Coastal Dynamics 2013
MODELING UNDERTOW DUE TO RANDOM WAVES
Pham Thanh Nam1, Hocine Oumeraci2, Magnus Larson3, Hans Hanson4
Abstract
A numerical model of undertow due to random waves is developed. The model includes three sub-models: (i) a model
for multi-directional and -frequency random wave transformation, (ii) a surface roller evolution model, and (iii) a
model for calculating the vertical distribution and the mean value of the undertow velocity. The calculation of wave
trough level is estimated based on theory of wave asymmetry. The model was successfully validated against small and
large-scale laboratory experiments. Thus, the model is expected to provide reliable input for the modeling of sediment
transport and morphological changes due to waves and currents.
Key words: undertow, random waves, surface roller, eddy viscosity, wave energy dissipation, wave asymmetry
1. Introduction
Wave-induced cross-shore currents play a key role in the calculation of both bed load and suspended
load in cross-shore sediment transport. Thus, an accurate prediction of cross-shore currents is a prerequisite
for any morphological evolution model.
More specifically, the wave-induced cross-shore current, called undertow, represents one of the
dominant forces behind sandbar formation, offshore sediment transport, and beach erosion. It is generated
as a result of the vertical imbalance between the depth-varying momentum flux and the depth-uniform
pressure gradient due to wave set-up. The undertow was first observed experimentally by Bagnold (1940),
and qualitatively described by Dyhr-Nielsen and Sørensen (1970), and then analytically modeled by
various researchers (e.g. Dally and Dean, 1984; Svendsen, 1984; Stive and Wind, 1986; Svendsen et al.,
1987). Deigaard et al. (1991) modeled undertow based on the shear stress contribution combined with a
one-equation turbulence model. Masselink and Black (1995) employed and compared several undertow
models based on the field measurements at two natural beaches in Australia. Their calculations showed that
the best agreement is achieved when the mass flux due to unbroken waves is incorporated and the shallow
water wave theory is employed. Cox and Kobayashi (1997) introduced a kinematic undertow model that
combines a logarithmic profile in the bottom boundary layer with a parabolic profile in the interior layer.
The model was calibrated and validated against laboratory data sets for regular waves with both rough and
smooth slopes. This model was then applied for irregular waves on a barred beach by Kennedy et al.
(1998). Kuriyama and Nakatsukasa (2000) followed the approach of Svendsen (1984) to calculate the
undertow, and validated the undertow model with field data observed at Hazaki Oceanographical Research
Station. Rattanapitikon and Shibayama (2000) introduced an explicit model based on the eddy viscosity
model for calculating the undertow velocity profile inside the surf zone. Several formulae for calculating
mean undertow current were also investigated and compared based on both small-scale and large-scale
experiments. Grasmeijer and Ruessink (2003) modeled the waves and currents in the nearshore based on
both parametric and probabilistic approaches. The model was validated against laboratory data and field
data observed at Duck, NC, and Egmond aan Zee, The Netherlands. Reniers et al. (2004) simulated the
vertical flow structure for the Sandy Duck data based on the three-layer concept given by De Vriend and
Stive (1987), and the obtained results were in good agreement with measurements. Tajima and Madsen
1
Dept. of Hydromechanics and Coastal Engineering, Technische Universität Braunschweig, Beethoventr. 51a, D-38106
Braunschweig, Germany. Email: [email protected]
2
Ditto, email: [email protected]
3
Dept. of Water Resources Engineering, Lund University, P.O. Box 118, S-22100 Lund, Sweden. Email:
[email protected]
4
Ditto, email: [email protected]
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Coastal Dynamics 2013
(2006) developed a nearshore hydrodynamics model in which the mean current model, following the
concept of Svendsen et al. (1987), includes two-layer 2DH momentum equations, integrated above the
wave trough and over the entire depth. Recently, Cambazoglu and Haas (2011) employed the NearCoM
model to simulate the hydrodynamics in the nearshore in which the wave conditions are calculated by the
REF/DIF model and the SHORECIRC model is used to determine the undertow in the nearshore.
In general, the differences among the aforementioned models are mainly related to the modeling of the
turbulent eddy viscosity, to the formulation of the boundary conditions, and to the simplification of the
depth-averaged momentum equations (Reniers et al., 2004, Tajima and Madsen, 2006). These models have
been validated with laboratory and field data. However, the hydrodynamic processes in the surf zone are
highly complicated, especially in shallow water, where waves become asymmetric and break. Nevertheless,
in most of the aforementioned models the wave asymmetry is not considered in undertow calculations, and
the wave trough level is approximately set to a half of the root-mean-square wave height. Furthermore, the
available data for model validation are still limited. Thus, modeling undertow due to random waves still
needs to be improved.
The objective of this study is to develop a robust and reliable numerical model for the vertical
distribution of the undertow velocity and the cross-shore evolution of the undertow velocity profile for
random waves. To achieve this objective, three sub-models are developed, including (i) a random wave
transformation model, (ii) a surface roller model, (iii) a model for the mean undertow velocity and the
vertical distribution of undertow velocity. The random wave transformation and surface roller models have
been developed by Mase (2001) and Nam et al. (2009). The mean undertow velocity is determined
following the work by Svendsen (1984), Kuriyama and Nakatsukasa (2000), and Grasmeijer and Ruessink
(2003), in which the water depth below the wave trough is calculated based on the wave asymmetry (Abreu
et al., 2010; Ruessink et al., 2012). A simple eddy viscosity model introduced by Rattanapitikon and
Shibayama (2000) is employed to determine the vertical distribution of the undertow velocity.
The model is validated against selected data sets from different laboratories.
2. Model Descriptions
2.1. Random wave transformation model
Mase (2001) developed a random wave transformation model based on the energy balance equation
including diffraction and dissipation terms (EBED). Subsequently, the model was modified and improved
by Nam et al. (2009) and Nam and Larson (2010), in which the calculation of the wave energy dissipation
after Dally et al. (1985) was employed. The modified energy balance equation is expressed as,
∂ ( vx S )
∂x
+
( ) + ∂ ( vθ S ) =
∂ vy S
∂y
∂θ
κ 
2
 CCg cos θ S y
2ω 
(
)
  Γh 2 
1
 K


