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Annual Transactions of IESL, 2005
© Institution of Engineers, Sri Lanka
Effect on Wave Run-Up of Scaling Method of Core
Material in Rubble-Mound Breakwater Model Testing
D. M. R. Sampath and J. J. Wijetunge
Abstract: The size of core material for rubble-mound breakwater models is usually found by
employing geometric scaling. However, Burcharth et al. [3] suggested that the diameter of the core
material in breakwater models should be determined based on Froude scale law for a characteristic
pore velocity. In this backdrop, the present paper attempts to quantify the potential reduction in wave
run-up associated with the new scaling method proposed by Burcharth et al. compared to the
traditional method of linear geometric scaling. Accordingly, a series of wave run-up measurements
were carried out in a 2D regular wave channel to examine the above for the type of rubble-mound
breakwaters typically found in Sri Lanka. The measurements were first carried out with a breakwater
model scaled using the traditional method, and thereafter, with the core of the model scaled using the
new method under identical wave conditions, for a range of the surf similarity parameter representing
both plunging and surging wave breakers. The results suggest that there is little reduction with the
new method for values of surf similarity parameter between 1.3 and 2.2, though a marginal reduction
of not more than 5% could be discernible for values of surf similarity beyond 2.2.
Keywords: Wave Run-up, Rubble-Mound Breakwaters, Froude Scaling, Linear Geometric
Scaling, Core Permeability, Notional Permeability
1. Introduction
A complex and strong interaction takes place
between wave field and wave damping
structures such as rubble-mound breakwaters
when they are exposed to wave attack. This
wave structure-interaction leads to several
hydraulic responses, namely, wave run-up,
wave overtopping, wave reflection, wave
transmission and wave breaking with
associated energy dissipation. Of these, wave
run-up, which determines the crest height, is a
very important phenomenon in the design of
wave-damping structures.
Although design charts and formulae are used
in the conceptual design of rubble-mound
breakwaters, the final design is often based on
extensive model testing to determine the crest
elevation as well as the weight of armour
stones. The permeability of the core material of
rubble-mound breakwaters influences the wave
run-up as well as overtopping and armour
stability (van der Meer [9]). The size of core
material for rubble-mound breakwater models
is usually found by employing geometric
scaling. However, Burcharth et al. [3] suggested
that the diameter of the core material in
breakwater models should be determined
based on Froude scale law for a characteristic
pore velocity. This is because, linear geometric
scaling usually results in smaller core material,
which reduces the flow in and out of the core,
probably resulting in relatively larger run-up.
Wave induced pore pressure in rubble-mound
breakwaters has also been investigated by
Jensen and Klinting [5], Oumeraci and
Partenscky [6], and Burcharth and Andersen
[2].
In this backdrop, the present paper attempts to
quantify the potential reduction in wave run-up
associated with the new scaling method
proposed by Burcharth et al. [3] compared to
the traditional method of linear geometric
scaling.
2. Preliminary Considerations
2.1. Typical Structural Details of Rubble-
Mound Breakwaters in Sri Lanka
Breakwaters in most of the small-craft fishery
harbours in Sri Lanka are of the rubble-mound
E
ng. D. M. R. Sampath, AMIE(SL), B.Sc. Eng. (Hons)
(Peradeniya), Temp. Lecturer, Department of Civi
l
E
ngineering, University of Peradeniya. He has fulfilled al
l
requirements for the award of the M. Sc. Engineering degree
f
rom University of Peradeniya.
Eng. (Dr) J. J. Wijetunge, AMIE(SL), B.Sc. Eng. (Hons)
(Moratuwa), Ph.D.(Cambridge), Senior Lecturer,
D
epartment of Civil Engineering, University of Peradeniya.
ENGINEER 1
type, consisting of an armor layer, a filter layer
and a core of quarry-run. Figure 1 shows a
typical section of the type of rubble-mound
breakwaters used in Sri Lanka.
The geometric parameters of the rubble-mound
breakwaters employed in some of the small-
craft fishery harbours in Sri Lanka were
obtained from design drawings, wherever
available, and from visual inspection at site (see
Table 1.). In Table 1, Da50 is the median
diameter of armour units; Wa is the weight of
armour units; Wf is the weight of filter units;
and Wc is the weight of core material.
