Article 3 rd place DPWA 2015

DPWA Winner 3rd place
NEW DESIGN METHOD OF ARMOUR UNITS COVERING
RUBBLE MOUND OF COMPOSITE BREAKWATER AGAINST
TSUNAMI OVERFLOW
MITSUI JUN
Fudo Tetra Corporation
e-mail: [email protected]
AKIRA MATSUMOTO and MINORU HANZAWA
Fudo Tetra Corporation
KAZUO NADAOKA
Tokyo Institute of Technology, Japan
KEYWORDS: tsunami, overflow, armour units, stability, breakwater, design method
MOTS-CLES: tsunami, surverse, éléments de car-
apaces, stabilité, digue, méthode de dimensionnement
1. INTRODUCTION
Numerous composite breakwaters were severely
damaged in the 2011 Off the Pacific Coast of Tohoku Earthquake Tsunami. One of the causes of
failure was scouring of the rubble foundation and
subsoil at the harbour-side of breakwaters due to
overflow. This was a formerly inconceivable type
of failure [Ministry of Land, Infrastructure, Transport
and Tourism, 2013]. After this disaster, it became
necessary to consider resilience against tsunamis.
One possible method was a placement of widened protection using additional rubble stones
behind the breakwater to prevent sliding of the
caisson.
Figure 1: Countermeasure against tsunami of
breakwaters
148
Installing armour units on the rubble mound on
the harbour-side would also be required to prevent scouring around the rubble mound (Figure
1). However, there are as yet few studies on the
stability of armour units against tsunami overflow.
1.1 Conventional Estimation Method of the
Required Mass of Armour Units
The Isbash formula [Coastal Engineering Research
Centre, 1977] has been applied previously as the
method to estimate the required mass of armour
units against tsunamis. This formula is expressed as
follows:
(1)
where, M is the mass of the armour unit, ρr is the
density of the armour unit, U is the flow velocity
near the armour unit, g is the gravitational acceleration, y is the Isbash constant, Sr is the specific
gravity of the armour unit with respect to water,
and θ is the angle of slope. The required mass calculated by this formula is proportional to the sixth
power of the flow velocity. This causes a practical problem that the required mass is too sensitive
to variations in the estimated flow velocity. Also,
it requires a lot of labour and time to obtain the
flow velocity by numerical computation. In addition, the Isbash constant is required to be set properly because the required mass varies in inverse
proportion to the sixth power of the Isbash constant. For concrete blocks, y = 1.08 has been applied previously regardless of the kind of the block
shape. This value was based on experiments using
tetrapods conducted by Iwasaki et al. (1984), and
it is not appropriate to use the same value to all
blocks. Moreover, the applicability of the Isbash
formula to the tsunami overflow in which a fast
flow acts locally on armour units has not been sufficiently validated.
[Hamaguchi et al., 2007 ; Kubota et al., 2008]. The
caisson model was made of wood and was fixed
with a weight so that it would not be moved by
tsunami action since this study was focused on the
stability of armour units.
In this context, the establishment of a more practical method to determine the mass of armour units
is one urgent issue towards the achievement of resilient breakwaters against tsunamis.
1.2 Aims of this Paper
This paper presents a new design method for armour units, which estimates the stability against
tsunami overflow using an overflow depth of tsunami instead of a flow velocity. Hydraulic model
experiments in a wide range of conditions were
conducted to extract key factors for armour stability. Numerical computation was also conducted to investigate the detailed failure mechanism.
Empirical formulae were then derived based on
these results.
Figure 2: Test setup in the flume
2. HYDRAULIC MODEL EXPERIMENTS
2.1 Experimental Setup
Experiments were conducted in a 50-m long, 1.0-m
wide, and 1.5-m deep wave flume, as shown in Figure 2. A horizontal mortar seabed was partitioned
into two sections along the length, and a breakwater model was installed in one 50-cm wide waterway. A submersible pump and discharge port
were located on the harbour-side and sea-side of
the breakwater model respectively to generate a
steady overflow. The capacity of the pump was 4
m3/min. A water level difference was generated
between the inside and outside of the breakwater by operating the pump. The height of the seaside water level could be changed by varying the
height of the overflow weir installed on the seaside of the breakwater model. The height of the
overflow weir could be varied in a range of 0 to
50 cm. A vent hole with a diameter of about 25
mm was provided in the partition wall close behind the caisson to maintain the space between
the caisson and the overflow nappe in ambient
atmospheric pressure conditions.
