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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 - PowerPoint PPT Presentation
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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
VOWSViolent Overtopping of Waves at Seawalls.
Outline
• Background
• Experimental set up
• Numerical simulation
• Results
• Conclusions
VOWSViolent Overtopping of Waves at Seawalls.
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
VOWSViolent Overtopping of Waves at Seawalls.
Experimental
Edinburgh, and SheffieldUniversities
• 2D wave flume testsIn Edinburgh.
• 3D wave basin tests atHR Wallingford.
Numerical
Manchester MetropolitanUniversity
• AMAZON-CC to helpexperimental design
• AMAZON-SC to simulate overtopping
VOWSViolent Overtopping of Waves at Seawalls.
VOWS Experimental Team:
William Allsop (Sheffield).
Tom Bruce, Jonathan Pearson
and Nicolas Napp (Edinburgh)
Funding:
EPSRC - Grant M/42428
VOWSViolent Overtopping of Waves at Seawalls.
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)
VOWSViolent Overtopping of Waves at Seawalls.
Edinburgh wave flume cross section
bed
seawall
Wave maker
Collectionsystem
Sloping beach
Shallow water simulations were reasonable …so go to wave basin
VOWSViolent Overtopping of Waves at Seawalls.
Experimental Investigation
Schematic of HR Wallingford wave basin
Wave guide
Seawall
Wavemaker
8m
21m
19m
10m
Water collection system
VOWSViolent Overtopping of Waves at Seawalls.
Experimental Investigation
• Wave maker: 2 blocks, 8, 0.5m units in each
• SWL: 0.425 - 0.525m• Elbow angle
• 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
VOWSViolent Overtopping of Waves at Seawalls.
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
VOWSViolent Overtopping of Waves at Seawalls.
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
VOWSViolent Overtopping of Waves at Seawalls.
Shallow Water Equations (SWE)
ASAdAQddAU
tSnH
y
x2
2
gbφ
gbφ
0
=Q,
2/φ+vφ
2/φ+uφ
φ
,
vφ
uφ
φ
U
jq
iq
q
H
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,
VOWSViolent Overtopping of Waves at Seawalls.
Semi-discrete approximation
ijm
1kkk
ij
ijQ
A
1
t
U
nH
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
VOWSViolent Overtopping of Waves at Seawalls.
2-step Numerical Scheme
Predictor step:
)Q
(A
Δt/2UU
nij1/2-ji,
Dij1/2+ji,
Uij
j1/2,-iLijj1/2,+i
Rij
ij
nij
1/2+nij
nHnH
nHnH
n1
n4
n3
n2
HU HR
HL HD grid cell ij showing interface fluxes and side vectors
VOWSViolent Overtopping of Waves at Seawalls.
)Q
(A
tUU
1/2nij1/2-ji,
*1/2-ji,1/2+ji,
*1/2+ji,
j1/2,-i*
j1/2,-ij1/2,+i*
j1/2,+iij
nij
1+nij
nHnH
nHnH
Corrector step:
U U L R U U i,j i+1,j
*j1/2,+iH : solution to Riemann problem at cell interface
H = H(U), find U at interface by MUSCL interpolation
VOWSViolent Overtopping of Waves at Seawalls.
MUSCL interpolation
UiR = Ui + 0.5 xi Ui
UiL = Ui - 0.5 xi Ui
Limited gradient : Ui
1ii
1ii
i1i
i1ii
xx
UU,
xx
UUfΔU
f : flux limiter function
VOWSViolent Overtopping of Waves at Seawalls.
Approximate Riemann Solver
HLL• robust
• efficient
• extends to dry bed - change wave speeds
LR
LRLRj1/2,+iL
j1,iLj1/2,+iR
ji,R
Rj1/2,+iL
j1,i
Lj1/2,+iR
ji,
j1/2,+i*
j1/2,i
q-q
)UU(qqqq
,0q,
,0q,
nHnH
nH
nH
nH
VOWSViolent Overtopping of Waves at Seawalls.
Cartesian Cut Cell Method
• Automatic mesh generation
• Boundary fitted
• Extends to moving boundaries
VOWSViolent Overtopping of Waves at Seawalls.
Method
solid boundary
Input vertices of solid boundary (and domain)
VOWSViolent Overtopping of Waves at Seawalls.
overlay Cartesian grid
VOWSViolent Overtopping of Waves at Seawalls.
Boundary fitting mesh
Compute solid boundary/cell intersection points and obtain cut cells
cut cell
VOWSViolent Overtopping of Waves at Seawalls.
Classical Cartesian grid gives saw tooth representation of body
VOWSViolent Overtopping of Waves at Seawalls.
y
x
(adaptive) cut cell grid
for a coastlinewave basin
Cut cells work for any domain
VOWSViolent Overtopping of Waves at Seawalls.
Also works for moving bodies:
e.g. wave maker
Independently moving wave
paddles
VOWSViolent Overtopping of Waves at Seawalls.
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
VOWSViolent Overtopping of Waves at Seawalls.
AMAZON-CC: generation of oblique waves using cut cells
VOWSViolent Overtopping of Waves at Seawalls.
Results
Numerical simulation showing effect of gap
between wave maker and guides
VOWSViolent Overtopping of Waves at Seawalls.
Results
VOWS: Numerical simulation of wave seawall interaction
VOWSViolent Overtopping of Waves at Seawalls.
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