Table 1 - Geometric parameters of typical
breakwaters in Sri Lanka.
Harbour Sea-
ward
Slope
Da50
(m)
Wa
(tons)
Wf
(tons)
Wc
(kg)
Puranawella
1:2 1.4 5-8 0.1-0.4 1-100
Ambalangod
a
1:2 1.3 6-8 0.5-1.0 1-250
Kudawella 1:2 1.4 4-6 0.5-1.0 1-250
Dodanduwa
1:2 1.4 6-8 0.4-1.0 1-200
Panadura 1:2 0.8 1-2 - 1-100
Dikkowita 1:2 1.2 2.5 0.1-0.2 -
2.2. Method of Burcharth et al. for Scaling
of Core Material
According to Burcharth et al. [3], the similitude
requires that the hydraulic gradients I in
geometrically similar points in the core to be
the same:
IP = IM (1)
where, the subscripts P and M refer to
prototype and model, respectively.
For one-dimensional flow in coarse granular
media, I may be estimated using:
t
U
cUUbaUI ∂
∂
++= (2)
where, U is a characteristic velocity, and a and b
are coefficients. In scaling porous flow in
breakwater cores the last term in eq (2) is small
and thus may be neglected [3]. Now, as a and b
are functions of the grain geometry, it is
possible to scale the core material using eqs (1)
and (2) provided that the prototype core
structure and prototype flow velocities are
known through use of Froude similitude.
However, the usual difficulty in the procedure
outlined above is owing to the fact that I and U
vary in space and time. Accordingly, the
method proposed by Burcharth et al. [3] is
based on knowledge about the wave induced
pore pressure distribution acquired through a
series of recent large-scale model as well as
prototype measurements. Further details of the
scaling procedure can be found in [3].
3. Experimental Set-Up
The wave run-up measurements were carried
out in a 2D regular wave flume in The Fluids
Laboratory of University of Peradeniya to
investigate the effect on wave-run-up of scaling
method of core material in the rubble-mound
breakwaters typically found in Sri Lanka. This
flume consists of a regular wave generator and
a 12.75 m long, 0.52 m wide and 0.70 m deep
Perspex walled channel (see Figure 2).
At the outset, a set of preliminary wave run-up
measurements was carried out on a smooth
slope to verify the reliability of the present
experimental set-up. For this, a wooden model
of a sloping structure with a Perspex sheet on
the face of the structure was used at the far end
of the channel.
A separate Perspex walled box (see Figure 3)
was used to construct a model of a typical
rubble-mound breakwater employed in Sri
Lanka, for which the notional permeability as
defined by van der Meer (1996) [x] is 0.4 as
shown in Figure 4.) . The sea-side slope of the
structure is 1 on 2 whilst a steel wire-mesh was
fixed to the backside to retain the material and
to allow water to flow.
The measurements were first carried out with a
breakwater model scaled using the traditional
method, and thereafter, with the core of the
model scaled using the new method proposed
in [x] under identical wave conditions, for a
range of the surf similarity parameter. The
stone sizes indicated in the model of length
scale 1:40 shown in Figure 3 were selected
based on data given in Table 1, and we see that
the core material is 5 mm in diameter. On the
other hand, if we follow the method suggested
by Burcharth et al. [3] , then the size of the core
material would be 6.7 mm, giving a model to
protype length scale ratio of about 1:30 for the
core material.
Table 2 gives a summary of the sieve analysis
data and the corresponding median diameters
of the stones used in constructing armour and
filter layers as well as the core of the structure.
ENGINEER 2
Furthermore, another structure consisting of
two layers of armour units placed on an
impermeable slope was used as a reference, for
which the notional permeability is taken as zero
(see Figure 4.).
Table 2 - Sieve analysis data.
D 50
(mm)
Sieve Size
(mm)
Cumulative
Percentage
Passin
g
(
%
)
50 100
37.5 50
37.5
(Armour)
32 0
25 100
19 50
18.8
(Filter)
12.5 0
9.5 100
4.75 50
5.0
(Core –
geometric
scaling) 2.36 50
9.5 100
6.7
(Core – Froude
scaling)
4.75 0
The wave parameters were recorded using an
Armfield H40, resistant type, twin-wire probe.
