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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

1. Ahrens, J.P., “Irregular wave run-up on

smooth slopes”, Tech. Aid No. 81-17, Coastal

Engineering Research Centre, Waterways

Experiment Station, Vicksburg, Miss., 1981.

2. Burcharth, H. F. and Andersen, O.H., “On

the one-dimensional steady and unsteady

porous flow equations”, Coastal

Engineering 24 Elsevier Science Publishers,

pp. 233 – 257, 1995.

3. Burcharth, H. F. and Liu, Z., and Troch, P.

(1999) ‘Scaling of core material in rubble

mound breakwater model tests” Proc. of

the 5th Int. Conf. on Coastal Eng. in

Developing Countries (COPEDEC),

Capetown, pp. 1518-1528.

4. Coastal Engineering Manual, Coastal

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The U.S. Government Printing Office,

Washington D.C., 2002.

5. Jansen, O.J. and Klinting, P., “Evaluation of

scale effects in hydraulic models by

analysis of laminar and turbulent flows” ,

Coastal Engineering 7, Elsevier Science

Publishers, pp. 319 – 329, 1983.

6. Oumeraci, H. and Partenscky., H.W.,

“Wave-induced pore pressure in rubble-

mound breakwaters”, Coastal Engineering,

Frazius Institute, University of Hannover,

Germany, Chaptr 100, pp. 1334 – 1347, 1990.

7. Peiris, D. A. “Influence of shallow water on

wave run-up over coastal structures”,

MPhil Thesis, Faculty of Engineering,

University of Peradeniya, 2004.

8. Sarma, A. K. “Wave run-up measurements

over smooth and rough slopes“, MPhil

Thesis, Faculty of Engineering, University

of Peradeniya, 2002.

9. Van der Meer, J.W., “Conceptual design of

rubble mound breakwaters”, pp. 221-315, in

Philip L.-F. Liu, Advances in Coastal and

Ocean Engineering, Vol. 1, World Scientific,

Singapore, 1996.

10. Van der Meer, J.W. and Stam, “Wave run-

up on smooth and rock slopes of coastal

structures”, ASCE J Waterway, Port,

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534 –550, 1992

ENGINEER 7