A schematic layout of the breakwater model is
shown in Figure 3. The model scale is 1/50. Experiments were conducted by changing the shape
of the harbour-side rubble mound, the harbourside water level, and the shape and mass of the
armour units. Two kinds of flat-type armour blocks
and wave-dissipating blocks shown in Figure 4
were used. The ‘armour block A’ is a recently developed block produced by refining the ‘armour
block B’. The large holes in the block have been
found to contribute to high stability against wave
action due to the reduction of the uplift force
Figure 3: Schematic layout of the tested breakwater
Figure 4: Armour units used in the experiments
The duration time of the steady overflow of tsunami
was set to 127 s (15 minutes in the prototype scale)
to simulate the actual event observed in Hachinohe port during the Tohoku tsunami on March
11, 2011. As it took about 60 s until the water level
achieved a steady state from the start of operating the pump, the total operation time of the pump
was set to 187 s. The stability limits of the armour
units were examined by increasing the overflow
depth in increments of 1 cm. The overflow depth
was defined as the difference between the seaside water level (measured at 2 m on the offshore
side from the front of the caisson) and the crest
height of the caisson. The harbour-side water level
was measured at 2 m on the onshore side from
the rear surface of the caisson. The section was
not rebuilt after tsunami attack with each overflow
depth. The number of the moved armour units
149
was counted as an accumulated number. The
damage to armour units were defined using the
relative damage N0, which is the actual number of
displaced units related to the width of one nominal diameter Dn [Van der Meer, 1988]. The nominal
diameter Dn is the cube root of the volume of the
armour unit. In this study, N0 = 0.3 was applied as
the criterion of damage.
2.2 Feature of the Damage by Tsunami
Overflow
Figure 5 shows snapshots of the tsunami overflow
in the experiment. As soon as the armour blocks at
the slope section were washed away, the scouring of the rubble mound progressed rapidly and
reached to the sea bottom within about 1 minute
(7 minutes in the prototype scale). Though the widened protection using additional stones exhibits a
function to delay scouring, the damage expands
rapidly if the armour units are washed away and
the rubble mound is exposed. This is one of the
features of damage by tsunami overflow. This also
suggests the importance of the accurate estimation of the armour stability.
on the crown section was one or two, whereas
it impinged on the crown section in the case of
more than four units on the crown section. The
cases in which the jet impinged on the slope section showed higher stability than the cases of impingement on the crown section. This shows that
impingement position largely affects the armour
stability. The effect of the impingement position
depended on the structural conditions, such as
the shape of the armour units and the presence
or absence of widened protection. Thus, it is necessary to incorporate properly this effect into the
estimation of the armour stability.
2.4 Influence of Harbour-Side Water Level
2.3 Influence of Impingement Position of
Overflow Jet
Figure 6: Influence of impingement position on the
overflow depth at stability limit
Figure 5: Snapshots of tsunami overflow.
(a) 27 seconds after the beginning of overflow; (b) 87
seconds after the beginning of overflow.
The impingement position of the overflow jet will
change with various factors such as the shape of
the harbour-side mound and the overflow depth.
The influence of the impingement position on armour stability was examined by changing the
crown width of the harbour-side mound. Figure
6 shows an example of the stability test results. In
this condition, the overflow jet impinged on the
slope section when the number of armour units
150
Figure 7: Influence of harbour-side water level on the
overflow depth at stability limit
When a tsunami overflows the caisson, the discharged water from the rear end of the caisson
accelerates during the freefall above the water
surface, and decelerates under the water surface
due to diffusion. Therefore, the stability of armour
units should decrease as the crown height of the
caisson above the harbour-side water level increases. Also, it should increase as the submerged
depth above the armour units increases. Figure 7
shows a comparison of the stability test results with
two different harbour-side water levels. On the
whole, the results of deep-water cases showed
higher stability than those of shallow-water cases.