The use of a single probe meant that the wave
parameters could not be obtained at the toe of
the structure as the incident waves at a location
so close to the structure get distorted by the
waves reflected from the structure in no time.
Therefore, the wave probe was positioned some
distance away from the structure so that target
wave was not disturbed with the reflected
wave due to the structure. Accordingly, after
several trial runs over a range of wave periods,
the wave probe was placed at a location 4 m in
front of the toe of the structure. The wave
records at this location indicated that the
reflected waves reach there only after about 5 –
7 incident waves have passed the probe.
Accordingly, the wave parameters and the
corresponding run-up were always recorded
for an incident wave that had not been affected
by the reflections from the structure (i.e.,
usually for the 5th or 6th wave). The wave
parameters obtained in this manner may be
considered as ‘deep water’ conditions.
A video camera which captures25 frames per
second was employed to obtain the wave run-
up on the slope. The video clips obtained in
this way were played on a Personal Computer
(PC) to obtain the maximum up rush and then
averaged 5 cm intervals across the slope
laterally to get the run-up levels. For a
particular wave setting test was repeated five
times and then average was taken.
Digital Video
Camer a
Core
Mean di ameter 5. 0 mm
0 .0 m SWL
Wave Maker Armor laye r
Two l ayer s of st ones
Mean di ameter 37.5 c m
Wave Gauge
Filter
Three la yer of st ones
Mean di ameter 18.8 mm
St eel wire mesh
Figure 2 - Experimental set-up.
Armor l ayer
Two layers of stones
Mean diameter 37.5 mm
Layer thicknes 75 mm
Core
Mean diameter 5.0 mm
Steel wir e mesh
Filter
Three layer of stones
Mean diameter 18.8 mm
Layer thi ckness 55 mm
Figure 3 - Cross Sectional view of the model.
P = 0.4
P = 0.0
2Da
n50
Impermeable surface
2Da
n50
1.5Da
n50
Df
n50 = 0.5Da
n50
Dc
n50 = 0.25Df
n50
ENGINEER 3
Figure 4 – Model structures with notional
permeability 0 and 0.4.
4. Dimensional Analysis
The wave run-up (R) over a sloping structure
depends on the incident wave height (H0), the
wave period (T), the structure slope (α), the
surface roughness (ks), the depth at the toe of
the structure (ds), the permeability (p), the
foreshore slope (β), and gravitational
acceleration (g). Thus, the relative run-up can
be expressed as a function of following non-
dimensional groups:
=p
d
k
H
d
H
gT
H
R
s
ss ,tan ,tan ,,,
00
2
0
βαφ
(3)
In the present study, no foreshore slope was
used; the surface roughness of armour layer
was maintained constant by using the same
stones for armour layer with median diameter
D50 equals to 37.5 cm throughout the study.
Further, the water depth at the toe of the
structure is also maintained constant for all the
test runs. Thus, the above expression may be
reduced as follows:
=p
H
d
H
gT
H
Rs,tan ,,
00
2
0
αφ
(4)
The combined effect of the non-dimensional
parameters of structure slope and wave
steepness is often represented by the surf
similarity parameter () defined as:
.
2
where,,
tan
2
0
0
0
0gT
H
s
s
π
=
α
=ζ (5)
It must, however, be added that the structure
slope is kept constant in the present study.
5. Test Conditions
The test ranges of the main parameters relevant
to the present study are summarized in Table 3.
There are two sets of data: the measurements in
Data Set A were made over a smooth
impermeable slope whilst those in Data Set B
were for three types of structures, namely,
impermeable reference structure (P = 0),
permeable (P=0.4) model based on geometric
scaling and the same based on Froude scaling
of pore velocity in the core.
Table 3 – Test conditions.
Study Range
Param
eter Data Set A
Smooth Slope
Data Set B
Rough Slopes
H0 3.75-12 cm 3.75-13.5 cm
T 0.77-1.00 s 0.77-1.00 s
ds 34.2 cm 34.2 cm
0 deg 0 deg
24.82 deg 24.82 deg
ds/H0 2.85- 10.00 2.85- 10.00
1.30 – 3.00 1.30 –3.00
Furthermore, Table 4 gives the specifications of
rubble-mound models: D85/D15 gives the
grading of stones and it was derived from the
particle size distribution curve. Material for
each layer of the rubble-mound model was
selected such that the grading is narrow and the
D85/D15 was about 1.25 to 1.5.
