LU Dong-qiang (盧東強)

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China, E-mail: dqlu@shu.edu.cn

SUN Cui-zhi (孫翠芝)

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

Transient flexural- and capillary-gravity waves due to disturbances in two-layer density-stratified fluid*

LU Dong-qiang (盧東強)

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China, E-mail: dqlu@shu.edu.cn

SUN Cui-zhi (孫翠芝)

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

(Received December 11, 2012, Revised January 18, 2013)

Generation of the transient flexural- and capillary-gravity waves by impulsive disturbances in a two-layer fluid is investigated analytically. The upper fluid is covered by a thin elastic plate or by an inertial surface with the capillary effect. The density of each of the two immiscible layers is constant. The fluids are assumed to be inviscid and incompressible and the motion be irrotational. A point force on the surface and simple mass sources in the upper and lower fluid layers are considered. A linear system is established within the framework of potential theory. The integral solutions for the surface and interfacial waves are obtained by means of the Laplace-Fourier transform. A new representation for the dispersion relation of flexural- and capillary-gravity waves in a twolayer fluid is derived. The asymptotic representations of the wave motions are derived for large time with a fixed distance-to-time ratio with the Stokes and Scorer methods of stationary phase. It is shown that there are two different modes, namely the surface and interfacial wave modes. The wave systems observed depend on the relation between the observer’s moving speed and the intrinsic minimal and maximal group velocities.

hydroelasticity, capillarity, wave generation, density stratification, two layers

Introduction

As is well known, the ocean is stratified in density due to the vertical variation of water temperature and salinity. A frequently-used model for the ocean is simply a two-layer fluid system, in which the thin pycnocline is just represented by an sharp and stable interface and each of the two layers is assumed to be inviscid, incompressible and homogeneous[1-8]. One fundamental problem of theoretical interest for this model is the generation of surface and interfacial waves by a submerged body which is idealized as a point force or a simple source to consider its far-field effects. It has been demonstrated that such a simple model is useful for the analytical purpose and the qualitative prediction.

The steady problems for the generation of pure gravity wave in two-layer fluid were well addressed in the last decades. Based on the method of Green’s function, Miloh et al.[1]considered the wave resistance and the interfacial elevation due to a steadily moving source submerged in the upper layer of two fluids of finite depth and presented numerical examples for a surface-piercing semi-submersible slender body with a constant forward speed. The Green function problem proposed by Miloh et al.[1]was further explored by Yeung and Nguyen[2]who performed an analytical prediction on the wave patterns using the Fourier transform for a double-integral solution and then the residue theorem and the stationary-phase method for the inverse Fourier transform. The critical Froude numbers were derived and their effects on the wave patterns were well studied. Wei et al.[3]investigated, both analytically and experimentally, the steady waves dueto a moving dipole submerged in the lower layer to show the qualitative agreement between the mathematical prediction and the laboratory observation.

In view of the time-dependent maneuverability, the unsteady wave problems for two-layer fluids have been attracting increasing attention among the community. Lu and Chen[4]investigated analytically the transient gravity waves caused by an impulsive source in a two-layer fluid with a lower fluid being either of infinite depth or bounded by a flat bottom and considered three kinds of upper boundary conditions, namely a free surface, a rigid lid, and a semi-infinite extent. For an oscillating and translating disturbance in a two-layer fluid, Alam et al.[5]derived explicit expressions for the Green functions of the linearized two- and three-dimensional potential flows and studied the far-field waves analytically and numerically. Nguyen and Yeung[6]obtained the velocity potentials due to a point source of time-dependent strength arbitrarily moving in a two-layer fluid of finite depth, on the basis of which the radiation and diffraction of waves by a floating structure were deduced.

All the above-mentioned are concerned with the pure gravity waves on the fluid with a clean free surface. Recently more complicated boundary conditions on the upper surface were taken into consideration. For the ice cover in the polar region[9]or the Very Large Floating Structures (VLFS) in the offshore region[10], a physical model that involves an inviscid fluid covered by a thin elastic plate is widely accepted[11-14]. As the flexural rigidity of the plate tends to zero, the inertial surface model will be employed[15]. Lu and Dai[12,14]studied the unsteady responses of an infinite elastic plate floating on a single layer fluid to a concentrated load and showed the dynamic similarities between the flexural- and capillary-gravity waves. Lu and Dai[14,15]also provided asymptotic solutions for the gravity and capillary-gravity waves on an inertial surface of a single fluid. For capillary-gravity wave motion in two-layer fluids, Mohapatra et al.[7]developed a model regarding the combined effects of surface tension and interfacial tension.

