3.4 Numerical Results
3.4.2 Cylinder submerged in the upper layer
Figure 3.5 shows the plot of trapped mode wavenumbers against the density ratio when the cylinder is placed in the upper layer. For both wavenumbers, different pairs of curves, each pair consisting of a thick line and a dashed line, correspond to the four different submergence depths of the cylinder: f /a = 1.05, 1.06, 1.07 and 1.10. A third mode exists corresponding only to f /a = 1.10 for very small density ratios, as is clearly evident from the figure. It is further observed that corresponding to each value of the submergence depth, there are at least two curves for each of the wavenumbersu1aandu2a. For the wavenumberu1a, the first modes remain constant while the second modes, the higher curves, decrease and appear to cross the
0 0.5 1 1.5 2 2.5 3 3.5 4 0
0.5 1 1.5 2 2.5 3 3.5 4
la u2a
f/a = −1.01 f/a = −1.02 f/a = −1.03 f/a = −1.05 f/a = −1.10 f/a = −1.30
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.5 1 1.5 2 2.5 3 3.5 4
la u2a
f/a = 1.01 f/a = 1.05 f/a = 1.15 f/a = 1.30
(b)
Figure 3.3: Dispersion curves for a cylinder of radius a (a) in the lower layer for different submergence depths f /a; ρ = 0.95, d/a = 3, h/a = 6, D/a4 = 0.001 and ε/a = 0.001 (b) in the upper layer for different submergence depths f /a; ρ = 0.95, h/a = 6, d/a = 3, D/a4 = 0.001 and ε/a= 0.001.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.5 1 1.5 2 2.5 3 3.5 4
la u2a
h/a = 2.05 h/a = 2.50 h/a = 3.00 h/a = 15.0
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.5 1 1.5 2 2.5 3 3.5 4
la u2a
d/a = 2.05 d/a = 2.50 d/a = 6.00
(b)
Figure 3.4: Dispersion curves for a cylinder of radiusa(a) in the lower layer for different depths h/a of the lower layer; ρ = 0.95, d/a = 3, f /a =−1.01, D/a4 = 0.001 and ε/a = 0.001 (b) in the upper layer for different depthsd/a of the upper layer; ρ= 0.95, h/a= 6, f /a= 1.01, D/a4 = 0.001 and ε/a= 0.001.
first modes at some points. However, we observe near crossing of the modes at these points.
If the density ratioρ increases further, the second modes remain constant and terminate at a certain value of ρ for different submergence depths whereas the first modes decrease to zero.
For the wavenumber u2a, the first modes increase and the second modes decrease with an increase in density ratio and they come very close to each other at near crossing points (Fig.
3.6). As ρ increases further, the modes interchange their properties, i.e., the second modes increase and terminate atla= 2 corresponding to a fixed value ofρfor all submergence depths while the first mode decreases to some fixed value for each submergence depth.
Figure 3.7 shows the trapped mode wavenumbers where the different curves correspond to four different depths of the upper fluid layer: d/a = 2.05, 2.15, 2.25 and 2.50. This figure corresponds to the case when the cylinder is submerged nearer the interface and it is observed
38 3.4. NUMERICAL RESULTS
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u1a
f/a = 1.05 f/a = 1.06 f/a = 1.07 f/a = 1.10
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u2a
f/a = 1.05 f/a = 1.06 f/a = 1.07 f/a = 1.10
(b)
Figure 3.5: Trapped mode wavenumbers plotted againstρfor a cylinder of radiusain the upper fluid layer for different submergence depths f /a; la = 2, d/a = 2.1, h/a = 6, D/a4 = 0.001 andε/a= 0.001.
0.25 0.3 0.35 0.4 0.45 0.5
1.2 1.3 1.4 1.5 1.6 1.7 1.8
ρ u2a
f/a = 1.05 f/a = 1.06 f/a = 1.07 f/a = 1.10
Figure 3.6: Trapped mode wavenumbers plotted againstρfor a cylinder of radiusain the upper fluid layer for different submergence depthsf /a, close-up of near crossing; la= 2, d/a= 2.1, h/a= 6, D/a4= 0.001 and ε/a= 0.001.
that two trapped modes exist for depths approximately upto 2.5 and hence near crossing points also exist. With an increase in the depth of the upper layer, these near crossing points shift upwards tola= 2 and when the depth increases further, only the first mode always exists and it becomes independent of the depth of the upper layer.
Figure 3.8 shows the plot of trapped mode wavenumbers for different depths of the lower layer: h/a = 0.5, 1.0, 2.0 and 6.0. The depth d/a of the upper layer is considered as 3.0 and hence based on the discussion of Fig. 3.7, there exists only one mode. The trapped mode wavenumber u1a increases with an increase in the depth of the lower layer for small density ratios. Then as ρ increases further towards 1, these modes cross each other at some points, and for the density ratios≥0.95, the wavenumberu1adecreases with an increase in the depth of the lower layer (Fig. 3.9). As the depth of the lower layer increases, the trapped mode wavenumber u2a decreases. As ρ → 1, the wavenumber u1a → 0 and each wavenumber u2a tends to some finite value for each depth of the lower layer.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u1a
d/a = 2.05 d/a = 2.15 d/a = 2.25 d/a = 2.50
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u2a
d/a = 2.05 d/a = 2.15 d/a = 2.25 d/a = 2.50
(b)
Figure 3.7: Trapped mode wavenumbers plotted against ρ for a cylinder of radius a in the upper fluid layer for different depths d/a of the upper layer; la = 2, f /a = 1.01, h/a = 6, D/a4 = 0.001 and ε/a= 0.001.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u1a
h/a = 0.5 h/a = 1.0 h/a = 2.0 h/a = 6.0
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u2a
h/a = 0.5 h/a = 1.0 h/a = 2.0 h/a = 6.0
(b)
Figure 3.8: Trapped mode wavenumbers plotted against ρ for a cylinder of radius a in the upper fluid layer for different depthsh/a of the lower layer h/a;la= 2, f /a= 1.01, d/a= 3, D/a4 = 0.001 and ε/a= 0.001.
