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Resúmenes sobre el VIII Simposio MIA15, Málaga del 21 al 23 de Septiembre de 2015
121
Sediment transport induced by skewness and asymmetry of the wave
Transporte sedimentar induzido pelo enviesamento e assimetria da onda
J. S. Antunes do Carmo
MARE/University of Coimbra, Department of Civil Engineering, 3030-788 Coimbra, Portugal, e-mail: jsacarmo@dec.uc.pt
Abstract: Numerical models are useful instruments for studying complex superposition of wave-
wave and wave-current interactions in coastal and estuarine regions, and to investigate the
interaction of waves with complex bathymetries or structures built in nearshore areas. The ability of
the standard Boussinesq and Serre or Green and Naghdi equations to reproduce these nonlinear
processes is well known. However, these models are restricted to shallow water conditions, and
addition of other terms of dispersive origin has been considered since the 90’s, particularly for
approximations of the Boussinesq-type. To allow applications in a greater range of
λ
0
h, other than
shallow waters, where 0
h is the water depth at rest and
λ
is the wavelength, a new set of extended
Serre equations, with additional terms of dispersive origin, has been developed and tested with
available data in the literature. The extended Serre model is briefly presented in this work. Then a
morphodynamic model composed of this hydrodynamic module and a sediment transport model is
proposed and discussed. The sediment transport model consists on a sediment conservation
equation and a dynamic equation. An improved version of Bailard model, incorporating various
sediment transport processes, is used as the dynamic equation of the solid-phase model. It is shown
that the wave asymmetry led to an increase of the sediment transport in the wave direction.
Key words: extended Serre equations, Bailard model, wave skewness, wave asymmetry, morphodynamics
1. INTRODUCTION
The theory shows, and the practice confirms that only
models of order 2
σ ( λhσ 0
=, where 0
h and
λ
represent, respectively, depth and wavelength
characteristics) or greater, of the Boussinesq (1872)
and Serre (1953) types, are able to reproduce effects
other than the dispersive effects, including the non-
linearities resulting from wave-wave and wave-
current interactions. Also the waves resulting from
sudden time-bed-level changes that cause tsunamis,
wherein submerged landslides in reservoirs or
landslides on reservoir banks are examples of such
changes, can only be reproduced by higher order
models.
In the last few years, the possibility of using more
powerful computational facilities, and the
technological evolution and sophistication of control
systems have required thorough theoretical and
experimental research designed to improve the
knowledge of coastal hydrodynamics. Numerical
methods aimed to applications in engineering fields,
more sophisticated and with a higher degree of
complexity, have also been developed.
The general shallow water wave theory has been
used to develop different mathematical
approximations, which are nowadays the basis of the
most sophisticated models in the ambit of
hydrodynamics and sedimentary dynamics.
An extension of the more general approach (Serre
model), for applications in intermediate water
conditions was performed recently (Antunes do
Carmo, 2013a,b).
This extended version is used herein as part of a
morphodynamic model, which incorporates two
more equations taking into account various processes
of sediment transport. The velocity-skewness and the
acceleration-asymmetry are taken into consideration
and discussed based on physical considerations and
numerical results.
