2.4 Results and discussion
2.4.2 Porous blocks with different porosity and friction
2.4. Results and discussion 49 Figure 2.15(a,b) presents the elevations η1/h1 at the free surface and η2/h1 at the interface, respectively, plotted against the non-dimensional distance x/h1 corresponding to different values of friction coefficients for f1 = f2, where x is the distance in the horizontal direction measured fromx= 0. Both the figures show that the elevation at the interface becomes higher than the elevation at the free surface. We can assume that this difference between the elevations of these two modes is observed due to the resonating interaction between the free surface waves and interfacial waves. The elevations in both propagating modes increase when the friction fj takes higher values. However, for both modes, we can observe a significant reduction of elevations in the second porous region.
0 0.5 1 1.5 2 2.5 3 3.5
K(L+D)
0 0.2 0.4 0.6 0.8 1
R SM
f1=0.5,f 2=1 f1=0.75,f
2=1 f1=1,f
2=1 f1=1.25,f
2=1
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
K(L+D)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
R IM
f1=0.5,f 2=1 f1=0.75,f
2=1 f1=1,f
2=1 f1=1.25,f 2=1
(b)
Figure 2.17: Variation of (a)RSM and (b) RIM against non-dimensional widthK(L+D) corresponding to various values of friction parameter f1 with f2 fixed, 1 =2 = 0.75and θ = 0◦.
in Figures 2.7 and 2.8. The changes in the reflection coefficients in SM and IM establish that, after a certain width of the structure, the reflection in both modes reach steady states and therefore, there is an effect of j and fj which is, however, not significant for wider structures. Although different porosity (1 6=2, Figure 2.16(a,b)) has no significant effect on total non-dimensional width but keeping the porosity lower for the first block results in lower reflection compared to the second one, and consequently, suitable width of the block should be chosen to prevent higher reflection in both modes. Alongside, the varying friction parameter also results in (i) lower reflection for f1 > f2 and (ii) higher reflection forf1 < f2than those for the casef1 =f2. On the basis of our graphical results, if we make a comparison between a simple structure (1 = 2, f1 =f2) and a composite structure (1 6=2, f1 6=f2), we observe that using a composite structure can have, on an average, a 18% higher reflection than that for the simple structure in SM. On the other hand, in IM, the composite structure can have nearly 12% higher reflection than that for the simple structure.
Figures 2.18(a) to 2.18(d) illustrate the effect of oblique angleθon the reflection coeffi- cient in surface mode and interface mode, respectively, for various porosityj and friction parameterfj keeping the porosity or friction of one of the blocks fixed. In Figure 2.18(a), the change of porosity does not affect the reflection in a significant way in SM, although in IM, all curves for different j and fj maintain a similar nature for all the cases (Fig- ure 2.18(b)). Both RSM and RIM attain their minimum values in the ranges 80◦-90◦ and 70◦-90◦ in surface mode and interfacial mode, respectively, for θ (for both cases).
As seen in Figures 2.9(a,b)-2.10(a,b), in SM, the incident angle for obtaining minimum differs to some extent for different porosity or friction but it is meanwhile the same in IM.
Therefore, porosity and friction parameter behave in a similar manner against the angle of incidence for both cases. It can be concluded that it is advisable to use composite blocks to construct barriers to get minimum wave reflection.
2.4. Results and discussion 51
0 10 20 30 40 50 60 70 80 90
0 0.2 0.4 0.6 0.8 1
RSM
1=0.8, 2=0.6 1=0.8,
2=0.7 1=0.8,
2=0.8 1=0.8,
2=0.9
(a)
0 10 20 30 40 50 60 70 80 90
0.99 0.992 0.994 0.996 0.998 1
R IM
1=0.8, 2=0.6 1=0.8,
2=0.7 1=0.8,
2=0.8 1=0.8,
2=0.9
(b)
0 10 20 30 40 50 60 70 80 90
0 0.2 0.4 0.6 0.8 1
RSM
f1=0.75,f 2=1 f1=1,f
2=1 f1=1.25,f
2=1 f1=1.5,f
2=1
(c)
0 10 20 30 40 50 60 70 80 90
0.99 0.992 0.994 0.996 0.998 1
R IM
f1=0.75,f 2=1 f1=1,f
2=1 f1=1.25,f
2=1 f1=1.5,f
2=1
(d)
Figure 2.18: Variation ofRSM andRIM against incident angleθ corresponding to various values of porosity and friction parameter keeping one of the parameter fixed,f1 =f2 = 1 (for (a,b)), 1 =2 = 0.8 (for (c,d)) and(L+D)/h1 = 1/4.
0 10 20 30 40 50 60 70 80 90
0 0.2 0.4 0.6 0.8 1
RSM
(L+D)/h 1=0.5 (L+D)/h
1=0.55 (L+D)/h1=0.6 (L+D)/h1=0.65 (L+D)/h1=0.7
(a)
0 10 20 30 40 50 60 70 80 90
0.99 0.992 0.994 0.996 0.998 1
R IM (L+D)/h1=0.5 (L+D)/h1=0.55 (L+D)/h1=0.6 (L+D)/h1=0.65 (L+D)/h1=0.7
(b)
Figure 2.19: Variation of (a)RSM and (b)RIM against incident angleθ corresponding to various values of (L+D)/h1 with f1 =f2 = 1 and 1 = 0.8,2 = 0.9.
