3.3 Trapping Problem
3.3.2 Numerical results and discussion
Cw, the horizontal wave force per unit length at the rigid sea-wall, is given by
Cw = iω
ρ2 0
Z
−h1
φ4(x, z)dz+ρ1 d
Z
0
φ4(x, z)dz
atx= (L1+L2 +L3), (3.58)
3.3. Trapping Problem 75 and the non-dimensional form of the hydrodynamic force coefficient, denoted by Kw, is evaluated from
Kw = |Cw|
ρ2gdh1. (3.59)
0 0.5 1 1.5
L3/ 1 0
0.2 0.4 0.6 0.8 1
|R1|
f2=0.8 f2=1 f2=1.2
(a)
0 0.5 1 1.5
L3/ 2 0
0.2 0.4 0.6 0.8 1
|R 2|
f2=0.8 f 2=1 f
2=1.2
(b)
Figure 3.8: Variation of (a)RSM and (b)RIM againstL3/λj forj = 1,2corresponding to different values off2 with ρ= 0.7, θ= 0◦, (L1+L2)/h1 = 0.2, (h1−h2) =h1/25, f3 = 1 and 2 = 0.45, 3 = 0.5.
In Figure 3.8(a,b), reflection coefficients are plotted versus normalized distance be- tween the breakwater and the rigid wallL3/λj forj = 1,2(λj = 2π/k1,j) in both propagat- ing modes for various values of friction parameterf2. In both propagating waves, the same oscillating behaviour is observed, and lower friction off2(≤f3)results in higher reflection.
For both modes, due to the composite thick breakwater, resonating pattern of reflection is observed in almost periodic intervals within the range (2n−1)λj/4< L3 <(2n+ 1)λj/4 for each integer value n. Furthermore, minimum reflection in both modes is achieved in the region(m−1)λj/2< L3 < mλj/2 forj = 1,2, m= 1,2, . . .. However, the minimum value depends on the friction factor of the structure. These minimum values in wave reflection are referred as wave trapping in the confined zone between the barrier and the sea-wall. Further, it is observed that the optima are obtained for the same normalized distance L3 for any friction factor f2. Therefore, to protect the shoreline and coastal areas, the use of block-wise composite breakwater is considered ideal for obtaining higher or lower reflection in both modes.
Figure 3.9(a,b) shows the impact of changing structural width (L1+L2) on reflection coefficients in which differences are observed for overall reflection in both propagating modes. Lower values of structural width result in higher reflection for both modes. Com- paring with Figure 3.8(a,b), we observe the similarity in obtaining the optimum values for certainL3/λj. Furthermore, the optimum value of wave reflection is dependent upon the structural width for both cases. As the width of the structure increases, more amount of wave energy gets reflected. The minimum value of reflection for a certain structural width is comparatively higher for SM than that for IM. For both propagating modes, certain
0 0.5 1 1.5
L3/ 1
0 0.2 0.4 0.6 0.8 1
|R 1|
(L1+L2)/h1=0.2 (L1+L2)/h1=0.4
(L1+L2)/h1=0.6 (L1+L2)/h1=0.8
(a)
0 0.5 1 1.5
L3/ 2
0 0.2 0.4 0.6 0.8 1 1.2
|R 2|
(L1+L2)/h1=0.2 (L1+L2)/h1=0.4 (L1+L2)/h1=0.6 (L1+L2)/h1=0.8
(b)
Figure 3.9: Variation of (a) RSM and (b) RIM against normalized distance L3/λj forj = 1,2corresponding to different values of (L1+L2)/h1 with L3/h1 = 0.6, f2 = 1.25, f3 = 1 and 2 = 0.7, 3 = 0.6.
shift toward left in the minimum value is observed with increasing values of (L1+L2)/h1 which may be due to the phase shift of the incoming and outgoing waves in surface mode and interface mode. Therefore, the overall reflection achieved due to the interface-piercing breakwater is mainly dependent on the physical properties of the structure.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
K(L1+L2)
0 0.2 0.4 0.6 0.8 1
|R 1|
h2/h 1=0.6 h2/h
1=0.7 h2/h
1=0.8 h2/h
1=0.9
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
K(L1+L2)
0 0.2 0.4 0.6 0.8 1
|R 2|
h2/h 1=0.6 h2/h
1=0.7 h2/h
1=0.8 h2/h
1=0.9
(b)
Figure 3.10: Variation of (a) RSM and (b) RIM against K(L1 +L2) corresponding to different values of h2/h1 with L3/h1 = 16, f2 = 0.75, f3 = 1 and 2 = 0.6, 3 = 0.7.
Figure 3.10(a,b) describes the effect of various values of depth ratioh2/h1 on reflection coefficients with depthh1 fixed. If we observe the patterns, lower depth ratio causes max- imum reflection although the difference in h2/h1 brings negligible difference in reflection for IM. An increase in the length of the rigid block implies a reduction of the size of the porous structure and therefore, relatively less amount of wave passes through the porous structure. This reduces the dissipation of energy by the porous block which may be the reason for occurrence of higher reflection in both modes for lowerh2/h1. It is also observed that higher step ratio (h1−h2)/h1 results in higher wave reflection in SM. Simillar result is observed at subsection 2.5.1.
