1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
185
ENUMERATION AND VALIDATION OF HYDRODYNAMIC
CHARACTERISTICS OVER PLANE AND SERRATED SLOPES
D. Swetha1
, K. Ramamohan Reddy2
1
PhD Scholar, Dept. of Civil Engineering, JNTUH College of Engineering, Hyderabad, Andhra
Pradesh, 500085, India
2
Professor and Head, Centre for Water Resources, Institute of Science and Technology,
Jawaharlal Nehru Technological University Hyderabad, Andhra Pradesh, 500085, India
ABSTRACT
Numerical simulation is carried out to study the generation, propagation and the run-up of
different types of waves such as the regular, random and solitary waves. Bed slopes of 30o
and 45o
are considered for the study with a variation of non-dimensionalised wave height ( )H d of the
generated waves. This numerical study is carried out by adapting FUNWAVE which uses a rigid
type piston type wave maker to generate waves from one end of the wave flume and the rear end
could be defined with various configurations such as the vertical walls or sloping beds that may be
the plane or projections like serrations and dentations. The non-dimensionalised parameters such as
maximum run-up as well as the reflection coefficient by two-probe method is computed and
compared with the published experimental results. The bed friction coefficient whose value ranges
from 0.0097 to 0.012 for the serrated slopes has been quantified through a Navier-Stokes solver. The
dimensionless run-up was found to be more for the bed slope of 30o
when compared to the bed slope
of 45o
since a lesser portion of energy is used in the run-up process. The slope with serrations will
reduce the run-up and reflection coefficient ( )rK by about 30% and 20% respectively.
Keywords: Boussinesq equation, plane and serrated slopes, wave maker, run-up, reflection
coefficient
1. INTRODUCTION
Various natural and man-made activities cause erosion of coastal areas which become a
serious threat to the marine environment. Different types of shore protection structures such as
seawalls, groins, offshore breakwaters and head lands are used for stabilisation of the shorelines
against wave induced erosion. Seawalls are shore defense structures that may effectively be used to
protect a shoreline from cross shore sediment transport. The seawalls may be classified as vertical or
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2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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sloped seawalls. Sloped seawalls are good dissipators compared to the vertical structures especially
when the slope is mild. Energy dissipators are commonly used for dissipation of kinetic energy in
coastal and hydraulic engineering. These dissipating blocks are proposed to be serrated or dented
which are usually arranged in staggered manner on the surface of plane sloping wall.
Substantial development in simulation of ocean surface waves has been made in the last few
decades. Major projects now demand not only careful estimates of wave conditions near the site but
also reliable predictions of the effects of the waves on the structures. Such problems can be analysed
experimentally with scale models in the wave tanks, or numerically by the application of a suitable
mathematical model and numerical method. The advantage of numerical models over experiments is
the ease with which different layouts can be constructed and tested compared to rebuilding of the
physical model within a shorter duration. The Boussinesq equations are the simplest class of
mathematical models that contain all the effects of wave refraction and diffraction, and nonlinear
wave interaction such as the generation of harmonic waves in a variable depth and shallow water
environment. In shallow water, weakly dispersive water waves with their linearized dispersion
characteristics approximate Stokes first order wave theory.
2. BACKGROUND
Interaction of water waves with seawalls and seafront slopes has been a topic of interest since
several centuries. Numerous literatures exist on various aspects of seawalls including their design,
analysis and construction. In view of the present topic and its focus, literature pertaining to
estimation of run-up and reflection coefficient is of interest. The standard Boussinesq equations for
variable water depth were first derived by [1] using the depth averaged velocity as a dependent
variable, which forms the basis of much of modern day work in this area. Although the extended
systems have improved dispersion characteristics they are all still formally of the same accuracy as
the original system and hence are restricted to a shallow water environment. The finite difference
methods, for solving the Boussinesq-type of equations, are simple to formulate, but difficulties in
modelling irregular geometries in two space dimensions with structured grids can lead to the
accuracy. The run-up over structures as well as the reflection characteristics due to the propagation
of waves are taken up, the review is reported in brief accordingly.
2.1 Wave Run-Up
The irregular wave run-up and overtopping on smooth straight slopes have been studied by
[2]. Empirical formulae to estimate the run-up and wave overtopping on smooth straight slopes were
proposed. They described new formulae for estimating the vertical run-up distance above the SWL
that will be exceeded by only 2%of the regular wave run-ups on smooth, impermeable slopes. These
formulae are based on the hypothesis that the weight of water above still water level at maximum
run-up is proportional to the maximum depth-integrated wave momentum flux occurring in a wave
just before it reaches the toe of the impermeable plane slope. They re-examined the existing wave
run-up for regular, irregular (random) and the solitary waves on smooth, impermeable plane slopes
[3].
2.2 Wave reflection
Researchers have carried out experimental investigations to assess wave reflection. Some of
the researchers have proposed the empirical formula by using the Surf similarity parameter as the
independent variable for plane and rough slopes. Some have compared the measured wave reflection
with the predictive formula.
[4] have experimentally and analytically studied the wave energy dissipation and the
reflection characteristics for beaches, revetments and breakwaters. Empirical formulae for predicting