2
− CCg cos θ S yy  − Cg S 1 − 
 
y
2
H
 h
  s  
(1)
where S = angular-frequency spectrum density; (x, y) = horizontal coordinates; θ = angle counter clockwise
from x axis; (vx, vx, vθ) = propagation velocities in their respective coordinate direction; ω = angular
frequency; C = phase speed; Cg = group speed; κ = free parameter; K = dimensionless decay coefficient; Γ
= dimensionless coefficient for stable wave height; h = water depth; and Hs = significant wave height.
The stable wave height and decay coefficients are proposed as functions of the bottom slope (Goda,
2006, Nam et al., 2009; Nam and Larson, 2010) as,
3

Γ = 0.45, K = 8 ( 0.3 − 19.2 β )

Γ = 0.45 + 1.5 β , K = 3 ( 0.3 − 0.5 β )

8
where β = bottom slope.
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:β <0
(2)
: 0 ≤ β ≤ 0.6
Coastal Dynamics 2013
The model output includes three main parameters: significant wave height Hs, significant wave period Ts,
and mean wave direction θ as,
H s = 4.0 m0 ; Ts = T0 m0 / m2 / T0 ; θ =
N
L
∑∑θ
i =1 j =1
j
Sij / m0
(3)
where T0 = offshore significant wave period, T0 = offshore mean wave period, and
mk =
N
L
∑∑ f
i =1 j =1
k
i
Sij , k = 0,1, 2
(4)
with N and L being the total number of frequency and angular components, respectively.
Based on these parameters, the radiation stresses can be determined as,
S xx =
E
E
E
 2n (1 + cos 2 θ ) − 1 ; S yy =  2n (1 + sin 2 θ ) − 1 ; S xy = n sin 2θ


