on drift motion of deformable ice sheets by nonlinear waves · 2020-01-07 · the 35th...

The 35th International Workshop on Water Waves and Floating Bodies, 26–29 April, Seul, South Korea, 2020.

On drift motion of deformable ice sheets by nonlinear waves

Vasily Kostikov1, Masoud Hayatdavoodi1,2, R. Cengiz Ertekin1,3

1 College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China2 School of Science and Engineering, University of Dundee, Dundee, UK

3 Ocean and Resources Engineering Department, University of Hawaii at Manoa, Honolulu, USAE-mail: [email protected]

The ice cover near the polar regions has been strongly affected as a result of global warming, particularly onits margins, such that it consists of a mixture of discrete ice floes and open water. The ocean waves propagatingin such regions contribute not only to the break-up of the ice, but also stimulate their drift. Therefore, in additionto melting, surface waves play a determining role in shaping the marginal ice zones [1].

The drift motion and hydroelastic response of a set of floating elastic ice sheets to incident nonlinear waves in

shallow water is studied by use of the Level I Green-Naghdi (GN hereafter) equations. The set of N deformable ice

sheets is presented by thin elastic plates of variable properties. The resulting governing equations together with

the appropriate boundary and matching conditions are solved in two-dimensions by the finite difference method.

Free surface elevation and ice sheet deformations are calculated. The effect of multiple plates on the wave-induced

loads and velocity field is investigated. Drift motion of the ice sheets is determined by calculating the instantaneous

wave-induced force on the bodies, and by solving Newton’s second law. Only horizontal motion is considered.

Problem formulation. The plane flow of inviscid fluid of constant depth h is considered in theCartesian coordinate system Oxy with the horizontal axis lying on the undisturbed free surface andvertical axis directed upwards. Incident waves propagate in the positive x–direction and excite themotion of a set of floating deformable sheets being initially at rest. The sketch of the problem is shownin Fig. 1. The ice sheets are homogenous and have arbitrary length Lk, mass per unit width mk andflexural rigidity Dk, where subscript k = 1, 2, . . . , N indicates each ice sheet.

RI RIRII

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RI RII RI RII RI RII RI

At the leading x = xkL and trailing x = xk

T edges of the ice sheets, the bending moments together with shear

stresses must vanish due to the free-free end boundary condition (⇣xx = ⇣xxx = 0). Applying this relations to

the first equation (1), we obtain another conditions under the edges of the plates:

3⌘xuxx + (1 � hk + ⌘)uxxx = 0, 4⌘xuxxx + (1 � hk + ⌘)uxxxx + ⌘xxxxu = 0, (x = xkL, xk

T ), (3)

The appropriate jump conditions are demanded by the theory because the fluid surface is singular or discon-

tinuous at the juncture of regions I and II. The matching conditions, demanded by the physics of the problem,

include continous surface elevation, mass flux and bottom pressure across the discontinuity curves x = xkL,

x = xkT .

The system of equations (1) and (2), subject to appropriate boundary conditions (3) in each region, along

with the matching and jump conditions prescribe the motion of the fluid in the entire domain and are solved

simultaneously for the unknowns. Spatial discretization of the equations is carried out by a central-di↵erence

method and time marching is obtained by use of the modified Euler’s method. The system of equations are to

be solved by use of a Gaussian elimination method.

The underlying principle of this technique is that at a given time, equations (2) can be solved explicitly for

⇣t once u is known. Then, ⇣t is substituted into the second equation of the system (2) to solve for ut.

(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1

3(1 � hk + ⇣)2(uxuxx � uuxxx) + uux + ⇣x,

Yp = m[(1 � hk + ⇣)(3uxuxx + uuxxx) + D⇣xxxxx + ⇣xxxu2 + 5⇣xxuux + 4⇣x(u2x + uuxx)].

(4)

The equation for ⌘ in RI can be obtained from (4) by simply equating the parameters mk, Dk and hk to zero.

The continuous variables ⌘(x, t) and u(x, t) in (4) are approximated by discrete variables ⌘ni and un

i , respec-

tively, where i is the mesh point in the spatial domain and n specifies a mesh point on the time axis. At each

given time, such discretization results in an array of N unknowns, and an N ⇥N coe�cient matrix, where N is

the total number of mesh points in the computational domain. The coe�cient matrix is a banded matrix and

the Gaussian elimination algorithm for a banded diagonal matrix is used to solve the system of equations.

The higher derivatives in the jump and boundary conditions require the involvement of fictitious points

between each pair of regions.

References

1. Green A.E., Naghdi P.M., 1976a. A derivation of equations for wave propagation in water of variable depth.

Journal of Fluid Mechanics 78, 237–246.

2. Ertekin R.C., 1984. Soliton Generation by Moving Disturbances in Shallow Water: Theory, Computation and

Experiment (Ph.D. thesis, Ph.D. dissertation). University of California at Berkeley, May, v+352 pp.

3. Xia D., Ertekin R.C., Kim J.W., 2008. Fluid–structure interaction between a two-dimensional mat-type VLFS

and solitary waves by the Green–Naghdi theory. Journal of Fluids and Structures 24, 527–540.

4. Hayatdavoodi M., Ertekin R.C., 2015. Wave forces on a submerged horizontal plate – Part I: Theory and

Modelling. Journal of Fluids and Structures.

2

Nonlinear dynamics of the set of floating elastic ice sheets

Vasily Kostikova, Masoud Hayatdavoodia,b, R. Cengiz Ertekina and Yongbo Chena

a College of Shipbuilding and Ocean Engineering, Harbin Engineering University, Harbin, Chinab School of Science and Engineering, University of Dundee, Dundee DD1 4HN, UK

E-mail: vasilii�[email protected]

Abstract The present research is dedicated to the interaction of waves with multiple deformable ice sheets floating

in marginal ice zones. The simultaneous motion of the ice sheets and the fluid is regarded within the framework of linear

beam theory for the structures and the nonlinear Level I Green-Naghdi theory for the fluid. The resulting governing

equations, subjected to appropriate boundary and jump conditions, are solved by finite di↵erence method.

Green-Naghdi theory Nonlinear waves of solitary and cnoidal types propagating from left to right and

exciting the motion of the set of N floating elastic ice sheets are considered. The ice sheets have arbitrary width

lk, mass mk and elasticity Dk.

Figure 1. Scheme of motion.

The mathematical model employed in this study is provided by the Level I Green-Naghdi theory (Green &

Nahgdi, 1976) which is mostly apliacable to the propagation of fairly long waves in shallow waters. This theory

is also known as the restricted theory and assumes a linear distribution of the vertical velocity along the water

column. This assumption, which is the only one made about the kinematics of the fluid sheet, results in the

horizontal velocities being invariant in the vertical direction for an incompressible fluid.

