M2/a2, as seen from Fig. 7.7, trapped modes for both the wavenumbers get affected. With an increase inM2/a2, the second trapped mode for both the wavenumbersu1aand u2aincreases more as compared to the first mode.
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
M2/a2 = 0 M2/a2 = 0.005 M2/a2 = 0.007 M2/a2 = 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
M2/a2 = 0 M2/a2 = 0.005 M2/a2 = 0.007 M2/a2 = 0.01
(b)
Figure 7.7: Trapped mode wavenumbers plotted against ρ for a cylinder of radius a in the lower fluid layer for different values ofM2/a2;la= 2, d/a= 3.0,h/a= 6.0, f /a=−1.01 and M1/a2 = 0.
Chapter 8
Summary and future work
This chapter is devoted to a brief summary of the results highlighting the contributions made by this thesis. It also provides information for the scope of possible extensions of the present work and future investigations.
8.1 Summary
In this thesis the trapping of a small amplitude harmonic surface water waves by submerged horizontal circular cylinder (cylinders) in two-layer and three-layer fluids has been investigated by using linear water wave theory.
In Chapter 2, we consider the trapped mode problem concerning a submerged cylinder entirely located within one of the layers of a two-layer fluid with both layers being of finite depth. The layers are bounded by upper and lower rigid surfaces, which are approximations of the free surface and the bottom surface, respectively, in a channel. Multipole potentials are constructed which, for frequencies less than the cut-off value, do not radiate energy away from the submerged body. These potentials, each of which satisfies all conditions except the body boundary condition, are singular on the axis of the cylinder but not in the fluid region. The trapped mode potential is then constructed from a linear combination of all possible multipoles.
Application of the body boundary condition results in an infinite system of homogenous linear equations. Then the trapped mode frequencies are obtained numerically by locating the zeros of the truncated determinant. Different modes are found by fixing the depth of both the layers (d/a and h/a for lower and upper layer respectively) and also the density ratio ρ, and then by varying the submergence depth f /a. When the cylinder is placed in either of the layers, trapped mode frequency decreases with an increase in the depth of the other layer. With an increase in density ratio, trapped mode frequency increases when the cylinder is placed in the lower layer but decreases when the cylinder is placed in the upper layer.
In Chapter 3, the previous work in Chapter 2 is extended to a problem where the upper fluid is bounded above by a thin ice-cover modeled as a thin elastic plate, which replaces the free surface and the lower fluid is bounded by a rigid horizontal bottom surface. The trapped mode supported by a submerged horizontal circular cylinder placed in such type of fluid region is handled by the same multipole expansion technique. Earlier, Linton and Cadby (2003) computed the trapped mode frequencies for a cylinder placed in either layer of a two-layer fluid, in which the upper layer had a free surface and the lower layer extended to infinite
depth. It is reasonable to compare the results obtained in this chapter with the work of Linton and Cadby (2003) as both the problems bear similarities to a large extent. For a cylinder placed in the upper layer, the trapped motion is confined only to the vicinity of the ice-cover and the interface. The trapped mode wavenumbers increase due to the presence of the ice-cover on the upper surface. For the case when the cylinder is placed in the lower layer, the motion is confined to the vicinity of the interface only. As the cylinder moves towards the horizontal rigid bottom away from the interface, the trapped mode ceases to exist.
Both problems mentioned above consist of a finite depth lower layer which is bounded below by a flat, horizontal, rigid bed. But the flexibility of the bed is also a very important aspect of study which has not been accounted for in these previous investigations. The understanding of the free vibration characteristics of the fluid-structure interaction plays a significant role in various branches of engineering, for example, the propellant in space vehicles can be free from resonance, large-capacity oil containers in the petrochemical industry can survive earthquakes, very large floating oil storage tanks, ships and submarines can avoid or be subjected to reduced localized vibrations. Therefore in Chapter 4 we examine the trapped modes supported by a submerged horizontal circular cylinder placed in either layer of a two-layer fluid flowing over an elastic bottom at a finite depth. By the same procedure as was followed in Chapter 2, we look into the effect of the variations of elastic plate parameters on the existence of trapped modes.
For the case of the cylinder placed in the lower layer, the presence of elastic bottom suppresses the existence of the second mode (among the two modes that exist for each wavenumber) which is a different conclusion as compared to the infinite depth lower layer. Both trapped mode wavenumbers increase due to the consideration of the elasticity of the sea-bed as compared to a flat and rigid bed.
Chapter 5 is concerned with the effect of width of the middle layer on the trapped modes due to the presence of a totally submerged cylinder placed in either the bottom layer or the uppermost layer of a three-layer fluid of infinite depth. It is observed that with an increase in the width of the middle layer, the trapped mode frequencies decrease for the case when the cylinder is placed in the lower layer. When either of the density ratios or both tend to 1, then it is not possible to recover either the two-layer or the single-layer case as is evident from the present investigation - contrary to what was concluded in Chakrabarti et al. (2005). That is, however small the width of the pycnocline may be, it still influences the trapped modes and the three-layer case cannot be considered as equivalent to the two-layer case in the limit. The presence of the middle layer in between two layers can be compared with the presence of a smooth pycnocline of constant density. The problem formulated here has the possibility of getting extended to other forms of upper surface conditions also.
Chapter 6 is concerned with the investigation of trapped water waves supported by a pair of horizontal circular cylinders submerged in either layer of a two-layer fluid in an ocean, where the upper layer is of finite depth and is bounded above by a thin ice-cover modeled as a thin elastic plate, which replaces the free surface, and the lower layer is of infinite depth. In this chapter we do not consider the trapped waves below the cut-off value but rather we consider the trapped waves which are embedded in a continuous spectrum. Hence, in addition to the multipole expansion method, the knowledge of contour integration is required to locate numer- ically the distance between these cylinders for which trapped wave exists. Although numerical