Numerical Simulation of Wave-Seawall Interaction

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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

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Outline

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

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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
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• 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

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

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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)

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Edinburgh wave flume cross section
Shallow water simulations were reasonable so go
to wave basin
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Experimental Investigation
19m
Seawall
Water collection system
j
Wave guide
21m
10m
Wave maker
8m

• Schematic of HR Wallingford wave basin

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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 110 battered wall
• Wave Climate Regular waves and JONSWAP
• period 1.5s, wave height 0.1m
• Variable wave guide length 5 10m

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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

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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

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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,
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Semi-discrete approximation
Aij area of cell ij 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
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2-step Numerical Scheme
Predictor step
grid cell ij showing interface fluxes and side
vectors
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Corrector step
solution to Riemann problem at cell interface
H H(U), find U at interface by MUSCL
interpolation
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MUSCL interpolation
UiR Ui 0.5 ?xi ?Ui UiL Ui - 0.5 ?xi ?Ui
Limited gradient ?Ui

f flux limiter function
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Approximate Riemann Solver

• HLL
• robust
• efficient
• extends to dry bed - change wave speeds

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Cartesian Cut Cell Method

• Automatic mesh generation
• Boundary fitted
• Extends to moving boundaries

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Method
Input vertices of solid boundary (and domain)
solid boundary
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overlay Cartesian grid
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Compute solid boundary/cell intersection points
and obtain cut cells
Boundary fitting mesh
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Classical Cartesian grid gives saw tooth
representation of body
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Cut cells work for any domain
(adaptive) cut cell grid for a coastline
wave basin
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Also works for moving bodies e.g. wave maker
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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

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AMAZON-CC generation of oblique waves using cut
cells
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Results
Numerical simulation showing effect of gap
between wave maker and guides
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Results

• VOWS Numerical simulation of wave
• seawall interaction

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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