The main object of the present work is to study the flexural- and capillary-gravity waves due to transient disturbances in a two-layer fluid. Based on the assumption of small-amplitude waves, a linear system is established. By virtue of the principle of superposition, we shall consider simultaneously three kinds of instantaneous loads, namely a point force on the surface and two sources each of which is immersed in the upper or lower fluid layers. Two types of upper surfaces are included in parallel: one is a thin elastic plate and the other the capillary inertial surface. By means of the Fourier-Laplace transform, the integral solutions for the surface and interfacial waves are obtained. A new representation for the dispersion relation of flexural- and capillary-gravity waves in a two-layer fluid is derived. The corresponding asymptotic representations are derived for large time with a fixed distanceto-time ratio by using the Stokes and Scorer methods of stationary phase.

The depths of the upper and lower fluids are denoted by h1and h2, respectively. Cartesian coordinatesoxyzare taken and the z-axis points vertically upward. The upper surface, the undisturbed interface and the flat bottom coincide withz=h1,z =0and z=-h2, respectively, as shown in Fig.1.

1. General mathematical formulation

Fig.1 Coordinate system

The two fluids are assumed to be inviscid and incompressible and the motion be irrotational. As is well known, the fundamental singularity for an inviscid flow is a simple source, which represents a point mass source in the fluid. For a general formation, we impose simultaneously one source into each of the two fluid layers. Therefore, the governing equation is

where Φmis the velocity potential for the perturbed flow in the upper(m=1)and lower(m=2)fluids, Δmnis the well-known Kronecker delta,Mnis the magnitude of the simple source in the upper (n=1) and lower(n=2)fluids,δ(?)is the Dirac delta function,x is an observation point while xna source point immersed in the fluidn. For two-dimensional case,x =(x, z)and xn=(0,zn), while for three-dimensional case,x =(x, y, z)and xn=(0,0,zn). It should be noted thatz1＞0and z2＜0.

It is assumed that the wave amplitudes generated are small in comparison with the wavelengths. So the linearized boundary conditions will be applied on the undisturbed upper surface and the interface. Denote the densities of the upper and lower fluids by ρ1and ρ2with ρ1＜ρ2, respectively. The kinematic and dynamic conditions on the upper surface (z=h1)and the interface(z=0)are given by

where ζ1and ζ2are the upper surface and interfacial elevations with respect to z=h1and z=0, respectively,D= Ed3/[12(1-ν2)]is the flexural rigidity of the plate,me=ρed,E,ρe,dandνare Young’s modulus, the density, the thickness, and Poisson’s ratio of the plate, respectively,P0is the magnitude of the strength of the applied load,z =(x, y,0)and z0=(0,0,0)are the field and source points at z=0. Equations (2) and (4) imply that fluid particles once on the upper surface or interface will always remain there. Equation (3) indicates the balance among hydrodynamic, elastic, inertial forces and the downward external load. Equations (5) and (6) represent the continuity of the normal velocity and the pressure on the interface, respectively.

The boundary condition at the flat bottom is

Since the finite disturbance caused by the impulsive source must die out at infinity,

which imposes a uniqueness on the problem concerned. It is assumed that the entire fluid is at rest for t＜0. Therefore, the initial conditions are given as

Another interesting model is the capillary-gravity waves on an inertial surface (z=h1), for which the mathematical formulation simply follows from the above-mentioned equations by replacing Eq.(3) with

and retaining the others, whereTis the surface tension of the upper fluid. For simplicity, the mathematical equations for the flexural-gravity and the capillarygravity waves are hereinafter referred to as Models (I) and (II), respectively. The solutions for Models (I) and (II) will be derived in parallel. The similarity between these two models for a single fluid of finite depth has been elucidated by Lu and Dai[14].

2. Integral solution

The assumption of linearity allows us to envisage the perturbed flow as the sum of the regular and singular flows. The former represents the influence of the boundaries while the latter the effect of the singularities. Taking the bottom boundary condition Eq.(7) into consideration, we introduce mathematically an artificial source at the point x3=(0,0,z3)which is defined, with z3=-2h2-z2, as the image point of x2with respect to the flat bottom(z=-h2). We denote the freespace potentials due to the simple source at xby ΦS

nn with n =1, 2, 3, for which the governing equation is

with M3=M2, and the corresponding solutions in a free space can be given by

Accordingly, the solution of Eq.(1) takes the form of

where ΦRis a harmonic function everywhere in the

m corresponding domain, satisfying

It follows from Eqs.(7) and (16) that

In order to obtain the formal solution of this initial-boundary-value problem, a combination of the Laplace transform with respect to tand the Fourier transform with respect to the spatial variables is introduced for{ΦmR,ζm}as

Applying the joint Laplace-Fourier transform to Eq.(15) and considering Eq.(17), we have

Obviously, the counterpart of Eq.(22) for the surface capillary-gravity waves reads

Through some mathematical manipulations, the solutions for ζmcan be given in a joint Laplace-Fourier integral form and the inverse Laplace transform can be performed exactly. Then we have

with ω02=gk,ε=1-γand tj=tanh(khj). It should

be noted that for Model (I)

With the change of variables (x, y )= R (cosθ, sin θ),(α, β)= k(cosφ, sin φ)(44) Equation (27) can be rewritten as

where J0(k R)is the zeroth-order Bessel function of the first kind.

