Mohammed Fazlul Karim, Katsutoshi Tanimoto and Phung Dang Hieu. Department of Civil & Environmental Engineering, Saitama University. Saitama, Japan.
Proceedings of The Thirteenth (2003) International Offshore and Polar Engineering Conference Honolulu, Hawaii, USA, May 25 –30, 2003 Copyright © 2003 by The International Society of Offshore and Polar Engineers ISBN 1 –880653 -60 –5 (Set); ISSN 1098 –6189 (Set)
Simulation of Wave Transformation in Vertical Permeable Structure Mohammed Fazlul Karim, Katsutoshi Tanimoto and Phung Dang Hieu Department of Civil & Environmental Engineering, Saitama University Saitama, Japan
ABSTRACT
limitations. Small-scale physical modeling is influenced by scale effect while large-scale modeling is relatively expensive. Moreover, the measurement inside the porous structure itself is highly challenging. Therefore, numerical models are the valuable design tools for studying wave motion in permeable structure.
Wave motion in porous structures and hydraulic performances of vertical permeable walls are investigated. A numerical hydrodynamic model for 2-D wave field is developed to simulate the wave transformation inside and outside the structure. Governing equations of porous flow are solved using SMAC algorithm for the time evolution of the velocity and pressure field. A two-phase model is incorporated to treat the prime variables at the air water interface. Experimental studies are done to calibrate the numerical model. The drag and inertia coefficients in the governing equations are calibrated for a typical porous material. The model is applied to assess the effect of structure width and porosity on the hydraulic performances for a vertical permeable structure. It is observed that there exist optimum values of width and porosity that can maximize hydraulic performances.
In the recent years, with the development of computational fluid dynamics and computing facilities, several numerical models are developed for the porous flow simulation (Isobe et al, 2000; Liu et al, 1999; Qiu and Wang, 1996; Sakakiyama and Kajima, 1992). Most of these models are developed based on the direct simulation of the Navier-Stokes equations in which volume of fluid (VOF) method is used for fluid advection algorithm. However, all of these studies are based on the one phase model, in which the effect of air movement above the water surface is ignored. As a result, trapped air inside the water and splash of water in the air are not treated fully. In addition to this problem, one phase model requires extrapolation or interpolation for physical variables (pressure and velocities) at the interface boundary between water and air. Occasionally, this approximation leads to the source of error in the solution domain. One alternative to avoid these problems is to use two-phase model considering both air and water movement. A few studies are done so far considering two-phases of air and water movement (Hieu and Tanimoto, 2002; Yuhi et al, 1998). However, wave motion in porous structure was not considered in those studies.
KEY WORDS: Permeable structure; porous flow; two phase model; large eddy simulation; VOF method, reflection; wave damping.
INTRODUCTION Many coastal structures constructed for the purpose of harbor securing and shore protection are porous. These structures are designed mainly to provide protection by reflection and/or dissipation of wave energy. For many years, rubble-mound type porous structures are extensively used. However, vertical permeable structures are seldom used due to limited information on the hydraulic performances. In general, a wave train through porous structure, irrespective of its regular or random nature, is reflected, transmitted and dissipated. All these processes induce significant changes in the wave properties such as height, energy and force inside the porous structure. All these changes depend on the structural parameters (width, material size, porosity) and incident wave conditions. Therefore, it is essential to investigate the influences of all these parameters on functional variables such as wave damping, reflection, dissipation and transmission. Due to the complexities of both porous flow and non-linear wave behavior, the problems related to it are extremely difficult to solve and are being investigated for many years. Wave motion in permeable structures is often studied experimentally (Losada et al 1995; Mizutani et al, 2001; Qiu and Wang, 1996). However, it is also understood that physical models have their natural
In the present paper, a VOF method based on the two-phases of water and air is developed for the simulation of wave transformation both inside and outside of the porous structure. The model is based on the continuity equation of an incompressible viscous fluid in the porous media, modified Navier-Stokes equations and convective equations of density and viscosity function. For the small-scale turbulence during wave breaking, large eddy simulation (LES) model is incorporated using Smagorinsky’s sub-grid scale model. For tracking the interface between air and water, VOF method originally proposed by Hirt and Nichols (1981) is used with new advection algorithm of Harvie and Fletcher (2001). Computed results of present model are compared with several sets of experimental data. The model is applied to investigate the influences of structure width and porosity on wave reflection and wave height at the rear wall.