3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
187
the wave reflection coefficients for beaches, revetments and breakwaters have been suggested. [5]
carried out experimental investigations to estimate the irregular reflection on a 1:3 rough
impermeable slope. The test results were used to compare results of a numerical model for predicting
the wave time series and spectral of the reflected waves on the slope.
3. THEORETICAL BACKGROUND
The system of equations which includes both the nonlinearity and the dispersion
characteristics are the Boussinesq equations which provide a general basis for studying the wave
propagation in two directions. These are the shallow water equations for non-dispersive linear wave
propagation. FUNWAVE involves the utilization of the Boussinesq equations, in which the physical
effects of frictional damping and wave breaking terms can be incorporated. Wave run-up on the
beach is simulated using a permeable-seabed technique. Instead of tracking the moving boundary
during wave run-up/run-down on the beach, the entire computational domain is treated as an active
fluid domain by employing an improved version of the slot or permeable-seabed technique proposed
by [6, 7] for simulation of run-up. The basic idea behind this technique is to replace the solid bottom,
where there is very little or no water covering the land by porous sea bed, or to assume that the solid
bottom contains narrow slots. This allows the water level to be below the beach elevation. In the
present numerical model, the run-up is estimated by the permeable sea-bed technique.
4. NUMERICAL SOLUTION
The numerical solution techniques used to obtain solutions of the system of model equations
are discussed in this section. In the present numerical work, an approach of using a higher-order
scheme is adopted in order to perform the computations. A composite 4th-order Adams- Bashforth-
Moulton scheme which uses a 3rd order Adams-Bashforth predictor step and a 4th-order Adams-
Moulton corrector step is used to step the model forward in time. Terms involving first-order spatial
derivatives are differenced to the fourth order accuracy utilizing a five-point formula. All the errors
involved in solving the underlying nonlinear shallow water equations are thus reduced to the 4th
order in grid spacing and the time step size. The computational domain involves the critical
discritisation of grid size along the horizontal direction ∆x=0.1 and the typical time step of ∆t=0.025.
Spatial and temporal differencing of the higher-order dispersion terms is done to second-order
accuracy, which reduces the truncation errors. For weakly nonlinear case, the scheme typically
requires no iteration unless problems arise from boundaries, or inappropriate values for ∆x, ∆y and
∆t are used. For strong nonlinearity, the model tends to make more iteration.
5. NUMERICAL INVESTIGATIONS
The present numerical study involves the measurement of run-up and the reflection
characteristics over the plane and the serrated slopes. The length of the tank is 25m and the typical
grid size is 0.1. Figure 1 shows the layout of generation and propagation of the waves which contains
a rigid type piston type wave maker at the left end from which the wave starts propagating at the
centre of the wave tank whereas the rear end is subjected to the various configurations such as the
vertical walls and the sloping beds which includes serrations also. The run-up along the slope and the
reflection coefficient for all the different types of waves (regular, random and solitary) is calculated.
The details of the various variables and the non-dimensionalised parameters used in the present study
are listed in Table 1.

4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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Table 1 Variables considered in the numerical model
Fig1 Computational Domain for the Fig2 Typical simulated time series of a
propagation of waves Regular wave
6. RESULTS AND DISCUSSIONS
A constant water depth of 0.5m is used for the entire numerical computational study. The
time histories for the various wave heights and periods are recorded. By using the two probe method
of [9], the reflection coefficient is calculated. The typical time history of a regular wave is shown in
Fig 2.
6.1 Effect of Relative Water-Depth on Run-Up
In the present numerical study, slopes of 30° and 45° are carried out in case of an inclined
planar slope and the serrated slopes. The run-up for the steeper slope is less compared to that for the
gentle slope in case of plane slopes. When compared to the serrated slope, the reflection coefficient
(Kr) is maximum for the plane wall for all d L tested. The present numerical results compares well
with that of the published experimental results of [8]. The comparison of the run-up between the
numerical and the experimental results of [8] with the relative water depth for the wave height of
H=0.04m for the two bed slopes of 30° and 45° for the plane slopes is shown in Fig 3(a) and Fig
3(b). The comparison of the run-up between the numerical and the experimental results of [8] with
the relative water depth for the wave height of H=0.04m for the two bed slopes of 30° and 45° for the
serrated slopes is shown in Fig 4(a) and Fig 4(b). The comparison of the run-up between the
numerical and the experimental results of [8] with surf similarity parameter for the wave height of
H=0.04m for the plane slope and the serrated slope is shown in Fig 5(a) and Fig 5 (b).
S. No. Parameters Measured Range
1.
Relative wave height (H/d)
0.07-0.09
0.10-0.14
0.15-0.18
2. Relative water depth (d/L) 0.19-0.64
3. Slope of beach 30o
and 45o
4. Size of serrations 100mm X 50mm X 50mm
@100mm c/c
40 44 48 52 56 60
Time (s)
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
SurfaceElevation(m)
d
1 12
LengthoftheTank
θ
T,H
WavePaddle
WaveGauges

