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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 Clive Mingham, Derek Causon, David Ingram and Stephen Richardson Centre for Mathematical Modelling and Flow Analysis, Manchester Metropolitan University, UK
Outline • Background • Experimental set up • Numerical simulation • Results • Conclusions
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
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
VOWS Experimental Team: William Allsop (Sheffield). Tom Bruce, Jonathan Pearson and Nicolas Napp (Edinburgh) Funding: EPSRC - Grant M/42428
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)
seawall Collection system Wave maker Sloping beach bed Edinburgh wave flume cross section Shallow water simulations were reasonable … so go to wave basin
Experimental Investigation 19m Schematic of HR Wallingford wave basin Water collection system Seawall j Wave guide 21m 10m Wave maker 8m
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
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
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
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,
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
2-step Numerical Scheme Predictor step: grid cell ij showing interface fluxes and side vectors
Corrector step: : solution to Riemann problem at cell interface H = H(U), find U at interface by MUSCL interpolation
MUSCL interpolation UiR = Ui + 0.5 xi Ui UiL = Ui - 0.5 xiUi Limited gradient : Ui f : flux limiter function
Approximate Riemann Solver • HLL • robust • efficient • extends to dry bed - change wave speeds
Cartesian Cut Cell Method • Automatic mesh generation • Boundary fitted • Extends to moving boundaries
Method Input vertices of solid boundary (and domain) solid boundary
Compute solid boundary/cell intersection points and obtain cut cells Boundary fitting mesh
Classical Cartesian grid gives saw tooth representationof body
Cut cells work for any domain (adaptive) cut cell grid for a coastline wave basin
Independently moving wave paddles Also works for moving bodies: e.g. wave maker
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
Results Numerical simulation showing effect of gap between wave maker and guides
Results VOWS: Numerical simulation of wave seawall interaction
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