Journal of Applied Mechanics and Technical Physics, Vol. 52, No. 3, pp. 415–426, 2011
HYDRODYNAMIC LOADS ACTING ON AN OSCILLATING CYLINDER SUBMERGED IN A STRATIFIED FLUID WITH ICE COVER I. V. Sturova
UDC 532.59:539.3
The two-dimensional problem of steady oscillations of a horizontal cylinder submerged in a linearly stratified fluid layer whose upper boundary is ice cover is considered in a linear treatment using the Boussinesq approximation. The method of mass sources distributed along the body contour is used for the internal wave generation regime, and the integral equation for the disturbed pressure in the fluid is used for the regime of no internal waves. The hydrodynamic load acting on the body was calculated as a function of the oscillation frequency for the case of a continuous ice cover and for special cases (broken ice, free surface, and rigid lid). Keywords: linear wave theory, stratified fluid, ice cover, oscillations of a submerged cylinder, hydrodynamic load. DOI: 10.1134/S0021894411030126 In a linear treatment, the problem of oscillations of a body under a free fluid surface and the resulting hydrodynamic loads have been thoroughly studied for the case of a homogeneous fluid and for some cases of a density-stratified fluid (a review of these studies is given in [1]). However, the effect of ice cover on the hydrodynamic characteristics of a submerged body have not been sufficiently studied. The added mass and damping coefficients for a sphere submerged in a uniform and two-layer fluid have been determined only recently [2, 3]. A problem similar to the radiation problem of forced oscillations of a submerged body is the diffraction problem of scattering of a system of periodic waves incident on a fixed body. Das and Mandal [4, 5] studied the oblique incidence of waves on a horizontal circular cylinder submerged in a uniform and a two-layer fluid. (In the latter case, external disturbance can be caused by both surface and internal waves.) These studies have been motivated by the active exploration of the polar regions of the oceans. In [2–5], the method of multipole expansions was used, which gives the most exact solutions in the case of potential flows and bodies of simple geometry: for a circular cylinder in the two-dimensional case and for a sphere in the three-dimensional case. In the present paper, a method is proposed to solve the two-dimensional problem of small oscillations of a horizontal cylinder of arbitrary cross section submerged in a linearly stratified fluid layer whose upper boundary is ice cover. In addition, the problem of scattering of surface waves is considered for the special case of a homogeneous fluid. The method used here has been previously employed to study oscillations of a cylinder piercing the middle layer of a linearly stratified fluid in an infinite three-layer fluid with uniform upper and lower layers [6]. 1. Formulation of the Problem. In the undisturbed state, a fluid layer of thickness H occupies a region −∞ < x < ∞, −H < y < 0 (x and y are the horizontal and vertical coordinates). It is assumed that the fluid is inviscid and incompressible and that the density ρ0 (y) increases linearly with increasing depth: ρ0 (y) = ρs (1 − αy), where α > 0 and ρs = ρ0 (0). The fluid is bounded from below by a solid horizontal bottom and from above by floating ice cover which is treated as a thin elastic plate of constant thickness. Wave motion in the initially calm fluid is caused by small horizontal, vertical, and rotational oscillations of the submerged body.
Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090;
[email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 52, No. 3, pp. 102–115, May–June, 2011. Original article submitted March 3, 2010; revision submitted April 27, 2010. c 2011 by Pleiades Publishing, Ltd. 0021-8944/11/5203-0415
415
Under the assumption that the disturbed motion of the fluid and ice is steady within the framework of linear wave theory, the total disturbed pressure in the fluid can be expressed as 3 ηj pj (x, y) , P (x, y, t) = ρs Re exp (iωt)
(1.1)
j=1
where the complex functions pj (x, y) (j = 1, 2, 3) characterizes the radiation pressure due to the oscillations of the body with frequency ω for three degrees of freedom with amplitudes ηj ; t is time. In the Boussinesq approximation for a linearly stratified fluid, the functions pj (x, y) satisfy the equation (see e.g., [7]) Δpj =
N 2 ∂ 2 pj ω 2 ∂x2
(−∞ < x < ∞, −
N=
−H < y < 0), (1.2)
√ g dρ0 = αg, ρs dy
where N = const is the buoyancy frequency and g is the acceleration due to gravity. In the general case of unsteady motion, the equation of small bending oscillations of the ice cover is written as [8] M
∂4ζ ∂2ζ P ∂2ζ + B + Q + gζ = 2 4 2 ∂t ∂x ∂x ρs
M=
ρ1 h 1 , ρs
B=
Eh31 , 12(1 − ν 2 )ρs
(y = 0),
Q=
Kh1 , ρs
where ζ(x, t) is the vertical displacement of ice; E, ν, K, ρ1 , and h1 are the Young’s modulus, Poisson’s ratio, compression force, density, and thickness of ice, respectively. In the special case where B = Q = 0, the upper surface of the fluid is broken ice. In this case, if M = 0, the upper boundary of the fluid becomes a clean free surface. Using the kinematic relation on the lower surface of ice, we obtain the following boundary condition for pj (x, y) [9, 10]: ∂p ∂4 ∂2 j B 4 + Q 2 + g − M ω2 + (N 2 − ω 2 )pj = 0 (y = 0). (1.3) ∂x ∂x ∂y On the closed contour of the submerged body S, the no-flow condition is imposed: nx
ny ∂pj ∂pj − 2 = ω 2 nj ∂x β ∂y
N2 − 1. ω2
(1.4)
n3 = (y − y0 )n1 − (x − x0 )n2 ,
(1.5)
(x, y ∈ S),
β2 =
Here, n = (nx , ny ) is the inward normal to the contour S; n1 = nx ,
n2 = ny ,
x0 and y0 are the coordinates of the center of roll oscillations. The boundary condition on the bottom has the form ∂pj =0 ∂y
(y = −H).
(1.6)
In the far field, the radiation condition should be imposed that requires that the radiated waves to be outgoing. The hydrodynamic load acting on the oscillating cylinder is determined by the force F = (F1 , F2 ) and the momentum F3 , which, without accounting for the hydrostatic term, have the form 3 ηj τkj , τkj = ρs pj nk ds = ω 2 μkj − iωλkj (k = 1, 2, 3). (1.7) Fk = j=1
S
Here μkj and λkj are the added mass and damping coefficients, respectively. 416
The solution of problem (1.2)–(1.4), (1.6) depends significantly on the oscillation frequency of the body. For ω < N (β 2 > 0), Eq. (1.2) is hyperbolic and the oscillations of the body generate both surface and internal waves in the fluid. For ω > N (β 2 < 0), Eq. (1.2) becomes elliptic, internal waves are absent, and only surface waves are generated in the fluid. Below, these cases are considered in greater detail. 2. Case ω < N . To solve problem (1.2)–(1.4), (1.6) for the corresponding oscillation mode of the body, we introduce an unknown distribution σj (x, y) of mass sources on the contour S. Then, the pressure disturbance at any point of the fluid can be represented as (2.1) pj (x, y) = σj (ξ, η)G(x, y; ξ, η) ds. S
Here G(x, y; ξ, η) is the Green’s function of the problem considered, which determines the pressure field in the fluid caused by oscillations of the mass source of unit intensity. To determine the Green’s function, we need to solve the equation 2 ∂2G 2 ∂ G − β = 2πδ(x − ξ)δ(y − η) ∂y 2 ∂x2
(δ is the Dirac delta function) with boundary conditions similar to (1.3), (1.6), and the radiation condition in the far field. Be means of an integral Fourier transform, the Green’s function can be represented as ∞ iπ cos kn β(y + H) G=− 2 cos kn β(η + H) exp (−ikn |x − ξ|), β n=0 kn Dn where kn (n = 0, 1, 2, . . .) are real positive roots of the transcendental equation tan (kβH) =
C , kβΛ(k)
C = N 2 − ω2,
Λ(k) = Bk 4 − Qk 2 + g − M ω 2 , (2.2)
1 sin (2kn βH) Dn = +H . 2 2kn β
Equation (2.2) is the dispersion relation for gravitational waves in a linearly stratified fluid under an ice sheet of constant thickness [9, 10]. This equation has a countable number of simple roots kn (k0 < k1 < . . .) for given frequency ω. The smallest wavenumber k0 corresponds to a surface wave, and the remaining wavenumbers kn (n 1) to internal waves that exist only for ω < N ; furthermore kn → ∞ as ω → N . Using boundary condition (1.4) on the body surface S, we obtain the integral equation for the functions σj (x, y) ∂G n ∂G 2 − 2 ds = ω 2 nj . (2.3) πσj (x, y) − σj (ξ, η) n1 ∂x β ∂y S
Once the distribution of the singularities σj (x, y) has been calculated, we determine the pressure (2.1) and the hydrodynamic load (1.7). It is of interest to compare the solutions of the problem with the simpler case where the upper boundary of the fluid is a rigid lid. In this case, boundary condition (1.3) is replaced by the no-flow condition ∂pj =0 ∂y
(y = 0).
