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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 972650, 15 pages
http://dx.doi.org/10.1155/2012/972650
Research Article

Characteristics of Wave Reflection for Vertical and Slit Caissons with Porous Structures

1Department of Civil Engineering, Hanbat National University, San 16-1, Duckmyoung-dong, Daejeon 305-719, Republic of Korea
2Department of Civil and Environmental Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Republic of Korea

Received 3 January 2012; Accepted 16 March 2012

Academic Editor: M. F. El-Amin

Copyright © 2012 Tae-Hwa Jung et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Offshore structures are occasionally located at a relatively deep water region, the outside of breakwater. In this case, these structures may be damaged by the supposition of incident and reflected waves from a vertical breakwater. To prevent the damage, the reflected waves are controlled by installing porous structures at the face of the vertical breakwater. In this study, numerical experiments are carried out to identify the characteristics of wave reflection from the porous structures installing in front of a vertical or slit caisson.

1. Introduction

Offshore structures such as oil buoys or floating structures are sometimes required to be located outside of a breakwater. These structures may be damaged by the reflected waves from a vertical breakwater as well as incident waves. Thus, it is important to reduce the energy of reflected waves from a vertical breakwater to secure the stability of offshore structures. One possible way to reduce reflected waves is dissipating wave energy by adding porous structures at the face of a vertical breakwater.

Few studies regarding numerical experiments have been paying attention to the wave reflection from a vertical breakwater with a frontally porous structure because of complexities of interaction between porous flow and nonlinear waves. The mechanism of wave energy dissipation and reflection due to a porous structure is not yet clearly understood. Therefore, the energy dissipation efficiency of a permeable structure was assessed by measuring the wave reflection from the breakwater. Sollit and Cross [1] performed the study on wave reflection and transmission through a porous structure using the linear wave theory and the linearized friction equation for flows in porous media. Following Sollit and Cross [1], several researches were carried out on wave and porous structure interaction [2, 3]. However, these studies were less practical because they assumed linear wave, constant depth, or integrated over the depth. The applicability of numerical study on water waves and porous structures was much improved by adopting the Navier-Stokes equation [4].

In this study, a numerical experiment was carried out to investigate the characteristics of a reflected wave from a porous structure located in front of a slit caisson. For numerical experiment, CADMAS-SURF [5, 6] was used. The irregular wave based on Bretschneider-Mitsuyasu’s frequency spectrum was used, and the three-point method was used to decompose the incident and reflected waves.

2. Numerical Model

2.1. Governing Equations

CADMAS-SURF based on the Navier-Stokes equations is composed of the continuity, momentum, VOF, and turbulence equations.

Continuity equation:
𝜕𝛾𝑥𝑢+𝜕𝑥𝜕𝛾𝑧𝑤𝜕𝑧=𝑆𝜌.(2.1)

Momentum equation:
𝜆𝑣𝜕𝑢+𝜕𝑡𝜕𝜆𝑥𝑢𝑢+𝜕𝑥𝜕𝜆𝑧𝑤𝑢𝛾𝜕𝑧=𝑣𝜌𝜕𝑝+𝜕𝜕𝑥𝛾𝜕𝑥𝑥𝑣𝑒2𝜕𝑢+𝜕𝜕𝑥𝛾𝜕𝑧𝑧𝑣𝑒𝜕𝑢+𝜕𝑧𝜕𝑤𝜕𝑥𝐷𝑥𝑢+𝑆𝑣𝑅𝑥,𝜆𝑣𝜕𝑤+𝜕𝑡𝜕𝜆𝑥𝑢𝑤+𝜕𝑥𝜕𝜆𝑧𝑤𝑤𝛾𝜕𝑧=𝑣𝜌𝜕𝑝+𝜕𝜕𝑥𝛾𝜕𝑥𝑥𝑣𝑒𝜕𝑤+𝜕𝑥𝜕𝑢+𝜕𝜕𝑧𝛾𝜕𝑧𝑧𝑣𝑒2𝜕𝑤𝜕𝑧𝐷𝑧𝑤+𝑆𝑤𝑅𝑧𝛾𝑣𝑞,(2.2) where 𝑡 is the time, 𝑥 and 𝑧 are the horizontal and vertical coordinates, 𝑢,𝑤 are the horizontal and vertical velocity components, 𝜌 is the density of fluid, 𝑝 is the pressure, 𝑣𝑒 is the summation of molecular kinematic viscosity and eddy kinematic viscosity, 𝑔 is the gravitational acceleration, 𝛾𝑣 is the volume porosity (fraction of the volume of voids over the total volume), 𝛾𝑥  and𝛾𝑧 are the surface porosity (fraction of the area of voids over the total area) components in the 𝑥 and 𝑧 projections, 𝑆𝜌,𝑆𝑢, and 𝑆𝑤 are wave generation source, and 𝐷𝑥 and 𝐷𝑧 are the coefficients for sponge layer.

