On the initial values of kinematic and dynamic free-surface boundary conditions for water wave problems ＊

Dong-qiang Lu

Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science,Shanghai University, Shanghai 200072, China

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China

Abstract: The kinematic and dynamic boundary conditions on the free surface of a fluid should be posed for water wave problems.In the framework of potential theory for an inviscid and incompressible fluid with an irrotational motion, the combined boundary condition, which involves the velocity potential only, is often used by eliminating the elevation terms mathematically. Such a combination is correct for the solutions in the frequency domain, and is not feasible for an initial-boundary-value problem in the time domain since it leads to a totally different physical formulation. The correct initial conditions for pure gravity waves and hydroelastic waves are presented.

Key words: Initial values, boundary conditions, water wave, hydroelasticity

Water wave problem, an interesting topic in fluid mechanics, is traditionally studied in the framework of the irrotational motion in an inviscid, incompressible and homogenous fluid. For an incompressible fluid,the conservation law of mass is represented by?·u= 0, whereuis the velocity vector. Thusucan be expressed in terms of a potential functionφ(x,t) asu=?φunder the assumption of an irrotational motion, wherex=(x,y,z)in the Cartesian coordinates andtthe time. Furthermore,for an inviscid fluid, the conservation law of momentum yields the Euler equation without the viscosity term as follows

wherePe(x,t) the total pressure,ρthe density,gthe gravitational acceleration,ζ(z,t) the surface elevation,B(t) the Bernoulli constant, andz= (x,y).B(t) can usually be set as zero by re-defining the potential without affecting the velocity vector. By introducing

Therefore,Φ(x,t) is used as the velocity potential hereinafter. For a moving surface, Eq. (1)reformulated in terms ofΦis employed as the dynamic boundary condition onz=ζ. The kinematic boundary condition onz=ζreads

Equation (3) indicates that the fluid particles on the surface move only tangentially.

Another approach for the boundary conditions is a combination of Eqs. (1) and (3) by eliminating mathematically all the elevation terms withζ. One can take the total derivative of Eq. (1) and use Eq. (3)to derive (Mei et al., 2005, p. 8, Eq. (1.1.16))[1]

Equations (1) and (3) form the complete and original boundary conditions for wave problems.These two equations involve two unknownsζandΦ. To pose properly the initial value problem with Eqs. (1) and (3), we should prescribe

whereζ0andI0are the given initial values.Obviouslyζ0is the initial profile of the surface, andI0physically represents an impulse acting on the surface at an instant slightly earlier thant= 0+(Mei et al., 2005, p. 25)[1].

WhenePrepresents the atmospheric pressure, it can be treated as a constant. Then Eq. (4) is a partial differential equation with one unknownΦ. To pose properly an initial value problem with Eq. (4), we have to take

where1Φis a known initial value. The well-posedness of Eq. (6) can also be justified by the Laplace transform, L[·], of the second derivative ofΦwith respect totas follows

wheresis the complex variable in the Laplace transform. The transform is possible if all the required initial valuesare prescribed.

Historically, there have been two approaches available to pose initial conditions for gravity wave problems. The first one involves the initial values ofΦand its first time derivative, namely Eq. (6), as formulated by Stoker (1957, p. 177, Eq. (6.7.4))[2].

Another involves the initial values ofΦandζ,namely Eq. (5), as proposed by Miles (1962, Eq.(1.6))[3]and Mei et al. (2005, p. 36, Eq. (2.1.43))[1].

With the small-amplitude wave assumption, the nonlinear terms in Eqs. (1) and (3) will be omitted.Then the linearized dynamic boundary condition reads

For a linear gravity wave problem, these two approaches in Eqs. (5) and (6) are equivalent if and only ifΦ1= -gζ0+Pe(x,y,0, 0)/ρis satisfied, wherePe(x,y,0, 0 ) represents the initial surface pressure onz= 0 att=0.

For the unsteady waves created by a disturbance on the surface of a running stream, for example,U ex,the main convective effect is taken to linearize Eq. (1)as

whereUandexare a constant and the unit vector along thexaxis, respectively. For this case, Stoker(1957, p. 210, Eq. (7.4.4))[2]still used Eq. (6) withΦ1= 0 andΦ0=0 for the undisturbed uniform flow.Stoker's initial condition was regarded by Miles (1962,p. 146)[3]who stated from the physical point of view that “Stoker (1957) also pose the spurious initial conditions in his analysis”. The meaningful initial conditions for this case should read, forPe=0

Obviously, one can see from Eq. (10) that the zero initial values in Eq. (5) (ζ0=0 andΦ0=0) do not lead toΦ1=0 in Eq. (6). For the initially undisturbed state, we should use Eq. (5) withζ0=0 andΦ0= 0, or Eq. (6) withζ0=0 andΦ1= -U2/2.