(2.4)
The Green’s function G1 (x, y; ξ, η) of problem (1.2), (1.6), (2.4) has the form G1 (x, y; ξ, η) = −
∞ 2iπ cos (kn βy) cos (kn βη) exp (−ikn |x − ξ|), Hβ 2 n=1 kn
kn =
nπ . βH
(2.5)
In the presence of a rigid lid, only internal waves exist in the fluid. 3. Case ω > N . In this case, we can also use the method of distributed singularities, but for a linearly stratified fluid, it is more convenient to solve the integral equation for disturbed pressure. Equation (1.2) can be written as 417
∂ 2 pj 1 ∂ 2 pj + = 0, ∂x2 γ 2 ∂y 2
γ 2 = −β 2 = 1 −
N2 . ω2
(3.1)
If we introduce the transformation of the vertical coordinates y¯ = γy, then Eq. (3.1) in the coordinates x and y¯ reduces to the Laplace equation and the boundary condition (1.4) on S to the value of the normal derivative on the deformed contour up to a factor that depends on the geometry of the body. This makes it possible to apply affine similitude to determine the hydrodynamic load acting on an arbitrary contour oscillating in an unbounded uniformly stratified fluid [11]. Using Green’s identity and the conditions in the far field, we obtain an integral equation, which for points of the deformed body contour S¯ has the form ∂pj 1 ∂G pj (x, y¯) = pj (ξ, η¯) − G(x, y¯; ξ, η¯) d¯ s. (3.2) π ∂n ¯ ∂n ¯ ¯ S
Here the overbar denotes quantities represented in deformable coordinates. The Green’s function G(x, y¯; ξ, η¯) is determined by solving the following problem: ∂2G ∂2G + = 2πδ(x − ξ)δ(¯ y − η¯) ∂x2 ∂ y¯2
(−∞ < x < ∞,
¯ < y¯ < 0), −H
∂p ∂4 ∂2 j γ B 4 + Q 2 + g − M ω2 + (N 2 − ω 2 )pj = 0 ∂x ∂x ∂ y¯
(¯ y = 0),
∂pj ¯ ¯ = γH. = 0 (¯ y = −H), η¯ = γη, H ∂ y¯ The solution for the Green’s function has the form ∞ cos k(x − ξ) r¯ A(k, η¯) exp (−k¯ y) + B(k, η¯) exp (k¯ y) dk G = ln + pv r¯1 Ω(k) 0
cos k (x − ξ) 0 , − iπ A(k0 , η¯) exp (−k0 y¯) + B(k0 , η¯) exp (k0 y¯) Ω (k0 )
(3.3)
where pv is a principal value integral, r¯ = (x − ξ)2 + (¯ y − η¯)2 , r¯1 = (x − ξ)2 + (¯ y + η¯)2 , ¯ ¯ A(k, η¯) = C/k − γΛ(k) ek(¯η −2H) − C/k + γΛ(k) e−k(¯η +2H) , ¯ ¯ B(k, η¯) = − 2γΛ(k) ekη¯ + C/k − γΛ(k) ek(¯η −2H) − e−k(¯η +2H) , dΩ
.