2
2
2
(5)
2
where E = ρ gH rms
/ 8 is the wave energy per unit area with H rms = H s / 2 being the root-mean-square
wave height, n = Cg/C is the wave index, ρ is the water density and g the acceleration due to gravity.
2.2. Surface roller model
The surface roller is modeled based on the energy balance equation (Dally and Brown, 1995; Larson and
Kraus, 2002) as,
Db +
∂ 1
 ∂ 1

2
2
2
2
 MCr cos θ  +  MCr sin θ  = g β D M
∂x  2
∂
y
2



(6)
where Db = wave energy dissipation, M = period-averaged mass flux, Cr = roller speed ( ≈ C), and βD =
roller dissipation coefficient.
The wave energy dissipation is calculated after Dally et al. (1985) as,
Db =
KCg ρ g
8h
 H 2 − ( Γh )2 
 rms

(7)
The stresses due to the roller are determined by the following relations:
Rxx = MCr cos 2 θ ; Ryy = MCr sin 2 θ ; Rxy = MCr sin 2θ
(8)
2.3. Mean undertow velocity
Based on the radiation stresses derived from the random wave transformation model and the surface
roller model, the longshore current and the wave setup/setdown are calculated by the continuity and
momentum equations (for details see Nam and Larson, 2010). The wave set-up and set-down are used for
calculating the cross-shore wave-induced currents.
The mean undertow velocity can be determined based on the mass fluxes due to wave motion and roller
evolution (Svendsen, 1984; Kuriyama and Nakatsukasa, 2000; Grasmeijer and Ruessink, 2003) as,
Um = −
Qw + Qr
ht
(9)
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Coastal Dynamics 2013
where Qw = mass flux due to wave motion, Qr = mass flux due to roller, and ht = water depth below the
wave trough.
Using linear theory, the mass flux due to the wave motion can be calculated as,
Qw =
1 g 2
H rms cos θ
8 Cg
(10)
The mass flux due to the roller is determined as,
Qr =
M
ρ
cos θ
(11)
where M is derived from Eq. (6).
The water depth below the wave trough can be estimated as,
ht = h + ζ − η rms
(12)
where ζ = wave setup/setdown, and η rms = rms value of the water elevation in the wave trough when
η (t ) ≤ 0 during the trough period, determined by the asymmetry from the wave theory below.
We assumed that the water elevation is a function similar to the expression for the free-stream near-bed
horizontal orbital motion introduced by Abreu et al. (2010),
sin(ωt ) +
1
2
η (t ) = H rms f
r sin φ
1+ 1− r2
1 − r cos(ωt + φ )
(13)
where r = index of skewness or nonlinearity, f = dimensionless factor ( = 1 − r 2 ), φ = wave form
parameter.
The index of skewness is determined based on the Ursell number following the studies of Ruessink et al.
(2012), Malarkey and Davies (2012), and Doering and Bowen (1995) as,
r=
2 b
(1 + b)
(14)
where b is related to the total non-linearity B as,
B=
3b
(15)
(1 − b 2 )
in which B is a function of the Ursell number in the form of a Boltzmann sigmoid given by,
p2 − p1
(16)
p − log U r
1 + exp 3
p4
where pi , i =1, 2, 3, 4 are fitting parameters. Using a large number of field data sets, Ruesink et al. (2012)
recommended that p1 = 0, p2 = 0.857 ± 0.016, p3 = −0.471 ± 0.025, p4 = 0.297 ± 0.021 , where the range
described by the ± values is the 95% confidence interval, and the Ursell number is calculated as,
B = p1 +
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Coastal Dynamics 2013
Ur =
H rms λ 2
(17)
( h + ζ )3
where λ = wave length.
The wave form parameter is also a function of the Ursell number (Ruessink et al., 2012; Doering and
Bowen, 1995),
φ=−
π
2
tanh( p5 / U rp6 )
(18)
where p5 and p6 are also fitting parameters, given as 0.815 ± 0.055 and 0.672 ± 0.073, respectively.
2.4. Vertical distribution of undertow velocity
The vertical distribution of undertow velocity can be derived from a simple eddy viscosity model
(Rattanapitikon and Shibayama, 2000) as,
1/3
D 
U ( z ) = α1  total 
 ρ 
  z 1