The different pairs of curves, each pair consisting of a thick line and a dashed line in Figs.
3.10(a) and 3.10(b), correspond to different sets of ice parameters (D/a4 = 0.001;ε/a= 0.001), (D/a4 = 0.01;ε/a = 0.01), (D/a4 = 0.1;ε/a = 0.01) and (D/a4 = 1.0;ε/a = 0.01). For the first two sets, there exist two modes and hence near crossing points also exist. For the last two sets of ice-parameters, there exists only one mode. For d/a = 2.1, the existence of near crossing points, when the upper layer is bounded above by a thin ice-cover, has already been observed. If the value of ice-parameter D/a4 is increased, then the second mode for both wavenumbers ceases to exist. In other words, with an increase in flexural rigidity, the ice-cover behaves more like a rigid lid and consequently there exists only one mode corresponding to both wavenumbers which is very similar to the modes found in the rigid lid problem discussed in Chapter 2.
When the cylinder is placed in the upper layer, the limit ρ→ 0 corresponds toρII → ∞.
Consequently the dispersion relation reduces tou(Du4+1−εK) =Kcothudand the multipoles
40 3.4. NUMERICAL RESULTS
0.950 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ρ u1a
h/a = 0.5 h/a = 1.0 h/a = 2.0 h/a = 6.0
Figure 3.9: Trapped mode wavenumbers plotted against ρ for a cylinder of radius a in the upper fluid layer for different depths h/a of the lower layer, close-up; la = 2, f /a = 1.01, d/a= 3,D/a4= 0.001 and ε/a= 0.001.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u1a
D/a4 = 0.001, ε/a = 0.001 D/a4 = 0.01, ε/a = 0.01 D/a4 = 0.10, ε/a = 0.01 D/a4 = 1.0, ε/a = 0.01
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ρ u2a
D/a4 = 0.001, ε/a = 0.001 D/a4 = 0.01, ε/a = 0.01 D/a4 = 0.10, ε/a = 0.01 D/a4 = 1.0, ε/a = 0.01
(b)
Figure 3.10: Trapped mode wavenumbers plotted against ρ for a cylinder of radius a in the upper fluid layer for different ice-parameters;la= 2, f /a= 1.01,d/a= 2.1 and h/a= 6.
become those for a finite depth homogenous fluid bounded above by a thin ice-cover. This equation has exactly one positive real rootu=k(say). For trapped modes it is required that l > k. To the best of our knowledge, these modes have not been computed earlier by anyone else. In Figs. 3.11(a) and 3.11(b), the trapped mode wavenumberkais plotted against la for fixed values of ice-parametersD/a4 = 0.001 andε/a= 0.001. In Fig. 3.11(a), the wavenumber is plotted for different submergence depths of the cylinder: f /a= 1.09, 1.05, 1.03 and 1.01, where the depth d/a of the upper layer is 2.1. In Fig. 3.11(b), the wavenumber is plotted for different depths d/a with f /a also varying accordingly so that (d−f)/a = 1.01, i.e, the cylinder is submerged nearer to the thin ice-cover.
Additional numerical computations confirm that as the cylinder moves towards the ice- cover, i.e., (d−f)/a→1, the trapped mode wavenumberkadecreases and as it comes further closer, a second mode appears. When the cylinder is very close to the ice-cover, i.e., say f /a= 1.09, there exist two modes for 1.9≤la≤2.1. But from the work of Linton and Cadby (2003), it is observed that when the cylinder is placed near the free surface, a second mode
0 0.5 1 1.5 2 2.5 3 3.5 4 0
0.5 1 1.5 2 2.5 3 3.5 4
la ka
f/a = 1.09 f/a = 1.05 f/a = 1.03 f/a = 1.01
(a)
0 0.5 1 1.5 2 2.5 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
la ka
d/a = 2.1, f/a = 1.09 d/a = 2.5, f/a = 1.49 d/a = 3.0, f/a = 1.99 d/a = 15, f/a = 13.99
(b)
Figure 3.11: Trapped mode wavenumbers in a single-layer fluid; D/a4 = 0.001; ε/a = 0.001 and h/a = 6 (a) for different submergence depths f /a; d/a = 2.1 (b) for different depths of the layerd/a; (d−f)/a= 1.01.
exists only for those values oflawhich are approximately greater than or equal to 0.8. When the upper surface of the cylinder just touches the ice cover, i.e,f /a= 1.1, then a second mode exists for 0.8≤la≤3.2. The trapped mode wavenumberkadecreases by a small amount with an increase in depth. These results bear similarities to those corresponding to the first mode in Linton and Cadby (2003).