Li Chunhua, Wang Yongxue and Xing Dianlu
Dalial University of Technology,
Dalian, China
E-mail: leech@student.dlut.edu.cn
Abstract: Simplifying the ice-wave interaction as a two-dimensional problem and applying the theory of floating body, the displacements of ice floe and forces induced by the waves are computed by finite element method (FEM). The limited wave height of ice breakup is derived and the factors that affect ice floe fractured are discussed.
Keywords: ice floe, wave, breakup
Understanding the mechanism of sea-ice breakup on waves is important for investigating the formation of overlapped ice, ice riding up and piling up. Furthermore, it can provide the background for predicting ice loads and ice induced vibration on structures. Squire (1995) reported that the Scott Polar Research Institute of the University of Cambridge, England, carried out a series of experiments in the sea of East Greenland and Bering Sea during the 1970s and 1980s to study the movement and flexure of single ice floe in waves. Goodman et al (1980) and Squire (1983) did a few experiments for single ice floe and concluded that ice floe would be easily fractured when the floe size to wavelength ratio was sufficient large. However, the papers gave out less quantificational relationship of the ice thickness and the minimum wave height to break the floe.
In present paper,
the effect of wave parameters on the ice floe breakup is investigated in more
details and some quantificational results are obtained. Firstly, the ice floe is
assumed to be a rigid floating body for consideration of hydrodynamic force on
the fact that the elastic deformation of ice floe can be neglected compared with
ice floe displacements on waves. Simplifying the ice-wave interaction as a
two-dimensional problem and applying the theory of floating body, the
displacements of ice floe and forces induced by the waves are computed by finite
element method (FEM). Next, the interior stress distributions of ice floe are
obtained with the elastic beam theory and the minimum wave height for ice floe
breakup can be determined. Finally, the effecting factors on the motion and
breakup of ice floes are discussed.
The sketch
of the interaction between ice floe and waves is shown in Figure 1. The origin
of coordinate is in the still water level and the middle of the ice floe with
length of Li. The
boundaries are wave surface Sf, seabed Sb, surface of ice
floe Sw, and far field, S∞ and S-∞.
Fig. 1 Sketch of the interaction of ice floe with waves
Under the assumption that the fluid is non-viscous, incompressible and irrotational, the flow field can be described by function of velocity potential that satisfies the Laplace equation. If only the linear wave with small amplitude is considered, the scattering process can be decomposed as radiation and diffraction problems.
According to the theory referred by
Mei (1984), the complex velocity potential,
and complex displacement of ice
floe,
are defined as
(1)
Here,
denotes complex velocity potential
which is independent of time t;
is complex velocity of the ice floe
and can be written as
( with j=1, 2 and 3 corresponding to surge,
heave and pitch respectively);
, is the amplitude vector of ice floe displacements, in which
the horizontal displacement,
the vertical displacement, and
the angular displacement. The
velocity potential,
, satisfies the following boundary problems,
(In
)
(2)
(On
Sf)
(3)
(On Sb)
(4)
(On Sw)
(5)
(on S∞、S-∞)
(6)
where, k is the wave number, d the water depth,
is the unit normal vector of ice
floe surface(
and
the component of normal vector with respect to the x and y directions respectively, and
).
Because the problem is considered as a
linear system, the velocity potential,
, can be decomposed into 5 components, that is:
(7)
Here,
is the incident wave potential,
the diffraction potential,
( j=1,2,3), the radiation potential at
j direction.
,
and
satisfy the same boundary problem with
except the boundary condition on
ice floe surface.
According to linear wave theory, the incident potential,
, is given by
(8)
in which,
is the wave amplitude, d the water depth, w the circular frequency.
and
satisfy the Eq.(9) and Eq.(10) on Sw respectively and are solved by
finite element method.
(On
Sw) (9)
(On
Sw)
(10)
and
Because the solution procedure of
is the same as
, the calculation of forced potential
is only presented by using finite
element method (Zhou, 1990). For convenience,
is replaced by
in following discussion.
The computational domain is divided into m grid cells with total k nodes. The velocity potential in the n-node cell, e can be approximately expressed as
(11)
here,
is the potential function at
node i in cell e,
the interpolating function, n the
node number of cell e. The eight-node
2-D isoparametric element is used in this paper.
The forced potential,
can be obtained by summing all
in the domain and given by,
(12)
with
and
(13)
Applying the
method of weighted residuals, the equation for potential function
can be obtained,
(14)
By using subsection integral and the Green formula, the Eq.(14) can be written as,
(15)
here,
is the boundary of computation
domain, which includes free surface Sf, seabed Sb, surface
of the floating body Sw, and far field, S∞
and S-∞. Applying the boundary conditions (3), (4), (6) and (10) to Eq.
(15), we have
(16)
Substituting Eq.(13) into Eq.(16) and applying the orthogonality of the interpolating function, the matrix form of finite element equations are obtained.
(17)
in which
(18)
(19)
(20)
(21)
Therefore, the forced potential
can be obtained by solving the Eq.(17),
after the interpolated function
is chosen.
The ice floe is considered as a free-floating body that oscillates in regular waves of small amplitude, and the governing equations for wave-induced motion of ice floe are applied based on the theory of floating body movements. The force on ice floe, which is related to the ice floe motion, can be decomposed as three components: the incident force, diffraction force to a restrained body on waves and the scatter force produced by a body forced to oscillate in still water.