dk k=k0 Under certain restrictions on the compression ratio Q (see Sec. 4), the integrand in (3.3) can have one simple positive pole at the point k0 , which is a root of the equation ¯
Ω(k) = C + γkΛ(k) + [C − γkΛ(k)] e−2kH ,
Ω(k0 ) = 0.
Ω (k0 ) ≡
(3.4)
This equation is equivalent to the dispersion relation for surface waves in a linearly stratified fluid for ω > N . The value k0 corresponds to the wavenumber of flexural-gravity waves for B = 0 or ordinary surface waves for B =Q= M = 0. However, in the case of broken ice (B = Q = 0 and M = 0), Eq. (3.4) has no positive roots for ω > g/M [8]. In this case, oscillations of the cylinder do not generate wave motion in the fluid, the damping coefficients are equal to zero, and the last term in (3.3) should be omitted. A similar situation occurs when the ice sheet is replaced with a rigid lid. In this case, in (3.3) one should set C = 0 and Λ(k) ≡ g. By solving the integral equation (3.2), one obtains the pressure distribution along the body contour and then the corresponding hydrodynamic load. A special case of the problem is wave motions in a fluid with constant density (N = 0) under ice cover. 418
4. Homogeneous Fluid. The flow of a homogeneous fluid will be considered potential. Similarly to the problem of generation of wave motions of a fluid by an oscillating body, we consider the diffraction problem for the scattering of an incident surface wave by a fixed solid body. The total velocity potential of the wave motion can be written as 4 Φ(x, y, t) = Re exp (iωt) ηj ψj (x, y) ; j=0
ψ0 (x, y) = exp (−ik0 x)
cosh k0 (y + H) ; cosh k0 H
(4.1)
k0 (Bk04 − Qk02 + g) tanh k0 H, (4.2) 1 + M k0 tanh k0 H where the functions ψj (x, y) (j = 1, 2, 3) characterize the radiation potentials due to oscillations of the body in the calm fluid in three degrees of freedom with amplitudes ηj , similarly to (1.1); ψ0 (x, y) is the potential of regular surface waves incidence from the left, ψ4 (x, y) is the diffraction potential which describes the wave motion resulting from wave scattering by the fixed body, and η0 = η4 is the amplitude of the incident wave. The wavenumber of the incident wave k0 is related to the frequency ω by the dispersion relation (4.2) which follows from (3.4). Within the fluid, the functions ψj (x, y) satisfy the Laplace equation ω2 =
Δψj = 0
(−∞ < x < ∞,
−H < y < 0).