Using the inertia coefficient 𝐶𝑚, and 𝛾𝑣,𝛾𝑥, and 𝛾𝑧 the inertia force effects from structure, 𝜆𝑣,𝜆𝑥, and 𝜆𝑧 are written as follows: 𝜆𝑣=𝛾𝑣+1𝛾𝑣𝐶𝑚,𝜆𝑥=𝛾𝑥+1𝛾𝑥𝐶𝑚,𝜆𝑧=𝛾𝑧+1𝛾𝑧𝐶𝑚.(2.3) With the drag coefficient, 𝐶𝐷, the resistance force due to porous media 𝑅𝑥 and 𝑅𝑧 are defined by the following equations: 𝑅𝑥=12𝐶𝐷Δ𝑥1𝛾𝑥𝑢𝑢2+𝑤2,𝑅𝑧=12𝐶𝐷Δ𝑧1𝛾𝑧𝑤𝑢2+𝑤2.(2.4)

Free Surface Equation (VOF)
The VOF method was used to represent the interface boundary between the water and air, known as free surface [7]. The method introduces a volume of fluid function 𝐹(𝑥,𝑧,𝑡) to define the fluid region. The physical meaning of 𝐹 is the fractional volume of a cell occupied by water. A unit value of 𝐹 corresponds to a cell full of water, and a zero value indicates that the cell contains no water. Cells with 𝐹 value between zero and one must then contain a free surface. The advection of free surface is represented by a convective equation of 𝐹 extended for porous media as follows: 𝛾𝑣𝜕𝐹+𝜕𝑡𝜕𝛾𝑥𝑢𝐹+𝜕𝑥𝜕𝛾𝑧𝑤𝐹𝜕𝑧=𝑆𝐹,(2.5) where 𝑆𝐹 is a source term for the wave generation source method.

Turbulence Model
In the turbulent model, a closure 𝑘-𝜀 turbulence model is adopted. In the closure 𝑘-𝜀 turbulence model, the turbulent kinetic energy 𝑘 and the rate of dissipation of turbulent kinetic energy 𝜀 are defined in (2.6) and (2.7) using the amount of fluctuation 𝑢;𝑤 and derived from the advection-diffusion equation shown in (2.8) and (2.9): 1𝑘=2𝑢2+𝑤2,2(2.6)𝜀=𝜈𝜕𝑢𝜕𝑥2+2𝜕𝑤𝜕𝑧2+𝜕𝑤+𝜕𝑥𝜕𝑢𝜕𝑧2𝛾,(2.7)𝑣𝜕𝑘+𝜕𝑡𝜕𝛾𝑥𝑢𝑘+𝜕𝑥𝜕𝛾𝑧𝑤𝑘=𝜕𝜕𝑧𝛾𝜕𝑥𝑥𝑣𝑘𝜕𝑘+𝜕𝜕𝑥𝛾𝜕𝑧𝑧𝑣𝑘𝜕𝑘𝜕𝑧+𝛾𝑣𝐺𝑠𝛾𝑣𝛾𝜀,(2.8)𝑣𝜕𝜀+𝜕𝑡𝜕𝛾𝑥𝑢𝜀+𝜕𝑥𝜕𝛾𝑧𝑤𝜀=𝜕𝜕𝑧𝛾𝜕𝑥𝑥𝑣𝜀𝜕𝜀+𝜕𝜕𝑥𝛾𝜕𝑧𝑧𝑣𝜀𝜕𝜀𝜕𝑧+𝛾𝑣𝐶1𝜀𝑘𝐺𝑠𝛾𝑣𝐶2𝜀2𝑘,𝐺(2.9)𝑠=𝜈𝑡2𝜕𝑢𝜕𝑥2+2𝜕𝑤𝜕𝑧2+𝜕𝑤+𝜕𝑥𝜕𝑢𝜕𝑧2,(2.10) where the eddy kinematic viscosity 𝜈𝑡 and the diffusion coefficient, and 𝜈𝑘 and 𝜈𝜀 are described as follows: 𝜈𝑡=𝐶𝜇𝑘2𝜀,𝜈𝑘𝜈=𝜈+𝑡𝜎𝑘,𝜈𝜀𝜈=𝜈+𝑡𝜎𝜀(2.11)

