Coastal Dynamics 2009
Paper No. 12

MODELING OF WAVES & CURRENTS AROUND POROUS SUBMERGED BREAKWATERS
Roshan Suminda Ranasinghe 1 , Shinji Sato 2 and Yoshimitsu Tajima 3
Abstract
The evolution of waves and wave-induced currents over porous submerged breakwaters is studied by applying a
numerical model based on a truncated version of Chen (2006) Boussinesq-type equations for waves and currents overporous bottoms. The equations of motion for the porous medium include an empirical Forchheimer-type term forlaminar and turbulent frictional losses and an inertial term for acceleration effects following Sollitt and Cross (1972).
Comparisons between simulations and new set of laboratory experimental data show excellent agreement for wave
field in one-dimensional horizontal wave propagation and qualitatively good agreement for wave and current field inhorizontal two-dimensional wave propagation.
Key words: Boussinesq-type model, porous media, porous damping, submerged breakwaters, turbulent kinetic energy
equation, nearshore currents

1. Introduction
The Boussinesq-type model equations for studying waves and currents around coastal structures have maderemarkable advance during the past few decades. However, contrast to the fast development in modeling ofhydrodynamics around impermeable structures, that of porous bottom or porous structures have been very
slow, inevitably due to the uncertainty on determination of a few empirical coefficients associated withporous media. Most of the coastal structures such as groins, seawalls and offshore breakwaters areconstructed with rubble, rocks or concrete blocks to withstand the forces generated by breaking waves andprovide sufficient dissipation by turbulence in the interstices. As submerged breakwaters result primarily inwave dissipation through wave breaking over the structure, bottom friction and turbulence in porous layer,
it is essential to include porous damping in modeling waves and currents around these structures.
A few mathematical models have been proposed to predict wave-current interaction with permeable
beds and structures recently. Kabayashi (1986) applied a permeable-bed model based on the nonlinear
shallow water wave equation and accordingly ignored frequency dispersion entirely. Rajanakamthorn et al.
(1990) in their work proposed a mathematical model for evolution of waves on permeable beds based on
the mild-slope equation. However their model is limited to small amplitude waves propagating over gentle
slopes of uniform porous beds. Water wave interactions with porous seabeds were investigatedtheoretically and experimentally by Gu and Wang (1991) extending the porous flow model developed by
Sollitt and Cross (1972). They pointed out the importance of including nonlinear and inertial resistanceforces in porous flow models and derived a complex wave dispersion equation for waves propagating overporous media.
Flaten and Rigg (1991) were the first to propose Boussinesq-type model for evolution of waves overporous bed in shallow water with an asymptotic approach. Cruz et al. (1997) derived a set of 2DH
Boussinesq equations over a porous bed of arbitrary thickness and tested their applicability on wave
transformation over porous submerged breakwater with an opening for non-breaking waves. They
incorporated frequency dispersion into the flow in porous media following the same approach of Madsen etal. (1991). Hsiao at al. (2002) presented a fully non-linear Boussinesq-type model, following the same
approach of Nwogu (1993), to improve the dispersion properties inside the porous media and compared
their model results with experimental data for the case of regular waves passing over the triangular porous
1

Doctoral Student, Dept. of Civil Engineering, 7-3-1, Hongo, Bunkyo-ku, The University of Tokyo. 113-8656, Japan.
sumindar@coastal.t.u-tokyo.ac.jp.
2
Professor, ditto, sato@coastal.t.u-tokyo.ac.jp.
3
Associate Professor, ditto, yoshitaji@coastal.t.u-tokyo.ac.jp.