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porosity in the real structure, however for the simplicity both volume and superficial porosities are considered to be equal.
NUMERICAL MODEL Model Equations
The governing equations of porous flow (Eqs. 1–3) are solved by standard finite difference technique. However, it is difficult to solve Eqs. 4-5 using existing finite difference scheme. The reason is that density and viscosity change abruptly at the air water interface. Consequently, numerical oscillation appears near the interface. To avoid this problem, Sussman et al (1994) proposed a level set method for the direct solution of Eqs. 4-5 by using a level set function, which defines the interface of water and air. The essential idea of his method is that density of fully air cell is equal to air density while for fully water cell is equal to water density. At the interface it is determined by averaging the water and air density. This idea can be represented by the VOF function, F. According to the definition of VOF function, F represents the amount of water in a cell and consequently amount of water can be calculated. Therefore, advection equations of density and viscosity (Eqs. 4-5) are replaced by the Eqs. 9-10 (Hieu and Tanimoto, 2002). However, numerical instability may occur from momentum equations due to high-density ratio between air and water. Therefore, the smoothing of density distribution near the interface is carried out.
Based on the Navier-Stokes equations, several researchers derived the governing equations for the water wave motion in porous structure. The basic equations for the porous body model are the continuity equation (Eq. 1) of an incompressible fluid in the porous media and the modified Navier-Stokes equations (Eqs. 2-3). These equations are taken from Sakakiyama and Kajima (1992). For the two-phase model advection equations of fluid density and viscosity (Eqs. 4-5) are used in addition to continuity and momentum equations. ∂ (γ x u ) ∂ (γ z w) + =0 ∂x ∂z
λv
(1)
∂u ∂ (λ x u 2 ) ∂ (λ z uw) ∂ψ + + = −γ v ∂t ∂x ∂z ∂x ∂ ∂u ∂w ∂ ∂u γ xν e 2 + γ zν e + + ∂z ∂z ∂x ∂x ∂x
(2) − Rx
∂ψ ∂w ∂(λ x uw) ∂ (λ z w 2 ) λv + = −γ v + ∂t ∂x ∂z ∂z ∂ ∂w ∂u + γ xν e + ∂x ∂x ∂z
∂ ∂w + γ zν e 2 ∂z ∂z
(3) − Rz
(9)
ν = (1 − F ) ρ air + Fρ water
(10)
Free Surface Model
∂ (γ v ρ ) ∂ (γ x uρ ) ∂ (γ z wρ ) + + =0 ∂x ∂z ∂t
(4)
∂ (γ vν ) ∂ (γ x uν ) ∂(γ z wν ) + + =0 ∂z ∂t ∂x
(5)
The free surface is described by the VOF method. The VOF method introduces a volume of fluid function, F(x,z,t) to define the fluid region. The physical meaning of F function is the fractional volume of the cell occupied by fluid. In particular, a unit value of F corresponds to a cell full of fluid, while a zero value indicates the cell contains no fluid. Cells with F values between zero and one must then contain a free surface. The convective equation describing the time dependence of F function extended for porous media (Isobe et al, 2000) is expressed as follows:
where t :time; x, z: horizontal and vertical coordinates; u, w: horizontal and vertical velocity components; ψ = p/ρ+gz, ρ: density of fluid, p: pressure, g: gravitational acceleration; νe: kinematic viscosity (summation of molecular kinematic viscosity, ν and eddy kinematic viscosity, νt); γv: volume porosity; γx, γz: superficial porosity components in the x and z projections; Rx, Rz : drag/resistance force exerted by porous media.