In the case of a homogeneous fluid according to (1.3), (1.4), and (1.6), the boundary conditions are as follows: — on the upper boundary of the fluid, ∂ψ ∂4 ∂2 j = ω 2 ψj B 4 + Q 2 + g − M ω2 (y = 0); (4.3) ∂x ∂x ∂y — on the surface of the body, ∂ψj ∂ψ0 ∂ψ4 = iωnj (j = 1, 2, 3), =− (x, y ∈ S); (4.4) ∂n ∂n ∂n — at the bottom, ∂ψj =0 (y = −H). ∂y The potentials ψj (x, y) on the contour S can be calculated using an integral equation similar to (3.2). The Green’s function is determined from the integral representation (3.3) in which one should set N = 0 and γ = 1. Below we consider only such values of the compression ratio for which Eq. (4.2) has no more than one root. √ As shown in [12, 13], this is the case for Q < 1.4 gB. Using Green’s integral, it is possible to calculate the potentials in the far field. In this case, it is sufficient to retain the limiting values of the last two terms for x − ξ → ±∞ in the Green’s function: iπ A1 (k0 , η) exp (−k0 y) + B1 (k0 , η) exp (k0 y) exp [∓ik0 (x − ξ)]. G(x, y; ξ, η) ≈ − Ω1 (k0 ) Here
A1 (k, η) = ω 2 /k − Λ(k) e−k(η+2H) − ω 2 /k + Λ(k) ek(η−2H) , B1 (k, η) = ω 2 /k + Λ(k) ek(η−2H) − e−k(η+2H) − 2Λ(k) ekη , Ω1 (k) = [Λ(k) + kΛ (k)] 1 − e−2kH + 2H[kΛ(k) + ω 2 ] e−2kH . Consequently, for x → ±∞ using (4.1) and the conditions on the body contour (4.4), we obtain ψj = Cj± exp (∓ik0 x) cosh k0 (y + H)/ cosh k0 H
where
Cj±
(j = 1, 2, 3, 4),
(j = 1, 2, 3) are the coefficients of the radiation potentials: 419
Cj± = C4±
iΛ(k0 ) Ω1 (k0 )
e±ik0 ξ {k0 [n2 Z1 (k0 , η) ± in1 Z2 (k0 , η)]ψj − iωnj Z2 (k0 , η)} ds,
S
is the coefficient of the diffraction potential: iΛ(k0 ) ∂ψ0 ds, e±ik0 ξ k0 [n2 Z1 (k0 , η) ± in1 Z2 (k0 , η)]ψ4 + Z2 (k0 , η) C4± = Ω1 (k0 ) ∂n S
Z1,2 (k, η) = ekη ∓ e−k(η+2H) . In the diffraction problem, the transmission T and reflection R coefficients are T = 1+C4+ and R = C4− , respectively. For a body which is symmetric about the vertical axis, the following relations for the amplitudes of the radiation waves hold: C1+ = −C1− = S1 ,
C2+ = C2− = S2 ,
C3+ = −C3− = S3 .
In the case of single-mode wave motion in a homogeneous fluid under ice cover, one can use the results of [14] to obtain a solution of the diffraction problem using the characteristics of wave motion for the radiation problem for horizontal and vertical oscillations of the body. In particular, for a symmetric body, the transmission and reflection coefficients are expressed in terms of the amplitudes of radiation waves in the far field: 1 S1 S1 S2 1 S2 , R = (4.5) − + T = 2 S2∗ S1∗ 2 S1∗ S2∗ (asterisk denotes complex conjugation). A consequence of these relations is the well-known energy equality |T |2 + |R|2 = 1. For a homogeneous fluid with free surface, there are equivalence relations which link the amplitudes of wave motion in the far field and hydrodynamic load (see, e.g., [15]). These relations are also valid for broken ice. However, in the case of wave motion in a fluid under a continuous ice cover, these relations have a more complex form due to the presence of high-order derivatives on the horizontal coordinate in the boundary condition (4.3) for y = 0 [16]. 5. Results of Numerical Calculations. The calculations were performed for an elliptical contour x2 /a2 + (y + h)2 /b2 = 1, where a and b are the major and minor axes of the ellipse, respectively, and h is the depth of its center. Rotational oscillations occur with respect to the point x0 = 0, y0 = −h [see (1.5)]. The parameters have the following values: E = 5 · 109 Pa, ρs = 1025 kg/m3 , ρ1 = 922.5 kg/m3 , K = 104 N/m2 , ν = 0.3, N = 0.05 sec−1 , b = 10 m, and H = 500 m. The calculations were performed for ice thickness h1 = 2 m (unless otherwise specified). To solve the integral equations (2.3) and (3.2), the body contour S is divided into K elements. In each element, an additional middle point is introduced, and the distribution of the unknown quantity on each element is approximated by a quadratic function with respect to the arc coordinate. Thus, for each value j = 1, 2, 3, one should solve a system of linear equations of order 2K to determine the values of σj and pj at all nodal points of the contour S, and only the right side of this system depends on the number j. In the calculations for ω < N , 300 modes of internal waves were taken into account and the number of elements in all calculations was K = 20. Figure 1 shows the dispersion curves for the surface mode k0 and the first five modes of internal waves k1 , . . . , k5 . Solid