In the analysis of this problem, it becomes necessary to consider two types of fluid regions: type I, for

which the top curve of the fluid sheet is free and on the top pressure p̂ is constant; and type II, for which the

upper pressure p̂ remains unknown. In type I regions, the governing equations for the motion of the fluid in

dimensionless variables can be written as follows (Ertekin, 1984)

⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)

The system (1) can be solved to find the two unknown functions, such as surface elevation ⌘(x, t) and depth

averaged horizontal velocity u(x, t). In type II region the motion of the fluid under the plate is prescribed by

equations where fluid and plate are coupled through the dynamic free-surface condition from the linear beam

theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1

3[2⇣x⇣̈ + (1 � hk + ⇣)⇣̈x], p̂ = mk⇣tt + Dk⇣xxxx + m. (2)

The system (2) is integrated for the unknown deformation of the plate ⇣, depth averaged horizontal velocity u

and pressure under the ice p̂. Then the pressure at the bottom surface of the fluid sheet and vertical component

of particle velocity can be determined from equations

p =1

2

((1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

8>><>>:

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

((1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

8>><>>:

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1

4 V. Kostikov, M. Hayatdavoodi and R. C. Ertekin

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ice sheet I ice sheet II ice sheet N






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-


given time, such discretization results in an array of N unknowns, and an N �N coe�cient matrix, where N is





x1L x2

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T x2T xN

T

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2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







References









2








equations, subjected to appropriate boundary and jump conditions, are solved by finite di�erence method.














⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1


The system (2) is integrated for the unknown deformation of the plate ⇣(x, t), depth averaged horizontal velocity

u and pressure under the ice p̂. Then the pressure at the bottom surface of the fluid sheet and vertical component


p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1Figure 1. Scheme of motion.

2. Formulation of the problem

The plane irrotational flow of heavy inviscid fluid of constant depth h is considered inthe coordinate system Oxy with the horizontal axis lying on the undisturbed free surfaceand vertical axis directed upwards. Incident waves propagate in the positive x–directionand excite the motion of the set of N floating deformable sheets being initially at rest.The k-th sheet has the width Lk, draft dk, mass per unit width mk and elasticity Dk.Let Lk,k+1 denote the distance between adjacent sheets. It is useful to introduce theclearence between the bottom of the plate and the sea floor as hk = h � dk. A sketchof the problem is shown in Fig. 1. It is assumed that elastic sheets are restricted to thevertical motion only and are always in contact with the fluid. The loss of energy due tothe structural damping is neglected and plate overwashing is disregarded.

The dimensionless variables use the fluid depth h as a length scale, the ratiop

h/gas a time unit, the quantity ⇢h as a mass unit. Here ⇢ and g are respectively the fluiddensity and gravity acceleration.

The governing equations for the motion of the fluid are provided by the Level I Green-Naghdi (GN) theory. They can be written in a compact form

It is conventional to divide the fluid domain into subregions of two types (RI and RII).Subregion of the first type is formed by the flat floor at the bottom and by the freesurface at the top. The pressure on the top free surface is constant atmospheric pressure.Subregion of the second type is formed by the flat floor at the bottom and by elasticplate at the top. As opposed to region RI, the pressure on the top of region RII dependson vertical displacement of the ice sheet. The Level I GN theory of the directed fluidsheets will be adopted to treat the fluid problem and to use it with suitable matchingconditions on the juncture boundary between the subdomains RI and RII.

⌘t + [(hk + ⌘)u]x = 0, (2.1)

u̇ + ⌘x + p̂x = �1

3[2⌘x⌘̈ + (hk + ⌘)⌘̈x] (2.2)

for k-th sheet, k = 1, . . . , N . The motion of the fluid and the plate is coupled throughthe dynamic free-surface condition. The vertical displacement of the plate ⌘ is governedby the thin elastic plate theory:

p̂ = mk⌘tt + Dk⌘xxxx + mk, (2.3)

The system (2.1)-(2.3) is integrated for the unknowns ⌘(x, t), u(x, t) and upper pressurep̂(x, t). Then the pressure at the sea-floor (y = �1) and vertical velocity can bedetermined from equations:

p̄ =1

2(hk + ⌘)(⌘̈ + 2) + p̂, (2.4)

v(y) =1 + y

hk + ⇣⌘̇. (2.5)

The formula (2.5) implies zero vertical velocity along the bottom (y = �1). The equation

Analysis of nonlinear wave interaction with elastic surfaces 13

method developed demonstrates high e�ciency, performing calculation with multiplesheets in seconds. Although the theory adopted in the paper was initally constructed forstudying the interaction between water waves and very large floating objects, the studyhas shown that this method is also valid for small- and middle-sized floating surfaces.The e↵ect of the multiple sheets and the distance between them is studied. The interplaybetween individual e↵ects of the sheet properties can be explored more thoroughly than itis attempted here. The obtained solution is validated by comparing with other numericalsolutions and experimental data obtained with use of alternative methods.

L1 L12

REFERENCES

Ertekin, R. C. & Xia, D. 2014 Hydroelastic response of a floating runway to cnoidal waves.Phys. Fluids 26.

Evans, D. V. & Davies, T. V. 1968 Wave-ice interaction. Rep. 1313. Davidson Lab StevensInstitute of Technology, New Jersey.

Evans, D. V. & Porter, R. 2003 Wave scattering by narrow cracks in ice sheets floating onwater of finite depth. J. Fluid Mech. 484, 143-165.

Fox, C. & Squire, V. A. 1991 Coupling between the ocean and an ice shelf. J. Ann. Glaciol.15, 101–108.

Green, A. E. & Naghdi, P. M. 1976a A derivation of equations for wave propagation in waterof variable depth. J. Fluid Mech. 78, 237–246.

Grue, J. 1992 Nonlinear water waves at a submerged obstacle or bottom topography. J. FluidMech. 244, 455–476.

Hayatdavoodi, M. & Ertekin, R. C. 2015 Wave forces on a submerged horizontal plate -Part I: Theory and modelling. J. Fluid Struct. 54, 566–579.

Kashiwagi, M. 1998 A B-spline Galerkin scheme for calculating hydroelastic response of a verylarge floating structure in waves. J. Marine Sci. and Tech. 3, 37–49.

Khabakhpasheva, T. I. & Korobkin, A. A. 2002 Hydroelastic behaviour of compound floatingplate in waves. J. Eng. Math. 44, 21–40.

Kohout, A. L., Meylan, M. H., Sakai, S., Hanai, K., Leman, P. & Brossard, D.2007 Linear water wave propagation through multiple floating elastic plates of variableproperties. J. Fluids Struct. 23, 649–663.

Korobkin, A. A. 2000 Numerical and asymptotic study of the two-dimensional problem of thehydroelastic behaviour of a floating plate in waves. J. Appl. Mech. Tech. Phys. 41(2),286–292.

Kim, J. W. & Ertekin, R. C. 2002 Hydroelasticity of an infinitely long plate in oblique waves:linear Green-Naghdi theory. Proc. Instn. Mech. Engrs. 216, 179–197.

Liu, X. & Sakai, S. 2000 Nonlinear analysis on the interaction of waves and flexible floatingstructure. Proc. Tenth Int. O↵shore and Polar Engineering Conf. 1,101–108.

