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Numerical Simulation of Wave-Seawall Interaction

Numerical Simulation of Wave-Seawall Interaction. Clive Mingham, Derek Causon, David Ingram and Stephen Richardson C entre for M athematical M odelling and F low A nalysis, Manchester Metropolitan University, UK. Outline. Background Experimental set up Numerical simulation

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Numerical Simulation of Wave-Seawall Interaction

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  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. Outline • Background • Experimental set up • Numerical simulation • Results • Conclusions

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

  4. 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 Experimental Team: William Allsop (Sheffield). Tom Bruce, Jonathan Pearson and Nicolas Napp (Edinburgh) Funding: EPSRC - Grant M/42428

  6. 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. seawall Collection system Wave maker Sloping beach bed Edinburgh wave flume cross section Shallow water simulations were reasonable … so go to wave basin

  8. Experimental Investigation 19m Schematic of HR Wallingford wave basin Water collection system Seawall j Wave guide 21m 10m Wave maker 8m

  9. Experimental Investigation • Wave maker: 2 blocks, 8, 0.5m units in each • SWL: 0.425 - 0.525m • Elbow angle j = 0, 45, 120o • 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. 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. 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. 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, bx, by bed slopes,

  13. Semi-discrete approximation Aij : area of cellij Uij , Qij : averages of U, Q over cell ij defined at cell centre m : number of sides of cell ij nk : outward pointing normal vector to side k whose magnitude is the length of side k Hk : interface fluxes

  14. 2-step Numerical Scheme Predictor step: grid cell ij showing interface fluxes and side vectors

  15. Corrector step: : solution to Riemann problem at cell interface H = H(U), find U at interface by MUSCL interpolation

  16. MUSCL interpolation UiR = Ui + 0.5 xi Ui UiL = Ui - 0.5 xiUi Limited gradient : Ui f : flux limiter function

  17. Approximate Riemann Solver • HLL • robust • efficient • extends to dry bed - change wave speeds

  18. Cartesian Cut Cell Method • Automatic mesh generation • Boundary fitted • Extends to moving boundaries

  19. Method Input vertices of solid boundary (and domain) solid boundary

  20. overlay Cartesian grid

  21. Compute solid boundary/cell intersection points and obtain cut cells Boundary fitting mesh

  22. Classical Cartesian grid gives saw tooth representationof body

  23. Cut cells work for any domain (adaptive) cut cell grid for a coastline wave basin

  24. Independently moving wave paddles Also works for moving bodies: e.g. wave maker

  25. 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. AMAZON-CC: generation of oblique waves using cut cells

  27. Results Numerical simulation showing effect of gap between wave maker and guides

  28. Results VOWS: Numerical simulation of wave seawall interaction

  29. 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

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