Effect of compression on wave diffraction by a floating ...

Journal of Fluids and Structures 36 (2013) 124–135

Contents lists available at SciVerse ScienceDirect

Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs

Effect of compression on wave diffraction by a floating elastic plate S.C. Mohapatra a, R. Ghoshal b, T. Sahoo a,n a b

Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur 721 302, India Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721 302, India

a r t i c l e in f o

abstract

Article history: Received 23 June 2011 Accepted 20 July 2012 Available online 18 October 2012

In the present paper, the water wave diffraction by a two-dimensional floating elastic plate is analyzed in the presence of compressive force. The solutions in the cases of infinite and finite water depths are derived based on integro-differential equation method in the presence of compressive force under the assumption of small amplitude water wave theory and plate deflection. Further, wave diffraction by the floating elastic plate is analyzed under the assumption of shallow water approximation. The role of compressive force and its limiting values are obtained by using the hydroelastic analysis of the flexural gravity waves. The limiting values of oblique angle of incidence are obtained in different cases and the effect of compressive force on the oblique angle is analyzed. Effect of compressive force and angle of incidence on the hydroelastic behavior of the floating plate are studied by analyzing the reflection coefficients in different cases. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Floating elastic plate Compressive force Buckling limit Reflection coefficient Shallow water approximation

1. Introduction In the last two decades, there is a significant interest to study the hydroelastic behavior of very large floating structures which are aimed at ocean space utilization for various human activities. Because of the large size and multi-module configurations, the design, construction, installation and analysis of these structures differ significantly from the existing floating structures. To analyze the elastic response in regular waves, the VLFS is regarded as a thin floating elastic plate or a beam, and the dynamic equations for the structures and the fluid are solved together. In most of the studies, finite element method or mode expansion method is used for the structural region and the panel method for the fluid region. Often admissible functions satisfying geometric boundary conditions on the peripherals of the structure are used to represent the elastic deflection (see Newman, 1994). Kashiwagi (1998) used B-spline Galerkin scheme for representing both the pressure and structural deflection for easy coding. There are three major lines of approach to the analysis of the VLFS. In the first approach, the hydrodynamic problem is solved separately by specifying the deformation of the plate by a number of selected mode shapes. This approach is an extension of the hydrodynamic problem for the rigid-body motions, and is used by several authors with certain modifications to increase the numerical efficiency of the computation. In the second approach, the interaction between the motion of the plate and the fluid is coupled and the two problems are solved simultaneously. On the other hand, in the third approach, one uses the three-dimensional hydroelasticity theory by utilizing the structural-mode shapes obtained in vacuo and through a finite-element program and the source distribution method of the linear potential theory. Even though this method is the most complete for obtaining the dynamic behavior of elastic bodies in waves, the computational time required is enormous. These methods are based on two basic solution

n

Corresponding author. Fax: þ 91 3222 255303. E-mail addresses: [email protected], [email protected] (T. Sahoo).

