4.1 Scattering by the porous structure
4.1.6 Results and discussion
The system of equations given by (4.42) is solved by using the in-built program Matlab R2019a. It is to be noted that, in order to utilize the algebraic approach and analytical
4.1. Scattering by the porous structure 101 approach, the occurrence of ill-conditioned matrices must be avoided. In this context, it is very important to choose the appropriate value of the parameters so as to avoid such an occurrence. Since A and A1 are m×n (m ≤ n) complex matrices and B, B1 ∈ C, therefore the solutionsX, X1 ∈C. Therefore, finding an appropriate algorithm to obtain the inverse may be one of the key aspects in numerical approach. In both the methods, we finally have to solve a large-scale linear square system and hence we choose the in-built function algorithm which uses LU factorization with partial pivoting. Proper care must be taken for ill-conditioned systems since it may give some inaccurate solution.
To carry out the computation, some parameter values are fixed as follows: (h−a)/h=
1
25, d/h = 1; m = 1; ρ = 0.97; E/h4 = 105, δ/h = 10−4, θ = 30◦, Kh = 0.1, ν = 0.3 along with values off and in[0.25,2]and[0.3,0.9], respectively. A detailed justification for consideration of these parameter values can be found in Appendix D.
Kh = 0.28 Kh= 0.16 Kh= 0.10 Kh= 0.07 RSM TSM RSM TSM RSM TSM RSM TSM
N = 0 0.866 0.516 0.958 0.671 0.849 0.655 0.660 0.548 N = 2 0.650 0.478 0.712 0.574 0.607 0.520 0.455 0.407 N = 4 0.636 0.457 0.680 0.542 0.573 0.489 0.432 0.387 N = 6 0.609 0.444 0.644 0.518 0.546 0.469 0.417 0.374 N = 8 0.594 0.432 0.630 0.506 0.540 0.463 0.415 0.373 N = 10 0.578 0.422 0.615 0.496 0.532 0.457 0.411 0.369 N = 12 0.568 0.415 0.610 0.492 0.530 0.456 0.410 0.369 N = 14 0.559 0.409 0.603 0.486 0.526 0.453 0.409 0.367 N = 16 0.558 0.408 0.603 0.486 0.526 0.453 0.409 0.367
Table 4.3: Convergence study of RSM and TSM for different Kh with B/(d +h) = 0.75, E/h4 = 105, = 0.8 and γ = 1 + i.
At the outset, we perform the convergence study of the reflection coefficient and trans- mission coefficient against the non-dimensional depthKhcorresponding to different values of evanescent modes N. An analytical approach is considered to find these values. Ta- ble 4.3 shows that the evanescent modes contribute significantly since zero evanescent mode (N = 0) produces a significant difference when compared with other propagating modes. Further, when the number of evanescent modes increases, a very good numerical convergence is attained for N = 16. Therefore, Table 4.3 suggests the consideration of the number of evanescent modes N = 16 to be appropriate for all further computations.
Convergence ofN1, N2 and N3 (the number of points in different intervals) is verified by calculating the reflection and transmission coefficients in both propagating modes. In Table 4.4, values ofRSM, RIM, TSM and TIM are calculated for different values of N1, N2 and N3 by the algebraic least squares method. Table 4.4 suggests that the convergence of the coefficients up to four decimal places are obtained after N1 = N2 = N3 = 1900.
N1 ⇒ 100 400 700 1000 1300 1600 1900 2100 RSM 0.1838 0.1717 0.1693 0.1683 0.1678 0.1674 0.1671 0.1671 RIM 0.7932 0.7958 0.7963 0.7965 0.79656 0.7967 0.7967 0.7967 TSM 0.1149 0.1106 0.1098 0.1095 0.1094 0.1092 0.1092 0.10913 TIM 0.0195 0.0193 0.0193 0.0192 0.0192 0.0192 0.0192 0.0192 Table 4.4: Convergence study of N1 =N2 =N3 with respect to RSM, RIM, TSM and TIM with N = 16, B/(d+h) = 1, E/h4 = 105, = 0.5 and f = 0.75.
Consequently, for calculation by this approach, we consider N1 = N2 = N3 = 1900 throughout this study unless otherwise stated.
