3.2 Scattering problem
3.2.4 Numerical results and discussion
3.2. Scattering problem 69
evaluated from the following:
Kf,j = |Cf,j|
ρ2gdh1, andKb,j = |Cb,j|
ρ2gdh1 forj = 2,3. (3.48) Following the same procedure as in Section 2.2, amplitude of elevation at the free surface and interface, i.e.,ηi fori= 1,2, can be obtained as
η1 = ∂φj
∂z , atz =d, and η2 = ∂φj
∂z , atz = 0. (3.49)
0 1 2 3 4 5 6 7 8
k1,1L1
0 0.2 0.4 0.6 0.8 1
|R 1|,|T 1|
Behera and Sahoo Present Work Full wave solution Behera and Sahoo
Present Work Long-wave appr.
|T1|
|R1|
(a)
0 1 2 3 4 5 6 7 8
k1,1L1
0 0.2 0.4 0.6 0.8 1
|R 2|,|T 2|
Behera and Sahoo Present Work Full wave solution Behera and Sahoo
Present Work Long-wave appr.
|T2|
|R2|
(b)
Figure 3.2: Reflection and transmission coefficients against k1,1L1 at (a) free surface and (b) interface corresponding to present result and [8].
To validate our model, we compare one result with the corresponding result of Behera and Sahoo [8] in which they had examined water wave interaction with a porous block for a two-layer fluid flow over an impermeable sea-bed. As in Venkateswarlu et al. [108], two different cases (full wave solution and long-wave approximation) are compared to ascertain the complete efficiency of the model. Therefore, in order to have a proper comparison by converting our model to their model, we consider G = 0 so that the porous sea-bed becomes an impermeable one and assume the following parameter values:
h1 =h2 =d = 2.5meter, T = 8sec, g = 9.81m/s2, 2 = 0.5, f2 = 1, L2 = 0. An excellent agreement in the results follows when the reflection and transmission coefficients are plotted against non-dimensional k1,1L1 (Figure 3.2(a,b)). In other words, the successful validation confirms that our model can be considered as an efficient one for formulating and solving various problems of wave interaction by porous structures containing block- wise different porosity for a two-layer fluid flow.
0 1 2 3 4 5 6 7
k1,1(L1+L2)
0 0.2 0.4 0.6 0.8 1
|R 1|,|T 1|
f2=0.8 f2=1 f2=1.2
|T1|
|R1|
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
k1,1(L1+L2)
0 0.2 0.4 0.6 0.8 1
|R 2|,|T 2|
f2=0.8 f2=1 f2=1.2
|T2|
|R2|
(b)
Figure 3.3: Variation of reflection and transmission coefficients against non-dimensional width k1,1(L1 +L2) corresponding to various friction values f2 with f3 = 1 and 2 = 0.7, 3 = 0.8
In Figures 3.3(a) and 3.3(b) and Figures 3.4(a) and 3.4(b), the reflection and transmis-
3.2. Scattering problem 71
0 1 2 3 4 5 6 7
k1,1(L1+L2)
0 0.2 0.4 0.6 0.8 1
|R1|,|T1|
3=0.6
3=0.7
3=0.8
|T1|
|R1|
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
k1,1(L1+L2)
0 0.2 0.4 0.6 0.8 1
|R 2|,|T 2|
3=0.6
3=0.7
3=0.8
|T2|
|R2|
(b)
Figure 3.4: Variation of reflection and transmission coefficients against non-dimensional lengthk1,1(L1+L2)corresponding to various porosity3with2 = 0.7andf2 = 0.9, f3 = 1.
sion coefficients in SM and IM, respectively, are plotted against non-dimensional width k1,1(L1 +L2) of the structure corresponding to various values of porosity 3 and fric- tion parameter f2. Similar patterns are visible for both modes in Figure 3.3(a,b) while changing friction parameter f2. Further, reflection and transmission coefficients both re- main almost constant after some specific width. However, transmission in both modes reduces sharply with increasing(L1+L2)while the interfacial transmission vanishes after k1,1(L1 +L2)≥ 1. Consideration of different values of friction factor f2 results in differ- ence in the values of the reflection and transmission coefficients, where higher value of friction factor (f2 > f3) results in higher reflection and lower transmission while lower value (f2 ≤ f3) results in lower reflection and higher transmission in both propagating modes. In Figure 3.4, it is observed that change of porosity 3 does not lead to a major impact in the coefficientsTSM and TIM. However, differences are observed for reflection coefficients in both modes where the lower values of 3 (i.e., 2 > 3) results in higher reflection and higher values of3 (i.e.,2 ≤3) results in lower reflection. The same trend is observed for both cases of varying porosity and varying friction. It can be concluded that it is advisable to use a composite structure of appropriate structural length to obtain higher reflection and lower transmission. Both cases result in the convergence of reflecting modes after some structural width and the propagating modes also attain the optimum at nearly the same structural width. But both the propagating modes maintain a steady behaviour with respect to the structural width which establishes the fact that, after a certain value, higher structural width has no impact on reflection and transmission.
Figure 3.5(a,b) illustrates the effect of friction parameter on the reflection coefficients with respect to angle of incidence θ in surface mode and interface mode, respectively.
By varying the friction factor, the reflection coefficients are observed to start from their minimum values, and keep increasing to the maximum in both modes while transmission follows an opposite trend. In both propagating modes, maximum reflection and minimum transmission are obtained at the angle90◦. However, change of friction does not contribute
0 10 20 30 40 50 60 70 80 90 0
0.2 0.4 0.6 0.8 1
|R 1|,|T 1|
f2=f 3=0.5 f2=f
3=0.75 f3=f
3=1
|R1|
|T1|
(a)
0 10 20 30 40 50 60 70 80 90
0 0.2 0.4 0.6 0.8 1
|R 2|,|T 2|
f1=f 2=0.5 f1=f
2=0.75 f1=f
2=1
|T2|
|R2|
(b)
Figure 3.5: Variation of reflection and transmission coefficients against incident angle θ corresponding to various friction parameterfj forj = 2,3with f2 =f3, 2 = 0.7, 3 = 0.9 and (L1+L2)/h1 = 0.5.
to any difference in the angle of incidence for obtaining its optimum in SM but differences are observed for attaining minimum in IM. Therefore, the optimum values depend on the friction parameter also. In both modes, lower friction results in lower reflection and higher transmission. Thus, by studying the wave pattern, porous structures can be designed as breakwaters by appropriately incorporating structure width, step height and friction factor in order to achieve maximum wave reflection and minimum transmission which will in turn protect the marine facilities.
0 1 2 3 4 5 6 7 8 9
k1,1(L 1+L
2)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Kf,2,Kb,3 f
2=0.5,f 3=1 f2=1,f
3=1 f2=1.5,f
3=1 Kf,2
Kb,3
Figure 3.6: Variation wave forces Kf,2 andKb,3 against non-dimensional lengthk1,1(L1+ L2) corresponding to various values offj with 2 = 0.9, 3 = 0.8.
In Figure 3.6, waveloads Kf,2 and Kb,3 are, respectively, plotted against k1,1(L1+L2) corresponding to different values of friction parameter. The figure shows that wave force Kf,2 does not show any major difference due to the increase of the structure width but Kb,3 starts from its highest value and generally reduces with an increase in the width.
This may have happened due to the dissipation of a large amount of wave energy by the porous structure. Wave force Kf,1 produces a higher impact for higher friction f2 with f2 ≤ f3 but wave force Kb,3 exhibits the opposite. As observed earlier, the steady nature of the reflection and transmission modes for higher structural width is one of the
3.3. Trapping Problem 73