Figure 2 is the sketch of ice floe
along the wave propagation direction in two-dimensional case. Where
and
denote the ice thickness and the length respectively, and centroid coordinate of
ice floe is (0, –0.4
). The governing equations of ice floe motion are expressed as
Fig. 2 Sketch of Ice floe model
(22)
Here,
is the coefficient of mass matrix,
(23)
in which, M is
the mass of ice floe;
and
are the inertial moments about
origin O, and given by
respectively; s is the sectional area
of ice floe,
the ice density;
and
corresponding to the added mass and
scatter damping of ice floe are given by
here,
;
density of sea water,
the coefficients matrix of restored
force,
(24)
in which,
is the submerging contour line of
ice floe.
,
and
are defined as
,
,
Based on the given coefficient matrix,
the amplitude of the displacements,
, can be worked out from Eq.(22) and the total
potential can be obtained from Eq.(7). Thus, we can further evaluate the wave
exciting force on ice floe and determine the conditions of ice breakup.
According to the bending theory of elastic beam, the stress distribution within ice floe are dependent on its interior bending moment that can be calculated according the wave exciting force and the relative position between the wave and the ice floe. Wang(1995) referred that the maximum bending moment occurred near the midsection of the floe with its length less than the wavelength, when the wave crest or trough is appeared on the midsection.
If
is the vertically distributed load
that exerts on ice floe, which includes the wave force and the scatter force etc
as shown in Chap. 2.3, the moment in complex form at arbitrary
-section,
, is given by
(25)
Therefore, the
moment of
-section at a certain time t can be
obtained
(26)
Here,
, is the amplitude of the moment at
-section,
is the phase difference between
-section and midsection. So, the maximum stress can be obtained because it is
appeared at the same location of the maximum moment. The ice floe will be
fractured when the maximum stress reached its limited flexural strength.
In present computations, the physical
and mechanical parameters of sea ice in Bohai Sea are adopted. That is, the
flexural stress,
, is 600kPa, the elastic modulus, E, is from 2GPa to 4GPa, the sea ice density,
, is 920kg/m3, the range of the ice thickness
is from 20cm to 40cm. The wave
period,
, is ranged from 2s to 10s.
Figure 3
shows the computation results of the floe’s displacement with the ice
thickness to length ratio
and ice length to wavelength ratio
. It is seen that the non-dimensional horizontal displacement
, vertical displacement,
and rotational displacement,
, decrease with the increase of
. In other hand, the horizontal displacement increases when
decreases shown in Figure 3(a),
while the ratio of
has a little influence on the
vertical and rotational displacement as illustrated in Figure 3(b) and 3(c).
Fig. 3
Figure 4 shows the distribution of
non-dimensional moment amplitude
of the ice floe with 15m in length
when the ratios
are 0.5, 1.0, 1.5 and 2.0.
Fig. 4
It can be seen from the figure that
the moment amplitude reaches the maximum value when
is equal to 1.0. Therefore we can
figure out that the minimum wave height is required to fracture the ice floe
when the ice length and wavelength are the same. As the ratio
reduces, the position of the
maximum moment lies more close to the midsection of ice floe, in other words,
the ice floe will be fractured near midsection.
Figure 5 shows the results of
non-dimensional minimum wave height to break ice floe,
, versus
and
. It is obviously shown that for the floe with
less than 1, the larger wave is
needed to fracture the ice floe with the decrease of
. The
changes a little for the ice floe
with
more than 1. For the ice floe with
the same ratio of
,
increases with the increase of
.
Fig. 5
Finally, the calculated minimum wave height to fracture ice floe are compared with the experimental data (Wang, 2000) and shown in Figure 6. The experiments were conducted by using non-refrigerated breakable ice materials. It can be seen that the theoretical results are in good agreement with experimental data.
Fig. 6 Comparison of the theoretical results with the experimental data
From the present study on ice floe breakup on waves, it can be concluded that:
(1) The displacement of ice floe on
waves increases largely with the reduction of
.
(2) The ice breakup occurs near the
midsection for the ice floe with the ratio of floe length and wavelength
less than 1,
(3) For the ice floe with the same
flexural strength, the ratio
and
are the significant factors
affecting on ice floe fractured on waves. The minimum wave height to breakup ice
floe decreases with the reduction of
and the increase of
.
References
[1] Goodman, D. J., Wadhams, P., Squire, V. A., 1980. The flexural response of a tabular ice island to ocean swell. Annual Glaciology, 1980(1):23-27.
[2] Liu, Y. Z., Miu, G. P., et al, 1986. The motion theory of ship on waves. Shanghai: Shanghai Transport University Press. (In Chinese).
[3] Mei, Q. C., 1984. Water wave dynamic. Beijing: Science Press. (In Chinese).
[4] Squire, V. A., 1983. Dynamics of ice floes in sea waves. J. Soc. Underwater Tech. 9(1): 20-26.
[5] Wang, J. D., Yang, Y. Q., 1995. Ship strength and structural design. Beijing: National Defence Industry Press(In Chinese).
[6] Wang, Y. X., Li, G. W., Li, C. H., and Sheng, Z. W., 2000. The experiment investigation on breakup of ice floe on waves. J. Natural Science Progress, 10(6): 549-553. (In Chinese).
[7] Zhou, S. X., Xing, D., L., 1990. The calculation of ship hydrodynamics with finite element method. J. Ship Building of China, Supplement: 145-152.