Methods Analytical Approach⇓ Linear Algebraic Approach⇓ KB⇒ KB=0.02 KB=0.03 KB=0.05 KB=0.02 KB=0.03 KB=0.05 RSM = 0.44234 0.52568 0.51768 0.46933 0.51006 0.52551 TSM = 0.07652 0.04793 0.03337 0.07928 0.04872 0.03164 RIM = 0.38299 0.37429 0.37283 0.36715 0.37002 0.37115 TIM = 0.25926 0.09052 0.02830 0.18534 0.10637 0.06920 Err = 4.83e−16 4.69e−16 1.13e−16 2.7e−3 2.58e−3 2.53e−3 Table 4.5: Comparison of analytical and linear algebraic approaches with respect to various reflection, transmission coefficients and error Err as defined in (4.49) with B/(d+h) = 1, = 0.5 and γ = 1 + i.
Before analyzing the effect of various parameters, the key question that arises is whether these two approaches lead to a correct result or not. Moreover, complications of badly scaled parameters is one of the drawbacks of linear algebraic procedure. The numerical values of the reflection and transmission coefficients and error Err as defined in (4.49) obtained by the present methods are compared for different values of KB in Table 4.5 from which it is observed that the results match up to two decimal places. This fact provides another check on the correctness of the present results. The error given by relation (4.49) is also computed for these different approaches in Table 4.5. However, it is noticed that a significant difference in error is observed and for both the methods, the error decreases as KB increases. Therefore, the method giving an analytic solution based on the eigenfunction approach provides a better solution than the least squares method based on the concepts of linear algebra of over-determined system of equations arising from the scattering of water waves by a porous structure. Therefore, the analytic approach can be used in all numerical works here. However, this kind of comparison may add an extra support in checking the accuracy while obtaining numerical results.
In Figure 4.6(a,b)-Figure 4.7(a,b), the reflection and transmission coefficients are plot- ted against non-dimensional widthk1,1B in SM and IM corresponding to various values of friction and porosity, respectively. In Figure 4.6(a,b), similar patterns can be observed for
4.1. Scattering by the porous structure 103
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k1,1B
0 0.2 0.4 0.6 0.8 1
RSM,TSM
f=0.75 f=1 f=1.25
RSM TSM
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
k1,1B
0 0.2 0.4 0.6 0.8 1
R IM,T IM
f=0.75 f=1 f=1.25 TIM
RIM
(b)
Figure 4.6: Reflection and transmission coefficients in (a) free surface mode and (b) interfacial mode against non-dimensional width k1,1B for various friction f with = 0.5.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k1,1B
0 0.2 0.4 0.6 0.8 1
RSM,TSM
=0.5
=0.7 T =0.9
SM
RSM
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
k1,1B
0 0.2 0.4 0.6 0.8 1
R IM,T IM
=0.5
=0.7
=0.9
TIM R
IM
(b)
Figure 4.7: Reflection and transmission coefficients in (a) free surface mode and (b) interfacial mode against non-dimensional length k1,1B corresponding to various porosity parameter with f = 1.
both propagating modes. Higher friction gives rise to higher reflection and lower trans- mission. But TSM attains higher values for the lower values of f only in k1,1B < 1.5 whereas the patterns are the opposite for k1,1B ≥ 1.5. Consideration of different values of porositybrings a difference in the behaviour of the reflection and transmission coeffi- cients between two propagating modes. We observe that higher porosity leads to higher reflection and higher transmission for SM whereas lower reflection is observed for IM.
However, no major effect can be observed inTIM for different porosity values. Therefore, higher reflection and lower transmission can be achieved by choosing appropriate porosity and friction values.
It is observed that, after a specific width, the reflection and transmission coefficients change sharply and remain almost constant. The same trend follows for the cases of varying porosity and varying friction. For both cases, the convergence of reflection and transmission modes is attained after some structural width. It is also observed that the propagating modes have an optimum for almost the same structural width. An important observation is that a steady behaviour is noticed for both the propagating modes corresponding to the structural width and subsequently, it may be concluded that,
beyond a specific value, an increase in the width of the structure cannot influence either reflection or transmission.
0 10 20 30 40 50 60 70 80 90
0 0.2 0.4 0.6 0.8 1
R SM,T SM
=0.5
=0.7
=0.9
RSM TSM
(a)
0 10 20 30 40 50 60 70 80 90
0 0.2 0.4 0.6 0.8 1
RIM,TIM
=0.5
=0.7
=0.9
RIM TIM
(b)
Figure 4.8: Reflection and transmission coefficients in (a) free surface mode and (b) interfacial mode versus angle of incidence for different porosity parameters with f = 1 and B/(d+h) = 1.