0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2012.07.005

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

125

techniques, the mode expansion and the direct method. In the mode expansion method, the amplitude of each mode is determined by solving the vibration equation of a plate with the added mass and damping force corresponding to specified mode shapes. In the direct method, the dynamic equations for the structure and fluid are solved simultaneously. In the mode expansion method, which is the mostly used method, Newman (1994) proposed to employ different orthogonal polynomials to represent the corresponding mode expansions for different edge conditions. Using the conjugate gradient method, Sturova (1998) studied the effect of obliquely incident waves onto an elastic band, while Sahoo et al. (2001) investigated the scattering of surface waves by a semi-infinite plate by the direct application of eigenfunctions expansion method using a new orthogonal inner product. Hermans (2000) analyzed the wave interaction with a finite floating plate in the presence of current in infinite water depth. By considering the boundary element method for the fluid region, the mode expansion method for the plate covered region, Hermans (2000) derived an integro-differential equation to solvefor the deflection of the plate which has been further generalized to deal with floating elastic plates of non-rectangular geometry (see Andrianov and Hermans, 2005, 2006). Fox and Squire (1990) applied appropriate matching conditions across the interface to investigate the scattering of ocean waves by a shore fast sea-ice that is modeled as a semi-infinite elastic plate in finite water depth. Tkacheva (2003) studied wave interaction with floating elastic plate based on Wiener– Hopf technique. Recently, Karmakar et al. (2009) analyzed the effect of multiple articulation on wave scattering by floating elastic plate using a recently developed expansion formulae and associated orthogonal mode-coupling relations. The optimum location and rotational stiffness of the connectors for the two-floating beam system with the view to minimize the hydroelastic response was studied by Riyansyah et al. (2010). It may be noted that the global response of an interconnected floating structure can be effectively reduced by adjusting the connector properties. There is a recent review on the developments of various procedures to mitigate the hydroelastic response of VLFS under wave action (see Wang et al., 2010). There is a significant progress on the use of flexible structures as breakwater in the literature. Loukogeorgaki et al. (2012) numerically studied the hydroelastic performance of a mat shaped floating breakwater which consists of a grid of flexible floating modules. Using variational principle, Manam and Kaligatla (2012) studied the effect of varying bottom on membrane-coupled gravity waves caused by a floating membrane. Apart from the use of floating elastic plate model in the hydroelastic analysis of very large floating structures, floating elastic plate models having free edges are widely used in the study of wave interaction with floating ice sheets in polar regions (Fox and Squire, 1994). Recently, the overlap among hydroelastic analysis of very large floating structures and wave ice interaction problems are discussed by Squire (2008). The effect of compressive force on floating ice sheet is studied in the context of moving load on floating ice sheet (see Kerr, 1983; Squire et al., 1996 and the literature cited therein). Bukatov and Zav’yalov (1995) studied the impingement of surface waves on the edge of compressed ice in a basin of constant depth based on conjugate gradient method. Although, Bukatov and Zav’yalov considered the effect of compression in the ice sheet, free edge boundary conditions are used in the analysis without adding the compressive force term in the edge condition. The presence of thermal strain, surface friction due to wind and water flow beneath the ice are some of the natural sources of compressive stress on a floating ice sheet as in Schulkes et al. (1987). Further, the critical compressive force is expected to be a safety issue during landing of an air craft on the very large floating structures used as floating air ports, which are often designed by taking interconnected or multiple articulated structures (Karmakar et al., 2009; Riyansyah et al., 2010). In the present study, interaction of surface gravity wave with a floating elastic plate in water of infinite and finite depths, and under shallow water approximation in the presence of compressive force is studied in three dimensions. The boundary integral equation approach similar to Andrianov and Hermans (2003) is used to analyze the problems in finite and infinite water depths and mode matching approach as in Ohkusu and Namba (2004) is used in the case of shallow water approximation. The role of compressive force and its limiting values are obtained by using the hydroelastic analysis of the flexural gravity waves in cases of infinite, finite and shallow water depth. Further, limiting values of oblique angle of incidence are obtained in different cases and the effect of compressive force on these limiting values are analyzed from the analysis of the reflection and transmission coefficients. 2. Mathematical formulation Under the assumption of the linearized theory of water waves, the problem is considered in three-dimensional Cartesian co-ordinate system with x-axis being in the horizontal direction along length, y-axis in the vertically downward positive direction and z-axis is along the breadth. The plate is of uniform property and water is of a uniform density r. The fluid is assumed to be infinitely extended which occupies the region 1 ox o1, 1 oz o 1 along with 0 oy oh in case of finite water depth and 0 o yo 1 in case of infinite water depth. The plate is placed at y¼0, 0 o x o l at the water surface where l is the length of the plate and an incoming surface gravity wave is incident upon the plate with an oblique angle y as in Fig. 1. The free surface is considered as a combination of the open water region referred to as R1, the plate covered region referred as R2 and the interface region referred as @R. Assuming that the fluid is inviscid and incompressible, the motion is irrotational and simple harmonic in time with angular frequency o. Thus, there exists a velocity potential Fðx,y,z,tÞ of the form Fðx,y,z,tÞ ¼ Re½fðx,y,zÞeiot , which satisfies the Laplace equation as given by

r2 F ¼ 0 in the fluid region:

ð1Þ

126

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

R2 Floating elastic plate θ

R1

R1 z

x

x y

l Fig. 1. Schematic diagram of floating elastic plate on the free surface.

The linearized free surface boundary condition at the mean free surface y¼0 is given by @2 F @F ¼0 g @y @t 2

on y ¼ 0:

The dynamic pressure exerted on the elastic plate is given by (  ) 2  2  @2 @2 @ @2 @2 þ þQ þ þm 2 W ¼ Ps , D @x2 @z2 @x2 @z2 @t

ð2Þ

ð3Þ

3

where D ¼ Ed =12ð1n2 Þ is the flexural rigidity of the elastic plate, n is the Poisson’s ratio, E is Young’s modulus, r is the density of water, m ¼ ri d is the density of the elastic plate, g is the acceleration due to gravity. Assuming that the compressive force in x and z directions is same, we have Q x ¼ Q z ¼ Q is the compressive force and d is the thickness of the elastic plate, Wðx,z,tÞ ¼ Re½wðx,zÞeiot  denotes the deflection of the plate. Further, the hydrodynamic pressure exerted on the elastic plate at the free surface y ¼0 is given by P H ¼ r

@F þ rgW: @t

ð4Þ

The linearized kinematic condition on the free surface at y¼ 0 is given by @W @F ¼ @t @y

at y ¼ 0:

ð5Þ

Assuming P s ¼ P H and combining Eqs. (3), (4) and (5), the linearized boundary condition on the plate covered region x,z 2 R2 is given by (  ) 2  2  D @2 @2 Q @ @2 m @2 @F 1 @2 F  þ þ þ þ 1 ¼ 0 at y ¼ 0: ð6Þ þ 2 2 2 2 2 rg @x rg @x rg @t @y g @t 2 @z @z Assuming that the plate is having free edge, so the zero bending moment and zero shear force conditions are given by  3  @2 W @2 W @ W @3 W @W ¼ 0: ð7Þ þ n 2 ¼ 0 and D þð2nÞ þQ 2 3 2 @x @x @z @x @x@z It may be noted that the second condition in Eq. (7) is a generalization of the edge condition of Hermans (2000) including the in-plane compressive force Q. The rigid bottom boundary conditions are given by @F ¼0 @y

on y ¼ h in case of finite water depth,

ð8aÞ

F,9rF9!0 as y!1 in case of infinite water depth:

ð8bÞ

In the next section, we will analyze the effect of compression on phase velocity and group velocity in cases of infinite, finite and shallow water depths and relate the same with the buckling limit before discussing the wave diffraction. 3. Buckling limit based on hydroelastic analysis In case of obliquely incident surface waves, assuming the water surface wave profile Zðx,z,tÞ ¼ RefAeiðkx x þ kz zotÞ g where kx ¼ k0 cos y and kz ¼ k0 sin y, we obtain the spatial velocity potential f, satisfying the boundary conditions (1), (2) and (8a), is of the form