Figure 8: Relationship between the nominal diameter
and the overflow depth at the occurrence of damage
by each failure mode
2.5 Failure Modes of Armour Units
Two failure modes for flat-type armour blocks were
observed in the experiments. One was an overturning mode in which armour blocks near the impingement position overturned. The other was a
sliding mode in which all the blocks on the slope
section slid together. Figure 8 shows the relationship between the nominal diameter of the armour
block Dn and the overflow depth h1 on the occurrence of damage. In the cases of overturning
mode, overflow depth at the occurrence of damage was almost proportional to the nominal diameter Dn. On the other hand, in the cases of sliding
mode, it had only small dependence on Dn. These
results suggest that enlargement of the block size
causes an increase in the acting force as much as
the increase in the resistance force with regard to
the sliding mode. For the wave-dissipating blocks,
almost every failure pattern was that of blocks
near the impingement position being displaced
individually.
puting the hydrodynamic force acting on each
armour unit.
3.1 Computation Method
With regard to the numerical computation of
the tsunami overtopping the caisson, Mitsui et al.
(2012) adequately reproduced the laboratory
experiment of an impinging bore-like tsunami by
using the VOF method implemented in the OpenFOAM (OpenCFD Ltd.) CFD model. In the case of
the steady overflow of tsunami, however, computation result using VOF method did not reproduce
well due to the excessive entrainment of air into the
impinging jet. Bricker et al. (2013) pointed out that
this model overestimates the eddy viscosity at the
air-water interface, and that it can be improved
by neglecting all the turbulence in the air phase.
In this study, the overflow jet above the water surface and the flow field on the harbour-side were
solved separately to avoid excessive entrainment
of air. Firstly, the landing position and flow velocity of the overtopped water on the harbour-side
water surface were calculated theoretically using
some empirical formulae. The flow field under the
water surface and the hydrodynamic forces acting on each block were then calculated by using
a CFD model.
3.2 Calculation of the Trajectory of the
Overflow Nappe
This section describes how to calculate the trajectory and velocity of the overflow nappe. The
definition of each symbol is shown in Figure 9. The
overflow discharge per unit width q is calculated
by using the Hom-ma formula [Hom-ma, 1940b]:
(2)
where, h1 is the overflow depth. The application
condition in this formula is h1/Bc < 1/2. The effect of
the approaching velocity u1 can be disregarded
if h1/hd < 0.5 [Hom-ma, 1940a]. The water depth
above the caisson at the rear end of the caisson
h2 and the cross sectional averaged flow velocity
u2 are calculated according to Hom-ma (1940a)
as shown below.
3. NUMERICAL ANALYSIS
As mentioned above, two failure modes were observed in the experiments. The failure mechanism
was investigated in detail by numerical analysis.
First, the computation method of the flow field at
the harbour-side of the breakwater was investigated. The method was validated by comparing the
measured and computed flow field. The stability
of the armour units was then investigated by com-
Figure 9: Definition of the symbols used in the
calculation of the trajectory of the overflow nappe
151
Applying the Bernoulli’s theorem to Section I and II
yields following relation:
3.3 Calculation of the Flow Field under the
Water Surface
(3)
where, z is the height measured from the top of
the caisson, p(z) is the pressure, and u(z) is the flow
velocity. The overflow discharge q is obtained by
integrating the flow velocity u(z) as follows:
(4)
If the pressure distribution p(z) is obtained, h2 can
be calculated using Eq. (2) and Eq. (4). The pressure distributions were assumed as the following
triangle distributions:
(5)
Using Eq. (2), Eq. (4), and Eq. (5), one obtain Eq.
(6):
(11)
The flow field under the water surface on the harbour-side was solved by a single-phase numerical
model. An incompressible flow solver within the
OpenFOAM was used. The governing equations
were the Reynolds-Averaged Navier-Stokes (RANS)
equation and the continuity equation. The Finite
Volume Method with an unstructured grid was
used to reproduce the complicated shape of the
armour blocks. The computational domain shown
in Figure 10 was cross-sectional 2-dimensions, and
the standard grid size was set to 2 mm. In the cases
of computing the hydrodynamic forces acting on
the armour blocks, the grid was subdivided into 3
dimensions. The grid size around the block was set
to about 1 mm so that the block shapes could be
reproduced in detail.