z
α 2  −  + α 3 (ln + 1)  + U m
ht

  ht 2 
(19)
where z = upward vertical coordinate from the bed, Dtotal = total energy dissipation, and α1, α2, α3 =
calibration coefficients.
In the study for regular wave of Rattanapitikon and Shibayama (2000), the coefficient α3 is given as a
constant α3 =-0.21, whereas the coefficients α1, α2 depend on the locations which determine the transition
zone and inner surf zone. In this study, we simply employed α1 = 1, α2 = 1, and α3 = -0.21.
For the energy dissipation another formulation is proposed in this study. Rattanapitikon and Shibayama
(2000) employed the energy dissipation formulation based on the bore model (Thornton and Guza, 1983).
In this study, the total energy dissipation is described by the sum of the energy dissipation due to wave
breaking ( Db ), and due to bottom friction from current ( Dc ) and waves ( Dw ) (Battjies, 1983; Camenen
and Larson, 2008) as,
Dtotal = Db + Dc + Dw
(20)
The energy dissipation Db induced by wave breaking is calculated by Eq. (7). In the bottom boundary
layer, the energy dissipation Dc due to friction from current can be calculated as,
Dc = τ c u*c
(21)
where τ c and u*c are the bottom shear stress and the shear velocity from the current only, respectively.
The bottom shear stress due to current only is calculated as,
τc =
1
ρ f cU m2
2
(22)
where fc = the friction coefficient determined by Soulsby (1997) as,