Table 4 - Specifications of rubble-mound
models.
Notional
permeability
P
Type
of
layer
D 50
(cm)
D85/D15
Layer
Thick-
ness
(cm)
0.0
Impermeable Armor 37.5 1.25 7.5
Armor 37.5 1.25 7.5
Filter 18.8 1.5 5.5
0.4
Permeable
(Using
geometric
scaling) Core 5.0 1.5 -
Armor 37.5 1.25 7.5
Filter 18.8 1.5 5.5
0.4
Permeable
(Using
Froude
scaling) Core 6.7 1.5 -
6. Results and Discussion
First of all, let us examine the results from
smooth slope tests in Data Set A to verify the
reliability of the experimental set-up.
Accordingly, Figure 5. shows the variation of
relative wave run-up with surf similarity
parameter over the smooth impermeable slope
of 1 on 2. The measurements of Ahrens [1],
Peiris [8], Sarma [9], and Van der Meer [10] are
also shown to enable a comparison of present
ENGINEER 4
results with previous results for 1.3 < ζ0< 3.0.
The data points of Van der Meer shown in the
Figure 5 were derived from run-up data with
2% exceedence assuming that the waves and
the run-up are both Rayleigh distributed.
It can be seen in Figure 5 that the relative wave
run-up is increasing with ζ and reaches a
maximum value when ζ0 is approximately 2.6,
before beginning to decline with further
increase of ζ0. Clearly, the present results on
the whole show good agreement with the
previous measurements for 1.3 < ζ0 < 2.42.
0
0.5
1
1.5
2
2.5
3
012
R/H0
3
Van der Meer (1992)
Ahrens (1981)
Sarma (2003)
Peiris(2004)
Present
ζ0
Figure 5 - Variation of relative wave run-up
with surf similarity parameter over a smooth
impermeable slope.
Now, Figure 6 shows the variation of R/H0
with ζ0 for the three different rough slopes of
rubble-mound models tested. This includes the
results of wave run-up measurements over the
reference structure (P = 0), over the model of a
typical rubble-mound breakwater found in Sri
Lanka (P = 0.4) with core material scaled using
geometric scaling, and the same structure with
core material scaling based on Froude similarity
for a characteristic pore velocity.
These results show that the relative wave run-
up increases gradually with surf similarity
parameter for all three cases. Also, it is not
surprising to note that the relative wave run-up
over reference structure with P = 0 is larger
than the other two cases as impermeable layer
under the armour layer introduces maximum
resistance to the flow through the structure.
making a larger quantity of water available for
wave run-up leading to a higher run-up.
1.00
1.10
1.20
1.30
1.40
1.0 1.5 2.0 2.5 3.0 ζο
R/H0
P 0. 0
P 0. 4
P 0.4 (Froude Scale
)
Figure 6 - Variation of relative wave run-up
with surf similarity parameter over rough
slopes.
We see in Figure 6 that, for 1.3 < ζ0< 2.2, the
relative wave run-up on the structure with core
modelled by employing linear geometric
scaling falls almost on that for the structure
with core material size determined from Froude
scaling. However, for values of the surf
similarity parameter larger than about 2.2, the
run-up measurements for the structure with
core material size determined from Froude
scaling appear to fall consistently below that for
the structure with the size of core material
obtained form the traditional geometric scale. It
must be noted that the wave breaking in this
region of surf similarity parameter belongs
mostly to surging type. Therefore, there is
more time for water to flow through the
structure. As the flow through the structure
becomes important permeability of the core
plays an important role in deciding the wave
run-up. Since the dimeter of the core material
found from Froude scaling method is larger
than that derived from geometric scaling
method there is comparatively more space in
the core and hence the permeability is relatively
high in the former case. Consequently, the
relative wave run-up is less, albeit marginally,
on the structure with core material from Froude
scaling than on the structure with the size of the
core material determined from geometric scale.
