2.3 Solutions by multipoles
2.3.1 Cylinder submerged in the lower layer
Symmetric multipoles,φn(n≥0), are defined by (Taylor and Hu (1991)) φIn = −
Z ∞
0
coshnucos(lxsinhu) h
AL(v)evz+BL(v)e−vz i
du, (2.17)
φIIn = Kn(lr) cosnθ+− Z ∞
0
coshnucos(lxsinhu) h
CL(v)evz+DL(v)e−vz i
du, (2.18) where v = lcoshu, Kn(.) is modified Bessel function of the second kind of order n and the integrals are Cauchy Principal Value integrals.
Kn(lr) cosnθhas the following integral representation (Ursell (1951)):
Kn(lr) cosnθ=
− Z ∞
0
coshnucos(lxsinhu)ev(z−f)du for z < f, (−1)n−
Z ∞
0
coshnucos(lxsinhu)ev(f−z)du for z > f.
With the help of the boundary conditions at the rigid lid, the interface and the bottom, the coefficients AL(v), BL(v), CL(v) and DL(v) appearing in Eqs. (2.17) and (2.18) are obtained
as
AL(v) = K(1 +σ) G(v)
³
(−1)n+1evf−e−v(f+2h)
´ , BL(v) = K(1 +σ)
G(v)
³
(−1)n+1evf−e−v(f+2h)
´ ,
CL(v) =
³
(−1)n+1evf−e−v(f+2h)
´³
(v+Kσ)e−2vd−(v+K)
´
G(v) ,
DL(v) =
³
CL(v) +e−vf
´ e−2vh, where
G(v) = (v+Kσ)e−2v(d+h)+ (v−Kσ)−(v+K)e−2vh−(v−K)e−2vd. (2.19) The total velocity potential can now be written as (Linton and Cadby (2002))
φ= X∞
n=0
αnφIIn, (2.20)
with
φIIn =Kn(lr) cosnθ+ X∞
m=0
AmnIm(lr) cosmθ, (2.21) whereIm(.) is modified Bessel function of the first kind of orderm and
Amn=²n− Z ∞
0
coshmucoshnu h
(−1)nCL(v)evf+DL(v)e−vf i
du, (2.22)
with²0= 1, ²n= 2, n≥1.
We note that there are no singularities of the integrand on the real axis since we know thatv > l > k. Applying the body boundary condition (2.15), we obtain an infinite system of homogenous linear equations in the unknownsαm:
αn+ In0(la) Kn0(la)
X∞
m=0
αmAmn= 0, n= 0,1,2, . . . (2.23)
For a fixed geometrical configuration, the problem of finding the trapped mode frequencies is completely specified by the two non-dimensional parameterskaandla: one of the parameters may be fixed and the other varied until the value of the determinant becomes approximately zero. This is, in fact, the most crucial stage of the investigation. The largest computational expense occurs while computing the integrals in (2.22) the values of which are used in evaluating the determinant of the matrix in Eq. (2.23). Thus it seems appropriate to keeplafixed while kais varied up to la. For numerical evaluation of the zeros of the determinant, we truncate the system to a 32×32 system and the result presented next are obtained correct up to three decimal places.
20 2.3. SOLUTIONS BY MULTIPOLES
Numerical results
Figures 2.2–2.5 show the results obtained for trapped modes above a horizontal circular cylinder of radiusasubmerged in the lower layer of a two-layer fluid bounded above by a rigid lid. For all the cases, the depthh/aof the lower layer is taken as 6.0 while thatd/aof the upper layer is taken as 3.0 for Figs. 2.4 and 2.5. Submergence depthf /a of the cylinder is considered as
−1.01 for Figs. 2.2 and 2.3 which means that the cylinder is very close to the interface. 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. Hence, throughout our numerical results, the density ratio of the fluid is chosen as 0.95, wherever applicable, for clear observation of the features of the dispersion curves: for Figs. 2.2 and 2.4 and later for Figs. 2.6 and 2.8, to be precise. Trapped modes requirel > kso thatl=k(thin dashed line) is an upper bound for the figures in which frequencies are plotted againstla.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.5 1 1.5 2 2.5 3
la ka
d/a = 0.5 d/a = 1.0 d/a = 1.5 d/a = 2.0
Figure 2.2: Trapped mode frequencies plotted againstlafor a cylinder of radiusain the lower fluid layer for different depthsd/a of the upper layer; ρ= 0.95,h/a= 6 andf /a=−1.01.
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
d/a = 0.5 d/a = 0.7 d/a = 0.9 d/a = 1.3
Figure 2.3: Trapped mode frequencies plotted againstρ for a cylinder of radiusain the lower fluid layer for different depthsd/a of the upper layer; la= 2,h/a= 6 and f /a=−1.01.
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 = −1.09 f/a = −1.17 f/a = −1.34
Figure 2.4: Trapped mode frequencies plotted againstlafor a cylinder of radiusain the lower fluid layer for different submergence depthsf /a;ρ= 0.95,d/a= 3 and h/a= 6.
Figure 2.2 shows the plots of trapped mode frequencies where the different curves cor- respond to the four different depths of the upper fluid layer: d/a = 0.5, 1.0, 1.5 and 2.0.
Corresponding to each value ofd/a, there are two curves showing the first and second modes.
With an increase in the depth of the upper fluid layer, these trapped mode frequencies decrease.
Figure 2.3 shows the plot of trapped mode frequencies against ρ when la = 2. Corre- sponding to different depths of the upper fluid layer: d/a = 0.5, 0.7, 0.9 and 1.3, we observe two modes. For the limiting case ρ → 1, the wavenumber ka tends to some limit for each mode. However for the second mode, the wavenumber ka→ la = 2. In the far-field form of the potential, we have an exp(−√
l2−k2|x|) term. Thus the rate of decay of the exponential term decreases asρ takes values closer to unity and in this case no trapped mode exists. The single-layer finite depth results for depth h/a and cylinder submergence f /a are recovered whenρ= 0 (with the interface now playing the role of the free surface) which are discussed in Linton and Cadby (2003).
In Fig. 2.4, the different curves correspond to different submergence depths of the cylinder:
f /a = −1.01, −1.09, −1.17 and −1.34. From this figure we observe that with an increase in submergence depth, frequency of the trapped mode increases and almost coincides with the upper bound k = l for some particular value of f /a. In other words, if the depth of submergence increases beyond a particular value, then trapped mode frequencies cease to exist. As the cylinder approaches the interface, i.e., |f /a| →1, the curve folds out from the upper bound.
Figure 2.5 shows the plot of trapped mode frequencies against ρ with la= 2 for different submergence depths: f /a = −1.01, −1.02, −1.03, −1.05, −1.10, −1.30 and −1.50. From Fig. 2.5(a) it is observed that when the cylinder is submerged near the interface, there exist two modes. With an increase in submergence depth, trapped mode frequency corresponding to the second mode increases to la= 2 and the trapped modes disappear. For submergence depth approximately greater than or equal to 1.05, there exists only one mode, as observed from Fig. 2.5(b), which also ceases to exist as we further move the cylinder downwards. Furthermore, for a fixed submergence depth, the trapped mode frequency increases steadily as the density
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.