2. MATHEMATICAL FORMULATION
2.1 Hydrodynamic model
Starting from the fundamental equations of the Fluid
Mechanics, written in Euler’s variables, relating to a
three-dimensional and quasi-irrotational flow of a
perfect fluid [Euler equations, or Navier-Stokes
equations with the assumptions of non-
compressibility ( 0== vdiv dtdρ
r
), irrotationality
(0=vrot
r
, i.e. xz wu =; yz wv =; yx uv =) and perfect
fluid (dynamic viscosity,
0
=
μ
)], and proceeding with
suitable non-dimensional variables, after some
mathematical developments the following equations
of motion are obtained, in second approach (order 2
in
σ
, i.e. 2
σ) (Seabra-Santos, 1989):
122
(
)
(
)
[
]
(
)
[
]
011 =−++−++− yx
tvξεηuξεηεξη
( )
( )
( )
[ ]
( )
( )
{ }
( )
( )
[ ]
0121
1312132
42
2
=+−+++
−+++−+
+
+
+
σQξεηQεησ
P ξεηPξξεη σ
ηεvuεuuu
x
x
x
x
x
x
yxt
( )
( )
( )
[ ]
( )
( )
{ }
( )
( )
[ ]
0121
1312132
42
2
=+−+++
−+++−+
+
+
+
σQξεηQεησ
P ξεηPξξεη σ
ηεvvεuvv
y
y
y
y
y
y
yxt
(
)
(
)
tyx AεvAεuAεAξεηP −−−−+= 2
1
y
x
tvwεuwεwQ ++=
(
)
yxt
vξuξξεw ++= 1
yx vuA += (1)
In dimensional variables and with a solid/fixed
bottom ( 0=
t
ξ), the complete set of equations is
written, in second approach:
(
)
(
)
( ) ( )
[ ]
( ) ( )
( ) ( )
[ ]
( ) ( )
02131
2132
02131
2132
0
=+++
+++++
=+++
+++++
=++
y
yy
y
yyyxt
x
xx
x
xxyxt
yxt
hQQhhP
P ξhgηvvuvv
hQQhhP
P ξhgηvuuuu
hvhuh
(2)
(
)
tyx
AvAuAAhP −−−= 2
yxt vwuwwQ ++=
yx vξuξw +=
yx vuA +=
where ηξhh +−= 0 is total water depth. The one-
dimensional form (1HD) of the equation system (2) is
written, also with a fixed bottom:
(
)
0=+ xt uhh
( )
[ ]
( )
0223
2=++++
++
QPhξQPh
ηghhuuhu
x
x
x
xt (3)
(
)
2
xxxxt
uuuuhP −+−=
(
)
2
uξuuuξQ xxxtx ++=
The classical Serre (1953) equations (3) (or Green and
Naghdi, 1976) are fully-nonlinear and weakly
dispersive. As for the Boussinesq-type models, also
Serre’s equations are valid only for shallow water
conditions.
2.1.a Serre’s 1HD equations with improved dispersive
performance
From the equation system (3), by adding and
subtracting terms of dispersive origin, using the
approximation x
tηgu −= and considering the
parameters
α
,
β
and
γ
, with
γ
.
α
.
β
5
0
5
1
−
=
, allows
to obtain a new system of equations with improved
linear dispersion characteristics (Antunes do Carmo,
2013a,b):
(
)
0=+ xt uhh (4)
(
)
(
)
(
)
( ) ( ) ( )
( )
( )
( )
( ) ( )
( )
( )
0
2
33
3
1
1
2
2
2
22
2
=ρτ++++
++++
−+−+−
+−+++−
−+++++
huξ
h
uuξhΩ
ξhuξξhuh
uuuu
h
uuhhξh
h
gβ
ξhghhαξh Ωgαu
h
β
uhhuΩαξhguuu
bxxxx
xx
xxxxx
xxxxxxxxxxxx
xxxx
xxt
xtxt
x
xt
(5)
where
(
)
(
)
2
50 xxx
xx h.hxΩ ξ+ξ+ξ= .
After linearization of the equation system (4)-(5), the
following dispersion relation is obtained (Liu and Sun,
2005):
(
)
[
]
( ) ( )
61211
621
2222
22
2
/hk/hk
hk kh
gk
γα
γα
ω
+−++
−+
= (6)
Comparing (6) with the linear dispersion relation
(
)
khtanhgk =
2
ω
, values of
1308
0
.
α
=
and
0076
0
.
γ
−
=
are obtained, so that
20
0
.
β
=
. It should
be noted that the Serre’s equation system (3) is
recovered by setting in (4)-(5)
0
=
=
β
α
.
2.1.b 1HD numerical model and application
The equation system (4)-(5) was solved using an
efficient finite-difference method (Antunes do Carmo,
2013a,b,c). For this purpose, the terms containing
derivatives in time of u are grouped. The final system
of three equations is re-written according to the
following equivalent form:
(
)
0=+ xt uhh (7)
( )
( )
[
]
( )
( )
( )
[ ]
( )
[ ]
( )
[ ]
( )
0
33
2
3
1
21
2
1
21
2
1
2
2
22
2
22
2
2
22
=++
−+
−+−
−+++
−−+−
++−+
ρhτuu
h
βuuh
β
α
uuh ξuξhuh ξ
α
η
h
βgηuuαhαΩg
ηαghhuξhuξα
uhαuuqq
b
xxx
xxx
x
xxxx
x
x
xx
xxxx
xx
xxx
x
xx
x
x
x
t
(8)
( )
[ ]
( ) ( )
qu
h
uhhu xxxx
=+−+−Ω++
3
1111
2
βαα
(9)
where, as above,
(
)
2
50 xxx
xx h.
