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

VOWSViolent Overtopping of Waves at Seawalls.

Outline

• Background

• Experimental set up

• Numerical simulation

• Results

• Conclusions


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


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


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)


Edinburgh wave flume cross section

bed

seawall

Wave maker

Collectionsystem

Sloping beach

Shallow water simulations were reasonable …so go to wave basin


Experimental Investigation

Schematic of HR Wallingford wave basin

Wave guide

Seawall

Wavemaker

8m

21m

19m

10m

Water collection system


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


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)

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,


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


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


)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


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


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


Cartesian Cut Cell Method

• Automatic mesh generation

• Boundary fitted

• Extends to moving boundaries


Method

solid boundary

Input vertices of solid boundary (and domain)


overlay Cartesian grid


Boundary fitting mesh

Compute solid boundary/cell intersection points and obtain cut cells

cut cell


Classical Cartesian grid gives saw tooth representation of body


y

x

(adaptive) cut cell grid

for a coastlinewave basin

Cut cells work for any domain


Also works for moving bodies:

e.g. wave maker

Independently moving wave

paddles


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


AMAZON-CC: generation of oblique waves using cut cells


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

numerical simulation of wave-seawall interaction

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