Fðx,y,z,tÞ ¼ 

gA cosh k0 ðhyÞ ik0 ðx cos y þ z sin yÞiot e , io cosh k0 h

ð9Þ

where k0 is the wave number associated with the free surface gravity wave, which satisfies the dispersion relation in k as given by

o2 ¼ gk tanh kh:

ð10Þ

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

127

Assuming ri d 5 1 as in Schulkes et al. (1987), for the plane flexural gravity wave of the form zðx,z,tÞ ¼ RefAeiðpx x þ pz zotÞ g, propagating in the plate covered region, where A is the wave height of progressive flexural gravity wave and px ¼ p0 cos y, pz ¼ p0 sin y, the velocity potential Fðx,y,z,tÞ in the plate covered region can be obtained in the form

Fðx,y,z,tÞ ¼ 

AðDp40 Qp20 þ rgÞ cosh p0 ðhyÞ ip0 ðx cos y þ z sin yÞiot e , ior cosh p0 h

ð11Þ

where p0 is the wave number associated with the flexural gravity wave, which satisfies the dispersion relation in p as given by ðDp4 Qp2 þ rgÞp tanh ph ¼ ro2 :

ð12Þ

In case of infinite water depth, the dispersion relation in Eq. (12) reduces to ðDp4 Qp2 þ rgÞp ¼ ro2 ,

ð13Þ

which has a real root at p ¼ p0 (say). Proceeding in a similar pffiffiffiffiffiffiffiffiffiffimanner as in Schulkes et al. (1987), Eq. (13) yields that the phase velocity attends zero minimum for Q ¼ Q cr ¼ 2 Drg and p ¼ pcr ¼ ðrg=DÞ1=4 . Further, it may be noted that o2 becomes negative in Eq. (13) for Q 4Q cr , which contradicts the existence of a real frequency and in this situation, instability in the floating elastic plate may occur. Hence, for all practical purposes, in the present analysis, Q is limited to Q cr . In addition, for 0 r Q oQ cr , cmin ispaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi non-zero value for all values of the wave number p0. Further, the group velocity cg becomes positive definite for Q o 20Drg =3 ¼ Q cg . Hence, the group velocity cg becomes zero or negative for Q cg rQ r Q cr which implies that the wave crest and the wave group will propagate in opposite direction. On the other hand, for Q ¼ Q cg and p0 ¼ ðrg=5DÞ1=4 , the group velocity vanishes. Kerr (1983) observed that the critical value Q cr for which the phase velocity vanishes is the same as the buckling limit of the compressive force. On the other hand, in case of shallow water depth, the dispersion relation in Eq. (12) reduces to ðDp4 Qp2 þ rgÞp2 h ¼ ro2 , which yields ( )1=2 ðDp4 Qp2 þ rgÞh c¼ ,

r

ð14Þ

cg ¼

ph

ro

ð3Dp4 2Qp2 þ rgÞ:

pffiffiffiffiffiffiffiffiffiffi As in case of infinite water depth, in this case also, it is obvious that for Q ¼ Q cr ¼ 2 Drg , c ¼ cmin ¼ 0. Further, in case of finite water depth, the dispersion relation in Eq. (12) has a real root at p ¼ p0 . From Eq. (12), the phase velocity c and group velocity cg are obtained as ( )1=2 ðDp4 Qp2 þ rgÞtanh ph c¼ , ð15Þ pr cg ¼

1 2 fðDp5 Qp3 þprgÞh sec h ph þð5Dp4 3Qp2 þ rgÞtanh phg: 2ro

ð16Þ

Thus, it is clear that c ¼ cmin ¼ 0 for Dp4 Qp2 þ rg ¼ 0 for all water depth, which is same as the characteristic equation of the differential equation associated with elastic beam on an elastic foundation as in Hetenyi (1946). From Eq. (15), it is pffiffiffiffiffiffiffiffiffi ffi clear that Q ¼ Q cr ¼ 2 Drg in case of finite water depth. N.B. The initial disturbance in the floating structure arises when the group velocity vanishes and after that wave energy propagates in the negative direction whilst, the phase velocity remains positive until it attends the critical limiting value. On the other hand, the study based on the direct buckling analysis provides the buckling limit as in Kerr (1983). Thus, it is clear that the value of the compressive force for which the phase velocity vanishes is the same as the buckling limit in case of a floating plate which is under uniform compression. Further, the critical limit is independent of water depth. 4. Wave diffraction by floating elastic plate In this section, the effect of compressive force on the diffraction of surface gravity waves by a floating elastic plate is analyzed in water of infinite and finite depths. Further, under the assumption of linearized theory of water waves, wave scattering by a finite floating elastic plate is analyzed directly under the assumption of linearized shallow water approximation and small amplitude structural response in the presence of compressive force. The spatial velocity potential in the open water region is denoted as f1 which is written as a linear combination of the i incident wave potential f along with the scattered potential. Hence the spatial velocity potential fðx,y,zÞ as defined in Section 2 can be written as ( f1 in the open water region R1 , f¼ ð17aÞ f2 in the plate covered region R2 ,

128

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

with !

!