curves correspond to the ice cover, the dashed curve corresponds to broken ice, and the dot-anddashed curves to the free surface. The differences between these three cases are significant only for the surface mode at relatively high frequencies ω > 10N . The dispersion curves for internal waves practically do not depend on the conditions on the upper boundary of the fluid and coincide with high accuracy with the values obtained according to (2.5) in the rigid lid approximation (see [9, 10]). Figures 2 and 3 show dependences of the added mass coefficient Mjj = μjj /˜ μjj and the damping coefficient Ljj = λjj /(πρs gb3 ) (j = 1, 2) on the dimensionless frequency ω b/g for a circular cylinder submerged to a depth h = 20 and 50 m, respectively. In this case, nonzero values are observed only for the hydrodynamic load coefficients τ11 and τ22 in (1.7). The diagonal coefficients of added masses are related to their values in an infinite homogeneous fluid μ ˜jj , which in the case of an elliptic contour are μ ˜11 = πρs b2 , 420
μ ˜22 = πρs a2 ,
μ ˜33 = πρs (a2 − b2 )2 /8.
w/N 102
10
1 3 n= 0
2
-1
10
1
5 4
-2
10
10-4
10-3
10-2
10-1
1
10 knb
Fig. 1. Dispersion curves: the solid curves correspond to the ice cover, the dashed curve corresponds to broken ice, and the dot-and-dashed curve to the free surface.
In the calculations, the difference between the hydrodynamic loads for horizontal and vertical oscillations occurs only for the added mass coefficients for ω < N . At frequencies ω b/g < 0.15, the conditions on the upper boundary of the fluid have almost no effect: the solutions almost coincide with the solutions in the rigid lid approximation. It is only in this frequency range that the stratification effect is significant. In Figs. 2a and 3a, dashed curves show the added mass coefficient μ11 versus oscillation frequency for a homogeneous fluid under ice cover. In this case for a circular cylinder and great thickness of the fluid layer H, the coefficients μ11 and μ22 almost coincide. Figure 3 also shows the added mass and damping coefficients versus the oscillation frequency of a circular cylinder in an unbounded uniformly stratified fluid (curves 4). In this case, the solution for the hydrodynamic load has the form [17] μ11 = μ22 = 0, λ11 = λ22 = πρs b2 N 2 − ω 2 (ω < N ), (5.1) μ11 = μ22 = πρs b2 ω 2 − N 2 /ω, λ11 = λ22 = 0 (ω > N ). For a circular cylinder at a sufficiently great submergence, the obtained numerical solutions for ω < N approach relations (5.1) for the damping coefficient for both horizontal and vertical oscillations of the body and for the added mass coefficient in the case of vertical oscillations. A significant increase in the added mass coefficient for low-frequency horizontal oscillations of the cylinder can be explained by the blocking effect in the case of a stratified fluid of finite depth. A study [9, 10] of internal waves under ice cover revealed an increase in the deflection of ice at frequencies ω close to N . However, the calculations performed show that as ω → N , there is a sharp decrease in the damping coefficients which characterize the power expended by the body in generating waves. From Figs. 2 and 3, it is evident that in the range ω b/g < 0.15, where the stratification effect is significant, the hydrodynamic loads vary slightly with increasing depth of the body. However, in the range of high frequencies, where the wave loads are determined by the surface mode, they decrease sharply with increasing depth. Dependences of the added mass and damping coefficients versus the oscillation frequency of an elliptic cylinder with a = h = 2b for a homogeneous fluid are presented in Figs. 4 and 5, respectively. In the case of an elliptical cylinder, nonzero values are obtained only for the diagonal coefficients of the hydrodynamic load τjj (j = 1, 2, 3) and τ13 = τ31 in (1.7). In addition to the coefficients Mjj and Ljj (j = 1, 2), the ordinate shows the coefficients μ33 μ13 λ33 λ13 , M13 = , L33 = , L13 = . M33 = 3 7 μ ˜33 πρs b πρs gb πρs gb5 421
a
Mjj 1.2 0.8
1 2 3
0.4 0 -3 4 .10 10-2
w =N 10-1
Ljj 0.3
1
p 10 w b/g
b
0.2
0.1
0 -3 4 .10 10-2
w =N 10-1
1
10
p 20 w b/g
Fig. 2. Added mass (a) and damping coefficients (b) versus the oscillation frequency of a circular cylinder at h = 20 m: open points correspond to j = 1, and filled points to j = 2; 1) continuous ice cover; 2) broken ice; 3) free surface; the dashed curve corresponds to the dependence μ11 (ω b/g ) for a homogeneous fluid.