Figure 4.8(a,b) shows the effect of porosity on the reflection coefficient and trans- mission coefficient against θ in surface mode and interfacial mode, respectively. When the porosity is varied, the reflection coefficient begins from the minimum value to the maximum value in both propagating modes. An opposite behaviour is observed for trans- mission coefficients. With a change in porosity, a negligible difference in transmission coefficients for both SM and IM is observed. Moreover, all transmission graphs tend to 0 with rising θ (TSM for θ > 5◦ and TIM for θ > 25◦). However, in the case of the re- flection coefficients, a significant difference is observed in all graphs and they obtain their maximum corresponding to the angle 90◦. But a change in porosity does not take into account the difference in the incident angle in order to obtain its optimum. For the sur- face mode, lower porosity values give rise to lower reflection but for the interfacial mode, RIM initially exhibits the opposite nature and after attaining the minimum value, they follow the similar behaviour as SM. Subsequently, a study of the wave pattern allows the design of efficient breakwaters with an appropriate selection of the width, step height and friction factor of the structure so as to get the maximum wave reflection which can help in protecting the offshore structures.
Figure 4.9(a,b) shows waveloadsFf andFb, respectively, versusk1,1B for various values of friction. For the initial non-dimensional width (k1,1B ≈ 0), the wave forces on both walls x= 0 and x=B are the same and very much close to 0. But when the structural width is increased, Ff increases and attains its maximum value. Figure 4.9(b) shows that wave force Fb also increases and achieves its highest value but a sharp reduction is observed when the width is increased. This may be attributed to a substantial amount of wave energy getting dissipated by the porous structure. Moreover, the wave forces initially show the highest impact for the lower structural width, and different values of γ
4.1. Scattering by the porous structure 105
0 1 2 3 4 5 6 7
k1,1B
0 0.05 0.1 0.15 0.2
Ff
f=0.5 f=1 f=1.5
(a)
0 1 2 3 4 5 6 7
k1,1B
0 0.05 0.1 0.15 0.2
F b
f=0.5 f=1 f=1.5
(b)
Figure 4.9: Wave forces (a) Ff and (b)Fb against non-dimensional length corresponding to various values off with = 0.8.
do not introduce any difference in attaining optimum value of the waveload. However, it is observed that both the wave forces exhibit a steady behaviour after a specific width. Ff
is observed to have a higher impact corresponding to higher friction f butFb follows this pattern only for the initial width, and afterk1,1B >1.5, the pattern is altered. Moreover, the steady waveload justifies that the energy dissipation is fixed up to a specific structural width beyond which the energy loss does not depend on the structural width.
-10 -8 -6 -4 -2 0 2 4 6 8 10
x/h
-1 -0.5 0 0.5 1 1.5 2 2.5
1/h
f=0.75 f=1 f=1.25
(a)
-10 -8 -6 -4 -2 0 2 4 6 8 10
x/h
-3 -2 -1 0 1 2 3
2/h
f=0.75 f=1 f=1.25
(b)
Figure 4.10: Non-dimensional elevation amplitude for (a) free surface and (b) interfacial modes versus x/hcorresponding to different values of f for B/(d+h) = 2, 1 = 0.8.
Figure 4.10(a,b) shows the free surface elevations η1/h and interface elevation η2/h versus the non-dimensional distancex/hfor various values of friction withxdenoting the horizontal distance measured from x= 0. It shows that η2/h becomes higher than η1/h. We can assume that this difference between the elevations of these two modes arises due to the resonating interaction between the free surface waves and interfacial waves. The elevation reduces significantly due to the presence of the porous structure. The oscillations in both propagating modes increase when the friction coefficient f takes higher values.
However, for the interface, a negligible difference in elevations is observed corresponding to differentf.
-20 -10 0 10 20 30 40 50
x/h
-1 -0.5 0 0.5 1 1.5
1/h
Kh=0.6 Kh=0.3 Kh=0.15
(a)
-1 0 1 2 3 4 5 6
x/h
-2 -1 0 1 2 3
2/h
Kh=0.6 Kh=0.3 Kh=0.15
(b)
Figure 4.11: Non-dimensional elevation amplitude for (a) free surface and (b) interfacial modes versus x/h corresponding to different values of f for B/(d+h) = 2, 1 = 0.8.
Non-dimensional elevation with respect to xis plotted in Figure 4.11(a,b) for various values of Kh. We can easily observe the phase shift that occurs in the elevations in both propagating modes due to the presence of the structure. For the interface, it is observed that the number of oscillations is higher corresponding to the samexdistance on the free surface. There is a continuous phase lag in the vertical displacement response for the free surface, but in the interfacial mode, it is highly dependent on the structure.
Based on the outcome, a suitable choice of parameters pertaining to a porous struc- ture interacting with waves will allow the creation of a tranquillity zone by reducing the waveload on the rigid wall and the porous structure.