0.4
fc = 2 

 1 + ln(ks / 30(h + ζ )) 
2
(23)
where ks = the bottom roughness.
The bottom shear velocity from current only can be calculated as,
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Coastal Dynamics 2013
τc
ρ
u*c =
(24)
Similarly, the energy dissipation due to friction from waves can be determined as,
Dw = τ wu*w
(25)
where τ w and u*w are maximum bottom shear stress and shear velocity from the waves only, respectively.
The maximum bottom shear stress Dw from waves can be estimated as,
τw =
1
ρ f wU w2
2
(26)
where U w = the wave orbital velocity amplitude, and fw = the friction coefficient by waves, determined
from Swart (1974) as,
exp(5.21R 0.19 − 6)
fw = 
0.3
( R > 1.57)
( R ≤ 1.57)
(27)
where R = the relative roughness (= Aw / k s ), and Aw = the semi-orbital excursion ( = U wTs / 2π ).
The bottom shear velocity is calculated as,
u*w =
τw
ρ
(28)
3. Model Validations
3.1. Validation against laboratory data from Grasmeijer and van Rijn (1999)
Two data sets from the laboratory flume of Delft University of Technology were employed for
validating the cross-shore evolution of the significant wave height and the mean undertow velocity. The
model beach with a sand bar was formed by sand with medium grain size of 0.095 mm. The bed level
varies from 0.6 m in deeper water to 0.3 m at the bar crest with a slope of 1:20 (Fig 1d). The beach profile
continuously varies from the crest to the trough of the bar with a slope of -1:25 until the water depth at the
trough is 0.5 m. From the bar trough to the shoreline, the beach profile varies with a slope of 1:63. The test
series B1 and B2 were conducted with offshore significant wave heights of 0.16 m and 0.19 m, respectively.
The peak wave period was 2.3 s for both test series.
Fig. 1 shows the comparisons of calculated significant wave height and mean undertow velocities with
the measurements for test series B1. The calculation of the significant wave height is in very good
agreement with the measurements with an rms error of only 4.9 %. As can be seen in Fig. 1a, waves are
broken at the crest of the bar (x = 12 m). After that, they reform and then broken again around x = 30 m.
The calculated mean undertow velocity also agrees well with the measurements. The calculated mean
undertow velocity with wave asymmetry is presented by the blue line in Fig. 1b, whereas the red line
represents the calculation without wave asymmetry. As can be seen, the calculation of the mean undertow
velocity with wave asymmetry is better than without wave asymmetry. With the wave asymmetry, the rms
error for the mean velocity is 24.5 %. Without the wave asymmetry effect, the rms error for the mean
velocity is 30.6 %. The wave asymmetry effects are more pronounced when the waves are broken in the
vicinity of the sand bar, and particularly further shoreward in the shallow water area.
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Coastal Dynamics 2013
0.25
Hs (m)
0.2
0.15
0.1
a)
0.05
0
0
5
10
15
20
25
30
35
40
45
50
Um (m/s)
0
with wave asymmetry
without wave asymmetry
-0.025
-0.05
-0.075
-0.1
b)
0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
-3
10
x 10
c)
ζ (m)
5
0
-5
0
Bed level (m)
0.2
d)
0
-0.2
-0.4
-0.6
0
Cross-shore distance (m)
Figure 1. Comparisons of calculated and measured significant wave height (a), mean undertow velocity (b), wave
setup/setdown (c) and bed level (d) for test series B1 of Grasmeijer and van Rijn (1999).
The calculated significant wave height and the mean undertow velocity for test series B2 is compared to
measurements and presented in Fig. 2. As for test series B1, very good agreement between measurements
and calculation of the significant wave height is obtained with an rms error of 5.2 %. The calculated mean
undertow velocity with the wave asymmetry included is also in relatively good agreement with
measurements with an rms error of 32.3 % as compared to an rms error of 41.2 % without the wave
asymmetry effect. Like in test series B1, an undertow mean velocity peak occurs at the crest of the bar.
However, the peak of the measured velocity was observed shoreward of the bar crest (x = 13 m), causing a
significant difference between calculation and measurements near the crest of the bar. However, the model
reproduced well the measurements at remaining measured locations.
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Coastal Dynamics 2013
0.25
Hs (m)
0.2
0.15
0.1
a)
0.05
0
0
5
10
15
20
25
30
35
40
45
50
0
with wave asymmetry
without wave asymmetry
U m (m/s)
-0.025
-0.05
-0.075