!

f1 ð X Þ ¼ fi ð X Þ þ fsc ð X Þ,

ð17bÞ ! where X ¼ ðx,y,zÞ is any point in the open water region. Applying Green’s theorem to the velocity potentials f1 and f2 , it can be easily derived that (as in Hermans, 2000).  Z sc  @G @f sc 4pf ¼  dS, fsc G @n @n @R[R1   Z @G @f 0¼ f2 G 2 dS for x,z 2 R1 ð18Þ @n @n @R[R2 and 

sc  @G @f G dS, @n @n @R[R1   Z @G @f 4pf2 ¼ f2 G 2 dS @n @n @R[R2

Z 0¼

fsc

for x,z 2 R2 ,

where G is the free surface Green’s function which satisfies the governing equation ! ! r2 G ¼ 4pdð X  X Þ,

ð19Þ

ð20Þ

which satisfies the linearized free surface boundary condition, the bottom boundary condition and far field condition depending on water depth. Adding the two equations in Eq. (19) and using the free surface condition, continuity of pressure and velocity in the interface of plate and water and plate covered boundary conditions (proceeding in a similar manner as in Andrianov and Hermans, 2003), we obtain an integro-differential equation as given by 9 8 ( ) !2 !  2 2  2  Z = @ @2 @ @2 o2 < @2 @2 @2 @2 4p f2y gf2y þ D þ 2 f2y þ Q þ 2 f2y  gf2z D þ 2 f2z Q þ 2 f2z GdS 2 2 2 2 ; g R2 : @x @z @x @z @Z @Z @x @x ¼K

Z @R

fi

i

@G @f G @n @n

! i

dS ¼ 4pfy

on y ¼ 0,

ð21Þ

where D ¼ D=rg, Q ¼ Q =rg, g ¼ mo2 =rg and K ¼ o2 =g. Next, for clarity we will discuss the full solution in both the cases of infinite and finite depths separately. 4.1. Infinite water depth i

The incident wave potential f satisfying Eqs. (1) and (8b) and the dynamic condition as in Eq. (2) is of the form

fi ¼ 

gA ik0 ðx cos y þ z sin yÞk0 y e , io

ð22Þ

where A is the wave height, o is the frequency, h is the water depth and k0 is the wave number which satisfies the i dispersion relation o2 ¼ gk0 . Using the boundary conditions in Eqs. (2), (5) and putting the value of f as given by Eq. (22), from Eq. (21), an integro-differential equation for plate deflection wðx,zÞ is obtained as (  ) 2  2  @2 @2 @ @2 þ þ Q þ g þ 1 wðx,zÞ ¼ Aeik0 ðx cos y þ z sin yÞ  D @x2 @z2 @x2 @z2 8 !2 !9 Z < k0 @2 @2 @2 @2 = wðx, ZÞGðx,z, x, ZÞ dx dZ, þ gD þ 2 Q þ 2 ð23Þ 2 2 4p R2 : @Z @Z ; @x @x where G is the free surface Green’s function in case of infinite water depth and is given by Z kJ0 ðkrÞ dk at y ¼ z ¼ 0, Gðx,z; x, ZÞ ¼ 2 c kk0

ð24Þ

where c is the contour of integration from 0 to 1 underneath the singularity k ¼ k0 in the complex k-plane as in Fig. 2, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 0 ðkrÞ is Bessel’s function and r ¼ ðxxÞ2 þðzZÞ2 is the horizontal distance.

Fig. 2. Schematic diagram for contour of integration.

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

129

Since the plate is assumed to be of finite length, the plate deflection is approximated as a linear superposition of the horizontal eigenfunctions associated with the flexural gravity waves in the following form (as in Andrianov and Hermans, 2003) wðx,zÞ ¼

4 X

ðan eimn x þ bn eimn x Þeik0 z sin y

for 0 o x o l, 1 oz o 1

and

0 r y r p=2,

ð25Þ

n¼0

where an’s and bn’s are the unknown amplitudes to be determined. Further, the reduced wave numbers mn ’s are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 p2n k0 sin y ¼ mn where 7 pn s are roots of the dispersion relation in p ðDp4 Qp2 g þ1Þp ¼ K:

ð26Þ

It may be noted that Eq. (26) has one real root p0, four complex roots 7 p1 , 7p2 (for detail proof on the root characteristics, see Manam et al., 2006). However, three roots which are physically realistic and situated in the upper complex half plane are taken into account. Substituting Green’s function G as in Eq. (24) and the plate deflection wðx,zÞ as in Eq. (25) and proceeding in a similar manner as in Andrianov and Hermans (2003), two linear equations are obtained as   2 X an bn ðDp4n Qp2n gÞk0  þA ¼ 0 ð27Þ ðmn k0 cos yÞcos y ðmn þ k0 cos yÞcos y n¼0 and 2 X

ðDp4n Qp2n gÞk0

n¼0



 an eimn l bn eimn l þ ¼ 0: ðmn þk0 cos yÞcos y ðmn k0 cos yÞcos y

ð28Þ

Substituting for the plate deflection wðx,zÞ as in Eq. (25), from the condition of zero bending moment as in Eq. (7), we obtain 2 X

2

ðm2n þ nk0 sin2 yÞðan þ bn Þ ¼ 0

ð29Þ

n¼0

and 2 X

2

ðm2n þ nk0 sin2 yÞðan eimn l þ bn eimn l Þ ¼ 0:

ð30Þ

n¼0

In a similar manner, from the condition of zero shear force of Eq. (7), it can be easily derive that 2 X

2

fDfm3n þð2nÞmn k0 sin2 ygQ mn gðan bn Þ ¼ 0,

ð31Þ

n¼0

and 2 X

2

fDfm3n þð2nÞmn k0 sin2 ygQ mn gðan eimn l bn eimn l Þ ¼ 0:

ð32Þ

n¼0

Thus, Eqs. (27)–(32) yield six equations for the determination of the unknowns ans and bns. Using the relations in Eqs. (18), the amplitude of reflected and transmitted waves R, T in terms of ans, bns are obtained as (as in Andrianov and Hermans, 2003) R¼

2 2 X X ðDp4n Qp2n gÞan k0 iðk0 þ mn Þl ðDp4n Qp2n gÞbn k0 iðk0 mn Þl ðe 1Þ ðe 1Þ ð m þ k cos y Þcos y ð mn k0 cos yÞcos y 0 n n¼0 n¼0

ð33Þ

and T ¼ 1

2 2 X X ðDp4n Qp2n gÞan k0 iðk0 mn Þl ðDp4n Qp2n gÞbn k0 iðk0 þ mn Þl ðe 1Þ þ ðe 1Þ ðmn k0 cos yÞcos y ðmn þ k0 cos yÞcos y n¼0 n¼0

ð34Þ

from which the reflection and transmission coefficients K r ¼ 9R9 and K t ¼ 9T9 are obtained directly. It may be noted that hereafter 9A9 ¼ 1. Next, we will discuss the effect of compression on wave diffraction in case of finite water depth. 4.2. Finite water depth In case of finite water depth, the incident wave potential satisfying Eq. (1), the free surface boundary condition (2) and the bottom boundary condition (8a) is given by

fi ¼ 

gA cosh k0 ðhyÞ ik0 ðx cos y þ z sin yÞ e , io cosh k0 h

ð35Þ

130

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

where k0 is the same as defined in Eq. (10). Proceeding in a similar manner as in case of infinite water depth from Eqs. (2), (5), (21) and (35), an integro-differential equation for wðx,zÞ in case of water of finite depth is obtained as (  ) 2  2  @2 @2 @ @2 D þ þ Q þ g þ 1 wðx,zÞ ¼ Aeik0 ðx cos y þ z sin yÞ  @x2 @z2 @x2 @z2 8 !2 !9 Z < K @2 @2 @2 @2 = wðx, ZÞGðx,z, x, ZÞ dx dZ, þ gD þ 2 Q þ 2 ð36Þ 2 2 4p R2 : @Z @Z ; @x @x where K ¼ o2 =g and G is the free surface Green’s function in finite water depth at mean free surface is given by Z J0 ðkrÞk cosh kh dk at y ¼ z ¼ 0: Gðx,z; x, ZÞ ¼ 2 c k sinh khK cosh kh

ð37Þ

Proceeding in a similar manner as in the case of water of infinite depth, in the case of finite water depth also, the plate deflection is approximated as a linear superposition of the horizontal eigenfunctions associated with the flexural gravity waves in the form given by (as in Andrianov and Hermans, 2003) 1 X

wðx,zÞ ¼

ðan eimn x þbn eimn x Þeik0 z sin y

for 0 ox ol, 1 o z o1

and

0 r y r p=2,

ð38Þ

n ¼ 0,...,IV,1

where ans and bns are the unknown amplitudes to be determined. Further, the reduced wave numbers mn s are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p2n k0 sin2 y ¼ mn where 7pn s are roots of the dispersion relation in p as given by ðDp4n Qp2n g þ 1Þpn tanh pn h ¼ K:

ð39Þ

The dispersion relation in Eq. (39) has one real root, four complex roots and infinite number of imaginary roots (details about the root behavior has been shown in Manam et al., 2006). However, in the context of the present study, keeping the realistic nature of the physical problem, contributions from the positive real root and two complex roots lying in the upper complex half plane are taken into consideration in wðx,zÞ defined in Eq. (38), which is similar to the case of infinite water depth. However, a more accurate solution in water of finite depth can be obtained by considering the contribution from the imaginary roots of the dispersion relation, the details of which are differed here. Substituting for Green’s function G as in Eq. (37) and the plate deflection wðx,zÞ as in Eq. (38) in Eq. (36) and proceeding in a similar manner as in Andrianov and Hermans (2003), a system of two linear equations are derived as   2 X k0 K an bn ðDp4n Qp2n gÞ  þA ¼ 0, ð40aÞ 2 ðKð1KhÞ þ k0 hÞ ðmn k0 cos yÞcos y ðmn þk0 cos yÞcos y n¼0 2 X

ðDp4n Qp2n gÞ

n¼0

k0 K 2 ðKð1KhÞ þ k0 hÞ



 an eimn l bn eimn l þ ¼ 0: ðmn þ k0 cos yÞcos y ðmn k0 cos yÞcos y

ð40bÞ

Further, substituting for wðx,zÞ in the free edge conditions at the plate edges in Eq. (7), a system of four equations are obtained as in case of infinite water depth which are solved along with Eqs. (40a) and (40b) to obtain the unknown coefficients ans and bn s. Once, the amplitudes of reflected and transmitted waves R, T in terms of ans and bns are obtained as ( ) 2 2 X X k0 K ðDp4n Qp2n gÞan ðDp4n Qp2n gÞbn iðk0 þ mn Þl iðk0 mn Þl R¼ ðe 1Þ ðe 1Þ , ð41Þ 2 ðmn k0 cos yÞcos y ðKK 2 h þk hÞ n ¼ 0 ðmn þk0 cos yÞcos y n¼0 0

and T ¼ 1

(

k0 K 2

ðKK 2 hþ k0 hÞ

) 2 2 X X ðDp4n Qp2n gÞan ðDp4n Qp2n gÞbn ðeiðk0 mn Þl 1Þ ðeiðk0 þ mn Þl 1Þ ðmn k0 cos yÞcos y ðmn þk0 cos yÞcos y n¼0 n¼0