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Coastal Dynamics 2009
Paper No. 12
submerged breakwater in 1DH. Though they achieved reasonably good results, their model makes theassumption of weak vertical vorticity, a limited range of particle size, and weak spatial variation of
damping rates inside the porous layer. Very recently, Chen (2006) improved Hsiao et al. (2002) model by
eliminating weak vertical vorticity assumption and retaining higher order porous damping terms in
momentum equation. His model relaxes the assumption on the scaling of the linear damping term in the
momentum conservation equation for the porous layer, and therefore it is applicable to a wide range
relative porous layer thicknesses. Recognizing the importance of retaining the vertical vorticity inBoussinesq-type equations for the simulation of breaking induced nearshore circulations, Chen et al. (2003)
have eliminated the z dependency in the fully nonlinear Boussinesq-type equations for impermeable beds
by setting z  z α in momentum conservation equation. Instead of choosing an arbitrary z , Chen (2006)
opted to integrate momentum conservation equations for both water layer and porous layer. This resulted
elimination of z dependency, at the same time retention of both the leading order and second order verticalvorticity terms. The fundamental properties for wave propagation (wave celerity, porous damping rates) of
Chen (2006) model equations are investigated in Cruz and Chen (2006) for constant depth and horizontal
porous interface. One of the desirable features of Chen (2006) model is its potential applicability toheterogeneous porous beds. Cruz and Chen (2007) discussed the heterogeneity in the hydraulic propertiesof porous layers, which could lead to spatial variation in the porous resistance.
Though most of the models have been tested for non-breaking wave transformation over porous
submerged breakwaters; a few studies investigate the evolution of waves and wave-induced nearshore
currents around these structures. Some exception are Johnson et al. (2005), and Johnson (2006) who
validated a higher-order Boussinesq-type model (Karambas and Koutitas, 2002) with improved linear
dispersion properties together with nonlinear long wave equation for porous media (Cruz. et al., 1997)
using experimental data collected during the DELOS project. An eddy viscosity formulation is adopted inorder to simulate the wave breaking induced energy dissipation (Kennedy et al., 2000).
The goal of the present study is to discuss the evolution of waves and wave-induced currents aroundsubmerged porous breakwaters and to validate a numerical model based on a truncated version of Chen
(2006) equations coupled with a transport equation to compute turbulent kinetic energy hence energy
dissipation due to wave breaking.
2. Description of the governing equations for waves and currents over porous beds
Figure 1 shows the definition of the variables and the Cartesian coordinate system, where ( x , y ) denotethe horizontal coordinates, z is the vertical coordinate, and z  η( x , y ,t ) is the free-surface elevation. The
impermeable bottom boundary is at z  hb ( x , y ) and permeable interface at z  h( x , y ) .

Figure 1. Definition sketch of variables associated with wave transformation over porous beds.

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Coastal Dynamics 2009
Paper No. 12

Dropping higher-order nonlinear terms and higher order porous damping terms, the mass conservation
equation reads as:η .(M  nM s )  0
t

(1)


 z 2 h 2 
 
h

M  h  εηu α  μ 2  α  .u α    z α  .hu α   n.h s u β  
6 
2


 2
 

(2)


 z β2 h 2  hhb  hb2  h  hb      
.u β    z β 
M s  h s u β  μ 2   . hb u β  



6
2 



 
 2

(3)

where,The momentum conservation equation for the water layer can be written as:

z2
u α
 εu α . u α  η  μ 2  α .u αt   z α  .hu αt   n.h s u βt    0
t

2





(4)

And the momentum conservation equation for the porous layer can be written as:u β1 δu β . u β  η  Rβ u β 
δ
t
2

 zβ
 h2

 .u βt   z β .hu αt    .u αt   h.hu αt   n h.hs u βt  

2
 2

μ2 

2

 h .u    h.h u 
βt 
b βt

  2













(5)


 z β2 Rβ h 2   Rβ h.hb u β   




.
.
.
R
R
z
h








u
u
u

β
β
β
β
β
b
β
 2

2
μ2 



0

δ 
1

1 2

2

Rβ  6 hb  hhb  h .u β  2 h  hb .hb u β 













where u α  ( uα , vα ) is the water particle velocity vector at z  z α (inside water layer) and u β  (u β , v β ) is
the water particle velocity vector at z  z β (inside permeable layer). The symbol n and hs ( x, y ) are theporosity and thickness of the permeable layer respectively, which could be arbitrary in x and y . Rβ is theresistance force evaluated at z  z β , as follows:Rβ  a p  b p u s z z  C A
β




t

(6)



u s z z  uβ2  vβ2  μ 2 z β .uβ  .hs uβ 
β

2

(7)

where a p and b p are respectively the linear and nonlinear porous resistance coefficients, and C A is theadded mass coefficient. The porous resistance coefficients a p and b p are estimated from the following
relationships:

