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

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:
𝜕𝛾_𝑥𝑢𝜕𝑥+𝜕𝛾^𝑧𝑤𝜕𝑧=𝑆^𝜌.(2.1)

Momentum equation:
𝜆_𝑣𝜕𝑢𝜕𝑡+𝜕𝜆^𝑥𝑢𝑢𝜕𝑥+𝜕𝜆^𝑧𝑤𝑢𝜕𝑧=−𝛾𝜌^𝑣𝜕𝜕𝑥𝑝+𝜕𝜕𝑥𝛾^𝑥𝑣_𝑒2𝜕𝑢𝜕𝑥+𝜕𝜕𝑧𝛾^𝑧𝑣_𝑒𝜕𝑢−𝐷_𝑥𝜕𝑧+𝜕𝜕𝑤𝑥𝜆_𝑣𝑢+𝑆_𝑣−𝑅_𝑥,𝜕𝑤𝜕𝑡+𝜕𝜆^𝑥𝑢𝑤𝜕𝑥+𝜕𝜆^𝑧𝑤𝑤𝜕𝑧=−𝛾𝜌^𝑣𝜕𝜕𝑥𝑝+𝜕𝜕𝑥𝛾^𝑥𝑣_𝑒𝜕𝑤𝜕𝑥+𝜕𝜕𝑧𝑢+𝜕𝜕𝑧𝛾^𝑧𝑣_𝑒2𝜕𝑤−𝐷_𝑧𝜕𝑧𝑤+𝑆_𝑤−𝑅_𝑧−𝛾_𝑣𝑞,(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, 𝑣_𝑒 is the summation of molecular kinematic viscosity and eddy kinematic viscosity, 𝑔 is the gravitational acceleration, 𝛾_𝑣 is the volume porosity (fraction of the volume of voids over the total volume), 𝛾_𝑥  and𝛾_𝑧 are the surface porosity (fraction of the area of voids over the total area) components in the 𝑥 and 𝑧 projections, 𝑆_𝜌,𝑆_𝑢, and 𝑆_𝑤 are wave generation source, and 𝐷_𝑥 and 𝐷_𝑧 are the coefficients for sponge layer.

Using the inertia coefficient 𝐶_𝑚, and 𝛾_𝑣,𝛾_𝑥, and 𝛾_𝑧 the inertia force effects from structure, 𝜆_𝑣,𝜆_𝑥, and 𝜆_𝑧 are written as follows: 𝜆𝜆_𝑣𝑥=𝛾_𝑣+1−𝛾_𝑣𝐶_𝑚,𝜆_𝑧=𝛾_𝑥+1−𝛾_𝑥𝐶_𝑚,=𝛾_𝑧+1−𝛾_𝑧𝐶_𝑚.(2.3) With the drag coefficient, 𝐶_𝐷, the resistance force due to porous media 𝑅_𝑥 and 𝑅_𝑧 are defined by the following equations: 𝑅_𝑥=12𝐶Δ^𝐷𝑥1−𝛾^𝑥𝑢√𝑅_𝑧𝑢²+𝑤²,=12𝐶Δ^𝐷𝑧1−𝛾^𝑧𝑤√𝑢²+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: 𝛾_𝑣𝜕𝐹𝜕𝑡+𝜕𝛾^𝑥𝑢𝐹𝜕𝑥+𝜕𝛾^𝑧𝑤𝐹𝜕𝑧=𝑆^𝐹,(2.5) where 𝑆_𝐹 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): 𝑘=12𝑢^2𝜀=𝜈2𝜕𝑢′+𝑤^2,(2.6)𝜕𝑥²+2𝜕𝑤′𝜕𝑧²+𝜕𝑤′𝛾𝜕𝑥+𝜕𝜕𝑢𝑧′²,(2.7)_𝑣𝜕𝑘𝜕𝑡+𝜕𝛾^𝑥𝑢𝑘𝜕𝑥+𝜕𝛾^𝑧𝑤𝑘𝜕𝑧=𝜕𝜕𝑥𝛾^𝑥𝑣_𝑘𝜕𝑘𝜕𝑥+𝜕𝜕𝑧𝛾^𝑧𝑣_𝑘𝜕𝑘𝜕𝑧+𝛾^𝑣𝛾𝐺_𝑠−𝛾_𝑣𝜀,(2.8)_𝑣𝜕𝜀𝜕𝑡+𝜕𝛾^𝑥𝑢𝜀𝜕𝑥+𝜕𝛾^𝑧𝑤𝜀𝜕𝑧=𝜕𝜕𝑥𝛾^𝑥𝑣_𝜀𝜕𝜀𝜕𝑥+𝜕𝜕𝑧𝛾^𝑧𝑣_𝜀𝜕𝜀𝜕𝑧+𝛾^𝑣𝐶₁𝜀𝑘𝐺^𝑠−𝛾_𝑣𝐶₂𝜀²𝐺𝑘,(2.9)_𝑠=𝜈_𝑡2𝜕𝜕𝑥𝑢²+2𝜕𝑤𝜕𝑧²+𝜕𝑤𝜕𝑥+𝜕𝜕𝑧𝑢²,(2.10) where the eddy kinematic viscosity 𝜈_𝑡 and the diffusion coefficient, and 𝜈_𝑘 and 𝜈_𝜀 are described as follows: 𝜈_𝑡=𝐶^𝜇𝑘²𝜈_𝑘𝜀,=𝜈+𝜈^𝑡𝜎_𝑘𝜈_𝜀,=𝜈+𝜈^𝑡𝜎_𝜀(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𝜕𝜕𝑓𝑥=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 −𝐷_𝑥𝑢 and −𝐷_𝑧𝑢 in the momentum equations of (2.2). Coefficients 𝐷_𝑥 and 𝐷_𝑧 are defined as follows: 𝐷_𝑥=𝜃_𝑥𝑔ℎ(𝑁+1)𝑥−𝑙𝑥⁰^𝑁𝐷_𝑧,=𝜃_𝑧𝑞ℎ(𝑁+1)𝑥−𝑙𝑥⁰^𝑁,(2.13) where ℎ 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 𝜃_𝑥 and 𝜃_𝑧 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 ℎ 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 0.4ℎ and 0.8ℎ. The crown height ℎ_𝑐 of two kinds of vertical breakwater was set to 1.25𝐻_1/3. 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.