ð42Þ

from which the reflection and transmission coefficients K r ¼ 9R9 and K t ¼ 9T9 are computed. Next, we will discuss the wave scattering by a plate of finite width in the presence of compressive force under the assumption of shallow water approximation in a direct manner which is different from the procedure followed in case of water of infinite/finite water depth. 4.3. Shallow water approximation In this subsection, we will follow the same notation for potentials in the open water region and the plate covered region. We will discuss the wave scattering by a floating elastic plate of finite width in the presence of compressive force under the assumption of shallow water approximation. The equation of continuity for the two-dimensional long wave is given by 2

W t ¼ hrxz F,

ð43Þ

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

131

2

where W and F are as defined in Section 2 and rxz ¼ @2 =@x2 þ @2 =@z2 . From linearized long wave equations of motion, we have ut ¼ gW x ,

vt ¼ gW z ,

ð44Þ

with u ¼ Fx and v ¼ Fz . Eliminating W from Eqs. (43) and (44), we obtain

Ftt ¼ hr2xz F,

ð45Þ

which in terms of the spatial velocity potential fðx,zÞ can be written as

r2xz f1 þ k20 f1 ¼ 0 in the open water region,

ð46Þ 2

2

where k0 is the real root of the shallow water dispersion relation as given by o ¼ ghk . From Eqs. (43) and (5), it can be easily derived that

r2xz f2 þ

io w¼0 h

in the plate covered region:

ð47Þ

Using the kinematic condition given by Eq. (5), from Eq. (47), it can be derived that the spatial velocity potential f2 in the plate covered region satisfies

f2 ¼ iohw for 0 ox o l and 9z9o 1:

ð48Þ

Further, eliminating W from Eqs. (3) and (43), the linearized long wave in the plate covered region is obtained as 4

2

2

fDrxz þ Qrxz þ ð1gÞgrxz F2 ¼

F2tt gh

,

ð49Þ

where D, Q and g are same as in Eq. (21). Eliminating F from Eqs. (3) and (43), the long wave equation of motion in Eq. (49) can be rewritten as fDr4xz þ Qr2xz þ ð1gÞgr2xz W ¼

W tt , gh

ð50Þ

2

where m2n ¼ p2n k0 sin2 y and 7pn s are the roots of the dispersion relation as given by fDp4n Qp2n þ ð1gÞgp2n h ¼ K:

ð51Þ

From Eq. (46), the general form of velocity potential in the open water region is given by ( eik0 ðx cos y þ z sin yÞ þ Reik0 ðx cos y þ z sin yÞ for x o0, f1 ðx,zÞ ¼ for x 4l, Teik0 ðx cos y þ z sin yÞ

ð52Þ

where R and T are the amplitude of the reflected and transmitted waves respectively with k0 being the same as defined in Eq. (52). As discussed in case of infinite/finite water depth, in this case also, in the plate covered region wðx,zÞ is given by Eq. (25) with pns being roots of the dispersion relation in Eq. (51) which lies on the upper complex half plane. Thus, the general form of the velocity potential f2 in the plate covered region can be obtained from Eq. (48). Continuity of pressure and mass flux at both the edges x ¼ 0, x ¼ l yields

f2 ¼ f1 ,

@f2 @f1 ¼ , x,z 2 @R: @n @n

ð53Þ

Using Eqs. (48) and (52) in the conditions (53) at both the edges, the following four linear equations are obtained as 1 þR ¼ 

2 oh X

i

ðan þ bn Þ,

ð54Þ

n¼0

ik0 cos yð1RÞ ¼ oh

2 X

mn ðan bn Þ,

ð55Þ

n¼0

Teik0 l cos y ¼ 

2 oh X

i

ðan eimn l þbn eimn l Þ

ð56Þ

n¼0

and ik0 cos yTeik0 l cos y ¼ oh

2 X

mn ðan eimn l bn eimn l Þ:

ð57Þ

n¼0

Again, the free edge boundary conditions yield the four linear equations, Eqs. (29)–(32). Using the above mentioned eight system of linear equations, the eight unknowns ans, bns, R and T are obtained. Once the reflected wave amplitude R and transmitted wave amplitude T are determined, the reflection coefficient Kr and transmission coefficient Kt can be obtained from the relations K r ¼ 9R9 and K t ¼ 9T9.

132

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

4.4. Effect of oblique angle on water depth In case of wave scattering by a floating elastic plate, equality of the z-component of the open water region wave number k0 and its counterpart in the floating plate p0 yields the equivalent form of Snell’s law corresponding to wave structure interaction problem and is given by (as in Karmakar et al., 2010) k0 sin yi ¼ p0 sin yT ,

