CHAPTER
159
Numerical
Modelling
of
Bed
Evolution
Behind
a
Detached
Breakwater
Philippe
Pechon
1
and
Charles
Teisson
1
Abstract
The
sedimentological
impact
of
waves
on
a
sandy
beach
with
a
detached
breakwater
is
simulated
using
a
compound
system
of
models.
The
results
are
satisfying
since
a
salient
could
be
generated
behind
the
structure.
They
are
in
agreement
with
bed
evolutions
surveyed
in
experimental
facilities
and
in
nature.
A
quantitative
analysis
of
the
results
performed
in
the
framework
of
the
working
group
'Coastal
Area
Modelling'
of
the
project
MAST
G8M
shows
that
the
volume
of
accretion
computed
here
is
in
good
agreement
with
the
volumes
obtained
by
other
models
and
empirical
formula.
Introduction
A
numerical
system
has
been
developed
for
simulating
bed
evolution
due
to
breaking
waves.
Wave,
current,
sediment,
transport
and
bed
evolution
are
computed
successively.
Since
the
wave
field
is
affected
by
sea
bed
changes,
hydrodynamic
phenomena
are
up-dated
when
bed
evolution
is
significant.
The
system
of
models
is
illustrated
on
figure
1.
The
three
numerical
models
belong
to
the
library
TELEMAC
based
on
finite
element
technic.
They
are
coupled
within
an
automatic
procedure.
Comparisons
of
the
numerical
results
with
measurements
in
flume
cases
were
carried
out
in
the
past.
They
gave
satisfying
results
(Broker
Hedegaard
et
al
1992,
Pechon
1994),
undertow
as
well
as
bed
evolution
were
well
predicted.
Standard
parameters
were
used
for
these
simulations
except
for
the
sediment
transport
formula
where
the
suspended
load
was
increased
in
the
surf
zone
to
account
for
breaking
effect.
1
Laboratoire
National
d'Hydraulique,
EDF.
6,
quai
Watier,
78400
Chatou.
France
2050
NUMERICAL
MODELLING
OF
BED
EVOLUTION
2051
Input-
Bathymetry
Offshore
wave
condition
Wave-
•
wave
propagation
and
deformation
breaking
process
3D
Wave-driven
currents
•
-
circulation
induced
by
wave
driving
terms,
mixing
of
momentum
and
bottom
friction
sand
transport
and
bed
evolutions-
-
bed
load
and
suspended
sediment
transport
due
to
wave
and
current
Fig.l
Structure
of
models
The
present
article
deals
with
the
application
of
the
models
on
a
three-
dimensional
case
;
the
impact
of
a
detached
breakwater
built
along
a
rectilinear
beach
is
investigated.
In
a
previous
study
(Pechon
et
al.,
1995)
the
good
agreement
of
computed
currents
with
measurements
collected
in
a
basin
(Mory
and
Hamm,
1995)
was
exhibited
on
the
same
but
reduced
scaled
structure.
Accurate
data
of
seabed
movement
and
wave
climate
are
not
available
for
the
present
full
scaled
test.
However
the
effect
of
breakwaters
is
qualitatively
known
according
to
surveys
in
coastal
zones,
and
the
ability
of
models
to
reproduce
realistic
morphological
changes
can
be
checked.
Moreover
an
intercomparison
exercise
with
other
modellers
was
performed
(Nicholson
et
al
1995)
in
the
framework
of
the
European
program
MAST2
G8M.
The
numerical
models
The
wave
model
The
model
ARTEMIS
V1.0
solves
the
complete
mild-slope
equations
:
div
(C
C
G
gradcp)
+
(0^-
(p
=
0
with
C
=
w/k
C:
phase
celerity
C
G
_
I
(1
+
2kh
-)
C
Co-
wave
group
celerity
2
sh
2
kh
(p
:
complex
horizontal
part
of
the
potential
0(x,y,z,t)
=
Z(z)(p(x,y)e
-'«
ch(k(h
+
z))
where
Z(z)
=
ch
(kh)
2052
COASTAL
ENGINEERING
1996
According
to
the
rather
schematise
bathymetry
tested
here,
the
wave
height
decay
in
the
surf-zone
is
given
by
the
formula
Hb
=
0.8
h
where
Hb
is
the
breaking
wave
height
and
h
the
still
water
depth.
The
radiation
stress
can
be
calculated
knowing
the
wave
potential.
However
this
leads
to
a
very
irregular
radiation
stress
field
which
has
to
be
smoothen.
So
it
is
prefered
here
to
express
the
driving
terms
in
function
of
the
dissipation
of
breaking
waves.
The
computed
instantaneous
wave
velocity
follows
an
ellipse,
therefore
there
is
not
a
simple
incidence.
In
the
following
current
and
transport
models,
the
required
incidence
is
supposed
to
be
given
by
the
large
axis
of
the
ellipse.
The
time-averaged
current
model
The
model
TELEMAC-3D
V3.0
solves
the
time-averaged
three-
dimensional
equations
accounting
for
the
vertical
variability
of
the
forces
due
to
breaking
waves
and
especially
roller
effect
(Pechon
1994).
They
read,
in
the
case
of
a
flume
for
clarity
:
dU
2
dUW
d(uJ-W)
,
Buww^
_
9g
9^V~
(
dx
dz
dx
dz
dx
dz
3f/
dW
=Q
dx
dz
with
U,
W
:time-averaged
current
due
to
breaking
waves
u
w
,
w
w
:
instantaneous
wave
velocity
u',w':
turbulent
fluctuations
of
velocity
x
:
free
surface
level
t
:
contribution
of
the
roller
of
breaking
waves
The
overbar
indicates
time-averaged
quantities.