5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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Fig 3 Dimensionless run-up with relative water depth for a plane slope for a wave height of
H=0.04m and for a bed slope of (a) 30° (b) 45°
Fig 4 Dimensionless run-up with relative water depth for a serrated slope for a wave height of
H=0.04m and for a bed slope of (a) 30° (b) 45°
Fig 5 Dimensionless run-up with surf similarity parameter for a wave height of H=0.04m (a) Plane
(b) Serrated
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d,Angle:300
Present numerical
Sakthivel(2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d, Angle:450
Present numerical
Sakthivel(2006)
0 1 2 3 4 5 6 7 8 9
Surf similarity parameter (ξ)
0
0.5
1
1.5
2
2.5
Ru/Hi
H=0.08d,Angle:300
Present numerical
Sakthivel(2006)
0 1 2 3 4 5 6 7 8 9
Surf similarity parameter (ξ)
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d, Angle:450
Present numerical
Sakthivel(2006)
(a) (b)
(b)(a)
(a) (b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d,Angle:300
Present numerical
Sakthivel(2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d, Angle:450
Present numerical
Sakthivel(2006)

6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
190
The comparison of the run-up between the planar slope and the serrated slope with the
relative water depth for the wave height of H=0.04m for the two bed slopes of 30° and 45° are shown
in Fig 6(a) and Fig 6(b). Similarly, the comparison of run-up between the planar slope and the
serrated slope with the surf similarity parameter for the wave height of H=0.04m is shown in Figure
7.
Fig 6 Dimensionless run-up with relative water depth for plane and serrated slopes for a wave height
of H=0.04m and for a bed slope of (a) 30° (b) 45°
Fig7 Dimensionless run-up with surf similarity parameter for a wave height of H=0.04m
6.2 Impact of Relative Water Depth on Reflection Coefficient
In the present numerical study, when compared to the serrated slope, the reflection coefficient
(Kr) is maximum for the plane wall for all d L tested. The beach slope of 30° and 45° are used for the
calculation of reflection coefficient (Kr) similar to the calculation of run-up both for the plane and the
serrated slopes. The present numerical results compares well with that of the published experimental
results of [8]. The comparison of the reflection coefficient between the numerical and the
experimental results of [8] with the relative water depth for the wave height of H=0.04m for the two
bed slopes of 30° and 45° for the plane slopes is shown in Fig 8(a) and Fig 8(b). The comparison of
the reflection coefficient between the numerical and the experimental results with the relative water
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d,Angle:300
Plane
Serrated
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d, Angle:450
Plane
Serrated
2 3 4 5 6 7 8 9
Surf similarity parameter (ξ)
0
0.4
0.8
1.2
1.6
2
2.4
Ru/Hi
H=0.08d
Plane
Serrated
(a) (b)

7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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depth for the wave height of H=0.04m for the two bed slopes of 30° and 45° for the serrated slopes is
shown in Fig 9(a) and Fig 9(b). The comparison of the reflection coefficient between the numerical
and the experimental results of [8] with surf similarity parameter for the wave height of H=0.04m for
the plane and serrated slopes is shown in Fig 10(a) and Fig 10(b).
The reflection coefficient for the steeper slope is more compared to that for the gentle slopes.
The comparison of the reflection coefficient between the planar and serrated slopes with the relative
water depth for the wave height of H=0.04m for the two bed slopes of 30° and 45° is shown in Fig
11(a) and Fig 11(b). The comparison of the reflection coefficient between the planar and the serrated
slopes with the surf similarity parameter for the same wave height and for the bed slopes of 30° and
45° is shown in Fig 12. For the range of relative water depth, d L =0.19 and 0.64 and surf similarity
parameter ξ, 1.8 to 8, the reflection coefficient, Kr increases with an increase in ξ. The flatter the
slope more is the reduction in Kr. The effect of slope on Kr is found to slightly increase with an
increase in the d L .
Fig 8 Dimensionless reflection coefficient with relative water depth for plane slope for a wave height
of H=0.04m and for a bed slope of (a) 30° (b) 45°
Fig 9 Dimensionless reflection coefficient with relative water depth for serrated slope for a wave
height of H=0.04m and for a bed slope of (a) 30° (b) 45°
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d,Angle:300
Present numerical
Sakthivel(2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d, Angle:450
Present numerical
Sakthivel(2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d,Angle:300
Present numerical
Sakthivel(2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d, Angle:450
Present numerical
Sakthivel(2006)
(b)
(b)(a)
(a)