(6)
The relationship between h1 and h2 are solved numerically with Newton’s method as follows:
(7)
In this study, the following relationship was used
considering its suitability to the experimental results:
(8)
The centre of trajectory of the overtopped water
was then obtained under the following assumptions:
• The overtopped water discharges horizontally
from the rear end of the caisson at the flow velocity u2 = q/h2.
• The trajectory of the overflow nappe above the
water surface is a parabola.
The landing position of the overtopped water on
the harbour-side water surface, L0, and the flow
velocity u0x, u0z are calculated as follows:
(9)
(10)
The width of the water jet at the water surface, h0,
is calculated as:
152
Figure 10: Computation method of the flow field under
the water surface
The landing position of the overtopped water L0
and the flow velocity ux0, uz0 at the harbour-side
water surface were given as boundary conditions.
These values were obtained by preliminary calculation as shown in the previous section. The water
surface on the harbour-side was assumed as a
fixed boundary. The rubble mound was modelled
as a porous structure to reproduce the seepage
flow under the caisson. The hydraulic flow resistance R in the porous medium was expressed by a
Dupuit-Forchheimer relationship as shown below:
(12)
where, U is the flow velocity vector, α is the laminar resistance coefficient, and β is the turbulent
resistance coefficient. These coefficients were expressed using the empirical formulae by Engelund
(1953) as follows:
(13)
where, ν is the kinematic viscosity of water, d is the
characteristic diameter of the stone, n is the porosity, and α0 and β0 are the material constants.
The material constants were investigated by the
preliminary experiment. The relationship between
the pressure difference and the discharge of the
seepage was obtained in the experiment, and
the constants were determined as α0 = 2,100 and
β0 = 1.5. The pressure difference due to the water
level difference between the inside and outside of
breakwater was given at both ends of the computational domain.
experimental case was selected where the overflow jet impinged on the shoulder of the mound.
Figure 12 shows the experimental result of this case.
When the overflow depth was 5 cm, the blocks at
the shoulder (block No. 3) were overturned. Figure
13 shows the computed flow field and fluid force
acting on each block.
Figure 12: Experimental results of the armour block B
(m = 254 g)
Figure 11: Comparison of measured and computed
flow field.
h1 = 9 cm
A Reynolds stress model was used as a relatively
high accuracy turbulence model among the
RANS models, since preliminary computation results showed that the degree of diffusion of the
impinging jet was influenced by the turbulence
model. The Reynolds stress model improved the
diffusion of the jet comparing to the result with a
standard k-ε turbulence model. Also there was a
problem that excessive turbulence was generated at the surface of the rubble mound when the
jet flowed along the rubble mound. In this study,
the turbulence inside the rubble mound was set to
zero as a countermeasure for this problem. Figure
11 shows the comparison of the computed flow
field with the measured one in a steady state. The
measured data was obtained by using an electromagnetic current meter, and was averaged for
20 seconds. The computed result with the countermeasures mentioned above adequately reproduced the measured flow field.
3.4 Stability Analysis of Armour Blocks against
Overturning Mode
The stability of the armour blocks was analysed,
based on the fluid force acting on each block. An
Figure 13: Computed flow field and fluid force acting
on each block. h1 = 5 cm
A large force is acting on the block at the shoulder
(block No. 3). The stability of this block was judged
by the balance of moment. In this analysis, only
the fluid force, buoyant force, and self-weight
were considered, but other forces such as the friction force between blocks were disregarded. The
condition of the occurrence of overturning was
expressed as follows:
(14)
where, Fx is the horizontal hydrodynamic force, Fz
is the vertical hydrodynamic force, My is the moment due to the hydrodynamic force, aH and aV
are the arm length, ρw is the density of the water,
and V is the volume of the armour block (see Figure 14). The resistance moment, which is the right
hand side of Eq. (14) was calculated to be 49.0
N•mm in this case. Meanwhile, the acting moment, which is the left hand side of Eq. (14), was
calculated to be 42.4 N•mm when the overflow depth was 4 cm, and 54.8 N•mm when the
153
overflow depth was 5 cm. Thus, this result agreed
with the experimental one. Further validation is required, but this analysis suggested that the failure
of the armor units in the overturning mode could
be explained by the balance of the moment acting on the block.