∂ (γ v F ) ∂ (γ x uF ) ∂ (γ z wF ) + + =0 ∂t ∂x ∂z
(6)
Rx =
1 CD (1 − γ x ) u u 2 + w 2 2 ∆x
(7)
Rz =
1 CD (1 − γ z ) w u 2 + w 2 2 ∆z
(8)
(11)
According to definition, F is a step function and the standard finite difference technique leads to smearing of the F function and interfaces lose their definition. Therefore, special treatment is necessary for the time evolution of F function. The earliest special treatment is called the donor-acceptor method proposed by Hirt and Nichols (1981). However, Hirt and Nichols scheme doesn’t conserve fluid volume correctly as it has common donating region for the adjacent donating boundaries. Harvie and Fletcher (2001) proposed a new VOF algorithm in which volume of fluid conservation is assured. In this scheme, actual donating region for each face of a cell is determined first and then volume of fluid passing through each face is calculated. In the present study, we followed Harvie and Fletcher (2001) scheme to calculate outgoing and incoming flux for each cell. Finally, new F value is calculated.
λx , λz , λv , Rx and Rz are defined from the following relationships:
λ x = γ x + (1 − γ x )C M λ z = γ z + (1 − γ z )C M λv = γ v + (1 − γ v )C M
ρ = (1 − F ) ρ air + Fρ water
Turbulence Model To simulate the turbulent flow, the concept of LES is used in which the effect of small-scale turbulence is represented. The basis for the LES simulation is a spatial filtering of the Navier-Stokes’ equation. For finite difference method, a top-hat filter (cell volume) is generally used. The width of the filter depends on the grid size, which means that for a finer grid a large part of the turbulent motion is represented directly in the simulation. For the present simulation, the governing Navier-Stokes’
where CD and CM are the drag and inertial coefficients. The values of CD and CM depend on the porous material properties and incident wave condition. Very often, these coefficients are determined by a numerical model using trial and error process. In the above equations, porosity is the measurable geometrical characteristics and it is determined on the volume basis. Superficial porosities may differ from the volume
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Solution Algorithm
equation is filtered using Smagorinsky’s scheme. In that scheme, an eddy viscosity term is introduced to represent the momentum exchange by the sub grid scale turbulence. This eddy viscosity is then added to the molecular kinematic viscosity. The resultant viscosity represents the total kinematic viscosity of the fluid. Eddy viscosity is determined from the strain rate tensor (S) in the flow filed that is resolved by the model. In this way, the model works as a mixing length model on the sub grid scale level. For the two dimensional problems the eddy viscosity (νt) is calculated using Eqs. 12-14.
ν t = (C s ∆ )2 2 S .S S=
1 ∂u ∂w + 2 ∂z ∂x
The governing equations are discretized using finite difference technique on a staggered grid system. The variables pi,k, Fi,k, ρi,k, νi,k are located at cell center, while ui,k, wi,k are defined at the cell faces. The porous parameter γv, is assumed at cell center and γx, γz are defined at cell faces. At all other locations variables and parameters are estimated by averaging adjacent values. The momentum equations are discretized based on the finite control volume to satisfy the conservation equations accurately. Setting position of control volume is different for different physical variables in staggered grid system. All the spatial derivatives are discretized using centered difference technique and upwind scheme. The convective equation of the F-function (Eq. 11) is discretized by forward difference technique. To solve for pressure field, a new equation called Poisson’s pressure equation is derived using continuity and momentum equations. The second order five points scheme is used to discretize the Poisson’s equation
(12)
(13)
where ∆ is a length scale dependent on the grid size:
∆ = (∆x∆z )
(14)
The simplified marker and cell (SMAC) method is applied to solve the continuity and momentum equations. In the solution algorithm of SMAC method, there are three steps: (a) explicit approximation of new time level velocities using the initial conditions or the previous time level values for all advective, pressure and viscous terms, (b) solution of Poisson’s pressure equation and (c) modification of transient velocities by adopting pressure difference. Euler explicit method is used to obtain transient values of the velocities from the momentum equations (Eqs. 23). Poisson’s pressure equation is solved for the pressure differences. The simultaneous equations of pressure equation are solved by biconjugate gradient stabilize (BIGSTAB) method (van der Vorst, 1992). Transient velocities are then modified to get the new time level velocities. After getting velocities, convective equation of F function is solved for new time level F values using Harvie and Fletcher (2001) scheme. The convective equations of density and kinematic viscosity are solved in terms of F-function (Eqs. 9-10).
The parameter, Cs is the Smagorinsky constant. It is an empirical constant and is determined from comparison between experiments and model simulations. In general, value of Cs from 0.1 to 0.2 gives reasonable results. In the present simulation a constant value of 0.12 is used for all the case studies.