Mjj 1.2
a
Ljj
b
0.05
1.0
0.04
0.8
1 2 3 4
0.6 0.4
0.03 0.02 0.01
0.2 0 -3 4 .10 10-2
w=N 10-1
p 1 w b/g
0 -3 4 .10 10-2
w =N 10-1
p 1 w b/g
Fig. 3. Added mass (a) and damping (b) coefficients versus the oscillation frequency of a circular cylinder at h = 50 m: the dashed curve corresponds to the dependence μ11 (ω b/g ) for a homogeneous fluid (notation the same as in Fig. 2; curve 4 refer to the unbounded homogeneous stratified fluid).
422
a
M11 1.4 1.2 1.0 0.8 0.6 0.06
0.1
1.0
p 10.0 w b/g
1.0
p 10.0 w b/g
1.0
p 10.0 w b/g
1.0
p 10.0 w b/g
b
M22 1.4 1.2 1.0 0.8 0.6 0.06
0.1
c
M33 1.2 1.1 1.0 0.9 0.8 0.06
0.1
d
M13 0.2 0.1 0 _0.1 _0.2 _0.3 0.06
0.1
Fig. 4. Added mass coefficients M11 (a), M22 (b), M33 (c), and M13 (d) versus the oscillation frequency of an elliptic cylinder for a = h = 20 m: solid curves correspond to the free surface, dotted curves to broken ice, dashed curves to continuous ice (h1 = 2 m), and dot-and-dashed curves to continuous ice (h1 = 1 m).
423
a
L11 0.4 0.3 0.2 0.1 0 0.1
p 10.0 w b/g
1.0
b
L22 1.6 1.2 0.8 0.4 0 0.1
p 10.0 w b/g
1.0
c L33 0.3 0.2 0.1 0 0.1
p 10.0 w b/g
1.0
d
L13 0.3 0.2 0.1 0 0.1
1.0
p 10.0 w b/g
Fig. 5. Damping coefficients L11 (a), L22 (b), L33 (c), and L13 (d) versus the oscillation frequency of an elliptic cylinder for a = h = 20 m (notation the same as in Fig. 4).
424
R 0.25 0.20 0.15 0.10 0.05 0 0.1
1.0
p 10.0 w b/g
Fig. 6. Reflection coefficient versus the frequency of the incident wave for an elliptic cylinder with a = h = 20 m (notation the same as in Fig. 4).
From Figs. 2–5, it follows that the effect of different conditions on the upper boundary of the fluid is significant only at relatively high frequencies. In the presence of broken ice, the hydrodynamic loads differ only slightly from the loads in the case of free surface. In the case of a continuous ice cover, the added mass coefficients vary more smoothly than in the cases of free surface and broken ice, whereas the maximum values of the damping coefficients and their corresponding frequency vary significantly. Figure 6 shows the modulus of the reflection coefficient |R| defined by (4.5) in the diffraction problem versus the frequency of incident surface wave for an elliptic cylinder. The results of the study show that the maximum values of the reflection coefficient are obtained in cases of free surface and broken ice. As the thickness of a continuous ice cover increases, the maximum value of |R| decreases but the range of frequencies in which |R| is nonzero increases. In the case of a deeply submerged elliptical cylinder, the hydrodynamic load in the diffraction problem can be calculated using approximate solutions [18]. The ice compression force has little effect on the hydrodynamic load in the range 0 < K < 104 N/m2 . A similar conclusion on the effect of only very large values of K on the propagation of flexural-gravity waves was made in [19], where it is also noted that this range includes all practically possible values of this parameter. This work was supported by Program No. 20.4 of the Presidium of Russian Academy of Sciences.