Figure 14: Definition of the symbols used for the analysis
of the balance of moment of the block
3.5 Stability Analysis of Armour Blocks against
Sliding Mode
The stability of armour blocks against sliding mode
was examined. An example of computed flow field
and hydrodynamic force acting on each block is
shown in Figure 15. A large force in the tangential
direction of the mound is acting on the block No.5
which locates near the impingement position of
the overflow jet. Also, a large force is acting on
the block No. 11 which locates at the toe of the
mound. The stability of the armour blocks was investigated using these forces.
Figure 15: Computed flow field and fluid force acting
on each block. Armour block A (m = 33 g), h1 = 6 cm.
(15)
where, Ni is the hydrodynamic force in the normal
direction acting on i th block, W’ is the underwater
weight of the block. If the block does not uplift, the
difference between the left and right side of Eq.
(15) becomes the reaction force from the mound,
Ri.
A balance of the total tangential force of all the
blocks located below the impingement position
was then considered. The condition where the
blocks slides is expressed as follows:
(16)
where, Ti is the hydrodynamic force in the tangential direction acting on i th block, μi is the friction
coefficient. The resistance due to the interlocking
between the block and stones was included in
the friction force. The friction coefficient of each
block was determined by tuning. The coefficient
of the block at the toe of the mound was set to 0.6
regardless of the block shape as the block at the
toe was placed on the mortal seabed in the experiments. The left and right side of Eq. (16) show
the total sliding force and the total resistance
force, respectively. Computed sliding force and
resistance force of each block are shown in Table
1. The sliding forces are larger than the resistance
forces for the blocks No. 5 to No. 7 which locate
near the impingement position and the block No.
11 at the toe, meanwhile the resistance forces are
larger for the other blocks. Therefore, the stability
against sliding should be judged by the balance
of the total force of the blocks No.5 to No. 10. The
total sliding force was calculated to be 0.952 N
and the total resistance force was calculated to
be 0.753 N, which means sliding occurs in this calculation. Actually, the blocks on the slope section
did not slide in the experiment where the overflow
depth was 6 cm, though the blocks at the toe slid.
When the overflow depth was 8 cm, all the blocks
below the impingement position slid together (Figure 16).
Table 1: Computed sliding force and resistance force
of each block.
Firstly, a balance of force in the normal direction
of the mound was considered for each block. The
condition where the block uplifts is expressed as
follows:
154
Figure 16: Failure situation of the armour block A
(m = 33 g).
h1 = 8 cm
Figure 17 shows a comparison of the computed
and experimented overflow depth at the occurrence of damage. The computed results almost
agreed with the experimental ones. Therefore, this
analysis suggested that the failure of the armour
units in the sliding mode could be explained by
the balance of the total force of the blocks on the
slope section.
tory of the water below the water surface is a
straight line:
(19)
in which u0x, u0z, and L0 are calculated by the method mentioned in the section 3.2. Stability numbers
NS1 and NS2 are functions of B/L and d2/d1, which
are the parameters representing the influence
of the impingement position and the harbor-side
water level respectively. The stability is determined
only by Eq. (17) if B/L is larger than 1.1 since failure
by sliding mode does not occur when the overflow jet impinges on the crown section. Similarly,
the stability of wave-dissipating blocks is also determined only by Eq. (17).
Figure 17: Comparison of the computed
and experimented overflow depth at the occurrence
of damage
4. STABILITY ESTIMATION METHOD
4.1 Derivation of Stability Formulae
Empirical formulae were developed based on
the findings of experiments and numerical analysis mentioned above. In this method, the overflow
depth h1 was used in the formulae to represent the
external force. The overflow depth of the stability
limit corresponding to each failure mode was obtained by two formulae. The final stability limit was
determined by the severer one. The formulae for
the overturning mode and sliding mode are expressed as follows:
Figure 18: Influence of B/L on NS1
Overturning mode :
(17)
Figure 19: Influence of d2/d1 on NS1
Sliding mode :
(18)
where, S is the slope length of the harbour-side rubble mound, NS1 and NS2 are the stability numbers,
B is the crown width of the harbour-side mound,
L is the impingement position of the overflow jet,
d1 is the crown height of the caisson above the
harbour-side water level, and d2 is the submerged
depth above the armour units (regarding the definition of symbols, see Figure 3). The impingement
position L is calculated assuming that the trajec-
For the overturning mode, the overflow depth
h1 represents the acting force on armour units,
whereas the nominal diameter of armour units Dn
represents the resistance force as shown in Eq. (17).