Boundary Conditions The governing equations with prime variables u, w and p are numerically solved with appropriate boundary and initial conditions. In the present study, boundary conditions for resolved field are categorized into two kinds, namely, inflow boundary and the mesh boundary conditions. The inflow boundary represents the wave source while mesh boundary represents the variables at the fictitious cells adjacent to the solution domain.
HYDRAULIC MODEL
At the inflow boundary, a piston type wave maker is introduced to generate the incident wave. The wave paddle is driven by the second order Stokes wave theory and an absorbing wave maker system, which is essentially same as the physical experiment. The non-reflective wave source proposed by Zhao and Tanimoto (1998) is adopted here. Two types of mesh boundaries are incorporated. At the top boundary, continuative condition is used. The essential idea of continuative boundary is that the velocities at the fictitious cells are equal to the velocities of adjacent computational cells. Free slip condition is adopted for all other boundaries except top boundary. The free slip condition poses that the normal velocity (un) to the boundary is zero and the tangential velocity (ut) is free (Eqs. 15-16). For pressure, Eq. 17 represents left and right boundaries, while Eq. 18 represents top and bottom boundaries. It is worthy mentioning that no boundary condition is needed at the surface boundary of porous structure because the continuity equation reserves the continuity of variables at local cells. However, all the porous parameters are averaged at the interfaces of structure with water and air. un = 0 ∂u t =0 ∂n ∂p =0 ∂x ∂p = − ρg ∂z
The experiments were conducted in the hydraulic laboratory of Saitama University. The flume, which was used for the experiments, is 18 m long, 0.40 m wide and 0.75m high with glass wall at both sides. A rectangular permeable wall was built at a distance of 12.32 m from the wave maker. This structure was build by assembling the irregular stone of mean diameter 2.5 cm and porosity 45%. The porous part is 50 cm high over the impermeable bottom of the flume and is 138 cm long in the direction of wave propagation, while the still water level is kept constant at 37.5 cm. Backside of the wall is supported by impermeable wooden frame while wire mesh was used to hold the stone at the front wall. In this study only regular waves were generated using piston type wave maker. The experiments were conducted for three wave periods 1.2, 1.6 and 2.0 s and several incident wave heights ranged from 3.68 to 12.30 cm. However, for the comparison of numerical model only a typical wave period (T=1.6 s) was considered. Capacity type wave gauges were set along the wave tank to record the water surface levels inside and outside of the structure. A total of 17 gages were set on the centerline of the flume for the free surface measurement. The wave gauges were connected to a personal computer to save the records of water surface levels.
(15) (16)
NUMERICAL MODEL VALIDATION
(17)
A numerical wave channel similar to the experimental condition is developed with appropriate inflow and mesh boundaries. The length and height of the numerical channel are 14.1 and 0.80 m respectively. The top boundary is set 0.425 m above from the still water level to avoid
(18)
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boundary effect. The top level of the structure is selected for nonovertopping condition. Different widths are considered for case studies for the fixed height (0.50 m) of the structure. A typical view of numerical model setup is shown in Fig. 1. Computational grid size is 2 cm in the horizontal direction and 1 cm in the vertical direction. Time increment is set by satisfying CFL and stability conditions.
0.4 m
12.32 m Air
Air
1.38 m
Still water level
Water
2.4
0.50m
0.60m
0.375 m
1.6
x=0
1.2
x
Fig. 1 Schematic view of numerical model setup
0.8 0.4
The inflow boundary is based on Stokes second order theory with wave absorbing mechanism. To examine the efficiency of wave maker source as a non-reflecting wave maker boundary, computations are continued until re-reflected wave propagates in the progressive wave direction. Fig. 2 shows the time profile of water surface at an intermediate section inside numerical wave tank where η represents water surface elevation above mean water level, HI: incident wave height, T: wave period and t: computational time. It is observed that wave is fairly stable before and after the arrival of reflected wave. It shows that effect of re-reflected wave from the wave maker is little, which implies the satisfactory treatment of reflected wave in the solution domain. It also shows the stable solution with time.
