Flexural-gravity Waves Due To Transient Disturbances In An Inviscid Fluid Of Finite Depth

131

2008,20(2):131-136

FLEXURAL-GRAVITY WAVES DUE TO TRANSIENT DISTURBANCES IN AN INVISCID FLUID OF FINITE DEPTH* LU Dong-qiang, LE Jia-chun Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China DAI Shi-qiang Shanghai Key Laboratory of Mechanics in Energy and Environment Engineering, Shanghai 200072, China, E-mail: [email protected]

(Received August 13 2007, Revised February 25, 2008)

Abstract: The dynamic response of an ice-covered fluid to transient disturbances was analytically investigated by means of integral transforms and the generalized method of stationary phase. The initially quiescent fluid of finite depth was assumed to be inviscid, incompressible, and homogenous. The thin ice-cover was modeled as a homogeneous elastic plate. The disturbances were idealized as the fundamental singularities. A linearized initial-boundary-value problem was formulated within the framework of potential flow. The perturbed flow was decomposed into the regular and the singular components. An image system was introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the vertical deflexion at the ice-water interface were obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motion were explicitly derived for large time with a fixed distance-to-time ratio. The effects of the finite depth of fluid on the resultant wave patterns were discussed in detail. As the depth increases from zero, the critical wave number and the minimal group velocity first increase to their peak values and then decrease to constants. Key words: waves, ice-cover, transient disturbances, asymptotic, group velocity

1. Introduction  Very large floating structures in the offshore region and the ice-cover in the polar region are usually idealized as thin elastic plates in the theoretical investigations [1-14]. Flexural-gravity waves occur at the surface of a fluid covered by an elastic plate. Asymptotic solutions for the transient flexural waves due to disturbances in an inviscid fluid of infinite depth were considered by Maiti and Mandal[10], Lu and Dai [11]. A well-posed initial-boundary-value problem was formulated by Lu and Dai [11]. The 

* Project supported by the National Natural Science Foundation of China (Grant No.10602032), the Shanghai Rising-Star Program (Grant No. 07QA14022), and the Shanghai Leading Academic Discipline Project (Grant No. Y0103). Biography: LU Dong-qiang (1972- ), Male, Ph. D., Associate Professor

dispersion relationship for flexural-gravity waves was studied in detail in comparison with that for pure gravity waves. It was found that there exists a minimal group velocity and the wave system observed depends on the moving speed of the observer. For an observer moving with the speed larger than the minimal group velocity, there exist two trains of waves, namely, the long gravity waves and the short flexural waves, the latter riding on the former. In this article, the flexural-gravity waves generated by an instantaneous line and point source submerged in an inviscid fluid of finite depth are considered. The asymptotic representations of the wave motions for large time with a fixed distance-to-time ratio are derived for different regions. The partition of region depends on the intrinsic group velocities of the system under consideration. The effect of finite depth on the wave system is studied. The other kinds of impulsive concentrated disturbances, for example, an instantaneous dynamic

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load on the plate, an initial impulse on the surface of the fluid, and an initial displacement of the ice plate, can be dealt with by the method presented in this article. The case with a point mass source can be studied by a straightforward application of the procedure provided in Sections 2-3 and the method given in Ref. [11].

2. Mathematical formulation Without loss of generality, a Cartesian coordinate system is used, in which the z axis points vertically upward while z = 0 represents the mean ice-water interface. Therefore, the governing equation is

’ ) = M G ( x  x0 )G (t ) 2

(1)

where ) ( x , t , x0 ) is the velocity potential for the perturbed flow, M is the constant strength of the simple source, x is a field point, x0 is the source point, and G ( ) is the Dirac delta function. For twodimensional cases, x = ( x, z ) and x0 = (0, z0 ) with while for three-dimensional cases, z0  0 x = ( x, y, z ) and x0 = (0, 0, z0 ) . The linearized kinematic and dynamic ice-cover conditions at z = 0 are given by

w] w) = wt wz

w) w 2] 4 U + U g] + D’ ] + U e h 2 = 0 wt wt

(2)

acceleration of gravity, D = Eh3 [12(1  Q 2 )] is the flexural rigidity of the plate, and E , h , and Q are the Young modulus, the thickness, and the Poisson ratio of the plate, respectively. The bottom condition at z =  H is given by

w] |t =0 = 0 wt

(5)

Moreover, since the finite disturbance must die out at infinity, it is required that ’) o 0 as z o f , which imposes the uniqueness on the problem considered. Now, let the entire solution be written as

) = ) S ( x , t , x0 ) + ) I ( x , t , x1 ) + ) R ( x , t ) (6) where ) S and ) I are the velocity potentials due to the simple source at x0 and at x1 , respectively,