Maiti, P. & Mandal B. N. 2014 Water wave scattering by an elastic plate floating in an oceanwith a porous bed. Applied Ocean Research. 47,73–84.

Ogasawara, T. & Sakai, S. 2006 Numerical analysis of the characteristics of waves propagatingin arbitrary ice-covered sea. Annals of Glaciology. 44, 95–100.

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Sun H., Song H. , Cui W. & Liu, Y. 2002 On the interaction of surface waves with an elasticplate of finite length in head seas. China Ocean Engineering. 16(1), 21–32.

Utsunomiya, T., Watanabe, E., Wu, C., Hayashi, N., Nakai, K. & Sekita, K. 1995 Waveresponse analysis of a flexible floating structure by BE-FE combination method. Proc.Fifth Int. O↵shore and Polar Engineering Conf. 3, 400–405.

Wu, C., Watanabe, E. & Utsunomiya, T. 1995 An eigenfunction expansion-matching method


RI RIRII

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surface 1 surface 2 surface N






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







References









2






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1


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T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

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2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







References









2






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.








⌘t + [(hk + ⌘)u]x = 0, (2.1)

u̇ + ⌘x + p̂x = �1

3[2⌘x⌘̈ + (hk + ⌘)⌘̈x] (2.2)




p̄ =1

2(hk + ⌘)(⌘̈ + 2) + p̂, (2.4)

v(y) =1 + y

hk + ⇣⌘̇. (2.5)




L1 L12

REFERENCES























L1 L12

REFERENCES























hkp

REFERENCES























A

REFERENCES





















Figure 1. Schematic of the problem of nonlinear wave interaction with N number of plates ofarbitrary size and location. Also shown in this figure are the RI and RII regions refereed to inthe text.

GN theory for a flat and stationary seafloor. They can be written in a compact form as(see Ertekin et al (1986)):

⌘,t +[(1 + ⌘)u],x = 0, (2.1)

u̇ +⌘,x = �1

3[2⌘,x⌘̈ + (1 + ⌘)⌘̈,x]. (2.2)

Similarly, the governing equations for the coupled motion of the fluid and the elasticsheet in RII can be written as:

⇣,t +[(hk + ⇣)u],x = 0, (2.3)

u̇ +⇣,x + p̂,x = �1

3[2⇣,x⇣̈ + (hk + ⇣)⇣̈,x], (2.4)

In our notation subscript with comma denotes the partial derivative and upper dotspecifies the total time derivative (⌘̇ = ⌘,t + u⌘,x). The wave-induced pressure under theplate is given by the thin elastic plate theory:

p̂ = mk⇣,tt + Dk⇣,xxxx + mk, (2.5)

where the flexural rigidity is defined by

Dk = Ek�3k/12(1 � ⌫2

k),

with �k, Ek, and ⌫k being the thickness, Young’s modulus and Poisson’s ratio of the k-thplate, respectively. Eqs. (2.1)–(2.5) can be integrated for the unknowns: free surface ele-vation ⌘(x, t), plate deflection ⇣(x, t), both measured from the still water level, horizontalcomponent of fluid velocity u(x, t) and pressure p̂(x, t) under the plate.

Ertekin (1984) provided explicit relations for the vertical velocity of the fluid andpressure on the bottom (y = �1), given as:

v (y) =1 + y

1 + ⌘⌘̇ (x, y) 2 RI, v(y) =

1 + y

hk + ⇣⇣̇ (x, y) 2 RII, (2.6)

p̄ =1

2(1 + ⌘)(⌘̈ + 2) (x, y) 2 RI, p̄ =

1

2(hk + ⇣)(⇣̈ + 2) + p̂ (x, y) 2 RII. (2.7)

The nonlinear kinematic and dynamic conditions on top and bottom boundaries of theflow domain are satisfied exactly by Eqs. (2.1)–(2.7).

In order to formulate the problem fully, suitable matching conditions at the fixedpositions of the leading (x = xL

k ) and traling edges (x = xTk ) of each floating sheet must

be specified. Since the elastic sheets are floating freely, the bending moment and the

This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1

On kinematic and dynamic analysisof nonlinear wave interactionwith floating elastic surfaces

Vasily Kostikov1, Masoud Hayatdavoodi1,2 and R. C. Ertekin 1,3†1College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China

2School of Science and Engineering, University of Dundee, DD1 4HN, UK3Ocean and Resources Engineering Department, University of Hawaii at Manoa, 2540 Dole St.,

Honolulu, HI 96822, USA

(Received xx; revised xx; accepted xx)

This paper presents the analytical method developed to model the hydroelastic responseof a set of floating elastic surfaces to incident nonlinear waves in shallow water. Thedeformable surfaces are presented by thin elastic plates of variable properties and theremay be N number of them of di↵erent size and elasticity. The coupled motion of elasticsurfaces and the fluid are solved simultaneously within the framework of linear beam the-ory for the structures and the nonlinear Level I Green-Naghdi (GN) theory for the fluid.The resulting governing equations together with the appropriate boundary conditions,and matching conditions, are solved in two-dimensions by the finite di↵erence method(FDM). Elevation of the free surface, the velocity and pressure fields are calculated andpresented. Results of the model developed here are compared with existing laboratorymeasurements and other numerical solutions. E↵ect of the size and number of floatingbodies on the deformation of the wave field is studied. In the absence of any restrictionon the nonlinearity of the wave field and the size and rigidity of the surfaces, a widerange of wave-structure conditions are considered. It is found that the Level I GNequations coupled with elastic beam theory can properly model the wave-structure field.It is found that increasing rigidity of the structures modifies the flow field significantly.Also, presence of small floating bodies upwave modifies remarkably the deformation andresponse of the downwave floating bodies.

Key words: hydroelasticity, Level I Green-Naghdi equations, multiple floating surfaces,floating ice sheets, elsatic beam theory

1. Introduction

�1Floating elastic surfaces are observed in nature and are widely utilized in industry.

Therefore the study of wave action on elastic floating bodies has received considerableattention. Floating ice sheets of di↵erent sizes, for example, are commonly observed inmarginal ice zones, see Squire et al (1995). Wave interaction with floating ice sheets isof interest due to the e↵ect that the ice sheets have on the wave field, and the impactof waves on the ice sheets, particularly when it results in breaking of the ice sheet to

† Email address for correspondence: [email protected]









1. Introduction

L2

Floating elastic surfaces are observed in nature and are widely utilized in industry.Therefore the study of wave action on elastic floating bodies has received considerableattention. Floating ice sheets of di↵erent sizes, for example, are commonly observed inmarginal ice zones, see Squire et al (1995). Wave interaction with floating ice sheets isof interest due to the e↵ect that the ice sheets have on the wave field, and the impactof waves on the ice sheets, particularly when it results in breaking of the ice sheet to










1. Introduction

LN











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N



Discussion of results. Results of the level I GN model for wave interaction with N number of floatingand deformable ice sheets are presented here. The results include plates deformations, time series of waveinduced forces, and variation of the forces with wave conditions. In these preliminary cases, we assumethat the plates are fixed in their horizontal locations.

