Results

6.4 Numerical results

6.4.2 Results

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|R₀|

Present work Liu and Li (2011)

Figure 6.5: Comparison between Liu and Li (2011a) and the current work

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|R₀|

Present work Sahoo et al.

(2000)

Figure 6.6: Comparison between Sahooet al. (2000b) and the current work

Another comparison is made with the work of Sahooet al. (2000b). With the consideration of d₂/h = 1 and G₂ = 0, the geometry of the problem exactly matches with one of the configurations considered by Sahoo et al. (2000b). The parameter values considered for the study are d₁/h= 1, νh= 1 and G₁ = 1 and Fig. 6.9 illustrates good agreement between the results.

Good agreement of the results shows the validation of the formulation of the current prob- lem and it confirms that further study on scattering process for such porous plate problems for different parameters can be carried out.

Wave damping by two thin vertical porous plates placed at finite distance from each other 113

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|R0|

d1/h= 0.2 d₁/h= 0.35 d1/h= 0.5 d₁/h= 0.75 d1/h= 1 G1=0.5; G

2=0.5; k 0h=1.6

(a) Variation of|R0|

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|T0|

d₁/h= 0.2 d₁/h= 0.35 d₁/h= 0.5 d₁/h= 0.75 d₁/h= 1 G1=0.5; G

2=0.5; k 0h=1.6

(b) Variation of|T0|

Figure 6.7: Effect of L/λ for different d1/h with G1 = G2 = 0.5, k0h = 1.6, d2 =d1, θ = 0⁰ andN = 30

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

|R0|

L/λ

d1/h=0.2 d₁/h=0.35 d₁/h=0.5 d1/h=0.75 d₁/h=1 G1=0.5+i; G

2=0.5; k 0h=1.6

(a) Variation of|R0|

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|T0|

d₁/h= 0.2 d₁/h= 0.35 d₁/h= 0.5 d₁/h= 0.75 d₁/h= 1 G₁=0.5+i; G₂= 0.5; k₀h=1.6

(b) Variation of|T0|

Figure 6.8: Effect of L/λ for different d₁/h with G₁ = 0.5 + i, G₂ = 0.5, k₀h= 1.6, d₂ =d₁, θ= 0⁰ and N = 30

6.4.2 Results

In figure 6.7, the reflection and transmission coefficients are plotted against dimensionless dis- tance L/λ (λ being the wavelength of incident wave) between the porous plates for different values of the dimensionless heightd₁/hof the first plate. Some specific values of other param- eters are taken, such as G₁ = 0.5, G₂ = 0.5, k₀h = 1.6 and d₂ = d₁. Oscillating natures of

|R₀|and |T₀| are observed for these values ofd₁/h. Both the minimum and maximum values of |R₀| are obtained at specific values of L/λ corresponding to all values of d₁/h but on the other hand, optimum values of|R₀|increase with an increase in the value ofd₁/h. In the case of|T₀|, reverse characteristic is observed, i.e., with an increase ind₁/h,|T₀|decreases with the addition of the fact that optimum values of |T₀| show a leftward movement with an increase Linear water wave damping by a bottom-mounted porous structure ... Ph.D. Thesis

114 6.4.2 Results

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|R0|

d₁/h= 0.2 d1/h= 0.35 d₁/h= 0.5 d1/h= 0.75 d1/h= 1 G₁=0.5; G₂=0.5+i; k₀h=1.6

(a) Variation of|R0|

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

L/λ

|T0|

d₁/h= 0.2 d₁/h= 0.35 d₁/h= 0.5 d₁/h= 0.75 d₁/h= 1 G1=0.5; G

2=0.5+i; k 0h=1.6

(b) Variation of|T0|

Figure 6.9: Effect of L/λ for different d₁/h with G₁ = 0.5, G₂ = 0.5 + i, k₀h= 1.6, d₂ =d₁, θ= 0⁰ and N = 30

0 0.2 0.4 0.6 0.8 1 1.2

0 10 20 30 40 50 60 70 80

L/λ Eloss(%)

d₁/h= 0.2 d₁/h= 0.35 d₁/h= 0.5 d₁/h= 0.75 d₁/h= 1 G₁=1; G₂=1+i;

k₀h=1.6; d₂=d₁

(a)G2= 1 + i

0 0.2 0.4 0.6 0.8 1 1.2

10 20 30 40 50 60 70 80

L/λ Eloss (%)

d1/h= 0.2 d

1/h= 0.35 d

1/h= 0.5 d

1/h= 0.75 d

1/h= 1 G₁=1; G₂=1;

k₀h=1.6; d₂=d₁

(b)G2= 1

Figure 6.10: Energy loss (%) againstL/λ for different values ofd1/hwithG1 = 1, k0h= 1.6, d₂ =d₁,θ= 0⁰ and N = 30

in the value ofd₁/h.