All the above-mentioned equations are for point sources in three dimensions and the formal solution for the wave elevations is represented by a single integral, as demonstrated by Eq.(45). For the case with a line source in two dimensions, one can simply follow the above equations by setting?/?y=0and ΦS=

n -Mnδ(t) log(1/rn)/2π. Accordingly theβintegral should be eliminated withf=iα x +stand k=α. Thus the solutions for surface and interfacial waves due to a line source can readily be represented by

3. Asymptotic solutions

In order to analyze the dynamic characteristics of the waves due to the transient singularities, it is necessary to derive, using the Stokes and Scorer methods of stationary phase, the asymptotic representation of the exact integral solutions Eqs.(45) and (46) for the large time with a fixed distance-to-time ratio. Next, we replace J0(k R)by its asymptotic formula for large kR

for a line source.

Next we shall take the point source for the mathematical presentation. For a line source, we may replaceRbyx and take the corresponding phase function. To perform thek integration in Eq.(49), the method of stationary phase is used for large twith a fixedR/ t.R/ tis referred to as the observer’s moving speed. It is easily seen that for a fixedj,θ12jand θ21jhave the same stationary points. For short θ12jis denoted by θjhereinafter. The solutions for the stationary points are determined by

where Cgj= ?ωj/?kis the group velocity for the surface(j=1)and the interfacial (j=2)wave modes. A graphical representation of Eq.(55) for the flexuralgravity waves is shown in Fig.2, where the physical parametersγ=0.7,E =5 GPa,ν=0.3,ρ1= 1 024 kg/m3,ρe=917 kg/m3, and g=9.8 m/s2are adopted[11].

Fig.2 Group velocities for flexural-gravity waves

From Fig.2, we can see that the group velocities of the surface(j=1)wave mode have the minimal and maximal values while those of the interfacial (j=2)wave mode have maximal values for a finite depthh2. For the surface(j=1)wave mode, the existence of the minimal group velocity Cgmin1is due to the presence of the elastic plate or the capillary surface. The wave number at whichCgmin1occurs is denoted by kc1, which is determined by ?2ω1/?k2=0.The analytical expressions for the maximal group velocitiesCgmaxjcan be readily given as

It can be seen from Fig.2 that, as Cgmin1＜R/ t＜Cgmax1, Eq.(55) has two distinct real roots k1jand k2jfor the surface wave mode(j=1), and as R/ t＜Cgmax2, Eq.(55) has one real root k1jfor the interfacial wave mode(j=2). The values of k11,k21and k12can be obtained numerically from Eq.(55). For simplicity, the root is denoted by kij, where the subscript i indicates the number of the roots. In these cases, the Stokes stationary-phase approximation will be used. The expansion for the phase function near kijis taken as

where i =1, 2 for j =1 and i =1 for j =2. For fixed iandj, the asymptotic solution contributed from the stationary pointkijto the wave profile is denoted by ξmij. It should be noted that the subscript mstands for the physical positions (m =1 and m=2 for surface and interfacial waves, respectively),ifor the wave components (i =1 and i=2 for gravity and flexural waves, respectively), andj for the wave modes (j= 1 and j=2 for surface and interfacial waves, respectively).

Afterwards the asymptotic solutions for the Eq.(49) are listed below

and the prime here denotes the differentiation with respect tok.

Ask tends to kc1, the approximation Eq.(58) breaks down and the expansion for the phase function nearkc1is taken as

where j=1. The asymptotic solution contributed from the stationary pointkc1to the wave profile is denoted by ηmc1. Then, according to Scorer[12], Eq.(58) can be approximately given as

andAi(?)is the Airy integral.

It is noted that as R/ t tends to Cgmaxj,k1jtends to zero. For the case of a point source, it should be noted that

Therefore,ζmtends to zero as R/ t approaches Cgmaxj, where m=1, 2. For the case with a line source, we may expand the phase function with smallkas follows

The asymptotic solution contributed from the neighbor of k =0 to the wave profile is denoted by ηm0j. Accordingly we have

The wave profiles depend on the solutions of Eq.(55) and the relation between R/ tand the group velocitiesCgj. Next, two cases will be studied.