0.0 -1.0
-0.8
-0.6
-0.4
0.0
Fig. 3 Spatial distribution of wave height
2.0
T =1.6 s H I =7.0 cm C M =2.0
M e a s ure d C o m (CD =2.0) C o m (CD =3.0 ) C o m (CD =4.0 ) C o m (CD =5.0 )
H/HI
1.6
T =1.6 s, H I =7.0 cm
0.6
-0.2
x/L
1.2
0.8
η/HI
M easured Computed
T =1.6 s H I =7.0 cm
2.0
H/HI
Wave maker
structure. In the governing equations there are several porous flow parameters, which must be known before the application of the model. The parameters are volume porosity, n drag coefficient, CD and inertia coefficient, CM. The porosity is a measurable geometrical characteristic of the porous media, which is known for specific porous material. However, CD and CM are unknown parameters, which must be set before application of the model. There is no empirical relationship to determine these parameters directly. In general, these are determined by trial and error process. However, it is essential to observe the influence of both parameters on flow characteristics independently. Fig. 4 shows the influence of CD on wave height distribution for the fixed value of CM (2.0) while Fig. 5 shows the influence of CM for fixed CD (4.0).
0.8
0.4 0.4
0.2 0.0
0.0 -0.50
-0.2
-0.25
-0.4 -0.6 0
5
10
t/T
15
20
x/L
0.00
0.25
0.50
Fig. 4 Wave height distribution for variable CD
25 2.0
Fig. 2 Surface profile inside the numerical wave tank
1.6
C D =4.0
H/HI
To verify the present model, results of model computations are compared with experimental data. At first, the model is applied to simulate the progressive and reflected waves in case of a vertical impermeable wall at the opposite end of the wave maker source. Fig. 3 shows the results of computed wave height distributions with experimental data in the non-dimensional form, where, L is wavelength and x is the distance from vertical wall. It shows that numerical results fairly agree with the experimental data. To evaluate the efficiency of model computation, statistical parameters namely root mean square (RMS) error and relative error (RE) are calculated. It is found that RMS error is only 1.3%, while RE is 3.7 %.
M e a s ure d Co m (CM =0.2 ) Co m (CM =0.5 ) Co m (CM =1.0 ) Co m (CM =2.0 )
T =1.6 s H I =7.0 cm
1.2 0.8 0.4 0.0 -0.50
-0.25
x/L
0.00
0.25
Fig. 5 Wave height distribution for variable CM
The model is calibrated for the simulation of wave motion in the porous
732
0.50
wave. As to the influential parameters, the structure width and porosity are considered. The incident wave conditions is T=1.6 s, HI=7.0 cm in the present application.
It is observed that wave damping inside the porous structure is influenced by CD considerably. The reason is that the drag coefficient directly represents the wave energy dissipation. In case of higher CD, rate of energy dissipation is higher and consequently wave damping is also higher. However, influence of CD on external flows is negligible. On the other hand, it is observed that the influence of CM is not significant in comparison with CD. In general, wave damping reduces with the increase of CM but the change is very small.
2.0 1.6 H/HI
0.4 0.0 -0.50
-0.25
x/L
0.00
0.25
0.50
Fig. 6 Comparison of normalized wave height distribution 1.2
T =1.6 s, H I =7.0
Measured Computed
η/HI
0.8 0.4 0.0 -0.4 -0.8 -1.0
-0.5
0.0
x /L
0.5
Fig. 7 Surface profile inside and outside structure 2.0
Computed (n=0.3) Exponential (n=0.3) Computed (n=0.6) Exponential (n=0.6)
y = Hw /HI
1.6
MODEL APPLICATIONS The validated numerical model is used to analyse the influences of various parameters on the hydraulic performance of the structure. The hydraulic performances that can be analysed for a porous structure are: reflection, transmission and energy dissipation of waves; maximum velocities on the bed near the structure; maximum wave forces on the structure and stability conditions. Any of these hydraulic performances can be represented by the function of following set of non-dimensional parameters (Eq. 19).