ð58Þ

where k0 and p0 are the wave numbers associated with surface gravity wave and flexural gravity wave, yi and yT are the angle of incidence and angle of transmission respectively. If k0 op0 then sin yT osin yi , which implies that the transmitted wave angle is always less than that of the incident wave angle. Thus, the transmitted wave is refracted towards the normal at the plate edge and full reflection never occurs. On the other hand, if k0 4 p0 then sin yT 4sin yi , which implies that the transmitted wave angle yT is always greater than the angle of incidence yi . Since 0 o yT r p=2, there exists a natural limit to the angle of incidence yi , beyond which wave propagation cannot take place into the plate covered region. Putting yT ¼ p=2 in relation (58), yi is obtained as

yi ¼ sin1 ðp0 =k0 Þ ¼ ycr ðsayÞ,

ð59Þ

which is called the critical angle of incidence. 4.5. Numerical results and discussion To understand the effect of compression on the diffraction of surface waves by the floating elastic plate, several results on the reflection coefficients are analyzed in different cases. Hereafter, all computation are carried out through out the paper by considering n ¼ 0:3, g ¼9.81 ms  2, flexural rigidity of the elastic plate D ¼ 105 N m, water density r ¼ 1025 kg m3 and time period T ¼5 s unless it is mentioned. In Fig. 3, the variation of reflection coefficient Kr versus angle of incidence for different values of compressive forces in case of infinite water depth are plotted. There is a resonating pattern in reflection coefficients as observed by Andrianov and Hermans (2003) in case of infinite water depth. However, beyond certain value of angle of incidence, Kr becomes one and the resonating pattern in the reflection coefficient disappear. This angle is referred as the critical angle as discussed in Section 4.4. Further, it is observed that the critical angle depends on the value of the compressive force.

Q=0

Q = (Dρg)1/2

Q = (1/3) (20Dρg)1/2

Q = 2 (Dρg)1/2 1 0.9 0.8 0.7

Kr

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15 20 25 Angle of incidence (θ)

30

35

40

Fig. 3. Variation of reflection coefficient Kr versus angle of incidence y for various values of compressive force Q with time period T¼ 5 s in case of infinite water depth.

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

133

1 0.9

Q=0 Q = (Dρg)1/2 Q = (1/3) (20Dρg)1/2 Q = 2 (Dρg)1/2

0.8 0.7

Kr

0.6 0.5 0.4 0.3 0.2 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 λ/l Fig. 4. Variation of reflection coefficient Kr versus wave length l=l for various values of compressive force Q with angle of incidence y ¼ 101 in case of infinite water depth.

1

Q=0

Q = (Dρg)1/2

Q = (1/3) (20Dρg)1/2

Q = 2(Dρg)1/2

0.9 0.8 0.7

Kr

0.6 0.5 0.4 0.3 0.2 0.1 0 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

λ/l Fig. 5. Variation of reflection coefficient Kr versus wave length l=l for various values of compressive force Q with angle of incidence y ¼ 101 and water depth h ¼5 m in case of finite water depth.

In Fig. 4, the variation of reflection coefficient Kr versus wavelength are plotted for different values of compressive force with y ¼ 101 in case of infinite water depth. In general, the reflection coefficient increases with an increase in compressive force. However, number of zeros in the reflection coefficient Kr decreases with an increase in compressive force. In Fig. 5, the variation of the reflection coefficient Kr versus wavelength for different values of compressive force Q with y ¼ 101 and h¼5 m in case of finite water depth are plotted. The pattern of reflection coefficient Kr is the same as in Fig. 4 as discussed in water of infinite depth. However, the number of zeros in the reflection coefficient has reduced in finite water depth. Further, in finite water depth, less reflection occurs compared to that in case of water of infinite depth. In Fig. 6, the variation of reflection coefficient Kr versus wavelength for different values of compressive forces with y ¼ 101 and h¼ 1 m are plotted in case of shallow water. The general patterns of the reflection coefficient Kr are similar to the result of Andrianov and Hermans (2003) for zero compressive force. However, in case of shallow water approximation, within the critical buckling limit, change in compression has negligible effect on the reflection coefficient. In Fig. 7, the variation of reflection coefficient Kr versus time period for different values of compressive forces with y ¼ 101 and h¼1 m are plotted. It is observed that for smaller time period within 3–5 s, resonating pattern occurs very frequently which decreases as the wave period increases beyond 5 s.

134

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

1 0.9 0.8 0.7

Kr

0.6 0.5 0.4

Q=0 Q = (Dρg)1/2

0.3

Q = (1/3) (20Dρg)1/2

0.2

Q = 2 (Dρg)1/2

0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 λ/l Fig. 6. Variation of reflection coefficient Kr versus wave length l=l for various values of compressive force Q with angle of incidence y ¼ 101 and water depth h ¼1 m in case of shallow water approximation.

1 0.9 0.8 0.7

Kr

0.6 0.5 0.4 0.3 Q=0 Q = (Dρg)1/2

0.2

Q = (1/3) (20Dρg)1/2

0.1

Q = 2(Dρg)1/2

0 0

5

10 Time period (T)

15

Fig. 7. Variation of reflection coefficient Kr versus time period T for various values of compressive force Q with angle of incidence y ¼ 101 and water depth h ¼ 1 m in case of shallow water approximation.

1 0.9 0.8 0.7

Kr

0.6 0.5 0.4 0.3 0.2

Finite Infinite Shallow

0.1 0

0

5

10

15 20 25 30 Angle of incidence (θ)

Fig. 8. Variation of reflection coefficient Kr versus angle of incidence y for Q ¼ ð1=3Þ

35

40

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20Drg for various water depths.