For the sliding mode, on the other hand, the slope
length S is used to represent the resistance force
as shown in Eq. (18). This is because the resistance
force should be represented by the total length of
the blocks on the slope as the whole blocks on the
slope section slide together in the sliding mode. As
a result, the overflow depth of the stability limit in
the sliding mode is not dependent on the block
size as can be seen from Eq. (18), whereas, that
155
in the overturning mode is proportional to the
block size. This corresponds with the experimental
results described above (see Figure 8).
4.2 Determination of Stability Numbers
The stability numbers NS1 and NS2 for each block
were determined through experimental results.
Figure 18 shows the influence of the impingement position by plotting the stability number NS1
against B/L. The conditions of water depth are almost at the same level (d2/d1 = 0.47 to 0.66). The
damage data with sliding mode is excluded in
the figure to reveal the stability limit of overturning
mode. The stability limit is expressed in a single line
as a function of B/L regardless of the mass of the
block. Also, the difference in the stability due to
the impingement position appears clearly. Figure
19 shows the influence of the harbour-side water
depth by plotting the NS1 against d2/d1. The data
on the conditions of B/L > 1.0 is used. The stability
tends to increase as d2/d1 increases.
0.8 and 1.0, the value is obtained by linear interpolation. The stability of the ‘armour block A’ is
higher than that of the ‘armour block B’ for both
failure modes. The stability number for the wavedissipating block is shown in Figure 21. In the case
of the wave-dissipating block, the influence of the
impingement position was different from the case
of the flat-type armour blocks. Namely, the cases
in which the jet impinged on the crown section
showed higher stability than the cases of impingement on the slope section. This result was reflected
in Figure 21.
Figure 21: Stability numbers for wave-dissipating block
4.3 Comparison with Experimental Results
Figure 22 shows a comparison of the estimated
overflow depth of stability limit with the damaged
overflow depth in the experiments. The estimated
results are on the safe side as a whole, and they
show good agreement for both failure modes.
Figure 20: Stability numbers for flat-type armour blocks
Figure 20 shows the stability numbers NS1 and NS2
for flat-type armour blocks determined through
all the test results. Different lines are used according to the B/L in Figure 20(a). When B/L is between
156
Figure 22: Calculated and experimented overflow
depth of the stability limit
5. CONCLUSIONS
To achieve resilient breakwaters against tsunami,
a practical design method for armour units to
cover a rubble mound at the rear side of a caisson breakwater against tsunami overflow was developed based on hydraulic model experiments
and numerical analysis. The features of this are as
follows:
1) The overflow depth is used to represent the external force. This enables the estimation of the
required mass of the armour units to be done
more robustly and easily than in the conventional method based on the flow velocity.
2) Two formulae are used corresponding to the
two failure modes, overturning and sliding.
3) This method takes into account two important
factors for armour stability, namely, influence
of the impingement position of overflow jet
and the influence of the harbour-side water
depth.
The stability numbers NS1 and NS2 for each armour
unit were determined through experiments conducted in a wide range of conditions. The validity
of this method was verified by comparing the estimated results with the experimental ones.
REFERENCES
Bricker, J.D., Takagi, H. and Mitsui, J. (2013): “Turbulence model effects on VOF analysis of Breakwater overtopping during the 2011 Great East
Japan Tsunami”, Proceedings of the 2013 IAHR
World Congress, Chengdu, Sichuan, China, paper
A10153.
Iwasaki, T., Mano, A., Nakamura, T. and Horikoshi,
N. (1984): “Experimental study on hydraulic force
acting on mound and pre-packed caisson of submerged breakwater under steady flows”, Annual
Journal of Coastal Engineering, JSCE, Vol. 31, 527531 (in Japanese).
Kubota, S., Hamaguchi, M., Matsumoto, A., Hanzawa, M. and Yamamoto, M. (2008): “Wave force
and stability of new flat type concrete block with
large openings for submerged breakwaters”,
Proceedings of 31st International Conference on
Coastal Engineering, ASCE, 3423-3435.