2.2. Boundary Conditions

To treat outgoing waves effectively, two boundary conditions are employed.

Radiation Boundary Condition
Sommerfeld’s open boundary condition is employed as follows: 𝜕𝑓𝜕𝑡+𝐶𝜕𝑓𝜕𝑥=0,(2.12) where 𝑓 is the wave property such as a mean velocity or mean free surface displacement, and 𝐶 is the phase velocity. In the case of a regular wave, it is easy to apply because the phase velocity 𝐶 is known in advance. In the case of irregular waves, however, it is difficult to satisfy a nonreflection condition because the phase velocity 𝐶 is not clear.

Absorbing Boundary Condition
Because it is difficult to satisfy Sommerfeld’s open boundary condition, sponge layer technique, which gradually attenuates wave energy, is used. The attenuation effect is represented by 𝐷𝑥𝑢 and 𝐷𝑧𝑢 in the momentum equations of (2.2). Coefficients 𝐷𝑥 and 𝐷𝑧 are defined as follows: 𝐷𝑥=𝜃𝑥𝑔(𝑁+1)𝑥𝑥0𝑙𝑁,𝐷𝑧=𝜃𝑧𝑞(𝑁+1)𝑥𝑥0𝑙𝑁,(2.13) where is the water depth, 𝑙 and 𝑥0 are the width of a sponge layer and starting point, 𝑁 is the order of the distribution function, and 𝜃𝑥 and 𝜃𝑧 are the nondimensional coefficients.

3. Numerical Experiments

3.1. Numerical Conditions

The numerical wave flume used in this study is shown in Figure 1. The length of wave flume changed according to the width of the sponge layer, equivalent to 2𝐿, where 𝐿 is the incident wave length. The water depth was set to 0.5m. In the sponge layer, the wave amplitude decreased exponentially, thus it became small enough to be applied Sommerfeld’s radiation condition at the both ends of wave flume. To generate expected target waves unaffected by the sponge layer, the waves are generated inside the computational domain (internal wave generation technique [8]). To analyze wave transformation, three wave gauges were placed as shown in Figure 2. The distances from the gauges to the porous structure were 1𝐿, 1𝐿+0.2m, and 1𝐿+0.56m. That is, the distances between the wave gauges were 0.2m and 0.36m. Four combination of the porous structure and the caisson were considered: (1) the vertical caisson only; (2) the vertical caisson and the porous structure; (3) the slit caisson only; (4) the slit caisson and the porous structure. When it comes to the shape of the porous structure, the rectangular, triangular, and trapezoidal shapes were used. The heights of the porous structure were set to 0.4 and 0.8. The crown height 𝑐 of two kinds of vertical breakwater was set to 1.25𝐻1/3. Since the ratio of distance of wave chamber to wave length 𝐵/𝐿=0.25 is known that it gave the minimum wave reflection [9], a constant value of 0.25 was applied to the wave chamber. In which, the wave chamber means the space between a slit and a vertical wall in the caisson. The significant wave periods of the incident waves were 1.5, 2.0, and 2.5sec. The significant wave heights of the incident waves were 2.0, 3.0, and 4.0cm. These conditions are summarized in Table 1.