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Coastal Dynamics 2009
Paper No. 12

2

ap 

Cf n
nν
, bp Kp
Kp(8)

The intrinsic permeability, K p , added-mass coefficient, C A , and nonlinear drag coefficient, C f , arecalculated by empirical formulae, suggested by van Gent (1995) as follows:Kp 

d 502 n 3

α p 1  n 

2, C A  Cm

1  n  , C  β 1  n
n

f

p

n

Kp
d 50

(9)

where d 50 is the median diameter of permeable materials, α p , β p and C m are empirical coefficients that,in principle, should be determined by experiments for given permeable material.

3. Properties of the numerical model
The terms (  2 ,  3 and  4 in Chen, 2006), which come from the contribution of the higher-order verticalvorticity in Rw s to the pressure field in the porous layer, and the terms, which stem from O(μ 4 ) terms ins
the polynomial expansion of the horizontal velocity vector, u are ignored in the present model. Only thehigher-order terms (  1 in Chen, 2006), which are yielded by the removal of z dependency (by depth
integration) are retained in the momentum conservation equation for the porous medium.
Following the same idea of Nwogu (1993), appropriate values for z α and z β are determined so that
optimal dispersion properties of the model are achieved over a wide range of depths. The properties of themodel are compared with those of the exact dispersion relation for linear water waves propagating in aconstant depth, h , overlying a permeable bed with a constant thickness, hs , is given by Gu and Wang
(1991) as follows:σ 2  gk tanh kh  intanh khs ( gk  σ 2 tanh kh)
a p σ  i1  C A (10)in which k is a complex number ( k  k r  ik i ), where the real part of k , denotes the wave number and theimaginary part represents the spatial porous damping rate.
η( x, t )  ae i ( kr xki xσt )  e  ki x ae i  kr xσt  (11)Detailed analysis of the model equations for their fundamental properties of wave motion over porous
media can be found in Cruz and Chen (2006).
4. Numerical model

The Boussinesq-type equations are solved in the time domain using a third-order Adams-Bashforth
predictor step and a forth-order Adams-Moulton corrector step. The computational domain is discretized asa rectangular grid with grid sizes x and y , in the x and y directions, respectively. The equation
variables η, u α , v α , u β and v β are defined at the grid points in a staggered mesh as shown in Fig. 2. The
water depth and surface elevations are defined at grid points (i, j ) , while the velocities are defined half agrid point on either side of the elevation grid points. The external boundaries of the computational domain
correspond to velocity grid points. The first-order spatial derivatives are differenced to O(x 4 ) , which
automatically eliminates the error terms that would be of the same form as the dispersive terms (Wei andKirby, 1995). The second-order spatial derivatives are discretized to O(x 2 ) , while the advection terms
are differenced with second-order upwind difference scheme.

4Coastal Dynamics 2009
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4.1. Offshore open boundary

As there is significant wave reflection from the submerged breakwaters, it is essential to absorb the waves,which propagate back to the wave incident boundary to prevent buildup of wave energy inside thecomputational domain. This has been achieved with line boundary method proposed by Ishii et al. (1994).

Figure 2. Definition sketch.

Here the neighboring grid points are divided into two types according to their position with respect to line
boundary. Inside the line boundary, incident and outgoing waves are to be dealt with, whereas only
outgoing waves exist outside the line boundary as shown in Fig.2. In the finite difference scheme, for thecomputation inside the line boundary, the parameters Ai ( uαi , vαi or η i ), which are coming from outside theline boundary are added with Ai ,in and for the computation outside the line boundary, the parameters Ai ,which are coming from inside the line boundary are subtracted with Ai ,in .
4.2. Wave breaking induced energy dissipation

Following Kennedy et al. (2000), a simple eddy viscosity type of formulation is adopted to simulate energy
dissipation due to wave breaking by introducing momentum-mixing terms into momentum conservation
equations in x and y directions;
Rbx 



(12)

1 
1

  t h   v y y   t h   v x   t h   u y x h    
2


(13)

1



1


  t h   u   x x   t h   u   y  t h   v  x y 2h    

Rby 









The magnitude of the eddy viscosity is computed using one equation turbulence closure model, in which
the eddy viscosity is related to turbulent kinetic energy and turbulent length scale:5