3.3. Trapping Problem 77
0 1 2 3 4 5 6 7 8 9
k1,1(L1+L2)
0.04 0.05 0.06 0.07 0.08 0.09
Kf,2
f2=1,f 3=0.5 f2=1,f
3=1 f1=2,f
3=1.5
(a)
0 1 2 3 4 5 6 7 8 9
k1,1(L1+L2)
0 0.02 0.04 0.06 0.08 0.1
K w
f2=1,f 3=0.5 f2=1,f
3=1 f2=1,f
3=1.5
(b)
Figure 3.11: Variation wave forces (a) Kf,2 and (b) Kw against non-dimensional length k1,1(L1 +L2) corresponding to various values of f3 with f2 = 1, L3/λ1 = 1, and 2 = 0.7, 3 = 0.6.
Figure 3.11(a,b) describes the effect of waveloads Kf,2 and Kw, respectively, corre- sponding to different values of friction parameter. Comparing Figures 3.11(a) and 3.11(b), we observe that the wave forceKwreduces sharply with an increase in the width and tends to negligible waveload for higher breakwater width. This amply justifies the dissipation of a large amount of wave energy by the porous structure. However, both the wave forces converge to a steady state after some width which is also justified by the occurrence of steady reflection for higher structural width which we observed earlier. Both wave forces produce a higher impact for lower friction f3 with f2 ≥ f3. Further, the wave forces on the first porous block initially show a difference due to the impact of different friction values but later on, all the graphs converge to one. But the steady nature of waveload also justifies that the dissipation by the structure is fixed up to some structural length beyond which the energy loss is independent of the structure length.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
L3/ 1 0.03
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
K w
L1/h 1=0.8,L
2/h 1=0 L1/h
1=L 2/h
1=0.4(Simple blocks) L1/h
1=L 2/h
1=0.4(Composite blocks) L1/h
1=0.2,L 2/h
1=0.6(Composite blocks)
Figure 3.12: Variation wave force Kw against normalized distance L3/λ1 corresponding to various values of L2/h1 with the cases (i) f2 = 1 and 2 = 0.6,(ii) f2 = f3 = 1 and 2 = 3 = 0.6,(iii) f2 = 1, f3 = 1.5 and 2 = 0.6, 3 = 0.7,(iv) f2 = 1, f3 = 1.5 and 2 = 0.6, 3 = 0.7.
In Figure 3.12, the impact of a fixed width breakwater ((L1 +L2)/h1 = 0.8) with various structural configurations is examined on the basis of the influence of waveload on
the sea-wall. Varying the normalized length, we can easily observe that the waveload is much higher in the case of a single block breakwater. As already explained in Section 3.2.2, the equal porosity and equal friction (i.e., 2 = 3, γ2 = γ3) merge Regions 2 and 3 into a single one due to which the two-block structure gets converted into a single block.
Therefore, the cases of a single block (L1/h1 = 0.8, L2/h1 = 0) and two blocks (L1/h1 = L2/h1 = 0.4) with same porosity and friction produce the same results. Moreover, it clearly shows that the two-block structure with different porosity and friction (in cases (iii) and (iv) in Figure 3.12) result in higher dissipation of energy and among them, the structure with unequal block widths (case (iv) (L1+L2)/h1 = 0.8, but L1/h1 6=L2/h1) results in the lowest waveload among all the cases. For a fixed set of parameters in the confined zone between the breakwater and the sea-wall, if the overall waveload is evaluated in L2 norm, then case (iii) results in nearly 2% less waveload compared to the cases (i) and (ii). For a composite structure with unequal block widths (case (iv)), nearly a 4%
lower waveload is observed than cases (i) and (ii). Therefore, composite structures are found to be comparatively more efficient as a wave-absorber than a single block structure, and an adjustment in the attributes of the composite structure results in better depletion.
-20 -15 -10 -5 0 5 10 15
x/h 1 -0.1
-0.05 0 0.05 0.1
1/h1
f2=f 3=0.6 f2=f
3=0.9 f2=f
3=1.2 f2=f
3=1.5
(a)
-20 -15 -10 -5 0 5 10 15
x/h 1 -0.2
-0.1 0 0.1 0.2
2/h1
f2=f 3=0.6 f2=f
3=0.9 f2=f
3=1.2 f2=f
3=1.5
(b)
Figure 3.13: Variation of the non-dimensional elevation amplitude for (a) free surface and (b) interface againstx/h1 corresponding to various values off2 =f3 with (L1+L2)/h1 = 0.8, L3/λ1 = 1 and2 = 0.7, 3 = 0.9.
Figure 3.13(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, wherexis the distance in the horizontal direction measured from x = 0. It shows that the elevation at the interface becomes higher than the elevation at the free surface. However, there is a significant reduction of elevation at both propagating modes in the confined zone between the porous structure and the wall.
On both sides of the breakwater, each propagating mode shows that lower friction results in higher oscillation. Before the penetration of the wave into the structure, the elevations exhibit a negligible difference between them but an appropriate dissipation effect on the elevation is clearly visible in the confined zone between the breakwater and the sea-wall.
3.4. Composite breakwater with a perforated front wall 79