Figure 2 shows velocity vectors of fluid particles plotted on contours of fluid velocity module(u2 + v2)1/2 in case of interaction of solitary wave with two plates, floating at a short distance �L = 5from each other, at three successive time moments. Small-amplitude leading waves propagate with higherspeed along the elastic surfaces, and hence the fluid domain feels the wave motion long before the waveitself reaches the end of the second plate. These elastic waves cause greater wave attenuation in a longerplate with higher rigidity. Another contribution to the wave attenuation is due to the e↵ect of multipleplates (see [8]).

Fig. 2. Vector field and module contours of dimensionless fluid velocity for the interaction of a solitary wave(A = 0.25) with the set of two plates with the same properties (Lk = 30, mk = 0.1, Dk = 5) divided by thefluid gap (�L = 5).

Figure 3 shows the interaction of a cnoidal wave with a set of three elastic plates of the same propertiesfor two values of plate rigidity. Also shown in figure 3 are the time series of the wave-induced horizontalcnoidal forces on these deformable ice sheets. The amplitudes of plates displacement and horizontal forcereduces downwave. The higher rigidity Dk of each plate contributes to the higher wave attenuation bythe floating system and to the lower drift of it’s elements.

Figure 4 shows the variation of the peak cnoidal wave horizontal force on the ice sheets F+1 with the

incoming wave length � and height H for the cases of a single plate and two plates located at di↵erentdistances �L from each other. The e↵ect of multiple plates is greater for shorter distance. The firstplate in the group of plates experiences greater drift force than a single plate of the same properties if thedistance is small. For larger distance, the second plate has less impact on the first plate and experiencesless horizontal force itself. Figure also 4 shows that drift force varies almost linearly with the wave heightand slight deviation of force from the straight line in the case of two plates can be observed.

More detailed discussion of results, including the interplay between individual e↵ects of each platein more complex plate systems and the drift motion of the plates, will be presented at the workshop.

�L1

References

[1] Melyan, M. & Squire, V.A. (1994) The response of ice floes to ocean waves. J. Geoph. Res. 99. 891-900.

RI RIRII

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x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-


given time, such discretization results in an array of N unknowns, and an N ⇥N coe�cient matrix, where N is





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⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

((1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

8>><>>:

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

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⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

((1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

8>><>>:

⌘̇

⌘ + 1, in RI

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x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







References









2






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.








⌘t + [(hk + ⌘)u]x = 0, (2.1)

u̇ + ⌘x + p̂x = �1

3[2⌘x⌘̈ + (hk + ⌘)⌘̈x] (2.2)




p̄ =1

2(hk + ⌘)(⌘̈ + 2) + p̂, (2.4)

v(y) =1 + y

hk + ⇣⌘̇. (2.5)




L1 L12

REFERENCES






















RI RIRII

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T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







References









2






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1


flat sea-floorwav

e m

aker

wav

e ab

sorb

er








T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







x1L x2

L xNL x1

T x2T xN

T

References









2






T ), (3)




x = xkT .








(1 � m⌘xx)ut � [⌘x(1 + ⌘) + 2m⌘x]uxt � [1

3(1 + ⌘)2 + m(1 + ⌘)]uxxt = �Yf � Yp,

Yf = ⌘x(1 � hk + ⇣)(u2x � uuxx) +

1



(4)



i , respec-







References









2






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.

1






















⌘t + [(1 + ⌘)u]x = 0, u̇ + ⌘x = �1

3[2⌘x⌘̈ + (1 + ⌘)⌘̈x]. (1)




theory

⇣t + [(1 � hk + ⇣)u]x = 0, u̇ + ⇣x + p̂x = �1





p =1

2

�(1 + ⌘)(⌘̈ + 2), in RI

(1 � hk + ⇣)(⇣̈ + 2) + p̂, in RIIv(y) = (y + 1)

��

⌘̇

⌘ + 1, in RI

⇣̇

⇣ + 1, in RII.








⌘t + [(hk + ⌘)u]x = 0, (2.1)

u̇ + ⌘x + p̂x = �1

3[2⌘x⌘̈ + (hk + ⌘)⌘̈x] (2.2)




p̄ =1

2(hk + ⌘)(⌘̈ + 2) + p̂, (2.4)

v(y) =1 + y

hk + ⇣⌘̇. (2.5)




L1 L12

REFERENCES























L1 L12

REFERENCES























hkp

REFERENCES























A

REFERENCES





















Figure 1. Schematic of the problem of nonlinear wave interaction with N number of plates ofarbitrary size and location. Also shown in this figure are the RI and RII regions refereed to inthe text.

GN theory for a flat and stationary seafloor. They can be written in a compact form as(see Ertekin et al (1986)):

⌘,t +[(1 + ⌘)u],x = 0, (2.1)

u̇ +⌘,x = �1

3[2⌘,x⌘̈ + (1 + ⌘)⌘̈,x]. (2.2)

Similarly, the governing equations for the coupled motion of the fluid and the elasticsheet in RII can be written as:

⇣,t +[(hk + ⇣)u],x = 0, (2.3)

u̇ +⇣,x + p̂,x = �1

3[2⇣,x⇣̈ + (hk + ⇣)⇣̈,x], (2.4)

In our notation subscript with comma denotes the partial derivative and upper dotspecifies the total time derivative (⌘̇ = ⌘,t + u⌘,x). The wave-induced pressure under theplate is given by the thin elastic plate theory:

p̂ = mk⇣,tt + Dk⇣,xxxx + mk, (2.5)

where the flexural rigidity is defined by

Dk = Ek�3k/12(1 � ⌫2

k),

with �k, Ek, and ⌫k being the thickness, Young’s modulus and Poisson’s ratio of the k-thplate, respectively. Eqs. (2.1)–(2.5) can be integrated for the unknowns: free surface ele-vation ⌘(x, t), plate deflection ⇣(x, t), both measured from the still water level, horizontalcomponent of fluid velocity u(x, t) and pressure p̂(x, t) under the plate.

Ertekin (1984) provided explicit relations for the vertical velocity of the fluid andpressure on the bottom (y = �1), given as:

v (y) =1 + y

1 + ⌘⌘̇ (x, y) 2 RI, v(y) =

1 + y

hk + ⇣⇣̇ (x, y) 2 RII, (2.6)

p̄ =1

2(1 + ⌘)(⌘̈ + 2) (x, y) 2 RI, p̄ =

1

2(hk + ⇣)(⇣̈ + 2) + p̂ (x, y) 2 RII. (2.7)

The nonlinear kinematic and dynamic conditions on top and bottom boundaries of theflow domain are satisfied exactly by Eqs. (2.1)–(2.7).

In order to formulate the problem fully, suitable matching conditions at the fixedpositions of the leading (x = xL

k ) and traling edges (x = xTk ) of each floating sheet must

be specified. Since the elastic sheets are floating freely, the bending moment and the









1. Introduction

�1Floating elastic surfaces are observed in nature and are widely utilized in industry.