Table 1 

Numerical conditions.

(a) Combination of the vertical caisson and the porous structure

(b) Combination of the slit caisson and the porous structure

Figure 1 

Schematic diagram of the numerical wave flume.

(a)

(b)

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)=^𝑀_𝑚₌₁𝑢(𝑧,𝑡)=_𝑀𝑎_𝑚cos2𝜋𝑓_𝑚𝑡−𝜀_𝑚,_𝑚₌₁2𝜋𝑓_𝑚cosh𝑘_𝑚(ℎ+𝑧)sinh𝑘_𝑚ℎ𝑎^𝑚𝑤(𝑧,𝑡)=_𝑀cos2𝜋𝑓_𝑚𝑡−𝜀_𝑚,_𝑚₌₁2𝜋𝑓_𝑚sinh𝑘_𝑚(ℎ+𝑧)sinh𝑘_𝑚ℎ𝑎^𝑚sin2𝜋𝑓_𝑚𝑡−𝜀_𝑚,(3.1) where 𝑀 is a number of frequency, subscript 𝑚 is 𝑚th component, 𝑓_𝑚 is the corresponding frequency, 𝜀_𝑚 is the corresponding random phase angle, 𝑘_𝑚 is the corresponding wavenumber, and 𝑎_𝑚 is the corresponding wave amplitude obtained from (3.2): 𝑎_𝑚=√2𝑆(𝑓)Δ𝑓,(3.2) where Δ𝑓 is the frequency interval, and 𝑆(𝑓) is the Bretschneider-Mitsuyasus frequency spectrum given by: 𝑆(𝑓)=0.257𝐻^21/3𝑇_1/3𝑇_1/3𝑓⁻⁵exp−1.03𝑇_1/3𝑓⁻⁴,(3.3) where 𝑓 is wave frequency, and 𝐻_1/3 and 𝑇_1/3 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 𝜂_𝑖(𝑡)=𝐻^𝑖2cos𝑘^𝑖𝜂_𝑟𝑥−𝜔𝑡+𝜙_𝑖,(𝑡)=𝐻^𝑟2cos𝑘^𝑟𝑥+𝜔𝑡+𝜙_𝑟,(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: 𝜂_𝑛(𝑡)=𝜂_𝑖(𝑡)+𝜂_𝑟(𝑡)=𝐻^𝑖2cos𝑘^𝑖𝑥_𝑛−𝜔𝑡+𝜙_𝑖+𝐻^𝑟2cos𝑘^𝑟𝑥_𝑛+𝜔𝑡+𝜙_𝑟+e_𝑛(𝑡)(3.5) or 𝜂_𝑛(𝑡)=𝑋+₁𝑋co₃s𝜔𝑡−𝑘_𝑖𝑥_𝑛+𝑋₂cos𝜔𝑡−𝑘_𝑟𝑥_𝑛sin𝜔𝑡−𝑘_𝑖𝑥_𝑛−𝑋₄sin𝜔𝑡−𝑘_𝑟𝑥_𝑛+𝑒_𝑛(𝑡),(3.6) where 𝑥_𝑛 is the distance from the first measuring point to the 𝑛th location, 𝑒_𝑛 is error due to signal noise, 𝑋_𝑖(𝑖=1∼4) are unknown coefficients expressed in terms of the height and phase of the incident and reflected waves. The squared error is 𝜀²=^𝑁_𝑛=1^𝑇0𝑒_𝑛(𝑡)²𝑑𝑡(3.7) and, the unknowns in (3.6) can be determined by using the least-square method: 𝜕𝜀²𝜕𝑋_𝑗=0,𝑗=1,2,3,4.(3.8)

Finally, the incident and reflected wave heights can be calculated using following relation: 𝐻_𝑖=2𝑋¹+𝑋₃cos𝜙_𝑖+sin𝜙_𝑖,𝐻_𝑟=2𝑋²+𝑋₄cos𝜙_𝑟+sin𝜙_𝑟.(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=𝐸_𝑟𝐸_𝑖,(3.10) where 𝐸_𝑖 and 𝐸_𝑟 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.

(a)

(b)

(c)

Figure 3 

Reflection coefficient for vertical caisson with different significant wave heights.

(a) 𝑇1/3=1.5 sec

(b) 𝑇1/3=2.0 sec

(c) 𝑇1/3=2.5 sec

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.

(a) 2 cm

(b) 3 cm

(c) 4 cm

Figure 5 

Reflection coefficient for vertical caisson with different significant wave periods.

(a)

(b)

(c)

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.

(a) 𝑇1/3=1.5 sec

(b) 𝑇1/3=2.0 sec

(c) 𝑇1/3=2.5 sec

Figure 7 

Reflection coefficient for vertical caisson with different shapes of porous structure.

(a) 𝑇1/3=1.5 sec

(b) 𝑇1/3=2.0 sec

(c) 𝑇1/3=2.5 sec

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).