Fig.3 The surface and interfacial waves due to a line source. Curves: (1-ζmfor 0 ＜|x|/ t ＜Cgmin1, 2-ζmfor|x|/ t≈Cgmin1and C gmin1 ＜|x|/ t＜Cgmax2, 3-ζmfor |x|/ t≈Cgmax2and C gmax2 ＜|x|/ t＜Cgmax1, 4-ζmfor |x|/ t≈Cgmax1and |x|/ t＞Cgmax1)

First we consider the case with Cgmin1＜Cgmax2.

For the case of line source,

Figure 3 shows the graphical representation for surface and interfacial waves due to a line source in the upper fluid covered by a thin elastic plate, where t= 5 s,M1=0.5 m2,z1=5 m and the other parameters are the same as those for Fig.2(a). For the upper surface, there exist two wave systems, namely the long gravity one and the short flexural one, the latter riding on the former.

For the case of point source,

For the case of point source,

4. Conclusion

The transient wave response to an instantaneous concentrated load has been analytically studied within the framework of linear potential theory. Asymptotic representations are derived, by means of the Stokes and Scorer methods of stationary phase, for three kinds of loads (namely a point force on the surface and two sources each of which is immersed in the upper or lower fluid layers) and two types of upper surfaces (namely a thin elastic plate and the capillary inertial surface). It is found that the wave profiles observed depend on the relation between the observer’s moving speed and the group velocities. The analytical solutions show that there are two different modes, namely the surface and interfacial wave modes. For the surface mode, there exist two wave systems: one is the long gravity component while the other the short flexural (capillary) one.

[1] MILOH T., TULIN M. P. and ZILMAN G. Dead-water effects of a ship moving in stratified seas[J]. Journal of Offshore Mechanics and Arctic Engineering, 1993, 115(2): 105-110.

[2] YEUNG R. W., NGUYEN T. C. Waves generated by a moving source in a two-layer ocean of finite depth[J]. Journal of Engineering Mathematics, 1999, 35(1-2): 85-107.

[3] WEI Gang, LU Dong-qiang and DAI Shi-qiang. Waves induced by a submerged moving dipole in a two-layer fluid of finite depth[J]. Acta Mechanica Sinica, 2005, 21(1): 24-31.

[4] LU Dong-qiang, CHEN Ting-ting. Surface and interfacial gravity waves induced by an impulsive disturbance in a two-layer inviscid fluid[J]. Journal of Hydrodyna- mics, 2009, 21(1): 26-33.

[5] ALAM M. R., LIU Y. M. and YUE D. K. P. Waves due to an oscillating and translating disturbance in a twolayer density-stratified fluid[J]. Journal of Enginee- ring Mathematics, 2009, 65(2): 179-200.

[6] NGUYEN T. C., YEUNG R. W. Unsteady three-dimensional sources for a two-layer fluid of finite depth and their applications[J]. Journal of Engineering Mathe- matics, 2011, 70(1-3): 67-91.

[7] MOHAPATRA S. C., KARMAKAR D. and SAHOO T. On capillary gravity-wave motion in two-layer fluids[J]. Journal of Engineering Mathematics, 2011, 71(3): 253-277.

[8] LU D. Q., SUN C. Z. Waves due to a disturbance in a two-layer fluid covered by an elastic plate[C]. Proceedings of the 6th International Conference on Asian and Pacific Coasts. Hong Kong, China, 2011, 1930- 1937.

[9] SQUIRE V. A. Of ocean waves and sea-ice revisited[J]. Cold Regions Science and Technology, 2007, 49(2): 110-133.

[10] KASHIWAGI M. Research on hydroelastic responses of VLFS: Recent progress and future work[J]. International Journal of Offshore and Polar Engineering, 2000, 10(2): 81-90.

[11] SQUIRE V. A., HOSKING R. J. and KERR A. D. et al. Moving loads on ice plates[M]. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1996, 105.

[12] LU D. Q., DAI S. Q. Generation of transient waves by impulsive disturbances in an inviscid fluid with an icecover[J]. Archive of Applied Mechanics, 2006, 76(1- 2): 49-63.

[13] SQUIRE V. A. Synergies between VLFS hydroelasticity and sea ice research[J]. International Journal of Offshore and Polar Engineering, 2008, 18(4): 241- 253.

[14] LU D. Q., DAI S. Q. Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth[J]. International Journal of Engi- neering Science, 2008, 46(11): 1183-1193.

[15] LU Dong-qiang, DAI Shi-qiang. Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface[J]. Acta Mechanica Sinica, 2008, 24(3): 267-275.

10.1016/S1001-6058(11)60372-8

* Project supported by the National Natural Science Foundation of China (Grant No. 11072140) the State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University (Grant No. 0803) and the Shanghai Program for Innovative Research Team in Universities.

Biography: LU Dong-qiang (1972-), Male, Ph. D., Professor