Coastal Dynamics 2009
Paper No. 12
3/ 2 u  2  v  2 
k 3/2
k t  u s .k  . t k   B       C D
lt
C D  z   z   z (14)

 t  k lt

(15)

l t2

The parameter B is introduced to ensure that turbulence is produced only when horizontal velocity at thewave crest, u s , exceeds the phase velocity of the wave calculated from the linear wave theory. The lengthscale, l t is set to be of the order of deep-water wave height, while an empirical coefficient, C D , is set to be
0.02. We employed anisotropic eddy viscosity concept in 2DH model following Tajima et al. (2007). The
model assumes that the eddy viscosity in the wave propagating direction should be characterized by theeddy viscosity generated by breaking waves while eddy viscosity features should not be that significant in
the direction of wave crests. Based on such assumption they determined anisotropic eddy viscosity by: ν xxν yx

ν xy   ν t cos 2 θν LK ν yy   ν LK
ν t sin 2 θ 

(16)

where, θ , is the angle between wave propagating direction and x axis. The square of wave angle( cos 2 θ, sin 2 θ ) is introduced considering both the shear velocity and the turbulent length scale. v LK is thewell-known mixing term proposed by Larson and Kraus (1991).
ν LK  Hu m(17)

where,  is a non-dimensional coefficient, H : local wave height, u m : bottom wave orbital velocity.

5. Laboratory Experiments
Laboratory experiments were performed in a 30m long, 0.6m wide, 0.8m deep wave-current flume and a
11m long, 6.5m wide, 0.3m deep wave basin at the University of Tokyo. Two different permeable materials
were used for the submerged breakwaters and the structural dimensions and the wave parameters
associated with all the experiments are shown in Table.1.
Table 1. Structural dimensions and wave parameters associated with experiments.

Case No.

Ex01_A01
Ex02_B03
Ex03_A01
Ex03_A02
Ex04_A02
Ex05_A01
Ex06_A01

Deep water
wave
height (m)
0.033
0.034
0.041
0.041
0.039
0.038
0.028

Wave
period
(s)
1.00
1.25
1.00
1.00
1.25
1.00
1.00

Offshore
water
depth (m)
0.255
0.248
0.300
0.300
0.275
0.275
0.218

Freeboard
(m)
0.055
0.148
0.028
0.028
0.03
0.03
0.025

Mean diameterofpermeablematerial (m)
0.02
0.02
0.012
Impermeable
0.012
0.012
0.012

Remarks

Non-breaking, 1DH
Non-breaking, 1DH
Breaking, 1DH
Breaking, 1DH
Breaking, 1DH
Breaking, 1DH
Breaking, 2DH

The layouts of six different experimental set-ups are shown in Fig. 3. The length of the constant depth of
the flume is 11.45m and the slope of the fixed bed is 1:30, whereas those are 1.5m and 1/20 of the wave
basin. All the experiments were conducted with normal incident regular waves and the submerged
breakwaters were not allowed to subject to wetting and drying coexisting field in any of the cases.
The water surface fluctuations were measured using capacitance-type wave gauges with a frequency of
100Hz, from which the wave heights and mean water levels are determined at those locations. The
nearshore currents were visualized using particle tracking velocimetry (PTV) technique in wave basin
experiments.

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Coastal Dynamics 2009
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Figure 2. Layouts of the 6 different experimental setups.

6. Results and Discussion
The model results are firstly verified with data obtained from the wave-current flume experiments. Figure 4
shows the time series of observed (yellow circles) and simulated (blue line) free surface elevations at 9
different locations over a trapezoidal porous submerged mound placed under constant water depth of
0.255m and having a minimum freeboard of 0.055m (case Ex01_A01). The mean diameter and the porosityof the permeable material were found to be around 0.02m and 0.44 respectively. van Gent (1995)
concluded that, though the coefficients α p , β p depend on parameters like shape, aspect ratio or theorientation of the permeable material, the values 1000 and 1.1 can be used for α p , β p respectively, if thecharacteristic length-scale the d 50 is used. In addition, he suggested using a constant value for theempirical coefficient C m of 0.34 for simplicity neglecting the complex dependency of coefficient C A on
the flow field. The following parameters are computed for the porous material used in cases Ex01_A01
and Ex02_B03 as suggested by van Gent (1995).
Intrinsic permeability, K p  1.087  10 7 m 2
Added-mass coefficient, C A  0.43
Nonlinear drag coefficient, C f  0.023
Then the corresponding dimensional drag coefficients become:a p  4.066 s-1, b p  13.552 m-1(18)