The
following
closures
are
taken
:
3
(u£-wj)
=
JLO_
with
dx
ph
C
j)
•
energy
dissipation
du
w
w,
C
:
wave
celerity
w
w
_
1
D_
h
:
mean
water
depth
dz
2
P
h
C
V
t
^-
v
t
=
Mh
(£
)i/3
,
constant
M
=
0.4
dz
'
>
14
_D_
T
:
wave
period
pgHT
H:
wave
height
NUMERICAL
MODELLING
OF
BED
EVOLUTION
2053
The
sediment
transport
model
The
sediment
transport
is
calculated
by
a
satellite
version
of
the
numerical
model
TSEF
V3.0
using
Bailard
formula
(1981)
in
terms
of
the
near-bed
velocity
field
resulting
from
the
previous
model.
The
bed
load
and
suspended
load
transport
rates
are
expressed
as:
q
b
=
fcw£b
AgtgQ
~X
_
Jew
&s
AgW
s
<|if|
2
M>--^—
<|M|
3
>;
<\u\'
i
u>-^-tge<\u\
5
>i
in
which
qb
bed-load
transport
rate
m
2
/s
q
s
suspended
load
transport
rate
few
friction
factor
wave+current
£b
efficiency
factor
for
bed-load
transport
e
b
=0.13
£,
efficiency
factor
for
suspended
load
transport
£,
=
0.024
i
unit
vector
directed
downslope
<j>
internal
angle
of
friction
of
the
sediment
W
s
fall
velocity
A
apparent
density
u
instantaneous
nearbed
velocity
vector
tg
8
bed
slope
<.>
time-averaged
quantity
This
formula
is
expected
to
give
the
transport
rate
inside
and
outside
the
surf-
zone.
The
friction
factor
is
adopted
for
the
wave
alone
:
f
w
=
exp{
-6.+
5.2
(<W.i9)
k
s
with
A
w
=
"
orbital
excursion
2
sin
kh
k
w
=
3
D
s
roughness
coefficient
D
s
grain
size
diameter
In
the
present
model
the
velocity
u
is
the
summation
of
the
time-averaged
velocity
and
the
instantaneous
orbital
wave
velocity
near
the
bed.
At
the
present
state
of
knowledge,
sediment
transport
formulae
have
broad
correlation
between
predicted
and
observed
values.
Soulsby
et
al
(1995)
compared
Bailard
formula
with
data
and
they
showed
that
only
57%
of
the
predicted
transport
rate
lay
within
a
factor
of
5
of
the
data.
In
the
following
application,
it
has
been
observed
that
the
original
formula
overestimates
the
transport
rate.
So
it
is
divided
by
10
here,
which
is
equivalent
to
increase
the
time
scale.
2054
COASTAL
ENGINEERING
1996
Bed
evolution
model
The
mass
conservation
of
sand
is
also
computed
by
the
previous
model
which
solves
the
equation
for
mass
conservation
of
sediment.
Wave
and
current
fields
are
updated
when
the
bed
evolution
is
greater
than
0.5
time
the
water
depth
at
3
nodes
of
the
mesh
or
0.3
time
at
10
nodes.
In
the
following
application
this
criteria
occurs
about
every
30
hours.
Application
Description
of
the
case
The
domain
of
computation
is
presented
on
figure
2.
At
the
initial
time
the
bottom
slope
is
1:50
nearshore
and
the
bed
is
horizontal
offshore
where
the
bottom
level
is
-7.4
m.
The
four
lateral
boundaries
are
closed
for
current
and
sediment
transport.
Only
the
offshore
boundary
is
open
for
waves.
The
generated
wave
at
the
offshore
boundary
is
perpendicular
to
the
shoreline.
In
the
present
model
the
wave
is
supposed
to
be
regular,
with
a
period
of
8.0
s.
The
wave
height
is
1.2
m.
The
median
grain
size
is
250
u..
The
mesh
grid
for
wave
computation
contains
10500
nodes.
It
is
refined
in
order
to
have
at
least
10
nodes
per
wave
length.
The
horizontal
mesh
grid
for
current
and
sediment
transport
(fig
3)
has
1450
nodes
and
each
vertical
profile
has
9
nodes
in
the
three-dimensional
computation
of
currents.
Results
For
the
initial
bottom,
along
a
current
cross-shore
profile
the
wave
height
increases
and
reaches
1.6
m
at
the
distance
of
the
structure
y
=
-110
m
where
the
still
water
depth
is
-2.0
m
(fig.
4),
then
it
breaks.
Wave
pattern
diffracts
behind
the
structure.
In
spite
of
the
account
for
three-dimensional
effects,
the
velocity
field
is
nearly
homogeneous
over
the
water
depth.
In
fact
in
this
case
the
currents
are
mainly
generated
by
alongshore
gradient
of
surface
elevation
because
the
wave
direction
is
nearly
normal
to
the
shoreline,
and
this
gradient
is
constant
over
the
vertical.
The
3D
effect
would
be
stronger
with
oblique
wave
direction
because
the
current
would
be
also
generated
directly
by
driving
terms
with
non-uniform
vertical
profile.
The
near-bed
velocity
field
displays
a
large
eddy
behind
the
breakwater
(fig.
5).
In
the
shallower
part
the
intensity
reaches
1.0
m/s
whereas
it
is
less
along
the
structure
in
deeper
part.
In
the
open
area
the
velocity
field
is
very
irregular
because
of
variations
of
wave
height.
A
cell
takes
place
in
the
right
hand
side
of
the
beach
because
of
the
closed
boundaries
at
this
corner.
2DV
undertows
are
not