-0.1
b)
0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
-3
10
x 10
c)
ζ (m)
5
0
-5
0
Bed level (m)
0.2
d)
0
-0.2
-0.4
-0.6
0
Cross-shore distance (m)
Figure 2. Comparisons of calculated and measured significant wave height (a) and mean undertow velocity (b), wave
setup/setdown (c), and bed level (d) for test series B2 of Grasmeijer and van Rijn (1999).
3.2. Validation against laboratory data from Sultan (1995)
The tests were conducted in a flume with slope of 1:35. The vertical distribution of undertow velocity
was observed at 12 locations from offshore to nearshore, called hereafter S1 to S12, for offshore wave
conditions with Hmo = 7.1 cm, Tp = 3 s, and θ = 00.
The comparison between calculated undertow velocity profiles and measurements by Sultan (1995) is
showed at 12 locations in Fig. 3. The calculations of undertow velocity profiles with asymmetry wave
effects are represented by the blue line, whereas the red line describes the calculated undertow velocity
profiles without asymmetry effect. Generally, the calculated undertow velocity profiles agree well with the
measurements, although they are somewhat underestimated in deeper water (locations S1-S3) and
overestimated in shallower water (locations S10 and S11). Waves are more symmetric in deeper water, thus
explaining why the calculations with and without wave asymmetry effects are similar at locations S1, S2,
and S3. The significant difference between observations and calculations can be seen at location S5 where
the measured velocities near the water surface are much higher than the calculated values. The data near
the water surface at S5 might contain some errors because the tendency of observations was not presented
at locations S4 and S6. The rms error for undertow at all 12 locations is 35.3 % and 38.4 % for the
calculations with and without wave asymmetry effects, respectively.
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Coastal Dynamics 2013
S1, h = 0.29 m
z/h
S2, h = 0.233 m
S3, h = 0.208 m
S4, h = 0.177 m
S5, h = 0.161 m
S6, h = 0.144 m
1
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0
-0.1 -0.05
0
0.05
z/h
S7, h = 0.131 m
0
-0.1 -0.05
0
0.05
S8, h = 0.121 m
0
-0.1 -0.05
0
0
-0.1 -0.05
0.05
S9, h = 0.097 m
0
0.05
S10, h = 0.077 m
0
-0.1 -0.05
0
0
-0.1 -0.05
0.05
1
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0
-0.1 -0.05
0
0.05
0
-0.1 -0.05
0
0
-0.1 -0.05
0.05
0
0
-0.1 -0.05
0.05
0
0.05
0
-0.1 -0.05
0
0
0.05
S12, h = 0.031 m
S11, h = 0.056 m
0.05
0
-0.1 -0.05
0
0.05
U (m/s)
Figure 3. Comparison of calculated and measured undertow velocity profiles at 12 locations (data by Sultan, 1995),
with wave asymmetry effects (blue line) and without wave asymmetry effects (red line)
3.3. Validation against laboratory data from Roelvink and Reniers (1995)
1.75
1.5
1.25
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1.25
-1.5
-1.75
0
25
50
75
100
125
150
175
5.25
4.5
3.75
3
2.25
1.5
0.75
0
-0.75
-1.5
-2.25
-3
-3.75
-4.5
-5.25
200
Bed level (m)
Hs (m)
Two data sets, Test 1A and 1B, from the large-scale wave flume of Delft Hydraulics (Roelvink and
Reniers, 1995) are employed to validate the cross-shore evolution of the significant wave height and of the
undertow velocity profiles. The model beach in the flume consisted of fine sand with a median grain size of
0.22 mm. Test 1A was carried out under slightly erosive wave conditions with Hmo = 0.9 m, Tp = 5 s, θ = 0
deg, and the still-water level at offshore is 4.1 m. The vertical distribution of the undertow velocity was
measured at 11 locations, represented by A1 to A11 in this study. The highly erosive wave conditions were
generated for Test 1B with Hmo = 1.4 m, Tp = 5 s, θ = 0 deg., and the undertow velocity was measured at 9
locations, denoted hereafter by B1 to B9.
Fig. 4 shows the comparison between calculated significant wave height and measurements for Test 1A.
The cross-shore evolution of the wave conditions is well reproduced by the numerical model. Although the
calculated significant wave height is slightly underestimated in the surf zone, the rms error is only 6.4%.
Cross-shore distance (m)
Figure 4. Comparison of calculated and measured significant wave height for Test 1A, Hmo = 0.9 m, Tp = 5 s,
θ = 0 deg. ( Roelvink and Renier, 1995)
The vertical distribution of undertow velocities and measurements at 11 locations are presented in Fig. 5.
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Coastal Dynamics 2013
z/h
The calculations with and without asymmetry wave effects are quite similar for this test case. The