The period of incident waves is 1.0s for the case Ex01_A01 and numerical computations are carried out
with a space step of 0.0125m and a time step of 0.00667s. These are the space step and time step used for
all the breaking and non-breaking 1.0s period waves. Numerical results show very good agreement with the7Coastal Dynamics 2009
Paper No. 12
measured data except in the region behind the porous mound, where there exists a slight overestimation of
the free water surface elevation. This could be probably due to the ignorance of higher-order porous
damping rate terms, as the maximum porous layer thickness ratio, r  hs h is as high as 3.63. It should
also be noted that, in lee of the submerged mound nonlinear waves are decomposed into shorter waves as aresult of energy transfer from primary wave component into its higher harmonics. Therefore wave
dispersivity as well as nonlinearity could be influential in the simulations. Simulated in-phase water surface
fluctuations (red lines) over impermeable submerged breakwater are also depicted here for comparison. It
clearly explains the effectiveness of including porous damping terms in numerical simulation, when
dealing with submerged breakwaters constructed with permeable materials.

Figure 4. Simulated free water surface fluctuations at 8.0m(a), 8.4m(b), 8.6m(c), 8.8m(d), 9.2m(e), 9.4m(f), 9.6m(g),10.0m(h), 10.4m(i) from wave generator boundary for case Ex01_A01

Time series of simulated and observed free water surface fluctuations for case Ex02_B03 are shown in Fig.
5. The wave period and the deep-water wave height in this case are 1.25s and 0.034m respectively.
Numerical computations are carried out with a space step of 0.02m and a time step of 0.00833s for thiscase and the overall agreement of the simulated water level fluctuations with observed data seems to be
confirmed. The wave is longer than the previous case, which makes the frequency dispersion
characteristics become relatively less significant. Though the crest of the submerged mound of Ex02_B03
is wider than that of case Ex01_A01, the height of the mound is only 0.1m. On the other hand minimum
freeboard is found to be 0.148m resulting weak nonlinear profiles for the waves as it could be clearly seen

8

Coastal Dynamics 2009
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from Fig. 5. The higher-order porous damping rate terms also play less significant role as the maximum
porous layer thickness ratio, r  h s h is 0.68.

Figure 5. Simulated free water surface fluctuations at 8.0m(a), 8.4m(b), 8.8m(c), 9.2m(d), 9.6m(e), 10.0m(f) fromwave generator boundary for case Ex02_B03

Then the evolution of breaking (surf-zone) waves over submerged breakwater in 1DH (wave flume
experiments) is investigated. Four different experiment results were utilized for the verification (note that
Ex03_A02 is conducted with an impermeable breakwater to compare the characteristics). The mean
diameter and porosity of the permeable materials used in these experiments were found to be 0.012m and0.44 respectively. Similar to previous cases, the following parameters are computed for the numericalmodel.
Intrinsic permeability, K p  3.911 10 08 m 2
Added-mass coefficient, C A  0.43
Nonlinear drag coefficient, C f  0.023
The corresponding dimensional drag coefficients become:a p  6.972 s-1, b p  22.53 m-1(19)

Figure 6. Simulated wave height distributions (a), and mean water level distributions (b) for cases Ex03_A01 and
Ex03_A02

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Coastal Dynamics 2009
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Figure 6 shows the observed and simulated wave height distribution (a) and the mean water level
distribution (b) over a submerged breakwater (both permeable and impermeable) placed 2.0m offshore witha freeboard of 2.8cm. Prominent difference in the wave height distributions is not observed between
permeable and impermeable condition, as the breaking criterion and the formulation for energy dissipation
due to wave breaking is kept unchanged in the model. In addition to that narrow crest width (0.3m) makes
the wave height reduction due to spatial porous damping become less significant compared that due towave-breaking induced energy dissipation. Moreover, modeled wave heights are slightly underestimated in
the lee of the breakwater particularly for case Ex03_A02 (impermeable), though the mean water levels
show excellent agreement. Production of high turbulent kinetic energy leads to over-decaying of wave
energy especially just behind the breakwater; hence insufficient recovery of the waves shoreward in the
numerical model.

