2.3 Solutions by multipoles
2.3.2 Cylinder submerged in the upper layer
22 2.3. SOLUTIONS BY MULTIPOLES
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 ka
f/a = −1.01 f/a = −1.02 f/a = −1.03
(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 ka
f/a = −1.05 f/a = −1.10 f/a = −1.30 f/a = −1.50 (b)
Figure 2.5: Trapped mode frequencies plotted against ρ for a cylinder in the lower fluid layer for different submergence depthsf /a;la= 2,d/a= 3 and h/a= 6.
ratio increases.
Once again, the integrand has no singularities on the real axis. By applying the body boundary condition (2.15), we obtain a similar kind of system of equations like (2.23) forβm:
βn+ In0(la) Kn0(la)
X∞
m=0
βmBmn= 0, n= 0,1,2, . . . (2.28) Here also, as in the previous case, by varying the frequencies ka and fixing the other parameters, we conveniently locate the zeros of the truncated determinant. It is already known that those frequencies correspond to the trapped modes. The results presented below are obtained correct up to three decimal places where a 32×32 system has been used after truncating the system arising from (2.28).
Numerical results
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
h/a=0.5 h/a=1.0 h/a=1.5 h/a=2.0
Figure 2.6: Trapped mode frequencies plotted againstlafor a cylinder in the upper fluid layer for different depthsh/a of the lower layer; ρ= 0.95,f /a= 1.01 and d/a= 3.
Figures 2.6–2.9 show the plots of the non-dimensional trapped mode frequencies for a circular cylinder of radius a immersed in the upper layer of the two-layer fluid. In all cases, the depthd/a of the upper layer is taken as 3.0. For Figs. 2.6 and 2.7, the submergence depth f /ais taken as 1.01 which means that the cylinder is very close to the interface and the depth h/aof the lower layer is taken as 6.0 for Figs. 2.8 and 2.9.
Figure 2.6 shows the dispersion curves for four different depths of the lower layer: h/a= 0.5, 1.0, 1.5 and 2.0. For each set of parameter values, there are two curves corresponding to two modes which are displayed in the graphs. We clearly observe that the frequency of the trapped modes decreases when the depth of the lower layer increases, the first mode being affected more than the second mode.
Figure 2.7 shows the plot of trapped mode frequencies againstρ whenla= 2. Correspond- ing to different depths of the lower layer: h/a= 0.5, 1.0, 1.5 and 2.0, we observe two modes for density ratios ρ ≥0.5. For the limiting case ρ → 0, trapped mode frequency ceases to exist.
The limit ρ → 0 corresponds to ρII → ∞. Consequently the dispersion relation reduces to u= 0 which is the dispersion relation for a finite depth homogenous fluid bounded above by a rigid lid. We know that trapped modes do not exist in this case. By letting the density ratio tend to zero, we are being able to recover the same.
24 2.3. SOLUTIONS BY MULTIPOLES
0 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 2 2.5 3
ρ ka
h/a=0.5 h/a=1.0 h/a=1.5 h/a=2.0
Figure 2.7: Trapped mode frequencies plotted againstρfor a cylinder in the upper fluid layer for different depthsh/a of the lower layer ;la= 3, f /a= 1.01 and d/a= 3.
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 = 1.01 f/a = 1.05 f/a = 1.15 f/a = 1.35
Figure 2.8: Trapped mode frequencies plotted againstlafor a cylinder in the upper fluid layer for different submergence depthsf /a;ρ= 0.95, d/a= 3 and h/a= 6.
Figure 2.8 shows the dispersion curves for the case in which ρ = 0.95. There are four different curves corresponding to different submergence depths of the cylinder in the upper layer: f /a = 1.01, 1.05, 1.15 and 1.35. As the cylinder approaches the interface, the curves fold out from the interface. Similar effects were observed when the cylinder was placed in the lower layer.
In Fig. 2.9, the different curves correspond to the different submergence depths of the cylinder: f /a = 1.01, 1.03, 1.05 and 1.10 for la= 2. Different modes can be found by fixing the depth of both layers and also the density ratioρ, and then by varying the submergence depth. For the limiting case ρ → 1, the wavenumber ka tends to some fixed but different limit corresponding to different submergence depths. If we move the cylinder away from the interface towards the upper surface, these trapped mode frequencies increase and the trapped mode ceases to exist as the cylinder comes closer to the rigid lid. By varying the submergence depth and fixing all the parameters, it is observed that the trapped modeka increases up to la= 2 as ρdecreases. There exist no trapped modes for density ratio approximately less than
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 ka
f/a = 1.01 f/a = 1.03 f/a = 1.05 f/a = 1.10
Figure 2.9: Trapped mode frequencies plotted againstρfor a cylinder in the upper fluid layer for different submergence depthsf /a;la= 2, d/a= 3 and h/a= 6.
0.01 as observed from the figure. Hence asρ→0, it is possible to recover the problem in which a homogenous fluid is bounded above by a rigid lid.
Intuitively it is very likely to come to one’s mind that single-layer fluid results may be obtained if the density ratio ρ → 1. More precisely, one might expect that there will be no trapped modes when the cylinder is submerged in either layer of a two-layer fluid bounded above by a rigid lid for the limiting caseρ → 1 since in this case the upper and lower fluids have almost the same density. But from Figs. 2.3 and 2.7, where trapped mode frequencies are plotted against density ratio, we observe that for each set of parameter values there are two curves corresponding to two modes. In the limitρ → 1, the second mode ceases to exist but the first mode tends to some value which gives a trapped mode. To explain what takes place, we next consider the boundary value problem, given by Eqs. (2.5)–(2.8), with the limitρ→1.
Limit as ρ→1
In the limitρ→1, we find thatK/k→0 from the dispersion relation (2.12) and so ifl(> k) is fixed, thenK must tend to zero. For a clear insight we consider the boundary conditions in the limit asρ→1 andK →0 simultaneously.
We introduce small parameters ²andδ such that K=², ρ= 1−δ, K0 = K
1−ρ = ²
δ =O(1). (2.29)
In this limit, the continuity of the vertical velocity at the interface remains in the same form but that of pressure changes to
K0φII −φzII =K0φI on z= 0. (2.30) Oblique waves propagating in the fluid have the same form given by Eqs. (2.9)–(2.11). The dispersion relation will have only one solutionk which must satisfy
k(1−e−2kd)(1−e−2kh) = 2K0(1−e−2k(d+h)). (2.31)