6.4 Numerical results and discussion for two identical cylinders submerged in either
6.4.1 Cylinders submerged in the lower layer
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6.4. NUMERICAL RESULTS AND DISCUSSION FOR TWO IDENTICAL CYLINDERS
Note that the functionCL(v) now has a singularity on the real axis only at ν2 but not at ν1 and the path of integration is indented beneath this pole.
This system of equations can be written in matrix form as follows:
Ax=
µ P+I Q R T-I
¶ · a b
¸
= 0, (6.37)
where
P = [Im0 (la)Pnm/Km0 (la)], Q= [Im0 (la)Qnm/Km0 (la)], R = [Im0 (la)Rnm/Km0 (la)], T = [Im0 (la)Tnm/Km0 (la)],
a = [A1m], b= [Bm1 ], (6.38)
for m ≥ 0 and n ≥ 0. For non-trivial solutions, we need to find the frequencies for which the determinant of the complex matrixA vanishes. To find such frequencies, we truncate the matrixAto a 2M×2M one and calculate the determinant. This process can be repeated for symmetrical arrangements involving larger numbers of cylinders.
Numerical Results
We consider a wave with wavenumberu2 propagating fromx=−∞of the form exp[ib(x−ξ)], b = p
u22−l2 = u2cosαinc, making an angle αinc to the positive x-axis and incident upon the cylinder centred at (ξ, f) assuming that the other cylinder does not affect the interaction between the wave and the cylinder under consideration. For a fixed geometrical configuration and a fixed density ratio, the problem of finding the trapped mode frequencies is completely specified by the two non-dimensional parameters Ka and ξ/a. For a fixed value of ξ/a, the parameterKais varied to locate the zeros of the real part of the truncated determinant. Then corresponding to those values of Ka, the absolute values of the determinant are plotted. In all the figures, the fixed values considered are: M = 8, ρ = 0.50 and αinc = 0.34. Note that αincis such that there are no propagating waves in the upper layer for all Ka. For a two-layer fluid consisting of fresh water and salt water, the value ofρ would ideally be around 0.97. The same qualitative features are observed for such a density ratio, but the effects of the interface are not very distinct. Therefore we considerρ = 0.5 for our problem in order to have a clear observation.
When both the cylinders are placed in the lower layer, the values ofξ/aare considered up to 8.0 and then the variation of trapped frequencies is observed forξ/a∈[1,8]. In Figs. 6.2, 6.3 and 6.4, the depth d/a of the upper layer is taken as 2.0 and the submergence depth f /a as −1.1. In Fig. 6.2, Ka is plotted against ξ/a with the values of the ice parameters taken as zero which presents the free surface problem investigated in Linton and Cadby (2003). The curves in part (a) of this figure as well in all the subsequent figures will be treated as modes.
In Figs. 6.3 and 6.4, the flexural rigidityD/a4 is considered as 0.001 and 0.01, respectively – a change from Fig. 6.2 and this implies that the free surface is replaced by a thin ice-cover.
It is observed from Figs. 6.3 and 6.4, upon this replacement, that the first and second modes remain unchanged. With an increase inD/a4, the frequencyKafor the third mode decreases.
Within the specified range ofξ/a, the number of points for which trapped waves exist decreases with an increase inD/a4.
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6.4. NUMERICAL RESULTS AND DISCUSSION FOR TWO IDENTICAL CYLINDERS SUBMERGED IN EITHER OF THE LAYERS
1 2 3 4 5 6 7 8
0 0.2 0.4 0.6 0.8 1 1.2
ξ/a Ka
1st mode 2nd mode 3rd mode
1 2 3 4 5 6 7 8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ξ/a
|Det A|
(a) (b)
Figure 6.2: (a) Values of Ka at which the real part of the determinant vanishes and (b) the absolute values of the determinant of the complex matrix for two cylinders of equal radius a submerged in the lower layer; d/a = 2, f /a = −1.1, ρ = 0.5, αinc = 0.34, D/a4 = 0 and ε/a= 0.
1 2 3 4 5 6 7 8
0 0.2 0.4 0.6 0.8 1
ξ/a Ka
1st mode 2nd mode 3rd mode
1 2 3 4 5 6 7 8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ξ/a
|Det A|
(a) (b)
Figure 6.3: (a) Values of Ka at which the real part of the determinant vanishes and (b) the absolute values of the determinant of the complex matrix for two cylinders of equal radius a submerged in the lower layer; d/a = 2, f /a=−1.1, ρ = 0.5, αinc = 0.34, D/a4 = 0.001 and ε/a= 0.001.
1 2 3 4 5 6 7 8 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ξ/a Ka
1st mode 2nd mode 3rd mode
1 2 3 4 5 6 7 8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ξ/a
|Det A|
(a) (b)
Figure 6.4: (a) Values of Ka at which the real part of the determinant vanishes and (b) the absolute values of the determinant of the complex matrix for two cylinders of equal radius a submerged in the lower layer; d/a = 2, f /a =−1.1, ρ = 0.5, αinc = 0.34, D/a4 = 0.01 and ε/a= 0.001.
Figures 6.5 and 6.6 show the variation of these modes for an increase in the depth d/a of the upper layer. In both the figures, the submergence depth f /a is taken as −1.1 and the values of the ice parameters as D/a4 = 0.001 and ²/a = 0.001. In this case also, there is no variation in the first mode. For the second mode, the values ofξ/a for which trapped waves exist increase with an increase in the depth of the upper layer. For the third mode, as can be seen from Fig. 6.6(a), frequency Kadecreases with an increase in the depth of the upper layer. From Fig. 6.6(b), it is observed that with a decrease in the depth of the upper layer, more pointsξ/aoccur for which trapped waves exist.
Three different submergence depths are considered in Figs. 6.7 and 6.8: f /a=−1.05,−1.10,
−1.15. For both figures, the depth d/a of the upper layer is taken as 2.0 and the non- dimensionalized ice parameters are fixed at 0.001. With the variation of the submergence depthf /a, though the first mode varies but it still does not produce any point on (ξ/a)-axis for which the absolute value of the determinant vanishes. As a consequence, the first mode does not give rise to any trapped waves within the specified range ofξ/a. For the second mode, the values ofξ/a for which trapped waves exist increase as the submergence depth increases, as observed by comparing all the (b) parts of Fig. 6.7. Figure 6.8(a) shows that frequencies Ka for the third mode decreases as f /a increases. It can be seen from Fig. 6.8(b) that the values ofξ/a, for which trapped waves exist, increase with an increase in submergence depth.