calculated undertow in deeper water is overestimated. However, in the surf zone and shallow area, the
calculation is in very good agreement with the measurements. The rms error for all 11 locations is
relatively high, 45.9 % with wave asymmetry effects and 49.6 % without wave asymmetry effects, mainly
due to the significant difference between observations and calculations in deeper water.
A1, h = 2.262 m
A2, h = 1.697 m
A3, h = 1.687 m
A5, h = 1.239 m
A6, h = 0.983 m
1
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0
-0.4 -0.2
0
0.2
z/h
A7, h = 0.924 m
0
-0.4 -0.2
0
0.2
A8, h = 0.891 m
0
-0.4 -0.2
0
A4, h = 1.54 m
0
-0.4 -0.2
0.2
A9, h = 0.815 m
0
0.2
A10, h = 0.750 m
0
-0.4 -0.2
1
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0
0.2
0
-0.4 -0.2
0
0.2
0
-0.4 -0.2
0
0
-0.4 -0.2
0.2
0
0.2
0.2
0
-0.4 -0.2
0
0.2
A11, h = 0.655 m
0.8
0
-0.4 -0.2
0
0
-0.4 -0.2
0
0.2
U (m/s)
Figure 5. Comparison of calculated and measured undertow velocity profiles at 11 locations for Test 1A (Roelvink and
Renier, 1995), with wave asymmetry effects (blue line) and without wave asymmetry effects (red line)
1.75
1.5
1.25
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1.25
-1.5
-1.75
0
25
50
75
100
125
150
175
5.25
4.5
3.75
3
2.25
1.5
0.75
0
-0.75
-1.5
-2.25
-3
-3.75
-4.5
-5.25
200
Cross-shore distance (m)
Figure 6. Comparison of calculated and measured significant wave height for Test 1B, Hmo = 1.4 m, Tp = 5 s,
θ = 0 deg. (Roelvink and Renier, 1995)
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Bed level (m)
Hs (m)
The comparisons between calculations and measurements for the significant wave height and undertow
velocity profiles for Test 1B are presented in Figs. 6 and 7, respectively. As for Test 1A, the calculated
significant wave height is also in very good agreement with observations. The rms error for wave
calculation is 6.8 %. The calculated undertow velocities also overestimate the measurements in deeper
water. The calculations showed that at location B1 (x = 65 m), the waves are broken, thus resulting in a
stronger wave-induced current at this location. Furthermore, at locations B7 (x = 152 m) and B8 (x = 160
m), the waves are reformed, resulting a relatively small wave energy dissipation and thus in smaller
calculated undertow velocities. The measured values are relatively small at location B1 and relatively
strong at locations B7 and B8, leading to a significant difference between measurements and calculations.
The rms error for the undertow in this case is 38.5 % with wave asymmetry effects and about 40.9 %
without wave asymmetry effects.
Coastal Dynamics 2013
B1, h = 2.271 m
z/h
B3, h = 1.547 m
B4, h = 1.222 m
B5, h = 0.919 m
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
z/h
0
-0.4 -0.2
0
0.2
B2, h = 1.71 m
0
-0.4 -0.2
0
0.2
0
-0.4 -0.2
0
0.2
B6, h = 0.840 m
B7, h = 0.822 m
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
-0.4 -0.2
0
0.2
0
-0.4 -0.2
0
0.2
B8, h = 0.51 m
0
-0.4 -0.2
0
-0.4 -0.2
0
0.2
0
0.2
0
-0.4 -0.2
0
0.2
B9, h = 0.308 m
0
-0.4 -0.2
0
0.2
U (m/s)
Figure 7. Comparison of calculated and measured undertow velocity profiles at 9 locations for Test 1B (Roelvink and
Renier, 1995), with wave asymmetry effects (blue line) and without wave asymmetry effects (red line)
4. Concluding Remarks
A numerical model for undertow due to random waves has been developed and validated against
published data sets from different laboratories. The wave conditions are well predicted by the random wave
transformation model, producing accurate input for calculating undertow velocity profiles. The wave
trough level calculation is improved and estimated by employing recent formulae for wave asymmetry.
Thus, the calculated mean undertow and the vertical distribution of the undertow are improved. Although
the calculated undertow is overestimated in deeper water for the Roelvink and Renier (1995) data, the
prediction of undertow in the surf zone and shallow water is quite good. Therefore, the new undertow
model is expected to provide reliable input for calculating wave- and current-induced sediment transport
and morphological change.
Acknowledgements
This study is funded by the Alexander von Humboldt Foundation. The authors would like to thank Dr.
Yoshiaki Kuriyama at the Port and Airport Research Institute for valuable discussion. Also, the help from
Mr. Kaan Koca at LWI for providing published references related to the data used for model validation is
much appreciated.
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