ξξηξ
++=Ω .
To compute the solution of equation system (4)-(5)
(values of the variables h and u at time
t
t
∆
+
), a
numerical procedure based on the last equation
system (7)-(9) for variables h, q and u is used. For
details see Antunes do Carmo, 2013a,b,c).
Data of a benchmark test conducted by Beji and
Battjes (1993) in a flume 0.80 m wide, and with a
submerged trapezoidal bar having slopes 1:10
(upstream) and 1:20 (downstream), are available in
the literature and can be used for comparisons with
8º Simposio sobre el Margen Ibérico Atlántico MIA15 Málaga, del 21 al 23 de septiembre de 2015
123
results of our numerical model. The water depth before
and after the bar is 0.40 m, with a reduction to 0.10 m above
the bar, as shown in Fig. 1.
Fig. 1. Bathymetry for a periodic wave propagating over a bar (not
in scale).
A regular incident wave case with height H = 0.02 m,
period T = 2.02 s and wavelength
λ
= 3.73 m has
been simulated. The computational domain was
discretized with a uniform grid interval
025
0
.
x
=
∆
m.
A time step
0010
0
.
t
=
∆
s was used.
The Serre model (3) is only valid for shallow waters,
thus under conditions up to 050
0.h ≤
λ
. In this
experiment, the dispersion parameter (
λσ
0
h=) is
greater than 0.05 (about 0.11) in front and behind the
bar, and therefore affects the validity of the
numerical outcomes (Antunes do Carmo, 2013b). Due
to the fact that over the bar there are very shallow
water conditions (
03
0
.
≈
σ
), the standard Serre
equations (3) can be used considering the input
boundary located at section x = 13.5 m, where the
input signal is known (measured data). Fig. 2 shows a
comparison of test data with numerical results of
both models (3) and (4)-(5) in a probe at x = 17.3 m.
Fig. 2. Propagation of a regular wave (at the entrance) in a channel
with a bar. A record obtained in a probe at x = 17.3 m (see Fig. 1).
Comparison of test data (
__
____
__
) with numerical results of the
extended Serre model (4)-(5) (
L
) and the standard Serre
equations (3) ( _ _ _ ).
The influence of additional terms of dispersive origin
included in the extended Serre equations (4)-(5) is
clearly shown in Fig. 2. The classical Serre model
results (dashed line) are clearly of lesser quality.
2.2 Sediment transport model
The Bailard model (Bailard, 1981) does not consider
the contribution of the wave acceleration-asymmetry
in the sediment transport. As outlined in Dubarbier et
al. (2015), models frequently used to estimate the
evolution of beach profiles are inefficient with regard
to the simulation of bottom shapes and migration of
bars. This may be attributed to the absence of
transport induced by acceleration-asymmetry of the
wave.
In the following we use a 1HD model to compute the
sediment transport in a channel, over a bar and a
ditch, and examine its ability to generate and
propagate ripples and other bottom shapes. The
morphodynamic model consists of the hydrodynamic
equations (4)-(5) and the following sediment
conservation equation 10) and a dynamic equation
(11), in which four sediment transport processes are
incorporated:
(
)
01 =+− x
stt q p
ξ
(10)
syskssslst qqqqq +++= (11)
where
( )
−
−
=32 1
1u
tan
uu
tansg
c
qx
asl
sl
ξ
φφ
ε
( )
ξ
ε
−
ε
−
=53
1u
w
uu
wsg
c
qx
s
s
s
sss
ss
(
)
skorbpsksk AUTcq 2
= and
(
)
asyorbpsysy AUTcq 2
−= .
In equations (10) and (11), L represents mean
values of the arguments in the wave period, st
q is the
net sediment transport, which is composed of the
bedload transport, sl
q, the suspended load transport,
ss
q, the skewness related transport, sk
q, and the
transport related to wave asymmetry, sy
q;
u
is the
wave velocity,
p
is the sediment porosity,
φ
is the
internal angle of friction,
[
]
300100 . ,.
a∈
ε
and
[
]
030010 . ,.
s∈
ε
are efficiency coefficients, s
w is the
sediment fall velocity, sl
c and ss
c are global rugosity
coefficients, sk
cand sy
c are calibration coefficients.