Ministry of Land, Infrastructure, Transport and Tourism, Port and Harbor Bureau (2013): “Design guideline of breakwaters against tsunami” (in Japanese).
Mitsui, J., Maruyama, S., Matsumoto, A. and Hanzawa, M. (2012): “Stability of armor units covering
harbor-side rubble mound of composite breakwater against tsunami overflow”, Journal of JSCE, Ser.
B2 (Coastal Engineering), Vol. 68, No. 2, I_881-I_885
(in Japanese).
OpenCFD Ltd.: OpenFOAM User Guide version
2.0.1, http://www.openfoam.com
Van der Meer, J.W. (1988): “Stability of cubes, tetrapods and accropode”, Proceedings of Conf.
Breakwaters ’88, 71-80.
Coastal Engineering Research Center (1977):
“Shore Protection Manual, U.S. Army Corps of Engineers”, U.S. Government Printing Office, Vol. II,
7_213-7_216.
Engelund, F. (1953): “On the laminar and turbulent flows of ground water through homogeneous
sand”, Transactions of the Danish Academy of
Technical Sciences, Vol. 3, No. 4.
Hamaguchi, M., Kubota, S., Matsumoto, A., Hanzawa, M., Yamamoto, M., Moritaka, H. and Shimosako, K. (2007): “Hydraulic stability of new flat
type armor block with very large openings for
use in composite breakwater rubble mound protection”, Proceedings of Coastal Structures 2007,
ASCE, 116-127.
Hom-ma, M. (1940a): “Discharge coefficient on
overflow weir (part-1)”, JSCE Magazine, Civil Engineering, Vol. 26, No. 6, 635-645 (in Japanese).
Hom-ma, M. (1940b): “Discharge coefficient on
overflow weir (part-2)”, JSCE Magazine, Civil Engineering, Vol. 26, No. 9, 849-862 (in Japanese).
157
SUMMARY
This article presents a practical design method of armour units to cover a rubble mound at the rear side
of a caisson breakwater against tsunami overflow. In
this method, the overflow depth of tsunami is used to
represent the external force. This enables the estimation of the required mass of the armour units to be
done more robustly and easily than in the conventional method based on the flow velocity. Hydraulic
model experiments were conducted to investigate
the armour stability. We found two important factors
for armour stability, which were the impingement position of the overflow jet and the harbour-side water
level. These effects were taken into account in the
method. Numerical analysis was also conducted
RÉSUMÉ
Cet article présente une méthode pratique de dimensionnement des éléments de carapace du
parement côté terre d’une digue en caissons pour
résister à la surverse en cas de tsunami. Dans cette
méthode, la hauteur de surverse du tsunami est utilisée pour représenter la force extérieure. Ceci rend
possible une estimation plus robuste et facile de la
masse requise pour les éléments de carapace que
la méthode conventionnelle basée sur les débits
de surverse. Des essais sur modèle hydraulique ont
été menés pour tester la stabilité de la carapace.
Deux facteurs importants pour la stabilité de la carapace ont été mis en évidence, à savoir la position de
l’impact du jet de surverse sur le parement aval et la
hauteur d’eau côté terre. Ces effets ont été pris en
compte dans la méthode de dimensionnement. Des
analyses numériques ont également été menées
ZUSAMMENFASSUNG
Dieser Artikel präsentiert ein praktisches Bemessungsverfahren für Decksteine zur Abdeckung einer
Aufschüttung an der Rückseite eines SenkkastenWellenbrechers als Schutz gegen Tsunamis. In diesem
Verfahren wird die Überströmungstiefe bei einem
Tsunami verwendet, um die äußere Kraft darzustellen. Das ermöglicht eine sicherere und einfachere
Schätzung der benötigten Masse der Decksteine als
im konventionellen Verfahren, das auf der Fließgeschwindigkeit basiert. Um die Stabilität der Decksteine zu untersuchen, wurden hydraulische Modellversuche durchgeführt. Es wurden zwei wichtige
Faktoren für die Stabilität der Decksteine gefunden:
Die Aufprallposition des Überflutungsstrahls und der
hafenseitige Wasserstand. Diese Faktoren wurden im
Verfahren berücksichtigt. Um den Versagensmecha158
to investigate the failure mechanism for the failure
modes of overturning and sliding. The stability of the
armour blocks was evaluated by computing the fluid
force acting on each block. In the case of the overturning mode, the stability is predicted by the balance of the moment of a block. In the case of the
sliding mode, it is necessary to consider the balance
of forces of all the blocks on the slope. Empirical formulae for the stability estimation were then derived
based on the findings from experiments and numerical analysis. The overflow depth of the stability limit
corresponding to each failure mode was obtained
by two formulae. The stability numbers for each armour unit were determined through the experiments.