H I h Dm B , , , ,n h L HI L
Co m p (HI =7.0 c m )
0.8
A series of spatial profiles at various time phases are shown in Fig. 7 for one wave period. It is seen that abrupt change in the water surface elevation is occurred at the front face of porous structure. This is due to the sudden resistive force offered by the porous material. The measured wave crest and trough are also plotted in the same figure. The results show that model computations fairly agree with the experimental data. From those time changes of wave profile and the envelope, it is confirmed that partial standing wave is formed in front the porous structure. The partial standing wave is also noticed in front of the impermeable wall in the porous structure due to the reflection of damped incident wave.
y=F
C M =0.5
1.2
In the present study we determined the combination of CD and CM that best fits the experimental results, by a least square technique for a selected typical incident wave condition (T=1.6 s, HI=7.0 cm). For the porous material used in this study (Dm=2.5 cm, n=0.45), the values of CD and CM are 3.5 and 0.5 respectively. However, they are greatly influenced by the porous material properties and incident wave conditions. Therefore, more investigations are necessary to determine empirical relationship for CD and CM. The model is applied to the porous structure with selected CD and CM and computed results are compared with the experimental results. A typical comparison of wave height distributions inside and outside the porous structure is shown in Fig. 6 for the wave of T=1.6 s. All measured data for different incident wave heights are plotted. In general, higher incident wave gives lower non-dimensional wave height. However, the trend is same for all the cases. It can be said that the numerical model predicts some average relation of measured wave heights. The results also show that effect of incident wave height is more significant inside the structure than the outside of the structure. The first reason is that the effect of CD and CM on external flow is negligible and the second reason is that the resistance force of porous media depends on the incident wave condition.
M e a s ure d (HI =3.7 to 11.4 c m )
C D =3.5
1.2
y = 2e −2.9 x
0.8
y = 2e −5.2 x
0.4 0.0 0.0
0.1
0.2
x = B/L
0.3
0.4
0.5
Fig. 8 Wave height at rear wall for different B/L
Fig. 8 shows the wave heights at the impermeable rear wall with structure’s relative width. It is observed that wave height at the rear wall reduces sharply with the increase of structure width for some extent. For example, for the porosity of 0.3 wave height at the rear wall reduces considerably until B/L = 0.3. The reason is that the rate of energy dissipation reduces with distance along the structure and hence rate of wave damping also reduces with distance. This phenomenon can be explained in terms of velocity field and resistance force. The resistance force is proportional to the square of velocity (Eqs. 7-8). As the wave
(19)
where y: hydraulic performance, HI: incident wave height, h: still water depth, Dm: mean diameter of porous material, B: width of the structure, L: wave length, n: porosity. In this paper, however, the porous structure with an impermeable wall is investigated. Therefore, wave height at the impermeable rear wall is analysed in a similar fashion of transmitted
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propagates inside the structure, velocity reduces with distance due to resistance force. Consequently, energy loss also reduces with distance. For the fixed incident wave condition, velocity is very small after certain distance inside the structure and energy loss is also very small. Therefore, after this limit, rate of wave damping is not considerable with the increase in width. This implies that there is optimum structure width beyond which the increase in structure width is not justified from economical point of view. It is also shown that wave heights can be represented fairly by exponential curve. The representative equations for porosity 0.3 and 0.6 are shown in Fig. 8. It can be concluded that wave height at the rear wall follows a smooth trend with the increase in structure width. Therefore, empirical relationship can be developed considering fixed values for other parameters.
1.0
KR
0.6
0.2 0.0 0.0
∗ w i
∗ w p
∗ w i
0.1
0.2
B/L
0.3
0.4
0.5
Fig. 9 Variation of KR with structure width 2.0
Computed Exponential curve
1.6
y = Hw /HI
(H ) − (H ) (H )
n = 0.6
0.4
To assess the efficiency of structure width in wave damping two cases are considered for (a) B/L = 0.25 and (b) B/L = 0.5 for porosity 0.3. The wave height damping factor (ξ) is calculated using Eq. 20, where (H*w)i represents the non-dimensional wave height at the wall for impermeable wall condition (no porous structure) and (H*w)p represents wave height at the rear wall for porous structure. It is seen that for case (a) wave damping is 73% while for case (b) it is 91%, which is 1.25 times of case (a). However, width is double for case (b). The results imply that rate of wave damping reduces considerably with B/L. However, this limiting vale varies with other conditions.
