Computational Modelling of Free Surface Flows: wave impact and wave power

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Computational Modelling of Free Surface Flows
wave impact and wave power

• Professor Derek Causon, Clive Mingham and Dr
  David Ingram
• Centre for Mathematical Modelling and Flow
  Analysis
• Manchester Metropolitan University

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Outline

• Wave overtopping
• Cartesian cut cell method
• Surface capturing method
• Wave energy device simulation
• Future work

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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 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
21m

• 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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Cut Cell Method
Input vertices of solid boundary (and domain)
solid boundary
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overlay Cartesian grid
Cut Cell Method
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Cut Cell Method
Compute solid boundary/cell intersection points
and obtain cut cells
Boundary fitted mesh
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In contrast the classical Cartesian grid gives
saw tooth representation of the boundary
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Cut cells work for any domain
(adaptive) cut cell grid for a coastline
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Also work for moving bodies e.g. wave maker
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AMAZON-CC generation of oblique waves using cut
cells
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Numerical Wave basin
AMAZON-CC simulation of a wave basin
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Results

• Numerical simulation of wave seawall interaction

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AMAZON-SC Numerical Wave Flume

• Two fluid (air/water), boundary conforming, time
  accurate, conservation law based, flow code
  utilising the surface capturing approach.
• Cartesian cut cell techniques are used to
  represent coastal structures and bathymetry.

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Multi-fluid Model

• Incompressible Navier-Stokes solver Based on an
  artificial compressibility solver.
• Surface-capturing method
• Treats the free surface as a contact
  discontinuity in the density field, allowing the
  use of modern high resolution shock capturing
  methods.
• Fully two phase approach which solves in both the
  air and the water.

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Governing equations

• 2D incompressible, Euler equations with variable
  density.

b is the coefficient of artificial compressibility
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Discretisation

• The equations are discretised using a finite
  volume formulation
• Where Qi is the average value of Q in cell i
  (stored at the cell
• centre), Vi is the volume of the cell, Fij is the
  numerical flux
• across the interface between cells i and j and
  and Dlj is the
• length of side j.

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Convective fluxes

• The convective flux (Fij) is evaluated using
  Roes approximate Riemann solver.
• To ensure second order accuracy, MUSCL
  reconstruction is used
• where (x,y) is a point inside the cell ij, r is
  the coordinate
• vector of (x,y) relative to ij and DQij is the
  slope limited
• gradient.

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Time discretisation

• The implicit backward Euler scheme is used
  together with an artificial time variable t (to
  ensure a divergence free velocity field) and a
  linearised RHS.

The resulting system is solved using an
approximate LU factorisation.
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Computer Implementation

• A Jameson-type dual time iteration is used to
  eliminate ? at each real (outer) iteration.
• The code vectorises efficiently with simulations
  typically taking about three hours to run on an
  NEC SX6i deskside supercomputer.

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Test Cases

• Low amplitude sloshing
• Collapsing Cylinder
• Collapsing Cylinder with an obstacle
• Regular wave propagation

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Collapsing water column flow over an obstacle

• As the water flows to the right it hits the
• obstacle, flows over it and hits the opposite
• wall.
• The confined air escapes upwards and the
• water falls to the floor on the other side of
• the obstacle.
• Comparisons with experiments conducted
• by Koshizuka et al. (1995) are presented.

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Collapsing water column flow over an obstacle
computation
experiment
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Collapsing water column flow over an obstacle
computation
experiment
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Boundary Conditions

• Seaward boundary a velocity controlled
  procedure is used to generate waves, velocities
  are either computed from linear wave theory using
  JONSWAP spectra or specified based on shallow
  water simulations.
• Atmospheric boundary a constant atmospheric
  pressure gradient is applied. Spray and water
  passing out of this boundary are lost from the
  computation.
• Landward boundary a solid wall boundary
  condition is used for the landward end of the
  domain.
• Seawalls and beaches modelled using Cartesian
  cut cell techniques.

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VOWS Wave Flume experiments

• Experimental programme conducted in the 20m wave
  flume in Edinburgh and the 100m wave flume at
  UPC, Barcelona.

• Vertical and 101 battered walls tested
  (with/without berms and recurves) on a 110
  beach, for impulsive wave conditions
  (0.03ltHlt0.10) using 1000 random waves.

• Water is 0.09m deep at the toe of the wall and
  the wall has a freeboard of 0.17m.

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VOWS Test 1A

• Wave Climate Hs0.063, Tm1.23
• Sea Wall 101 batter, Rh0.1296
• Wave Conditions h0.0544
• Wave gauges, sampled at 100Hz, located 1.0, 2.0,
  3.0, 4.25, 5.5, 6.75, 8.0 and 11.21 metres from
  the seawall.
• Wall fitted with an overtopping detector and a
  load cell.