The estimated results by this method agreed well with
the experimental ones.
pour tester les mécanismes de rupture pour la défaillance par renversement des blocs et pour la défaillance par glissement des blocs. La stabilité des
blocs de carapace a été évaluée par le calcul des
forces hydrauliques agissant sur chaque bloc. Dans
le cas d’une rupture par renversement, la stabilité est
évaluée par le calcul de l’équilibre du moment d’un
bloc. Dans le cas d’une rupture par glissement, il est
nécessaire de considérer l’équilibre des forces sur
chaque bloc de la pente. Des formules empiriques
pour estimer la stabilité ont ensuite été déduites des
résultats de ces essais et simulations numériques. La
hauteur de surverse limite de stabilité correspondant à chaque mode de rupture est obtenue par
deux formules. Les facteurs de stabilité de chaque
type d’élément de carapace ont été déterminés au
grâce aux essais. Les résultats obtenus par cette méthode de dimensionnement concordent bien avec
les résultats expérimentaux.
nismus für die Versagensarten Kippen und Rutschen
zu untersuchen, wurden numerische Analyse durchgeführt. Die Stabilität der Decksteine wurde mittels
Berechnung der auf jeden Block einwirkenden hydraulischen Kraft untersucht. Für die Versagensart Kippen wird die Stabilität über die Impulsbilanz für einen
Block vorhergesagt. Für die Versagensart Rutschen ist
es erforderlich, das Kräftegleichgewicht aller Blöcke
auf dem Hang zu berücksichtigen. Basierend auf den
Ergebnissen der experimentellen und numerischen
Analysen wurden dann empirische Formeln für die
Stabilitätsberechnung entwickelt. Die kritische Überströmungstiefe wurde für jede Versagensart mit zwei
Formeln ermittelt. Die Stabilitätswerte für jeden Deckstein wurden in den Experimenten ermittelt. Die mit
Hilfe des vorgestellten Verfahrens geschätzten Ergebnisse stimmten gut mit denen aus den Experimenten
überein.
RESÚMEN
Este artículo presenta un método práctico para el
diseño de mantos de protección en la cara interior de diques de cajones que protegen frente a rebases derivados de tsunamis. En esta metodología
la altura del rebase se usa para representar una
fuerza externa. Esto permite que la estimación del
peso de las piezas de protección se realice de una
forma más robusta y sencilla que mediante el método tradicional basado en la velocidad de flujo.
Se han desarrollado modelos físicos para analizar
el comportamiento hidráulico del manto. En ellos
se han encontrado dos factores de importancia
para su estabilidad, como son la dirección del flujo del rebase y el nivel de agua en la cara interior
de la estructura. Ambos efectos se han tenido en
cuanta en la metodología planteada. También
se han llevado a cabo modelos numéricos para
evaluar el comportamiento frente a los modos de
fallo de vuelco y deslizamiento. La estabilidad de
las piezas del manto se ha evaluado modelizando
computacionalmente las fuerzas con que el flujo de agua afecta a cada pieza. En el caso del
modo de fallo frente a vuelco, la estabilidad se
evalúa a través del equilibrio de momentos que
afecta al bloque. En el caso del deslizamiento, es
necesario considerar el equilibrio de fuerzas sobre
cada bloque del manto. A continuación, se plantean fórmulas empíricas para la estimación de las
condiciones de estabilidad, basadas en las conclusiones obtenidas a partir de los modelos físicos
y numéricos llevados a cabo. La altura de rebase
que provoca unas condiciones de equilibrio límite
para cada uno de los modos de fallo considerados se obtiene a través de dos formulaciones. Los
resultados del método propuesto se ajustan de
manera satisfactoria a los obtenidos a través de
modelos experimentales.
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