) R is a continuous function everywhere in the corresponding domain, which represents the effect of ice-cover boundary, x1 is the image point of x0 with respect to the flat bottom z =  H and for two-dimensional case and x1 = (0, z1 ) x1 = (0, 0, z1 ) for three-dimensional one, where z1 = 2 H  z0 , and ) I is introduced to meet the requirement of Eq.(4). For a line source in the two-dimensional cases, there holds

^)

S

, )I` = 



(3)

where ] is the vertical deflexion of the ice-water interface, U and Ue are the uniform densities of the fluid and the plate, respectively, g is the

w) =0 wz

) |t =0 = ] |t =0 =

M­ 1 1½ ®log , log ¾ 2S ¯ r r1 ¿

+f M c +i f exp(i D x + st ) < ds ³ dD 2 ³ c i f f 8S i k

^ exp  k z  z ,exp  k z  z ` 0

1

(7)

where r = x  x0 , r1 = x  x1 , k = D , and c is the Laplace convergence abscissa. Substitution of Eq.(6) into Eq.(1) yields

’ 2) R = 0

(8)

The boundary conditions at z = 0 can be re-written as

(4)

where H is a positive constant. It is assumed that the entire fluid is at rest for t  0 . Therefore, the initial conditions at z = 0 are

w] w) R w) S w) I  = + wt wz wz wz

U

w) R w 2] + U g] + D’ 4] + Ue h 2 = wt wt

(9)

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§ w) S w) I · + U ¨ ¸ wt ¹ © wt

(10)

w ) S +) I =0  wz z= H

(11)

D Ug

,

V=

h Ue

U

To perform the k integration in Eq.(14), the method of stationary phase is used for large t with a fixed x t [10,11,15]. The solutions for the stationary points, denoted by k j , are determined by

From Eqs.(4), (6), and (11) it follows that

w4 mn =0 wk

R

=0

(19)

(12)

z = H

A straightforward derivation for Eq.(19) yields

3. Asymptotic solutions For two-dimensional cases, a combination of the Laplace transform with respect to t and a Fourier transform with respect to x is introduced for {) R , ] } as

{) R , ] } =

J=

,

(18)

It follows from Eq.(7) that

w) wz

Z0 = gk

c +i f f 1 ds ³ dD exp(i D x + st )< 2 ³ c i f f 4S i

^ )

R

cosh > k ( z + H ) @ , ]

`

(13)

By substituting Eq.(13) into the Laplace-Fourier transforms of boundary conditions (9) and (10), two simultaneous algebraic equations are set up for the unknown functions ) R and ] , which can be readily solved. Consequently, the formal integral expression for the displacement of ice-water interface can be given as

1 2 2 +f ] = ¦¦ A exp  i t4 mn dk 4S m =1 n =1 ³ 0

x t

 Cg =

g 1/ 2 {k0  k 1/ 2 < 2

[coth(kH ) + V k ] 3/ 2 (1+ J k 4 ) 1/ 2 < [(1+ 5J k 4 ) coth(kH ) + kH (1+ J k 4 )< csch 2 ( kH ) + 4V J k 5 ]} = 0

(20)

where C g (k , h, H ) = wZ wk is the group velocity and

k0 = gt 2 4 x 2 is the solution of Eq.(19) with h = 0 and H = +f , corresponding to the wave numbers of the pure gravity waves in a fluid of infinite depth.

(14)

where

A( k ) =

M [1+ coth(kH )][exp( kz0 ) + exp( kz1 )] 2[coth( kH ) + V k ] (15)

4 mn

x = (1) m +1 k + (1) n +1Z t º 1+ J k 4 » ¬ coth( kH ) + V k ¼ ª

Z (k ) = «

(16)

Fig.1 Group velocity curves C g with h = 0.01 m

For a graphical representation of the theoretical results, hereinafter, the following physical parameters given by Squire [1] are adopted: E = 5GPa , Q = 0.3 , kgm 3

U = 1024

,

U e = 917

kgm 3

,

and

g = 9.8ms . Figure 1 shows the curves for the group velocity Cg (k , h, H ) . It can be seen from Fig. 1 that 2

1/ 2

Z0

(17)

once h and H are given, there exists a minimal

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group velocity, denoted by Cg min = Cg (kc , h, H ) , at which Eq.(20) has only one real positive root kc and

Zccc = w 2Z (kc ) wk 2 = 0

.

It

is

noted

4 mn | 4 mn (kc ) +

that

Z0cc = lim w Z (k ) wk = 0 . When Cg min  x t  Cg max , 2

2

1 w 34 mn (kc ) ( k  kc )3 3 6 wk

k o0

where C g max = lim C g (k , h, H ) = gH , Eq.(20) has k o0

two

real

positive

k2 ( x t , h, H )

with

roots,

k1 ( x t , h, H )

0  k1  k2  +f

.

and

w4 mn (kc ) ( k  kc )  wk

Furthermore, according to Scorer [16], it is valid that

When

x t ! Cg max , Eq.(20) has only one real positive root, k2 . The values for k1 , k2 , and kc can be obtained numerically from Eq.(20). The effects of depth variation on the critical wave number kc and the minimal group velocity

Cg min are shown in Figs.2 and 3, respectively. It can be seen that as H increases from zero, kc and Cg min first increase to their peak values and then decrease to constants.