The same study is carried out by adding an imaginary quantity toG₁, i.e., G₁= 0.5 + i and the results are shown in figure 6.8. By comparing figure 6.8(a) with figure 6.7(a), a significant amount of change in the minimum value of|R0|is observed ford1/h= 0.35,0.5,0.75,1. Hence it can be deduced that the imaginary part of G₁, which causes the inertial effect, plays a significant role in reducing reflection by the porous plate. But by comparing figure 6.8(b) with figure 6.7(b), it is found that the value of|T₀|increases for those aforesaid values ofd₁/hwhich, on the other hand, present significantly reduced values of|R₀|.

Further study is carried out for the coefficients |R₀|and|T₀|by plotting them againstL/λ withG₁ = 0.5 and by adding an imaginary quantity toG₂: G₂= 0.5+i (figure 6.9). Comparing

Wave damping by two thin vertical porous plates placed at finite distance from each other 115

0 0.2 0.4 0.6 0.8 1 1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

L/λ

|R0|

d1/d 2= 0.5 d1/d

2= 1 d₁/d₂= 1.5 d1/d

2= 2 d₂/h=0.5; G₁=G₂=1;

k0 h=1.6

(a) Variation of|R0|

0 0.2 0.4 0.6 0.8 1 1.2

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

L/λ

|T0|

d₁/d₂= 0.5 d₁/d₂= 1 d₁/d₂= 1.5 d₁/d₂= 2

d2/h=0.5; G 1=G

2=1; k 0 h=1.6

(b) Variation of|T0|

Figure 6.11: Effect ofL/λfor different values ofd₁/hwithd₂/h= 0.5,G₁ =G₂= 1,k₀h= 1.6, θ= 0⁰ and N = 30

0 0.2 0.4 0.6 0.8 1 1.2

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

L/λ

|R0|

d₂/d₁= 0.5 d2/d

1= 1 d₂/d₁= 1.5 d2/d

1= 2 d1/h=0.5; G

1=G 2=1;

k₀h=1.6

(a) Variation of|R0|

0 0.2 0.4 0.6 0.8 1 1.2

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

L/λ

|T0|

d₂/d₁= 0.5 d₂/d₁= 1 d₂/d₁= 1.5 d₂/d₁= 2

d₁/h=0.5; G₁=G₂=1;

k0h=1.6

(b) Variation of|T0|

Figure 6.12: Effect ofL/λfor different values ofd₂/hwithd₁/h= 0.5,G₁ =G₂= 1,k₀h= 1.6, θ= 0⁰ and N = 30

figure 6.9 with figure 6.7, an increase and a decrease in the minimum and maximum values of

|R₀|, respectively, are observed with the value of |T₀| increasing significantly. One interesting observation made by comparing figures 6.8(b) and 6.9(b) is that interchanging the values of G₁ and G₂ does not lead to any change in |T₀|.

The energy loss is plotted against L/λ for different values of d₁/h in figure 6.10. It is observed that higher values ofd₁/hresult in higher energy loss. Also, more oscillation in the energy loss curve is found corresponding to higher values of d₁/h. But from comparison of these two curves, it is evident that, for a specific value ofd₁/h, real values ofG₂ show higher maximum value in energy loss as compared to that due to the complex values ofG2.