Shallow Water Model AMAZON-CC, showing test
section 2m from the seawall.
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Seaward boundary conditions

• Surface elevation
• experimental wave gauge 2m from wall.
• Used to specify r
• U Velocity
• Computed velocity from the Riemann invariant
  boundary condition of a shallow water simulation
• Applied uniformly to water column
• Other variables
• p and v are extrapolated

?
?
?
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AMAZON-SC simulation

• Overtopping event occurred 142 seconds into the
  experiment.
• Seaward boundary located 2m from wall.
• Landward boundary is transmissive.
• Assume flat water for initial conditions.

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Velocity vectors at t2.75s
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Wave Energy

• Simulation of wave energy devices (with Queens
  University, Belfast )

LIMPET Oscillating Water Column (OWC)
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Acknowledgements
Professor TJT Whittaker and Dr Matt Folley
School of Civil Engineering, Queens
University of Belfast Device design, experiments
and full scale trials
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Wave Energy
LIMPET 500KW Station, Islay, Scotland. First
wave power station connected to the national
grid, in the world.
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Wave Energy
LIMPET Supplies the town of Portnahaven and
local industry.
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Wave Paddle Test

• Waves simulated by a moving paddle using
  AMAZON-SC

Comparison with experimental results
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LIMPET OWC Simulation

• Simulation of waves interacting with LIMPET using
  AMAZON-SC

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LIMPET Comparison
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• Oscillating wave surge converter (OWSC)

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The Vane as a Wave Maker
Simulation of a vane with prescribed motion in
air and water using AMAZON-SC
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The Vane as a Wave Maker

• Animation showing velocity vectors and free
  surface position around the vane

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OWSC Device Modelling using AMAZON-SC

• The motion of the vane is derived from the forces
  exerted on it by
• the surrounding water.

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Future Work

• AMAZON-SC is easily extended to 3D.
• EPSRC Grant in preparation.
• Implementation of AMAZON-SC, incorporating AMR,
  on high performance computers.
• EPSRC Grant being reviewed.
• Apply MMU solvers to sediment transport
• Currently being developed
• Application of AMAZON-SC to deck slam

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Water Entry of a Rigid Wedge
Simulation of a wedge moving at 1m/s until total
immersion using AMAZON-SC
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Water Exit of a Rigid Wedge
Simulation of a wedge moving at 1m/s from water
into air using AMAZON-SC
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Water Entry of a Rigid Wedge
T 0.25s
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Water Entry of a Rigid Wedge
T 1.0s
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Water Entry of a Rigid Wedge
T 1.5s
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Water Entry of a Rigid Wedge
T 2.23s
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References

• JB Shiach, CG Mingham, DM Ingram and T Bruce
  (2004) The applicability of the shallow water
  equations for modelling violent wave overtopping,
  Coastal Engineering 51(1)1-15.ISSN 0378-3839
• JG Zhou, DM Causon, CG Mingham and DM Ingram
  (2004) Numerical Prediction of Dam-Break Flows in
  General Geometries with Complex Bed Topography,
  Journal of Hydraulic Engineering - ASCE 130
  (4)332-340.ISSN 0733-9429
• L Qian, DM Causon, DM Ingram and CG Mingham
  (2003) Cartesian Cut Cell Two-Fluid Solver for
  Hydraulic Flow Problems, Journal of Hydraulic
  Engineering 129(9)688-696.ISSN 0733-9429
• DM Causon, DM Ingram and CG Mingham (2001) A
  Cartesian Cut Cell Method for Shallow Water Flows
  with Moving Boundaries., Advances in Water
  Resources 24(2001)899--911.ISSN 0309--1708
• JG Zhou, DM Causon, CG Mingham and DM Ingram
  (2001) The Surface Gradient Method for the
  Treatment of Source Terms in the Shallow Water
  Equations, Journal of Computational Physics
  1681-25.ISSN 0021-9991
• K Hu, CG Mingham and DM Causon (2000) Numerical
  Simulation of Wave Overtopping of Coastal
  Structures Using the Non-linear Shallow Water
  Equations, Coastal Engineering 41433-465.ISSN
  0378-3839
• DM Causon, DM Ingram, CG Mingham, G Yang and RV
  Pearson (2000) Calculation of Shallow Water Flows
  Using a Cartesian Cut Cell Approach., Advances in
  Water Resources 23545-562.ISSN 0309-1780

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• Thank you for your Attention