REFERENCES 1. A. I. Korotkin, Added Masses of Ship Structures, Springer (2009). (Ser. Fluid Mechanics and Its Applications, Vol. 88.) 2. D. Das and B. N. Mandal, “Water wave radiation by a sphere submerged in water with an ice-cover,” Arch. Appl. Mech., 78, No. 8, 649–661 (2008). 3. S. Mohapatra and S. N. Bora, “Radiation of water waves by a sphere in an ice-covered two-layer fluid of finite depth,” J. Adv. Res. Appl. Math., 2, No. 1, 46–63 (2010). 4. D. Das and B. N. Mandal, “Oblique wave scattering by a circular cylinder submerged beneath an ice-cover,” Int. J. Eng. Sci., 44, Nos. 3/4, 166–179 (2006). 5. D. Das and B. N. Mandal, “Wave scattering by a horizontal circular cylinder in a two-layer fluid with an ice-cover,” Int. J. Eng. Sci., 45, No. 10, 842–872 (2007). 6. I. V. Sturova, “Oscillations of a cylinder piercing a linearly stratified fluid layer,” Fluid Dyn., 41, No. 4, 619–628 (2006). 425
7. Yu. Z. Miropol’sky, Dynamics of Internal Gravity Waves in the Ocean, Kluwer Academic (2001). 8. D. Ye. Kheysin, Dynamics of the Ice Cover [in Russian], Gidrometeoizdat, Leningrad (1967); English translation available at U.S. Army Foreign Science and Technology Center, Technical translation FSTC-HT-23-485-69 (1969). 9. C. B. Muzylev and L. N. Oleinikova, “On the theory of internal waves under ice cover,” Oceanology, 47, No. 2, 172–177 (2007). 10. S. V. Muzylev, “Internal waves under ice cover,” Doklady Earth Sciences, 418, No. 1, P. 2, 145–148 (2008). 11. E. V. Ermanyuk, “The rule of affine similitude for the force coefficients of a body oscillating in a uniformly stratified fluid,” Exp. Fluids, 32, No. 2, 242–251 (2002). 12. A. E. Bukatov, “Effect of longitudinal compression on unsteady oscillations of an elastic plate floating on a fluid surface,” Prikl. Mekh., 17, No. 1, 93–98 (1981). 13. A. E. Bukatov and D. D. Zavyalov, “Incidence of surface waves on the edge of compressed ice,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 121–126 (1995). 14. M. McIver, “Global relationships between two-dimensional water wave potentials,” J. Fluid Mech., 312, 299–309 (1996). 15. C. C. Mei, M. Stiassnie, and D. K.-P. Yue, Theory and Applications of Ocean Surface Waves. Part 1. Linear Aspects, World Sci., Singapore (2009). (Ser. Advanced Series on Ocean Engineering, Vol. 23.) 16. D. Das, B. N. Mandal, and A. Chakrabarti, “Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives,” Fluid Dyn. Res., 40, No. 4, 253–272 (2008). 17. D. G. Hurley, “The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution,” J. Fluid Mech., 351, 105–118 (1997). 18. I. V. Sturova, “Plane problem of hydrodynamic rocking of a body submerged in a two-layer fluid without forward speed,” Fluid Dyn., 23, No. 3, 414–423 (1994). 19. R. M. S. M. Schulkes, R. J. Hosking, and A. D. Sneyd, “Waves due to a steadily moving source on a floating ice plate. Part 2,” J. Fluid Mech., 180, 297–318 (1987).
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