Figure 7. Simulated wave height distribution (a), and mean water level distribution (b) for case Ex04_A02

Figure 8. Simulated free water surface fluctuations at 3.6m(a), 3.2m(b), 2.8m(c), 2.4m(d), 2.0m(e), 1.6m(f), 1.2m(g),0.8m(h), 0.4m(i) offshore from still water shoreline for case Ex04_A02

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Coastal Dynamics 2009
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Interestingly mean water level reduction of about 25% is observed if the breakwater is considered to beporous. This is mainly due to the flow through the breakwater (partial free-path to undertow throughporous medium), which declines piling-up of water behind the breakwater.
Figure 7 shows the wave height distribution (a) and mean water level distribution (b) over a broadcrested submerged breakwater under regular waves with a period of 1.25s). Fairly smooth decay of waves
was observed with little recovery behind the breakwater. Model predicted the surface water fluctuations
with excellent agreement with experimental data throughout the surf zone (Fig. 8). Even the generation ofsome higher harmonics was caught in the simulation though the waves could be considered as relatively
long.

Figure 9. Simulated wave height distribution (a), and mean water level distribution (b) for case Ex05_A01

Figure 10. Simulated free water surface fluctuations at 3.0m(a), 2.6m(b), 2.4m(c), 2.2m(d), 1.8m(e), 1.4m(f), 1.0m(g),0.6m(h), 0.2m(i) offshore from still water shoreline for case Ex05_A01

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Coastal Dynamics 2009
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Numerical computations are also carried out for the wave evolution over a step-type submerged breakwaterand the wave height distribution (a) and mean water level distribution (b) are shown in Fig. 9. Similar tocase Ex04_A02, wave heights are well modeled, but the mean water levels are slightly overestimated overthe stepped-crest of the breakwater. Time series of water level fluctuations also show some discrepancy
near the shoreline, but the overall agreement is considered to be satisfactory.

Figure 11. Simulated and observed wave height distribution (a), mean water level distribution (b), andnearshore current field (c) for case Ex06_A01
Simulated wave height distribution (a), mean water level distribution (b), and the nearshore current field (c)for case Ex06_A01 (infinite number of step-type submerged breakwaters 105cm long, having 55cm gaps
between them) are depicted in Fig. 11 and compared with observed data [note that the 1(c) shows thedepth-averaged current velocities, whereas 2(c) shows the nearbottom current velocities]. Significant
alongshore variation of wave heights, and mean water levels is observed particularly in laboratory
experiments, which drives the formulation of secondary circulation (converging) near the shoreline andinterestingly, the direction of currents is found to be almost parallel to the alongshore direction. However,waves tend to break slightly earlier through the gap in the numerical simulation due to strong mixing of
turbulent kinetic energy by advection and diffusion. This leads to relatively weaker alongshore variation in
the wave height and mean water level distributions compared to observation, which weakens the strengthof circulation currents behind the breakwater.

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Paper No. 12

7. Conclusions
A 2DH model has been developed to simulate waves and currents around porous submerged breakwaters
based on a truncated version of Chen (2006) Boussinesq-type equations for waves and currents over porous
bottoms. Numerical simulations are carried out maintaining water layer throughout the space domain at
every time step. i.e. porous breakwaters are not allowed to subject to wetting and drying coexisting field.
Numerical computation results are verified with a new set of data collected using wave-current flume andwave basin experiments. Wave set up is reduced with a considerable degree if the breakwaters are ofporous material, which is an essential characteristic for the formation of secondary circulation (converging)near the shoreline. Irrespective of the weakly nonlinear properties of the present model, 1DH model results
show excellent agreement and 2DH model results show qualitatively good agreement with the observed
data for variety of wave conditions and structural dimensions confirming selection of appropriate values
for the empirical coefficients of the porous media and free parameters of the wave breaking-induced energy
dissipation sub model.

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