(21)

1/ 3

§ 2 · ] = ] c ~ Ac ¨¨ ¸¸ © Zcccc t ¹

Ai( Z c ) cos  kc x  Zc t (22)

where

Ac = A(kc ) , Zc = Z (kc )

(23)

1/ 3

§ 2 · Z c =   x  Zcct ¨ ¸ © Zcccct ¹

(24)

and Ai( ) is the Airy function. When Cg min  x t  C g max , the expansion for the phase function near k j ( j 1, 2) is taken as

4 mn

Fig.2 Critical wave numbers

2 1 w 4 mn (k j ) (k  k j )2 | 4 mn (k j ) + 2 wk 2

(25)

By a straightforward application of the method of stationary phase, the asymptotic representation of Eq.(14) can be given as

] = ]1 + ] 2

(26)

where

]j ~

Aj

 2S Z cc t j

1/ 2

Sº ª cos « k j x  Z j t + (1) j » 4¼ ¬ (27)

Fig.3 Minimal group velocities

When x t  C g min , Eq.(20) has no real roots, as shown in Fig.1. However, as x t is lower than Cg min and is sufficiently close to Cg min , the phase function near kc may be expanded as

A j = A(k j ) , Z j = Z ( k j )

Z ccj =

(28)

w 2Z (k j )

(29)

wk 2

As x t is sufficiently close to C g max , k1 and

Z1cc tend to zero. According to Mei

[17]

, the

phase

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function for small k may be expanded as

x | xmax . The flexural-gravity waves at a given instant are shown in Fig.5. It can be seen that there exists two critical positions for the flexural-gravity waves: xmin = C g min t and xmax = Cg max t . For

x t

4 mn = (1) m +1 k + (1) n +1 ( gH )1/ 2 < ª º k 3 H (1+ V H ) k + "»  « 6 ¬ ¼

(30)

In this case, Eq.(25) is still used for k2 . Thus, for Eq.(14), there holds

] = ]1 + ] 2

(31)

x d xmin and x | xmin , the wave profile is predicted by Eq.(22). In this region, there is only one gravity wave. For xmin  x  xmax , the wave profile is predicted by Eq.(27). In this region, there are two wave systems, the short flexural one and the long gravity one. For xmax  x and x | xmax , the wave profile is predicted by Eq.(31). In this region, the two waves decay as x increases.

where

a Ai ^a [ x  ( gH )1/ 2 t ]` 2

(32)

a = 21/ 3 ( gH ) 1/ 6 [ H (1+ V H )t ] 1/ 3

(33)

]1 ~

and ] 2 is given by Eq.(27).

Fig.5 Flexural-gravity waves with h = 0.01 m, H = 5 m, M = 1 m, z0 = 0.1 m, and t = 5 s (1ʊ ] in Eq.(26), 2ʊ ] in Eq.(31), 3ʊ ] in Eq.(22), 4ʊ ] 1 in Eq.(27) with h

0 , 5ʊ ] 1 in Eq.(32) with h = 0 )

For three-dimensional component in Eq.(6) is Fig.4

Pure gravity waves with M = 0.5 m, z0 = 0.1 m,

)S = 

and t = 5 s (1ʊ ] 1 in Eq.(27), 2ʊ ]  in Eq.(32), 3ʊ ]  in Eq.(27))

4. Discussion and conclusions When h = 0 , the problem considered here reduces to that for pure gravity waves in an inviscid fluid of finite depth, for which Eq.(20) has only one real positive root and C g max is the maximal group velocity for the gravity waves. The gravity waves at a given instant are shown in Fig.4. It can be seen that there exists one critical position xmax = C g max t for pure gravity waves. For x  xmax , the wave profile is

predicted

by

Eq.(27).

For

x | xmax

and

x t xmax , the wave profile is predicted by Eq.(32). Two asymptotic solutions are in good agreement at

the

singular

c +i f M M G (t ) =  ds < 4S r 16S3 i ³ c i f

+f

+f

f

f

³ ³

cases,

d D dE

1 exp( K z  z0 + i F + st ) K (34)

where K = D + E and F = D x + E y . The asymptotic solutions for waves due to a simple point source can be obtained by a straightforward application of the procedure provided in Section 3 and the method presented in Ref. [11]. According to the relation between the moving speed of an observer and the minimal group velocity of waves on the fluids of finite depth with a thin elastic plate, three asymptotic schemes are proposed for the phase function to derive the asymptotic solutions for the waves by means of the stationary-phase method. The components and spatial 2

2

136

distribution of the wave system are analyzed.

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