So far our investigation is limited to the consideration of porous plates of the same height Linear water wave damping by a bottom-mounted porous structure ... Ph.D. Thesis

116 6.4.2 Results

0 0.2 0.4 0.6 0.8 1 1.2

35 40 45 50 55 60 65 70 75 80

L/λ Eloss (%)

d1/d 2= 0.5 d₁/d₂= 1 d1/d

2= 1.5 d₁/d₂= 2 d₂/h=0.5; G₁=G₂=1;

k₀h=1.6

(a) Against variousd1/d2 withd2/h= 0.5

0 0.2 0.4 0.6 0.8 1 1.2

35 40 45 50 55 60 65 70 75 80

L/λ Eloss(%)

d2/d 1= 0.5 d₂/d₁= 1 d2/d

1= 1.5 d₂/d₁= 2 d₁/h=0.5;

G1=G 2=1;

k₀h=1.6

(b) Against variousd2/d1 withd1/h= 0.5

Figure 6.13: Energy loss (%) against L/λwith G1 =G2 = 1,k0h= 1.6,θ= 0⁰ and N = 30

0 0.2 0.4 0.6 0.8 1 1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

L/λ

|R0|

θ= 0⁰ θ= 15⁰ θ= 30⁰ θ= 45⁰

d1=d 2=h; G

1=G 2=1; k

0h=1.6

(a) Variation of|R0|

0 0.2 0.4 0.6 0.8 1 1.2

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

L/λ

|T0|

θ= 0⁰ θ= 15⁰ θ= 30⁰ θ= 45⁰ d1=d

2=h; G 1=G

2=1; k 0h=1.6

(b) Variation of|T0|

Figure 6.14: Effect of L/λ for different values of angle of incidence θ with d₁ = d₂ = h/2, G₁ =G₂ = 1,k₀h= 1.6 andN = 30

only. Hence a separate investigation is carried out by taking different heights of the porous plates (d1 6= d2). Figure 6.11 shows the variation of |R0| and |T0| against L/λ for different values ofd₁/d₂ withd₂/h= 0.5. It is observed from figure 6.11(a) that optimum values of the reflection coefficient are obtained periodically with respect to the values ofL/λirrespective of the values of d₁/d₂ but higher values of d₁/d₂ result in higher reflection and all the optimum values are less than 0.45. Moreover, the optimum values tend to shift rightward for higher values of d₁/d₂. The transmission coefficient follows the same oscillation pattern as in |R₀|, but higher values are obtained corresponding to lower values ofd₁/d₂ (figure 6.11(b)).

Study on the reflection and transmission coefficients against L/λ for different values of d₂/d₁ is also carried out with the parameter d₁/h fixed at 0.5 (figure 6.12). Here both the minimum and maximum values of |R₀| occur corresponding to the maximum value of d₂/d₁

Wave damping by two thin vertical porous plates placed at finite distance from each other 117

0 0.2 0.4 0.6 0.8 1 1.2

10 20 30 40 50 60 70 80

L/λ Eloss(%)

θ= 0⁰ θ= 15⁰ θ= 30⁰ θ= 45⁰

d1=d 2=h; G

1=G 2=1; k

0h=1.6

Figure 6.15: Energy loss (%) against L/λ with d₁ =d₂ =h/2, G₁ =G₂ = 1, k₀h = 1.6 and N = 30

considered and they are obtained periodically with respect toL/λ. By comparing figure 6.11(b) with figure 6.12(b), it is observed that|T₀|is invariant under the interchange of the positions of the plates.

A study on energy loss (in %) against L/λ is carried out in figure 6.13. First, in figure 6.13(a), energy loss for different values of d₁/d₂ with d₂/h = 0.5 are plotted and periodic nature of the curves is observed. It is also observed that higher values of d₁/d₂ give rise to higher energy loss and the point of optimum values slightly shift towards right with an increase in the value of d₁/d₂. Figure 6.13(b) illustrates energy loss for different values of d₂/d₁ with d₁/h= 0.5. The same nature of the curves is observed as was the case in figure 6.13(a) with an exception that optimum values shift slightly towards left with an increase in the value of d2/d1. Moreover, by comparing figure 6.13(a) with figure 6.13(b), it is observed that greater height of second porous plate results in maximum energy loss in comparison with the energy loss caused by first porous plate with greater height.

An investigation on scattering process for various angles of incidence θ is also carried out in figure 6.14. It is observed that the reflection coefficient shows oscillation with respect to L/λbut with an increase in the value of θ, the period of oscillation increases. Moreover higher maximum reflection is found for lower angle of incidence (figure 6.14(a)). In the case of the transmission coefficient (figure 6.14(b)), the same nature of the curves is observed as in the case of the reflection coefficient with the exception that higher values of θ result in higher maximum transmission.

Energy loss is also plotted against L/λ for different values of θ in figure 6.15. Periodic Linear water wave damping by a bottom-mounted porous structure ... Ph.D. Thesis