tab1
Table 1: Numerical conditions.
fig1
Figure 1: Schematic diagram of the numerical wave flume.
fig2
Figure 2: Reflection coefficients of the regular and irregular waves.
3.2. Incident Wave

One regular wave and nine irregular waves were used as an incident wave condition. The irregular waves were generated by using following equations: 𝜂(𝑡)=𝑀𝑚=1𝑎𝑚cos2𝜋𝑓𝑚𝑡𝜀𝑚,𝑢(𝑧,𝑡)=𝑀𝑚=12𝜋𝑓𝑚cosh𝑘𝑚(+𝑧)sinh𝑘𝑚𝑎𝑚cos2𝜋𝑓𝑚𝑡𝜀𝑚,𝑤(𝑧,𝑡)=𝑀𝑚=12𝜋𝑓𝑚sinh𝑘𝑚(+𝑧)sinh𝑘𝑚𝑎𝑚sin2𝜋𝑓𝑚𝑡𝜀𝑚,(3.1) where 𝑀 is a number of frequency, subscript 𝑚 is 𝑚th component, 𝑓𝑚 is the corresponding frequency, 𝜀𝑚 is the corresponding random phase angle, 𝑘𝑚 is the corresponding wavenumber, and 𝑎𝑚 is the corresponding wave amplitude obtained from (3.2): 𝑎𝑚=2𝑆(𝑓)Δ𝑓,(3.2) where Δ𝑓 is the frequency interval, and 𝑆(𝑓) is the Bretschneider-Mitsuyasus frequency spectrum given by: 𝑆(𝑓)=0.257𝐻21/3𝑇1/3𝑇1/3𝑓5𝑇exp1.031/3𝑓4,(3.3) where 𝑓 is wave frequency, and 𝐻1/3 and 𝑇1/3 are the significant wave height and period, respectively.

3.3. Wave Decomposition

The three-point method suggested by Suh et al. [10] was used to decompose the waves into the incident and reflected waves. A brief description of this method is as follows. Incident and reflected waves can be given by 𝜂𝑖(𝐻𝑡)=𝑖2𝑘cos𝑖𝑥𝜔𝑡+𝜙𝑖,𝜂𝑟𝐻(𝑡)=𝑟2𝑘cos𝑟𝑥+𝜔𝑡+𝜙𝑟,(3.4) where 𝐻 is the wave height, 𝑘 is the wavenumber, 𝜙 is the random phase angle, and 𝜔 is the angular frequency. The surface elevation at the 𝑛-point can be expressed as follows: 𝜂𝑛(𝑡)=𝜂𝑖(𝑡)+𝜂𝑟(𝐻𝑡)=𝑖2𝑘cos𝑖𝑥𝑛𝜔𝑡+𝜙𝑖+𝐻𝑟2𝑘cos𝑟𝑥𝑛+𝜔𝑡+𝜙𝑟+e𝑛(𝑡)(3.5) or 𝜂𝑛(𝑡)=𝑋1cos𝜔𝑡𝑘𝑖𝑥𝑛+𝑋2cos𝜔𝑡𝑘𝑟𝑥𝑛+𝑋3sin𝜔𝑡𝑘𝑖𝑥𝑛𝑋4sin𝜔𝑡𝑘𝑟𝑥𝑛+𝑒𝑛(𝑡),(3.6) where 𝑥𝑛 is the distance from the first measuring point to the 𝑛th location, 𝑒𝑛 is error due to signal noise, 𝑋𝑖(𝑖=14) are unknown coefficients expressed in terms of the height and phase of the incident and reflected waves. The squared error is 𝜀2=𝑁𝑛=1𝑇0𝑒𝑛(𝑡)2𝑑𝑡(3.7) and, the unknowns in (3.6) can be determined by using the least-square method: 𝜕𝜀2𝜕𝑋𝑗=0,𝑗=1,2,3,4.(3.8)