(
)
[
]
khsinhTHπU
prms
orb = is the orbital velocity
amplitude, 23
23 uuAsk = is a measure of orbital
velocity-skewness, and
(
)
[
]
{
}
23
2
3uAasy tuH= is
the velocity asymmetry coefficient, where
(
)
[
]
tu
H
is
the Hilbert transform of
u
. The asymmetry
coefficient is here approximated by 33
rmsasy aaA =,
with 21
2
aarms =, being
a
the wave acceleration.
All calibration coefficients, in particular the
efficiencies ( a
ε,s
ε) and ( sk
c,sy
c), which represent
the incomplete knowledge in our understanding of
these processes, require a site-specific
morphodyamic calibration. Once properly calibrated
a comprehensive cross-shore profile model may
predict the bar dynamics on the time-scale of days (at
least). However, it must be pointed out that the
124
calibration process is non-trivial since a large number
of model coefficients is involved, typically requiring a
large number of computations and optimization
strategies. At a first approach, coefficients sk
c and
sy
c are of the order of 5
10− to 5
105 −
×, and are not
necessarily equal. We will use in this work sk
c = sy
c =
5
102 −
×. Anyway, it should be noted that the effects
are in a significant part determined by the calibration
coefficient settings that have been kept constant.
Bed slope-related transport is included according to
the Bailard equation increasing (decreasing) the
down-slope (up-slope) sediment transport. Equation
(10) is easily computed using the Weighted
Essentially Non-Oscillatory (WENO) scheme, as is
presented in Long et al. (2008).
3. NUMERICAL APPLICATIONS
A wave with the following characteristics: height H =
0.20 m, period T = 8, and wavelength
λ
= 24.8 m is
introduced at the upstream boundary and
propagated along a horizontal channel 1.0 m depth in
the first 28.75 m. From this point there is a bar or
ditch, with the upstream face having a positive/
negative slope 9.82% up/down to a maximum/
minimum
275
0
.
±
=
ξ
and left constant between
31.55 m and 32.175 m. Then the bar/ditch decreases/
increases up to
0
=
ξ
, having this face a negative/
positive slope 18.64%. A median diameter 01
50 .d =
mm is representative of the bottom grain size.
Fig. 3 shows the simulated wave along the channel
with a bar, between 15 m and 45 m. The wave
transformations that occurred are evident, increasing
the skewness and asymmetry of the wave. Figs. 4 and
5 show bottom configurations, having a bar or a
ditch, respectively, obtained 450 waves after,
corresponding to a simulation time of 60 minutes.
In both figures comparisons are shown considering
the first two terms of equation (11) ( sl
q and ss
q,
dashed line), and additionally the asymmetry of the
wave ( sl
q, ss
q and sy
q, dotted line). Although lacking
experimental evidence, the presented results seem to
translate the physical phenomena. Observing Fis. 4
and 5, a preliminary conclusion can be drawn.
Excluding the transport associated with the wave
asymmetry reduces the onshore sediment transport
(wave direction).
Fig. 3. Initial velocity of the wave propagating over the bar.
Fig. 4. Bottom profiles with bar after 60 minutes of simulation,
considering terms sl
q and ss
q of equation (11) (dashed line), and
terms sl
q, ss
q and sy
q of the same equation (dotted line).
Fig. 5. Bottom profiles with ditch aft er 60 minutes of si mulation,
considering terms sl
q and ss
q of equation (11) (dashed line), and
terms sl
q, ss
q and sy
q of the same equation (dotted line).
REFERENCES
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models with improved linear dispersion characteristics:
Applications. Journal of Hydraulic Research, 51(6), 719-
727.
Antunes do Carmo, J.S. (2013b). Extended Serre equations
for applications in intermediate water depths. The
Open Ocean Eng. Journal, 6, 16-25.
Antunes do Carmo, J.S. (2013c). Applications of Serre and
Boussinesq type models with improved linear
dispersion characteristics. Congreso de Métodos
Numéricos en Ingeniería, 25 June, Bilbao, Spain.
Bailard, J.A. (1981). An energetics total load sediment
transport model for a plane sloping beach. J.
Geophysical Research, 86, C11, 10938-10954.
Beji, S. & Battjes, J.A. (1993). Experimental investigations of
wave propagation over a bar. Coastal Engineering,
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