ξ=
n = 0.3
0.8
1.2
y =2
0.8
(20)
B/L =0.5
e ax − 1 ea −1
a = ln10
0.4
The influence of structure width on wave reflection is graphically represented in Fig. 9. The results are shown for two porosities of 0.3 and 0.6. In general, porosities for prototype structures vary from 0.3 to 0.6. Therefore, present conditions are selected to view the maximum and minimum porosity effect on wave reflection. The reflection coefficients are estimated using Healy’s method for small amplitude wave and then corrected by Goda and Abe (1968) method for finite amplitude waves. The results show that for smaller relative widths, reflection coefficient (KR) decreases until a minimum value of KR is obtained. For the present condition, wave reflection is minimum for the relative width nearly 0.2. However, this minimum width depends on the other parameters also. For relative widths greater than this minimum, reflection coefficient increases for some extent and then reaches a stable value. The reason is that waves transmitted to the rear wall becomes small and the reflected waves at the rear wall do not go out from the structure, when the relative width becomes large enough. In the example of Fig. 9, the wave reflection coefficient is stable when B/L is 0.4 or greater. The results also show that wave reflection is smaller for higher porosity for any relative width. However, the trend is almost same for both the porosities. For smaller B/L, reflection coefficient is not influenced significantly by porosity while considerable effect is seen for higher relative width. In Fig. 9, it seems that magnitude of B/L for minimum reflection coefficient is little higher for higher porosity.
0.0 0.0
0.2
0.4
x=n
0.6
0.8
1.0
Fig. 10 Wave height at vertical wall with porosity 1
B/L =0.50 0.8
KR
0.6 0.4 0.2 0 0.0
0.2
0.4
n
0.6
0.8
1.0
Fig. 11 Variation of reflection coefficient with porosity
Fig. 10 shows the wave height at the rear wall for different porosities and fixed structure width (B/L=0.5). It is observed that for small porosities wave height at the rear wall is very small. The reason is that major part of the wave is reflected by front wall and structure itself. Therefore, energy transmission through the structure is very small. For higher porosities, wave height increases rapidly. The relationships of wave heights at the rear wall can be represented by an exponential curve shown in the figure where ‘y’ represents the non-dimensional wave height at the rear wall (Hw/HI) and ‘x’ represents porosity (n). However, for higher porosities present curve shows some discrepancy from the computed results.
Fig. 11 shows the variation of the reflection coefficient with the material porosity for the fixed values of other parameters. It is seen that porosity has a great influence on the reflection. For very small porosities, the structure behaves as if it has an impermeable front wall and the reflection coefficient tends to be 1 (no dissipation). For another extreme toward to no porous materials with very high porosities, the behavior is similar, but with the reflecting point at the rear wall. In between, there is a point of optimal porosity, where the reflection coefficient is minimum and the maximum energy dissipation is occurred. It should be noted that the point of minimum reflection and maximum dissipation depends on another parameter B/L. In the
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example of Fig. 11, wave reflection is minimum when porosity is 80%. This value is not so realistic. The reason may be the large relative width of the structure (B/L = 0.5). In Fig. 9, we observed that wave reflection is minimum for B/L equal 0.2. Therefore, more investigations are necessary to observe the effect of porosity on reflection for different relative widths.
governing factors for the wave reflection. It is confirmed that wave reflection reduces significantly for certain combination of structure width and porosity. It is also concluded that there are optimum structure width and porosity for which wave reflection is minimum. However, all these parameters are inter-linked with each other. Hence, more investigations are necessary to develop empirical relationships among the parameters.
Two snapshots of velocity field are given in Fig. 12. The upper one is for t/T = t0 and lower figure is for t/T= t0+0.5, where t0 is chosen arbitrarily. The velocity in the porous structure is the bulk velocity. Therefore, the magnitudes of the velocities at the front wall (both inside and outside) are almost same. It can be confirmed that the velocity decreases rapidly in the porous structure.
0.6
ACKNOWLEDGEMENTS The results presented here is part of the Ph. D research of the first author. He is grateful to the ministry of education, science and culture, Government of Japan for granting scholarship for this study. The authors would like to thank Y. Sou and Y. Nishiwaki for their assistance during experiments.