Finally, the incident and reflected wave heights can be calculated using following relation: 𝐻𝑖=2𝑋1+𝑋3cos𝜙𝑖+sin𝜙𝑖,𝐻𝑟=2𝑋2+𝑋4cos𝜙𝑟+sin𝜙𝑟.(3.9) Since the energies of the incident and reflected waves are proportional to the squares of their heights, the reflection coefficient can be estimated from 𝑅=𝐸𝑟𝐸𝑖,(3.10) where 𝐸𝑖 and 𝐸𝑟 are the energies of the incident and reflected waves, respectively.

4. Results and Discussion

Figure 2 showed the reflection coefficients of the regular and irregular waves for the vertical and slit caissons, respectively. The reflection coefficient decreased as the height of the porous structure increased when the porous structure was installed in front of the vertical caisson or the slit caisson. In the case of the regular wave, the reflection coefficient was significantly reduced, whereas the reflection of the irregular waves was slightly reduced. This may be due to fixed width of wave chamber. That is, the width of wave chamber using in the present study referred to the previous study for regular waves. Therefore, it may not be effective for irregular waves because there are many regular components, and some of them are less affected by the width of wave chamber.

Figures 3 and 4 showed the reflection coefficients for different significant wave heights. It was observed that the reflection coefficient decreased as the height of porous structure increased when the porous structure was located in front of the vertical caisson or the slit caisson. As the height of the significant waves increased, the reflection coefficient decreased. The waves at the slit caisson were more dissipated than those at the vertical caisson.

fig3
Figure 3: Reflection coefficient for vertical caisson with different significant wave heights.
fig4
Figure 4: Reflection coefficient for slit caisson with different significant wave heights.

Figures 5 and 6 showed the reflection coefficients for different significant wave periods. As the height of a porous structure increased, the reflection coefficient decreased when the porous structure was located in front of the caisson. The reflection coefficients for a slit caisson were much smaller than those for a vertical caisson. However, the significant wave period rarely affected to the wave reflection. That is, the variation of reflection coefficient due to significant wave period was very small compared to other parameters.

fig5
Figure 5: Reflection coefficient for vertical caisson with different significant wave periods.
fig6
Figure 6: Reflection coefficient for slit caisson with different significant wave periods.

Figures 7 and 8 showed the reflection coefficient for different shapes of porous structure. As the height of porous structure increased, the reflection coefficient decreased. As for the estimated wave coefficients based on the shape of the porous structure, the rectangular and trapezoidal porous structure showed obvious energy dissipation. On the other hand, the triangular porous structure showed little reduction effect on the slit caisson and almost no reduction on the vertical caisson.

fig7
Figure 7: Reflection coefficient for vertical caisson with different shapes of porous structure.
fig8
Figure 8: Reflection coefficient for slit caisson with different shapes of porous structure.

5. Concluding Remarks

In this study, the wave reflection of a vertical and slit caissons with porous structures was analyzed using the number model based on the Navier-Stokes equations. Both of regular and irregular waves were used as incident waves. In the case of regular waves, the reflection coefficient was significantly reduced, whereas the reflection coefficient for irregular waves reduced by a relatively small amount. As the wave height increased, the reflection coefficient decreased for both vertical and slit caissons. The waves were more dissipated at the slit caisson than the vertical caisson. The reflection coefficient was rarely affected by the variation of significant wave period. The rectangular and trapezoidal porous structures showed obvious energy dissipation, whereas the triangular porous structure showed a little reduction effect on the slit caisson and almost no reduction on the vertical caisson. Because porous structure with low height is not able to dissipate wave energy effectively, a proper height is required for efficiency. Although rectangular and trapezoidal porous structures showed almost same energy dissipation, the trapezoidal structure is more preferred because it has superiority in the workability and stability.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (no. 2010-0022337).

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