3.4 Numerical Results
3.4.3 Cylinder submerged in the lower layer
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).
42 3.4. NUMERICAL RESULTS
Figure 3.12 shows the trapped mode wavenumbers againstρwhenla= 2. Corresponding to the different depths of the upper layer: d/a= 0.5, 1.0, 1.5 and 2.0, there are two curves which correspond to the two modes for each of the wavenumbersu1a and u2a. For the wavenumber u1a, these two modes tend to zero asρ→1 for any depth of the upper layer. With a change in the depth of the upper layer, there is no variation of the two modes for the wavenumber u1a but these two modes decrease with an increase in the depth of the upper layer for the wavenumberu2a. For the wavenumber u2a, as ρ → 1, the first mode increases marginally to a fixed value corresponding to each depth of the upper layer and the second mode tends to la= 2.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5
ρ u1a
f/a = −1.01 f/a = −1.05 f/a = −1.10 f/a = −1.20
(a)
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 = −1.01 f/a = −1.05 f/a = −1.10 f/a = −1.20
(b)
Figure 3.13: Trapped mode wavenumbers plotted against ρ for a cylinder of radius a in the lower fluid layer for different submergence depthsf /a;la= 2,h/a= 6,d/a= 3,D/a4 = 0.001 andε/a= 0.001.
In Fig. 3.13, the different curves correspond to the different submergence depths of the cylinder: f /a = −1.01, −1.05, −1.10 and −1.20 for la = 2. As the submergence depth increases, i.e., the cylinder moves away from the interface, the second mode ceases to exist but the first mode always exists with the same properties as discussed above. As the submergence depth increases, both the wavenumbers u1a and u2a increase. For the wavenumber u1a and for a small density ratio, the gap between two consecutive curves is more and as ρ increases towards the value 1, all the curves merge and tend to zero but for the wavenumberu2a, the gap between two consecutive curves remains almost the same as the density ratio increases towards 1. The single-layer finite depth results for the depthh/a and the cylinder submergence depth f /a are recovered when ρ = 0 (with the interface now playing the role of the free surface) which are discussed in Linton and Cadby (2003).
Ursell (1951) established the existence of a trapped surface wave mode in the vicinity of a long and totally submerged horizontal circular cylinder of small radius in deep water.
Throughout our numerical computation, we non-dimensionalize all the parameters with respect to the radiusaof the cylinder. Therefore, our present work reveals that a single trapped mode appears to exist for all values ofa <|f|and not just when the radius of the cylinder is small.
Limit as ρ→1
Like in the rigid lid problem considered in Chapter 2, it is very likely to come to one’s mind
that the single-layer fluid results may be obtained if the density ratio ρ → 1. But from all the figures where trapped mode wavenumbers are plotted against the density ratio, we observe something intriguingly different. It is observed that corresponding to each set of parameter values, there are two curves corresponding to two modes for the wavenumberu2a. In the limit ρ→1, the second mode ceases to exist but the first mode tends to some value which represents a trapped mode. For the wavenumberu1a, both modes tend to zero asρ→1. For the density ratios having values closer to 1, we have already seen that u2 > u1 > K. So, if u1a → 0, we must have Ka → 0 as ρ → 1. Thus, for a cylinder in either fluid layer, though the first mode tends to some value for the wavenumber u2a, it certainly does not correspond to the one for a single-layer fluid becauseKa→0. Hence, from our numerical investigations, we can conclude that it is not possible to recover the single-layer fluid results in the limitρ→ 1. To explain this occurrence we consider the boundary conditions in the limit asρ→1 andK→0 simultaneously.
Small parameters²and δ are introduced such that (same as Eq. (2.28)) K=², ρ= 1−δ, K0 = K
1−ρ = ²
δ =O(1). (3.31)
In this limit, the upper surface condition becomes
∂φI
∂z = 0 on z=d. (3.32)
The continuity condition of the vertical velocity at the interface and the bottom boundary condition remain the same but that of pressure at the interface changes and becomes
K0φII −∂φII
∂z =K0φI on z= 0. (3.33)
In the absence of any structure, oblique waves propagating in the fluid take the form φI = exp(±ixp
k2−l2)
³
ek(z−d)+e−k(z−d)
´
, (3.34)
φII = exp(±ixp
k2−l2) coshk(z+h)F(k), (3.35) with
F(k) = e−kd−ekd
sinhkh , (3.36)
wherekis the only root of the dispersion relation
k(1−e−2kd)(1−e−2kh) = 2K0(1−e−2k(d+h)). (3.37) In order that the trapped modes exist, it is required thatl > kso that the motion decays as
|x| → ∞. Thus for a fixedl, a boundary value problem is obtained in terms of a new spectral parameterK0. The trapped mode frequencies are computed by using the multipole expansion method. These results match with those observed from the figures in sections 3.4.2 and 3.4.3 when the density ratio approaches 1. Hence for a fixed l, the trapped mode problem in the limitρ→1 is related to the limits of the trapped mode curves in these figures. But if we fixK and letρ→1, thenu→ ∞from dispersion relation (3.15) and sincel > u for trapped modes, hencel→ ∞. Thus it is not possible to recover the single-layer fluid results in the limitρ→1.