1m/s
0.5
REFERENCES
0.4
Goda, Y, and Abe, Y (1968). “Apparent Coefficient of Partial Reflection of Finite Amplitude Waves, Report of the Port and Harbour Res. Inst., Japan, Vol 7, No 3, pp 1-58. Harvie, DJE, and Fletcher, DF (2001). “A New Volume of Fluid Advection Algorithm: The Defined Donating Region Scheme,” Int. J of Numerical Methods in Fluids, Vol 35, pp 151-172. Hieu, PD and Tanimoto, K (2002). “A Two-Phase Flow Model for Simulation of Wave Transformation in Shallow Water,” Proc 4th International Summer Symposium, JSCE, Japan, pp 179-182. Hirt, CW, and Nichols BD (1981). “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries,” J Computational Physics, Vol 39, pp 201-225. Isobe, M, Hanahara, Y, Yu, X, and Takahashi, S (2000). “A VOF-Based Numerical Model for Wave Transformation in Shallow Water, Proc International Workshop on Advanced Design of Maritime Structures in the 21st Century, PHRI, Japan, pp 200-204. Liu, PLF, Lin, P, Chang, K, and Sakakiyama, T (1999). “Numerical Modeling of Wave Interaction with Porous Structures, J Waterway, Port, Coastal and Ocean Engg, ASCE, Vol 125, No 6, pp 322-330. Losada, IJ, Losada, MA and Martin, FL (1995), “Experimental Study of Wave Induced Flow in a Porous Structure,” Coastal Engineering, Elsevier, Vol 26, pp 77-98. Mizutani, N, Golshani, A, and Hur, DS (2001), “Three Dimensional Study on the Wave-Induced Flow Inside and Around a Permeable Structure,” Proc 11th International Offshore and Polar Engineering Conference, ISOPE, Norway, pp 638-644. Qiu, D, and Wang, L (1996), “Numerical and Experimental Research for Wave Damping over Submerged Porous Breakwater, Proc 6th Int Offshore and Polar Engg Conf, ISOPE, pp 572-576. Sakakiyama, T, and Kajima, R (1992). “Numerical Simulation of Nonlinear Wave Interacting with Permeable Breakwaters, Proc 23rd Int Conference on Coastal Engineering, ASCE, pp1517-1530. Sussman, M, Smereka, P, and Osher, S (1994). “A Level Set Approach for Computing Solutions to Incompressible Two Phase Flow,” J Computational Physics, Vol 114, pp 146-159. van der Vorst, HA (1992). “BI-CGSTAB: A Fast and Smoothly Converging Variant of BI-CG for the solution of Nonsymmetric Linear Systems”, SIAM J Sci and Stat Comput, Vol 12, pp 631-644. Yuhi, M, Ishida, H, and Bochi, M (1998). “A Numerical Study of Strongly Interacting Two Phase Flows,” Proc Coastal Eng, JSCE, Vol 45, pp 61-65 (in Japanese). Zhao, Q, and Tanimoto, K (1998). “Numerical Simulation of Breaking Waves by Large Eddy Simulation and VOF Method,” Proc International Conference on Coastal Engg, ASCE, pp 892-905.
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Fig. 12 Snapshots of velocity field
CONCLUSIONS Based on Navier-Stokes equations, a two-phase numerical wave tank is developed to simulate wave motion in and out of the permeable structures. The model is calibrated and validated using measured data in the physical experiments for a vertical porous structure with a impermeable rear wall. The drag and inertia coefficients in the governing equations are calibrated for a typical porous material (stones of mean diameter 2.5 cm and porosity 0.45). The calibration is made by minimizing errors between the numerical results and experimental data for the wave height distributions inside the porous structure. In the calibration process it is observed that the influence of drag coefficient is much higher than that of inertia coefficient. In general the agreements between numerical and experimental results are fairly acceptable. After calibration, the numerical model is applied to investigate the influence of structural parameters namely width and porosity on the functional variables (reflection coefficient and wave height at the rear wall). For the permeable structure with rear impermeable wall, wave height at the rear wall depends considerably on structure width and porosity. It is confirmed that the wave height at the rear wall decreases exponentially with the increase in structure width while it increases with the increase in porosity. Structure width and porosity are the major
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