Therefore the study of wave action on elastic floating bodies has received considerableattention. Floating ice sheets of di↵erent sizes, for example, are commonly observed inmarginal ice zones, see Squire et al (1995). Wave interaction with floating ice sheets isof interest due to the e↵ect that the ice sheets have on the wave field, and the impactof waves on the ice sheets, particularly when it results in breaking of the ice sheet to










1. Introduction

L2











1. Introduction

LN











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N











1. Introduction

xL1 xT

1 xL2 xT

2 xLN xT

N



Discussion of results. Results of the level I GN model for wave interaction with N number of floatingand deformable ice sheets are presented here. The results include plates deformations, time series of waveinduced forces, and variation of the forces with wave conditions. In these preliminary cases, we assumethat the plates are fixed in their horizontal locations.

Figure 2 shows velocity vectors of fluid particles plotted on contours of fluid velocity module(u2 + v2)1/2 in case of interaction of solitary wave with two plates, floating at a short distance �L = 5from each other, at three successive time moments. Small-amplitude leading waves propagate with higherspeed along the elastic surfaces, and hence the fluid domain feels the wave motion long before the waveitself reaches the end of the second plate. These elastic waves cause greater wave attenuation in a longerplate with higher rigidity. Another contribution to the wave attenuation is due to the e↵ect of multipleplates (see [8]).


Figure 3 shows the interaction of a cnoidal wave with a set of three elastic plates of the same propertiesfor two values of plate rigidity. Also shown in figure 3 are the time series of the wave-induced horizontalcnoidal forces on these deformable ice sheets. The amplitudes of plates displacement and horizontal forcereduces downwave. The higher rigidity Dk of each plate contributes to the higher wave attenuation bythe floating system and to the lower drift of it’s elements.

Figure 4 shows the variation of the peak cnoidal wave horizontal force on the ice sheets F+1 with the


More detailed discussion of results, including the interplay between individual e↵ects of each platein more complex plate systems and the drift motion of the plates, will be presented at the workshop.

�L1

References

[1] Melyan, M. & Squire, V.A. (1994) The response of ice floes to ocean waves. J. Geoph. Res. 99. 891-900.RIRII

Discussion of results. Results of the level I GN model for wave interaction with N number of floatingand deformable ice sheets are presented here. The results include plate deformations, time series of waveinduced forces, and variation of the forces with wave conditions. In these preliminary cases, we assumethat the plates are fixed in their horizontal locations.

Figure 2 shows velocity vectors of fluid particles plotted on contours of fluid velocity module(u2 + v2)1/2 in case of interaction of a solitary wave with two plates, floating at a short distance�L = 5 from each other, at three successive time moments. Small-amplitude leading waves propagatewith higher speed along the elastic surfaces, and hence the fluid domain feels the wave motion long beforethe wave itself reaches the end of the second plate. These elastic waves cause greater wave attenuationin a longer plate with higher rigidity. Another contribution to the wave attenuation is due to the e↵ectof multiple plates (see [8]).


Figure 3 shows the interaction of a cnoidal wave with a set of three elastic plates of the same propertiesfor two values of plate rigidity. Also shown in figure 3 are the time series of the wave-induced horizontalcnoidal forces on these deformable ice sheets. The amplitudes of plates displacement and horizontal forcereduces downwave. The higher rigidity Dk of each plate contributes to the higher wave attenuation bythe floating system and to the lower drift of its elements.

Figure 4 shows the variation of the peak cnoidal wave horizontal force on the ice sheets F+k with the


More detailed discussion of results, including the interplay between individual e↵ects of each platein more complex plate systems and the drift motion of the plates, will be presented at the workshop.LN�1

References


[2] Green, A.E. & Naghdi, P.M. (1976a) A derivation of equations for wave propagation in water of variable depth. J. FluidMech. 78(2), 237–246.

[3] Green, A.E. & Naghdi, P.M. (1976b) Directed fluid sheets. Proc. of Royal Society of London. 347(1651), 447–473.







More detailed discussion of results, including the interplay between individual e↵ects of each platein more complex plate systems and the drift motion of the plates, will be presented at the workshop.xL

N�1 xTN�1

References










More detailed discussion of results, including the interplay between individual e↵ects of each platein more complex plate systems and the drift motion of the plates, will be presented at the workshop.xL

N�1 xTN�1

References




Fig. 1. Schematic of the problem of nonlinear wave interaction with N number of deformable plates ofarbitrary size and location. The plates may have horizontal motion due to the wave loads. Also shown in thisfigure are the RI and RII regions referred to in the text.

In the context of using the GN equations, the problem is best studied by dividing the fluid domaininto two types of regions. Region I (RI) is formed by a flat floor at the bottom and by a free surface atthe top, where the fluid pressure is the constant atmospheric pressure. Region II (RII) is formed by aflat floor at the bottom and by an elastic plate at the top, where, as opposed to Region RI, the fluidpressure is unknown. Solutions, obtained in each region, are connected through the proper matchingconditions at the interfaces.

The governing equations for the motion of the fluid in RI are provided by the Level I GN theory fora flat and stationary seafloor [2, 3]. Using density, ρ, gravitational acceleration, g, and fluid depth h,the Level I GN equations can be written in dimensionless and compact form as [4]:

η,t + [(1 + η)u],x = 0, u̇+ η,x = −1

3[2η,xη̈ + (1 + η)η̈,x], (1)

where η is the surface elevation measured from the still-water level and u is the horizontal particlevelocity. Subscripts after comma indicate differentiation, and superposed dots are the two-dimensionaltotal derivatives. Similarly, the governing equations for the coupled motion of the fluid and the elastic

sheet in RII Regions can be written as:

ζ,t + [(hk + ζ)u],x = 0, u̇+ ζ,x + p̂,x = −1

3[2ζ,xζ̈ + (hk + ζ)ζ̈,x], (2)

where ζ is the plate deformation, hk = 1 − dk is dimensionless fluid depth under the k-th plate. Thewave-induced pressure p̂ under the plate is given by the thin elastic plate theory [5], usually adopted tomodel ice floes and very large floating structures (VLFS), when they interact with surface waves:

p̂ = mkζ,tt +Dkζ,xxxx +mk. (3)

Here, the flexural rigidity is defined as Dk = Ekδ3k/12(1− ν2k), with δk, Ek, and νk being the thickness,

Young’s modulus and Poisson’s ratio of the k-th plate, respectively. Equations (1)–(3) are solved for freesurface elevation η, plate deformation ζ, horizontal component of fluid velocity u and pressure under theplates p̂.

Ertekin [6] provided explicit relations for the vertical velocity of the fluid along the water columnand pressure on the bottom (y = −1), given as:

v(x, y) =

{η̇ (1 + y)/(1 + η), (x, y) ∈ RIζ̇ (hk + y)/(hk + ζ), (x, y) ∈ RII,

p̄(x) =

{12(1 + η)(η̈ + 2), (x, y) ∈ RI12(hk + ζ)(ζ̈ + 2) + p̂, (x, y) ∈ RII.