E/h4 ⇓ f = 1 f = 1.5
RSM TSM RIM TIM RSM TSM RIM TIM
101 0.9043 0.7683 0.3719 0.2723 0.8658 0.6588 0.4117 0.2778 102 0.8729 0.7683 0.3719 0.2707 0.8369 0.65923 0.41167 0.2762 103 0.8493 0.9931 0.3718 0.2985 0.7671 0.8653 0.4116 0.3032 104 0.9007 0.5353 0.3718 0.2225 0.8592 0.4522 0.4115 0.2342 105 0.6227 0.4126 0.3718 0.2209 0.5971 0.3470 0.4115 0.2331 106 0.6045 0.3995 0.3718 0.2204 0.5795 0.3357 0.4115 0.2329 107 0.6028 0.3981 0.3718 0.2204 0.5779 0.3345 0.4115 0.2329 Table 4.6: Reflection and transmission coefficients against different f corresponding to different values of E with B/(d+h) = 1, = 0.8and f = 1.
The effect of flexural rigidity E of the elastic bottom is examined against different f in Table 4.6. A difference is observed only for the values of E/h4 = 101 to E/h4 = 105, otherwise, all the reflection and transmission coefficients converge to a fixed value in both propagating modes for higher E. Different rigidity E does not contribute to any major difference for reflection and transmission coefficients in interfacial mode. It was also observed in Table 4.1 that the change of flexural rigidity did not contribute any
4.1. Scattering by the porous structure 107
-6 -4 -2 0 2 4 6 8
x/h
-40 -20 0 20 40
S F
f=0.75 f=1 f=1.25
5.2 5.4 5.6 5.8 6 6.2
-10 -8 -6 -4
Zoom View
(a)
-6 -4 -2 0 2 4 6 8
x/h
-50 0 50
S F
B/h=0.8 B/h=2 B/h=3.2
8 9 10 11 12
-3 -2 -1 0 1
Zoom View
(b)
Figure 4.13: Shear force against non-dimensional x/h for (a) different values of friction f, (b) different values of non-dimensional structural width with = 0.8, E/h4 = 107 and (a) B/(d+h) = 1, (b) f = 1.
major difference in the dispersive roots in interfacial propagation. This may be one of the reasons for such behaviour ofRIM and TIM.
-40 -20 0 20 40 60
x/h
-4 -3 -2 -1 0 1 2
/h
f=0.75 f=1 f=1.25
(a)
-60 -40 -20 0 20 40 60 80 100
x/h
-4 -3 -2 -1 0 1
/h
E=105h4 E=106/h4 E=107h4
49.8549.949.95 50 50.0550.150.15 -0.7
-0.698 -0.696 -0.694 -0.692 -0.69
Zoom view
(b)
Figure 4.12: Non-dimensional bottom deflection amplitude against non-dimensional x/h for (a) various values of friction f, (b) various values of non-dimensional flexural rigidity with B/(d+h) = 2, = 0.8and (a) E/h4 = 107, (b) f = 1.
Furthermore, the impact of the friction parameter as well as of the flexural rigidity upon the elastic bottom deflection is also plotted against the non-dimensional distance in Figure 4.12(a,b). It can be observed that the elastic bottom attains the lowest amplitude of deflection corresponding to the lowest value of f. Further, ζ attains 0 value at both the edges of the elastic bottom plate which also satisfies the clamped edge conditions.
Moreover, a major reduction in the amplitude can be observed clearly for the bottom deflection due to the structure. Deflection for variousE maintains the same value which clearly shows that, after some value of the elastic rigidity, the bottom does not have any impact for such a wave interaction system.
Figure 4.13(a,b) presents the shear force on the elastic bottom for different values of the friction and different values of structural length. It reveals that the shear force
achieves the lowest value near the edge (0,−h). Further, the forces exhibit an oscillatory behaviour similar to the deflection pattern of the elastic bottom. Figure 4.13(a) shows the variation in the shear force acting on the elastic bottom for different values of the friction f. For the interaction of the waves with the structure after x > B/h, the amplitude of the shear force decreases corresponding to an increase in f. Lower f results in lower SF
and the oscillation reduces to the one with increasing x. Furthermore, Figure 4.13(b) shows that, for an increase in B/h - the width of the porous structure, the amplitude of shear force on the elastic bottom decreases. This fact can be understood clearly from the argument that a porous structure of larger width can cause greater dissipation of energy, which is responsible for lesser shear force acting on the bottom. The fact that an increase in the width B/h of the porous breakwater gives rise to a greater reflection and dissipation of wave energy, as observed in Figure 4.6(a,b) and Figure 4.9(b), supports this observation.