S.C. Mohapatra et al. / Journal of Fluids and Structures 36 (2013) 124–135

135

In Fig. 8, variation of reflection coefficient Kr versus angle of incidence for finite, infinite and shallow water depth with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi compressive force Q ¼ ð1=3Þ 20Drg are plotted. It is observed that, the critical angle is highest for infinite water depth and the least in case of shallow water approximation. Further, within the critical angle limit of shallow water approximation, resonating pattern in Kr occurs more frequently. 5. Conclusion In the present study, the diffraction of surface gravity waves with a floating finite elastic plate is analyzed based on the boundary integral equation method. The present formulation is a generalization of the formulation developed by Hermans (2000) in which the plate deflection is expanded in terms of the eigenvalues associated with the flexural gravity waves. Also, the linearized long wave equations are derived in direct manner to study the wave scattering by a finite plate in the presence of compressive force. The results are analyzed and compared in water of infinite and finite depths under small amplitude wave theory as well as under shallow water approximation in the presence of compressive force. Further, limiting values of compressive force are found by direct hydroelastic analysis in different water depths by analyzing the phase and group velocities of the flexural gravity waves. Within the critical limit, effect of compression changes the hydrodynamic characteristics. Also, the role of critical angle have been analyzed in case of wave scattering by floating plate. The novelty of the present approach is that, the free surface Green’s function is used to form the integro-differential equation and the method can be easily extended to deal with floating structures of different configurations and interconnected structures. The present study is likely to enhance the understanding of the hydroelastic analysis of floating elastic plates, which are of importance to Naval Architects and Ocean Engineers involved in the design of large floating structures, and researchers working on wave ice interaction problems.

Acknowledgement The authors acknowledge the financial support of Naval Research Board, New Delhi, Government of India for pursuing this research work. References Andrianov, A.D., Hermans, A.J., 2003. The influence of water depth on the hydroelastic response of a very large floating platform. Marine Structure 16 (5), 355–371. Andrianov, A.I., Hermans, A.J., 2005. Hydroelasticity of a circular plate on water of finite or infinite depth. Journal of Fluids and Structures 20 (5), 719–733. Andrianov, A.I., Hermans, A.J., 2006. Hydroelastic behavior of a floating ring-shaped plate. Journal of Engineering Mathematics 54 (1), 31–48. Bukatov, A.E., Zav’yalov, D.D., 1995. Impingement of surface waves on the edge of compressed ice. Fluid Dynamics 30 (3), 435–440. Fox, C., Squire, V.A., 1990. Reflection and transmission characteristics at the edge of shore fast sea ice. Journal of Geophysical Research 95 (C7), 11629–11639. Fox, C., Squire, V.A., 1994. On the oblique reflection and transmission of ocean waves at shore fast sea ice. Philosophical Transaction of Royal Society London, Series A 347, 185–218. Hermans, A.J., 2000. A boundary element method for the interaction of free-surface waves with a very large floating flexible platform. Journal of Fluids and Structures 14, 943–956. Hetenyi, M., 1946. Beams on Elastic Foundation. University of Michigan. Kashiwagi, M., 1998. A B-spline Galerkin scheme for calculating hydroelastic response of a very large floating structure in waves. Marine Science and Technology 3, 37–49. Karmakar, D., Bhattacharjee, J., Sahoo, T., 2009. Wave interaction with multiple articulated floating elastic plates. Journal of Fluids and Structures 25, 1065–1078. Karmakar, D., Bhattacharjee, J., Sahoo, T., 2010. Oblique flexural gravity-wave scattering due to changes in bottom topography. Journal of Engineering Mathematics 66, 325–341. Kerr, A.D., 1983. The critical velocities of a load moving on a floating ice plate that is subjected to in-plane forces. Cold Regions Science and Technology 6, 267–274. Loukogeorgaki, E., Michailides, C., Angelides, D.C., 2012. Hydroelastic analysis of a flexible mat-shaped floating breakwater under oblique wave action. Journal of Fluids and Structures 31, 103–124. Manam, S.R., Bhattacharjee, J., Sahoo, T., 2006. Expansion formulae in wave structure interaction problems. Proceeding of Royal Society of London, Series A 462, 263–287. Manam, S.R., Kaligatla, R.B., 2012. A mild-slope model for membrane-coupled gravity waves. Journal of Fluids and Structures 30, 173–187. Newman, N., 1994. Wave effects on deformable bodies. Applied Ocean Research 16 (1), 47–59. Ohkusu, M., Namba, Y., 2004. Hydroelastic analysis of a large floating structure. Journal of Fluids and Structures 19, 543–555. Riyansyah, M., Wang, C.M., Choo, Y.S., 2010. Connection design for two-floating beam system for minimum hydroelastic response. Marine Structures 23, 67–87. Sahoo, T., Yip, L., Chwang, A.T., 2001. Scattering of surface waves by a semi infinite floating elastic plate. Physics of Fluids 13 (11), 3215–3222. Schulkes, R.M.S.M., Hosking, R.J., Sneyd, A.D., 1987. Waves due to a steadily moving source on a floating ice plate. Part-2. Journal of Fluid Mechanics 180, 297–318. Squire, V.A., Hosking, R.J., Kerr, A.D., Laghorne, P.J., 1996. Moving Loads on Ice Plates. Kluwer Academic Publisher. Squire, V.A., 2008. Synergies between VLFS hydroelasticity and sea-ice research. International Journal of Offshore and Polar Engineering 18 (3), 1–13. Sturova, I.V., 1998. The oblique incidence of surface waves onto the elastic band. In: Proceedings of the 2nd International Conference on Hydroelasticity in Marine Technology, Fukuoka, Japan, pp. 239–245. Tkacheva, L.A., 2003. Plane problem of surface wave diffraction on a floating elastic plate. Fluid Dynamics 38 (3), 465–481. Wang, C.M., Tay1, Z.Y., Takagi, K., Utsunomiya, T., 2010. Literature review of methods for mitigating hydroelastic response of VLFS under wave action. Applied Mechanics Reviews vol. 63. 030802-1–18.