i U i L = U i - 0.5  x i  U i Limited gradient :  U i f : flux limiter function

17. VOWS Violent Overtopping of Waves at Seawalls. Approximate Riemann Solver HLL robust efficient extends to dry bed - change wave speeds

18. VOWS Violent Overtopping of Waves at Seawalls. Cartesian Cut Cell Method Automatic mesh generation Boundary fitted Extends to moving boundaries

19. VOWS Violent Overtopping of Waves at Seawalls. Method solid boundary Input vertices of solid boundary (and domain)

20. VOWS Violent Overtopping of Waves at Seawalls. overlay Cartesian grid

21. VOWS Violent Overtopping of Waves at Seawalls. Boundary fitting mesh Compute solid boundary/cell intersection points and obtain cut cells

22. VOWS Violent Overtopping of Waves at Seawalls. Classical Cartesian grid gives saw tooth representation of body

23. VOWS Violent Overtopping of Waves at Seawalls. (adaptive) cut cell grid for a coastline wave basin Cut cells work for any domain

24. VOWS Violent Overtopping of Waves at Seawalls. Also works for moving bodies: e.g. wave maker Independently moving wave paddles

25. VOWS Violent Overtopping of Waves at Seawalls. Cut cell treatment of moving body prescribe body (wave maker unit) velocity At each time step: - find the position of the body - re-cut the mesh - use generalised MUSCL reconstruction - use exact Riemann solution at moving interface

26. VOWS Violent Overtopping of Waves at Seawalls. AMAZON-CC: generation of oblique waves using cut cells

27. VOWS Violent Overtopping of Waves at Seawalls. Results Numerical simulation showing effect of gap between wave maker and guides

28. VOWS Violent Overtopping of Waves at Seawalls. Results VOWS: Numerical simulation of wave seawall interaction

29. VOWS Violent Overtopping of Waves at Seawalls. Conclusions The shallow water equations, although technically incorrect, can provide useful guidance to set up wave basin experiments More accurate simulation needs to include non-shallow water effects like dispersion AMAZON-CC with its automatic boundary fitted mesh generation and moving body capability is widely applicable