8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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Fig 10 Dimensionless reflection coefficient with surf similarity parameter for a wave height of
H=0.04m (a) Plane (b) Serrated
Fig 11 Dimensionless reflection coefficient with relative water depth for a wave height of H=0.04m
and for a bed slope of (a) 30° (b) 45°
Fig12 Dimensionless reflection coefficient with surf similarity parameter for a wave height of
H=0.04m
0 1 2 3 4 5 6 7 8 9
Surf similarity parameter ( ξ)
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d
Present numerical
Sakthivel(2006)
0 1 2 3 4 5 6 7 8 9
Surf similarity parameter (ξ)
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d
Present numerical
Sakthivel(2006)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d,Angle:300
Plane
Serrated
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d/L
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d, Angle:450
Plane
Serrated
2 4 6 8
Surf similarity parameter (ξ)
0
0.2
0.4
0.6
0.8
1
Reflectioncoefficient(Kr)
H=0.08d
Plane
Serrated
(a) (b)
(b)(a)

9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
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7. CONCLUSIONS
Using a Boussinesq model, the wave run-up on the plane and serrated slopes is calculated
using a permeable sea bed technique. In addition to the run-up, the reflection coefficient is also
calculated, using the two probe method [9]. The present numerical method uses a different approach
of finite difference method and the Adams-Bashforth scheme. The present numerical results are in
good agreement with the published experimental results of [8] by which the validation is carried out.
The dimensionless run-up was found to be more for the bed slope of 30° compared to that of the bed
slope of 45°. The reflection coefficient is observed to be more for the bed slope of 45°compared to
that of the bed slope of 30° since the major portion of energy is utilized in the dissipation while very
less portion of energy is used for the calculation of run-up. Among the plane and the serrated slopes,
plane slopes is observed to have more run-up as well as the reflection coefficient since the friction
value reduces the run-up. It is inferred that as friction value increases, run-up decreases. The
serrations have changed the flow characteristics and provided the improved results. The work can be
extended by providing the dentations onto the planar slope and the same hydrodynamic
characteristics like run-up and the reflection coefficient can be calculated.
8. REFERENCES
[1] Peregrine, D.H., (1967), Long waves on a beach, J. Fluid Mech., 27, 815-882.
[2] Van der Meer, J. W., and Jansen J.P.F.M (1994), Wave run-up and wave overtopping at dikes. In
and Vertical Structures, ASCE PP1-27.Also Delft Hydraulics Publication No.485.
[3] Steven A.Hughes (2004), Estimation of wave run-up on smooth, impermeable slopes using
momentum flux parameter, Coastal Engineering, 51, PP. 1085-1104.
[4] Ahrens, J.P.1981:Irregular wave run-up on smooth slopes, CETA No.81-17, U.S. Army Corps of
Engineers, Coastal Engineering Research Center, Fort Belvoir, V.A.
[5] Kobayashi (1990), Irregular wave reflection and run-up on rough impermeable slopes. J.
Waterways, Port, Coastal and Ocean Engineering, 116, PP. 708-725.
[6] Tao.J., (1983), Computation of wave run-up and wave breaking, Internal report, Danish
Hydraulic Institute, 40PP.
[7]Tao, J., (1984), Numerical modelling of wave run-up and breaking on the beach, Acta
Oceanologica Sinica, 6,692-700 in Chinese.
[8] Sakthivel, (2006), Wave induced pressures and run-up on plane, serrated and dented sea walls,”
M.Tech Thesis, Ocean Engineering Department, IIT Madras.
[9] Goda, Y., and Suzuki, Y. (1976), Estimation of incident and reflected waves in random wave
experiments, Proc. Of 15th Coastal Engineering Conference, PP. 828-845.
[10] B.T.P.Madhav, Habibulla Khan, Atluri Lakshmi Tejaswani, Kharahari Tripuraneni, Bhaskar
Teja Varada and Banda Krishna Chaitanya, “Reduction of Harmonics and Surface Wave Losses in
Serrated MSPA using 2d-Ebg Structures”, International Journal of Electronics and Communication
Engineering & Technology (IJECET), Volume 3, Issue 2, 2012, pp. 439 - 444, ISSN Print:
0976- 6464, ISSN Online: 0976 –6472.