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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

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

  • Background
  • Experimental set up
  • Numerical simulation
  • Results
  • Conclusions
the vows project v iolent o vertopping of w aves at s eawalls http www vows ac uk

To investigate the

violent overtopping

of seawalls and help

engineers design

better sea defences.

The VOWS Project(Violent Overtopping of Waves at Seawalls)

Photo by G. Motyker, HR Wallingford



  • 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)


EPSRC - Grant M/42428

vows numerical approach
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)




Wave maker

Sloping beach


Edinburgh wave flume cross section

Shallow water simulations were reasonable …

so go to wave basin

experimental investigation
Experimental Investigation


Schematic of HR Wallingford wave basin

Water collection system



Wave guide






experimental investigation1
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
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
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
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
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
MUSCL interpolation

UiR = Ui + 0.5 xi Ui

UiL = Ui - 0.5 xiUi

Limited gradient : Ui

f : flux limiter function

approximate riemann solver
Approximate Riemann Solver
  • HLL
  • robust
  • efficient
  • extends to dry bed - change wave speeds
cartesian cut cell method
Cartesian Cut Cell Method
  • Automatic mesh generation
  • Boundary fitted
  • Extends to moving boundaries

Input vertices of solid boundary (and domain)

solid boundary


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


Numerical simulation showing effect of gap

between wave maker and guides


VOWS: Numerical simulation of wave

seawall interaction

  • 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