(4)In order to obtain continuous solution, suitable matching conditions at the leading (x = xLk ) and

trailing (x = xTk ) edges of each floating sheet must be specified. Since the elastic sheets are floatingfreely, the bending moment and the shear stress should vanish at the edges, i.e. Dkη,xx = Dkη,xxx = 0.Moreover, we assume no gap between the bottom surface of the sheets and the top surface of the fluidlayer, and hence the mass continuity equation (2) together with vanishing bending moment conditionimply:

3ζ,xu,xx + (hk + ζ)u,xxx = 0, 4ζ,xu,xxx + (hk + ζ)u,xxxx + ζ,xxxxu = 0. (5)

In the approach discussed above, the floating elastic surfaces cause discontinuities of the fluid layerand velocity at the interfaces between regions. Consequently, the derivatives of η and u are also discon-tinuous. Appropriate jump conditions should be called to provide the matching of the solution at theinterfaces between regions. The theory demands the conservation of mass and momentum (achieved bycontinuity of bottom pressure across the discontinuity curves).

On the left side of the domain, numerical wavemaker capable of generating GN solitary and cnoidalwaves is installed. On the right side of the domain, Orlanski condition is used to minimise the wavereflection back to the domain.

The system of equations of the entire domain, subject to appropriate boundary conditions in eachregion, along with the matching and jump conditions, is solved simultaneously for the unknowns. Spatialdiscretization of the equations is carried out by a central-difference method, second order in space, andtime marching is obtained by use of the modified Euler’s method. The system of equations are solved byuse of a Guassian Elimination method. Hayatdavoodi & Ertekin [7] applied successfully the same modelto nonlinear problem of the wave scattering by a submerged rigid plate.

In two-dimensions, the horizontal wave-induced force on the floating plate is calculated by consideringthe pressure differential at the leading and trailing edges of each plate

Fk(t) = p̂(xLk , t)− p̂(xTk , t). (6)

We note here that since the plate is thin, the two-dimensional horizontal force is the force per unit width(into the page) and per unit thickness, and thus has the same dimension as pressure. The force Fk haspositive and negative components. Drift motion of the floating ice sheets is determined by solving thefollowing equation of motion:

mkak,tt = Fk(t), (7)

where ak,tt is the instantaneous horizontal acceleration of k-th plate. Spatial location of each plate isdetermined by integrating the horizontal acceleration twice.

Discussion of results. Results of the level I GN model for wave interaction with N number of floatingand deformable ice sheets are presented here. The results include plate deformations, time series of waveinduced forces, and variation of the forces with wave conditions. In these preliminary cases shown here,we assume that the plates are fixed in their horizontal locations.

Figure 2 shows velocity vectors of fluid particles plotted on contours of fluid velocity module(u2 + v2)1/2 in case of interaction of a solitary wave with two plates, floating at a short distance∆L = 5 from each other, at three successive time moments. Small-amplitude leading waves propagatewith higher speed along the elastic surfaces, and hence the fluid domain feels the wave motion long beforethe wave itself reaches the end of the second plate. These elastic waves cause greater wave attenuationin a longer plate with higher rigidity. Another contribution to the wave attenuation is due to the effectof multiple plates (see [8]).

Fig. 2. Vector field and module contours of dimensionless fluid velocity for the interaction of a solitary wave(A = 0.25) with the set of two plates with the same properties (Lk = 30, mk = 0.1, Dk = 5) separated bythe fluid gap (∆L = 5).

Figure 3 shows the interaction of a cnoidal wave with a set of three elastic plates of the same propertiesfor two values of plate rigidity. Also shown in Fig. 3 are the time series of the wave-induced horizontalcnoidal forces on these deformable ice sheets. The amplitudes of plates deformation and horizontal forcereduces downwave. The higher rigidity Dk of each plate contributes to the higher wave attenuation bythe floating system and to the lower drift of its elements.

Figure 4 shows the variation of the peak cnoidal wave horizontal force on the ice sheets F+k with

the incoming wave length λ and wave height H for the cases of a single plate and two plates located atdifferent distances ∆L from each other. The effect of multiple plates is greater for shorter distance. Thefirst plate in the group of plates experiences greater drift force than a single plate of the same propertiesif the distance is small. For larger distance, the second plate has less impact on the first plate andexperiences less horizontal force itself. Figure 4 also shows that drift force varies almost linearly with thewave height and slight deviation of force from the straight line in the case of two plates can be observed.

More detailed discussion of results, including the interplay between of each plate in more complexplate systems and the horizontal mean motion of the plates, will be presented at the workshop.

References




150 160 170 180t

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

F 1

150 160 170 180t

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

F 2

(c)

150 160 170 180t

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

F 3

Dk = 3Dk = 0.1

5 10 15 20 25 30 35 40-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40x

-0.1

0

0.1

y

(b)

Also shown in figure 3 are the time series of the wave-induced horizontal cnoidal forces on these

deformable ice sheets.

150 160 170 180t

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

F 1

150 160 170 180t

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

F 2

(c)

150 160 170 180t

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

F 3

Dk = 3Dk = 0.1

5 10 15 20 25 30 35 40-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40x

-0.1

0

0.1

y

(b)

(c)

Fig. 3. Surface elevation and plate displacement for the interaction of the cnoidal wave (H = 0.2,� = 5) with the set of three identical plates (Lk = 3, mk = 0.01), equally spaced, having therigidity (a) Dk = 3 and (b) Dk = 0.1 at the time moment t = 200.

Figure 4 shows the variation of the peak (F+1 ) cnoidal wave horizontal force on the ice sheets with

the incoming wave length � and height H for the cases of a single plate and two plates located at

di↵erent distances �L from each other. The e↵ect of multiple plates is greater for shorter distance.

The first plate in the group of plates experiences greater drift force than a single plate of the same

properties if the distance is small. For larger distance, the second plate has less impact on the first

plate and experiences less horizontal force itself. Figure also 4 shows that drift force varies almost

linearly with the wave height and slight deviation of force from the straight line in the case of two

plates can be observed.

More detailed discussion of results, including the interplay between individual e↵ects of each plate

in more complex plate systems and the drift motion of the plates, will be presented at the workshop.

References


[2] Green, A.E. & Naghdi, P.M. (1976a) A derivation of equations for wave propagation in water of variable depth. J.

Fluid Mech. 78(2), 237–246.


[4] Ertekin, R. C., Webster, W. C. & Wehausen, J. V. (1986) Waves caused by a moving disturbance in a shallow

channel of finite width. J. Fluid Mech. 169, 275–292.

[5] Timoshenko, S. P., Woinowsky-Krieger, S. (1959) Theory of plates and shells. McGraw-Hill, NY.

[6] Ertekin, R. C. (1984) Soliton generation by moving disturbances in shallow water: theory, computation and experi-

ment, Ph.D. dissertation. University of California at Berkeley.

Fig. 3. Surface elevation and plate deformation for the interaction of the cnoidal wave (H = 0.2, λ = 5)with a set of three identical plates (Lk = 3, mk = 0.01), equally spaced, having the rigidity of (a) Dk = 3 and(b) Dk = 0.1 at time t = 200. (c) Time series of horizontal force acting on each plate in the set for differentvalues of rigidity Dk. The peak F+

1 of horizontal force F1 is indicated in the figure.