Figure 2.19(a,b) presents the reflection coefficientsRSM andRIM plotted against inci- dent angleθ in SM and IM corresponding to various values of(L+D)/h1. The minimum in reflection for various values of (L+D)/h1 occurs for the same incident angle in any mode. As in Figure 2.11(a,b), here also we observe a similar nature of the reflection coef- ficients. Even the minimum of reflection also occurs for the same range of angles in both modes as seen in Figure 2.11(a,b). Higher values of(L+D)/h1 results in higher reflection
for SM. On the other hand, for initial incident angles, we observe an opposite nature for IM but as the incident angle increases, a similar nature is observed.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
K(L+D)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|RI|
=0.9
=0.8
=0.7
=0.6
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
K(L+D)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|RII|
=0.9
=0.8
=0.7
=0.6
(b)
Figure 2.20: Variation of (a) RSM and (b) RIM against non-dimensional total width of the structure K(L+ D) corresponding to various values of ρ with f1 = 1, f2 = 1.2, 1 = 0.9, 2 = 0.8and θ = 30◦.
Figure 2.20(a,b) shows reflection coefficients plotted against the non-dimensional total width of the blocks corresponding to different values of density ratioρ. A similar pattern is observed here also, as in Figure 2.12, for both propagating modes. In SM, a lower density ratio gives higher reflection and vice versa for IM. The same explanation provided for higher reflection in interface mode as in Figure 2.12(b) holds good for the case of Figure 2.20(b). For both propagating modes, the reduction of density ratio exhibits a convergent behaviour in wave reflection.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k1,I(L+D)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
F w 1=0.6,
2=0.8 1=0.7,
2=0.8 1=0.8,
2=0.8 1=0.9,
2=0.8 1=
2=1
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k1,I(L+D)
0 1 2 3 4 5 6 7 8
F RW 10-3
1=0.6, 2=0.8 1=0.7,
2=0.8 1=0.8,
2=0.8 1=0.9,
2=0.8
(b)
Figure 2.21: Variation of wave force (a)Fw and (b)FRW againstk1,I(L+D)corresponding to various values of porosity with f1 =f2 = 1 and θ = 30◦.
Figures 2.21(a,b) and 2.22(a,b) look into the behaviour of the wave force Fw and force reductionFRW againstk1,I(L+D)corresponding to different values of porosity and friction. The patterns of wave forceFwon the rigid wall is similar in general in comparison to those seen in Figure 2.14. In both cases, the reduction of wave force is very high for
2.4. Results and discussion 53
0 1 2 3 4 5 6
k1,I(L+D)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Fw f
1=0.5,f 2=1 f1=0.75,f
2=1 f1=1,f
2=1
f1=1.25,f 2=1 f1=f
2=0
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k1,I(L+D)
0 1 2 3 4 5 6 7 8
F RW 10-3
f1=0.5,f 2=1 f1=0.75,f
2=1 f1=1,f
2=1 f1=1.25,f
2=1
(b)
Figure 2.22: Variation of wave force (a)Fw and (a)FRW againstk1,I(L+D)corresponding to various values of friction parameter withθ = 30◦ and 1 =2 = 0.8.
lower structural width, although at some particular width, the reduction of force assumes a steady rate. The effect of the presence of the porous structure is also clearly observed by plotting for the numerical dataf1 =f2 = 0and 1 =2 = 1 which converts the porous region into a plain water one. It can be clearly observed that, without the presence of the structure, the waveload is much higher. It also signifies the effect of wave dissipation due to the presence of porous structure. From Figures 2.21(b) and 2.22(b), we can observe that using a composite structure(1 6=2, f1 6=f2) can result in a better reduction of wave force atx=L compared to that for a simple structure(1 =2, f1 =f2). Also, Figures 2.21(a) and 2.22(a), respectively, show that approximately2%and7%lower wave impact is made possible by using a composite structure rather than a simple structure. Even though this comparison is solely on the basis of the results found for some sample parameters, we are confident that it can support our conclusion to use composite structures as wave- absorbers. Moreover, we observe that the waveload Fw is lower for lower2 (1 ≥2) and lower f2 (f1 ≥f2). Reduction of wave force is also observed to be lower under the same consideration at the vertical boundary between the blocks.
-15 -10 -5 0 5 10 15
x/h1 -0.06
-0.04 -0.02 0 0.02 0.04 0.06
1/h 1
f1=0.5,f 2=1 f1=0.75,f
2=1 f1=1,f
2=1 f1=1.25,f
2=1
(a)
-15 -10 -5 0 5 10 15
x/h1 -0.08
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
2/h 1
f1=0.5,f2=1 f1=0.75,f2=1
f1=1,f2=1 f1=1.25,f2=1
(b)
Figure 2.23: Variation of non-dimensional elevation against x/h1 at (a) free surface and (b) interface corresponding to various values of 1 with L/h1 = D/h1 = 0.5 and 1 = 0.6, 2 = 0.8 and θ = 30◦.
Figure 2.23(a,b) presents free surface and interface elevations againstx/h1 correspond- ing to values of friction parameter f1 of the first porous region. Similar to Figure 2.15, here also, the interface elevations are comparatively higher than the free surface elevation.
Effect of the porous blocks brings a reduction in the elevations between the porous regions in front of the rigid wall. Therefore, the casef1 ≤f2 results in lower elevation compared to the case f1 > f2. It also justifies our conclusion to take lower friction in the first block along with other geometrical configurations to construct an efficient breakwater.
Subsequently, by an appropriate selection of physical parameters for a composite type structure interacting with waves, a tranquillity zone can be created which will reduce waveload on the rigid wall and the porous structure to a large extent.