We now examine the reduction of relative wave
run-up of the two permeable structures with
respect to the wave run-up over impermeable
(P = 0) structure. This will allow us to
differentiate the effect of the permeability of the
core material and the effect of scaling method
of the core material. The run-up reduction
factor for the under-layer permeability may be
defined as follows:
ENGINEER 5
,
)/(
)/(
00
0
=
=γ
P
P
PHR
HR (6)
i.e., as the ratio between the relative run-up
over a given breakwater with a permeable core
and that without a core (two layers of armour
stones placed on an impermeable.).
Figure 7 shows that for the structure with core
modelled from geometric scale, the variation of
the reduction factor with the surf similarity
parameter shows a general scatter around a
mean line. This implies that there is little or no
variation of the reduction factor with the surf
similarity parameter over the range of ζ0 tested.
However, reduction factor for the structure
with core scaled from Froude similarity for
pore velocity appears to fall below that for the
structure with core modelled from geometric
scale for values of ζ0 larger than about 2.3.
0.875
0.9
0.925
0.95
0.975
1
012
ζ0
γP
3
P 0.4 (24.8 deg)
P 0.4 (Froude Scale 24.8 deg)
Figure 7 - Variation of the reduction factor
with the surf similarity parameter.
We also consider in Figure 8, the reduction in
run-up associated with the method of Froude
similarity based core scaling compared to that
based on traditional geometric scaling.
Figure 8 suggests that the reduction of run-up
associated with Froude based scaling compared
to geometric scaling is less than 5% for the
present experimental conditions. However, it
must be emphasised that the above reduction in
run-up is for a model: prototype scale of about
1:40, and it is not entirely clear whether or not
there would be a larger reduction of run-up if
the structure were to be modelled at a larger
scale, for example, 1:20.
0.925
0.95
0.975
1
12
ζ0
γ'
3
Figure 8 - Reduction of wave run-up with surf
similarity parameter for core found from
Froude scaling with respect to the linear
geometric scaling.
Despite the marginal effect of the core scaling
method on the run-up, we fit two approximate
curves to describe the variation of the relative
wave run-up with surf similarity in Figure 9.
Equation (7) gives the approximate curve for
the results of reference structure; equation (8)
describes the variation of relative wave run-up
with surf similarity parameter over a rubble
mound model typically found in Sri Lanka with
the median diameter of the core material
determined by employing linear geometric
scaling; equation (9) is for a similar breakwater
model but the core of the structure is scaled
based on Froude similitude for a characteristic
pore velocity. All of these results are valid for a
structure for which the notional permeability
factor is about 0.4 and the sea-side slope is 1:2,
and the surf similarity parameter falling
between 1.3 to 2.4.
R/H0 = 1.00 ζ00.29 (7)
R/H0 = 0.96 ζ00.29 (8)
R/H0 = 0.98 ζ00.24 (9)
ENGINEER 6
1.00
1.10
1.20
1.30
1.40
1.50
1.00 1.50 2.00 2.50 3.00
ζ o
R/H0
P 0.0
P 0.4
P 0.4 (Froude Scale)
Figure 9 - Approximate curves for variation of
relative wave run-up with surf similarity
parameter for the three cases tested: - - - - eq
(7); …….. eq (8); ____ eq (9).
7. Conclusions
Following conclusions can be drawn for the
range of conditions covered in the present
study of the effect of the scaling method of core
material in the model testing of the type of
rubble-mound breakwaters found in Sri Lanka.
A set of preliminary run-up measurements
carried out over a smooth surface to verify the
reliability of the present experimental set-up
found to give good agreement with some of the
previous results.
The results appear to suggest that the relative
wave run-up is slightly less on the structure
with core material from Froude scaling than on
the structure with the size of the core material
determined from geometric scaling. However,
this reduction in run-up over the structure with
core material scaled using Froude similituide is
almost negligible for 1.3 < ζ0< 2.2, though a
marginal reduction of not more than 5% could
be discernible for values of ζ0 beyond 2.2.
Acknowledgment
The first author gratefully acknowledges the
financial support from the Science &
Technology Personnel Development Project of
the Ministry of Science & Technology, Sri
Lanka, which enabled him to carry out the
experimental work described in the present
paper.
References
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smooth slopes”, Tech. Aid No. 81-17, Coastal
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3. Burcharth, H. F. and Liu, Z., and Troch, P.
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ENGINEER 7