4 6 8 10 12 140

0.02

0.04

0.06

0.08

F x +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)

single plate1st plate2nd plate

4 6 8 10 12 140

0.02

0.04

0.06

0.08

F x +

(c)

0 0.05 0.1 0.15 0.2 0.25H

0

0.02

0.04

0.06

0.08(d)



5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)


Figure 4 shows the variation of the peak (F+x ) cnoidal wave horizontal force on the ice sheets with








H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)

Fig. 4. Variation of the peak of cnoidal wave horizontal force on floating ice sheets with wavelength and wave height, for two configurations of a single plate (L = 5, m = 0.01, D = 0.1) andtwo identical plates of the same properties. In sub-plots (a) and (b), the distance between the twoplates is �L = L/2, and in (c) and (d) �L = L.





5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)






5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)






5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)






5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)






5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)






5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)






5 10 15 20 25 30 35 40

-0.1

0

0.1

y

(a)

5 10 15 20 25 30 35 40

x

-0.1

0

0.1

y

(b)










H = 0.2 � = 5

4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(a)

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08(b)


4 6 8 10 12 140

0.02

0.04

0.06

0.08

Fx +

(c)

0 0.05 0.1 0.15 0.2 0.25

H

0

0.02

0.04

0.06

0.08(d)




Dis

cuss

ion

ofre

sult

s.R

esult

softh

ele

velIG

Nm

odel

forw

ave

inte

ract

ion

wit

hN

num

ber

offloa

ting

and

def

orm

able

ice

shee

tsar

epre

sente

dher

e.T

he

resu

ltsin

clude

pla

tesdef

orm

atio

ns,

tim

ese

ries

ofw

ave

induce

dfo

rces

,an

dva

riat

ion

ofth

efo

rces

wit

hw

ave

condit

ions.

Inth

ese

pre

lim

inary

case

s,w

eass

um

eth

atth

epla

tes

are

fixed

inth

eir

hor

izon

tallo

cati

ons.

Fig

ure

2sh

ows

velo

city

vect

ors

offluid

part

icle

splo

tted

on

conto

urs

of

fluid

velo

city

module

(u2

+v

2)1

/2

inca

seof

inte

ract

ion

ofso

lita

ryw

ave

wit

htw

opla

tes,

floati

ng

ata

shor

tdis

tance

�L

=5

from

each

other

,at

thre

esu

cces

sive

tim

em

omen

ts.

Sm

all-a

mplitu

de

leadin

gw

aves

pro

paga

tew

ith

hig

her

spee

dal

ong

the

elas

tic

surf

aces

,an

dhen

ceth

efluid

dom

ain

feel

sth

ew

ave

moti

onlo

ng

bef

ore

the

wav

eit

self

reac

hes

the

end

ofth

ese

cond

pla

te.

Thes

eel

ast

icw

aves

cause

grea

ter

wav

eatt

enuati

onin

alo

nge

rpla

tew

ith

hig

her

rigi

dity.

Anot

her

contr

ibuti

onto

the

wav

eatt

enuati

onis

due

toth

ee↵

ect

ofm

ult

iple

pla

tes

(see

[8])

.

Fig

.2.

Vec

tor

fiel

dan

dm

odule

conto

urs

ofdim

ensi

onle

ssfluid

vel

oci

tyfo

rth

ein

tera

ctio

nof

aso

lita

ryw

ave

(A=

0.2

5)w

ith

the

set

oftw

opla

tes

wit

hth

esa

me

pro

per

ties

(Lk

=30

,m

k=

0.1,

Dk

=5)

div

ided

by

the

fluid

gap

(�L

=5)

.

Fig

ure

3sh

owsth

ein

tera

ctio

nofa

cnoi

dal

wav

ew

ith

ase

tof

thre

eel

asti

cpla

tesof

the

sam

epro

per

ties

for

two

valu

esof

pla

teri

gidity.

Als

osh

own

infigure

3are

the

tim

ese

ries

ofth

ew

ave-

induce

dhori

zonta

lcn

oidal

forc

eson

thes

edef

orm

able

ice

shee

ts.

The

am

plitu

des

ofpla

tesdis

pla

cem

entan

dhori

zonta

lfo

rce

reduce

sdow

nw

ave.

The

hig

her

rigi

dity

Dk

ofea

chpla

teco

ntr

ibute

sto

the

hig

her

wav

eatt

enuati

onby

the

floa

ting

syst

eman

dto

the

low

erdri

ftof

it’s

elem

ents

.Fig

ure

4sh

ows

the

vari

atio

nofth

epea

kcn

oidal

wav

ehori

zonta

lfo

rce

on

the

ice

shee

tsF

+ 1w

ith

the

inco

min

gw

ave

lengt

h�

and

hei

ght

Hfo

rth

eca

ses

ofa

singl

epla

tean

dtw

opla

tes

loca

ted

at

di↵

eren

tdis

tance

s�

Lfr

omea

chot

her

.T

he

e↵ec

tof

mult

iple

pla

tes

isgre

ater

for

short

erdis

tance

.T

he

firs

tpla

tein

the

grou

pof

pla

tesex

per

ience

sgr

eate

rdri

ftfo

rce

than

asi

ngle

pla

teofth

esa

me

pro

per

ties

ifth

edis

tance

issm

all.

For

larg

erdis

tance

,th

ese

cond

pla

tehas

less

impac

ton

the

firs

tpla

tean

dex

per

ience

sle

sshor

izon

talfo

rce

itse

lf.

Fig

ure

als

o4

show

sth

atdri

ftfo

rce

vari

esal

most

linea

rly

wit

hth

ew

ave

hei

ght

and

slig

ht

dev

iati

onof

forc

efr

omth

est

raig

ht

line

inth

eca

seoftw

opla

tes

can

be

obse

rved

.M

ore

det

aile

ddis

cuss

ion

ofre

sult

s,in

cludin

gth

ein

terp

lay

bet

wee

nin

div

idual

e↵ec

tsof

each

pla

tein

mor

eco

mple

xpla

tesy

stem

san

dth

edri

ftm

otio

nof

the

pla

tes,

will

be

pre

sente

dat

the

wor

ksh

op.

F+ 1,F

+ 2

Refe

rence

s

[1]

Mel

yan,M

.&

Squir

e,V

.A.(1

994)

The

resp

onse

ofic

efloes

tooce

an

wav

es.J.G

eoph.Res

.99.891-9

00.

[2]

Gre

en,A

.E.&

Naghdi,

P.M

.(1

976a)

Ader

ivati

on

ofeq

uati

ons

for

wav

epro

pagati

on

inw

ate

rofva

riable

dep

th.J.Flu

idM

ech.78(2

),237–246.

[3]

Gre

en,A

.E.&

Naghdi,

P.M

.(1

976b)

Dir

ecte

dfluid

shee

ts.Pro

c.ofRoya

lSoc

iety

ofLondon.347(1

651),

447–473.

Dis

cuss

ion

ofre

sult

s.R

esult

sof

the

leve

lIG

Nm

odel

forw

ave

inte

ract

ion

wit

hN

num

ber

offloa

ting

and

def

orm

able

ice

shee

tsar

epre

sente

dher

e.T

he

resu

ltsin

clude

pla

tesdef

orm

atio

ns,

tim

ese

ries

ofw

ave

induce

dfo

rces

,an

dva

riat

ion

ofth

efo

rces

wit

hw

ave

condit

ions.

Inth

ese

pre

lim

inary

case

s,w

eass

um

eth

atth

epla

tes

are

fixed

inth

eir

hor

izon

tallo

cati

ons.

Fig

ure

2sh

ows

velo

city

vect

ors

offluid

par

ticl

esplo

tted

on

conto

urs

of

fluid

velo

city

module

(u2

+v

2)1

/2

inca

seof

inte

ract

ion

ofso

lita

ryw

ave

wit

htw

opla

tes,

floati

ng

ata

shor

tdis

tance

�L

=5

from

each

other

,at

thre

esu

cces

sive

tim

em

omen

ts.

Sm

all-am

plitu

de

leadin

gw

aves

pro

paga

tew

ith

hig

her

spee

dalo

ng

the

elas

tic

surf

aces

,an

dhen

ceth

efluid

dom

ain

feel

sth

ew

ave

moti

onlo

ng

bef

ore

the

wav

eit

self

reac

hes

the

end

ofth

ese

cond

pla

te.

Thes

eel

ast

icw

aves

cause

grea

ter

wav

eatt

enuati

onin

alo

nge

rpla

tew

ith

hig

her

rigi

dity.

Anot

her

contr

ibuti

onto

the

wav

eatt

enuati

onis

due

toth

ee↵

ect

ofm

ult

iple

pla

tes

(see

[8])

.

Fig

.2.

Vec

tor

fiel

dan

dm

odule

conto

urs

ofdim

ensi

onle

ssfluid

vel

oci

tyfo

rth

ein

tera

ctio

nof

aso

lita

ryw

ave

(A=

0.2

5)w

ith

the

set

oftw

opla

tes

wit

hth

esa

me

pro

per

ties

(Lk

=30

,m

k=

0.1,

Dk

=5)

div

ided

by

the

fluid

gap

(�L

=5)

.

Fig

ure

3sh

owsth

ein

tera

ctio

nof

acn

oidal

wav

ew

ith

ase

tof

thre

eel

asti

cpla

tesof

the

sam

epro

per

ties

for

two

valu

esof

pla

teri

gidity.

Als

osh

own

infigure

3are

the

tim

ese

ries

ofth

ew

ave-

induce

dhori

zonta

lcn

oid

alfo

rces

onth

ese

def

orm

able

ice

shee

ts.

The

am

plitu

des

ofpla

tesdis

pla

cem

entan

dhori

zonta

lfo

rce

reduce

sdow

nw

ave.

The

hig

her

rigi

dity

Dk

ofea

chpla

teco

ntr

ibute

sto

the

hig

her

wav

eatt

enuati

onby

the

float

ing

syst

eman

dto

the

low

erdri

ftof

it’s

elem

ents

.Fig

ure

4sh

ows

the

vari

atio

nof

the

pea

kcn

oidal

wav

ehori

zonta

lfo

rce

on

the

ice

shee

tsF

+ 1w

ith

the

inco

min

gw

ave

lengt

h�

and

hei

ght

Hfo

rth

eca

ses

ofa

singl

epla

tean

dtw

opla

tes

loca

ted

at

di↵

eren

tdis

tance

s�

Lfr

omea

chot

her

.T

he

e↵ec

tof

mult

iple

pla

tes

isgre

ater

for

short

erdis

tance

.T

he

firs

tpla

tein

the

grou

pof

pla

tesex

per

ience

sgr

eate

rdri

ftfo

rce

than

asi

ngle

pla

teofth

esa

me

pro

per

ties

ifth

edis

tance

issm

all.

For

larg

erdis

tance

,th

ese

cond

pla

tehas

less

impac

ton

the

firs

tpla

tean

dex

per

ience

sle

sshori

zonta

lfo

rce

itse

lf.

Fig

ure

also

4sh

owsth

atdri

ftfo

rce

vari

esal

most

linea

rly

wit

hth

ew

ave

hei

ght

and

slig

ht

dev

iati

onof

forc

efr

omth

est

raig

ht

line

inth

eca

seoftw

opla

tes

can

be

obse

rved

.M

ore

det

aile

ddis

cuss

ion

ofre

sult

s,in

cludin

gth

ein

terp

lay

bet

wee

nin

div

idual

e↵ec

tsof

each

pla

tein

mor

eco

mple

xpla

tesy

stem

san

dth

edri

ftm

otio

nof

the

pla

tes,

will

be

pre

sente

dat

the

wor

ksh

op.

F+ 1,F

+ 2

Refe

rence

s

[1]

Mel

yan,M

.&

Squir

e,V

.A.(1

994)

The

resp

onse

ofic

efloes

tooce

an

wav

es.J.G

eoph.Res

.99.891-9

00.

[2]

Gre

en,A

.E.&

Naghdi,

P.M

.(1

976a)

Ader

ivati

on

ofeq

uati

ons

for

wav

epro

pagati

on

inw

ate

rofva

riable

dep

th.J.Flu

idM

ech.78(2

),237–246.

[3]

Gre

en,A

.E.&

Naghdi,

P.M

.(1

976b)

Dir

ecte

dfluid

shee

ts.Pro

c.ofRoya

lSoc

iety

ofLondon.347(1

651),

447–473.

Fig. 4. Variation of the peak of cnoidal wave horizontal force on floating ice sheets with wave length andwave height, for two configurations of a single plate (L = 5, m = 0.01, D = 0.1) and two identical plates ofthe same properties. In sub-plots (a) and (b), the distance between the two plates is ∆L = L/2, and in (c)and (d) ∆L = L.

[4] Ertekin, R. C., Webster, W. C. & Wehausen, J. V. (1986) Waves caused by a moving disturbance in a shallow channelof finite width. J. Fluid Mech. 169, 275–292.

[5] Timoshenko, S. P., Woinowsky-Krieger, S. (1959) Theory of plates and shells. McGraw-Hill, NY.

[6] Ertekin, R. C. (1984) Soliton generation by moving disturbances in shallow water: theory, computation and experiment,Ph.D. dissertation. University of California at Berkeley.

[7] Hayatdavoodi, M. & Ertekin, R.C. (2015) Wave forces on a submerged horizontal plate. Part I: Theory and modeling.J. Fluids and Structures. 54, 566-589.

[8] Kohout, A.L. & Meylan, M.H. (2008) An elastic plate model for wave attenuation and ice floe breaking in the marginal

ice zone. J. Geoph. Res.: Oceans. 113. C09016.

on drift motion of deformable ice sheets by nonlinear waves · 2020-01-07 · the 35th...

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