For the example presented in Eq. (9), one can draw the conclusion that zero initial values in Eqs. (5)and (6) correspond to two different mathematical problems. Generally speaking, we can prescribe any initial values mathematically. However for a specified problem with physical significance, there is one choice only. This statement also holds for a nonlinear problem.

Let us consider hydroelastic (flexural-gravity)waves. The ice cover in the polar region and the very large floating structures in the offshore region are usually idealized as thin elastic beams/plates floating on an inviscid incompressible fluid in the theoretical investigations. It is assumed that there is no gap between the fluid and the floating structure.Hydroelastic waves will appear as the flexible structure is heavily subjected to incident wave action or an external moving load, and the deflection of the structure is the same as the elevation of fluid surface.

For a thin homogeneous elastic plate with a uniform mass densityeρand a constant thicknessd, the relationship between a plate deflectionζand the pressureePconsists on the Euler-Bernoulli beam/plate theory which can be written as

whereD=Ed3/ [12(1 -υ2)] ,M=ρed,υPoisson’s ratio andEYoung’s modulus of the plate.Qis associated to the lateral stress of the plate (with compression atQ＞0 and stretch atQ＜0). The first, second and third terms in the right-hand side of Eq. (11) are the elastic, compressive and inertial forces, respectively. Substitution of Eq. (11) into Eq.(4) yields the dynamic boundary condition for the hydroelastic wave, which indicates the balance between the hydrodynamic pressure of the fluid and the resulting forces due to the structural deformation.

Equation (11) can be regarded as a general model for water wave problems. AsePis a constant, Eq. (1)is for pure gravity waves. AsD=0,Q=-TandM= 0 for Eq. (11), Eq. (1) is used for the capillarygravity waves, whereT＞0 is the coefficient of surface tension. AsD=0,Q=0 andM＞0 for Eq. (11), Eq. (1) is for gravity waves on an inertial surface. AsD＞0,Q=0 andM＞0, Eq. (1) is conventionally employed for hydroelastic waves. The model withD＞0,Q≠0 andM＞0 is also recently investigated[4-5].

Equation (11) involes the second derivative ofζwith respect tot. So the initial conditions for Eqs.(1) and (3) with Eq. (11) should read[6-8]

V0is the initial speed of the surface. For a thin elastic plate, the inertial effect can be neglected. Under this assumption, the third term in the right-hand side of Eq.(11) is eliminated, and the initial conditions still follow Eq. (5).

Although Stoker’s formulation was corrected by Miles in 1962[3], one can still find some recent publications which follow Stoker's formulation, for example, Refs. [4, 9-11]. The author is motivated by this situation and corrections are made for some previously published articles on transient linear hydroelastic waves, for which the linearized kinematic and dynamic conditions read, for waves in a uniform current,

On basis of the model presented in Eqs. (13) and(14), exactly analytical solution of the dispersion relation for the hydroelastic waves in a two-layer fluid with a uniform current was explicitly derived by Lu[12].Considering Eqs. (13) and (14), we have the linearized combined boundary conditions for hydroelastic waves

Waves generated by initial axisymmetric disturbances in water with an ice cover were considered by Maiti and Mandal[9]who used the combined dynamic boundary condition. Equation (4)in Ref. [9] is a special case of Eq. (15) withU=0 andQ=0. Maiti and Mandal[9]imposed the initial values (See Eqs. (5) and (6) in Ref. [9]) for

is not purely the initial impulse as Maiti and Mandal[9]expected. The second term of Eq. (16)

is related to the initial profile.

With the aid of Eq. (15), Mohanty et al.[4]studied the time-dependent flexural-gravity waves in the presence of current for a single layer fluid, and they employed Eq. (6) withΦ1=0 andI0=0 as the initial conditions for the Green function (See Eq. (25)in Ref. [4]). Such a treatment leads to spurious conditions for the physical problem under consideration. These initial conditions were also used by Mohanty et al.[10]for the time-dependent capillarygravity waves in the presence of current, which can be regarded as a special model of Eq. (15) withD=0 andM=0.

More recently, Mohanty[11]considered the transient linear hydroelastic waves due to initial disturbances in a two-layer fluid with a current, in which the dynamic condition is a special case of Eq.(15) withM=0. The initial conditions (See Eqs.(12)-(15) in Ref. [11]) follow Eq. (6), which leads to a pure mathematical problem without physical significances. To have correct mathematical formulation with physical meanings for the flexural- and capillary-gravity waves, Eqs. (5) and (12) should be applied for the cases withM=0 andM≠0,respectively.

The initial-boundary-value problems are often concerned with the wave generation due to an unsteady moving load or an initial profile/impulse.The wave-structure interaction problems are usually solved in the frequency domain by separating the time with exp(iωt), whereωis the frequency, and thus a pure boundary-value problem is mathematically formulated without initial conditions.

We assume the solutions in the frequency domain take the form ofζ=ηe x p(iωt) andΦ=φe x p(iωt).Substitution of these forms into Eqs. (13) and (14) and the combination the resultant equations yields

We can also substitute the time-harmonic forms of into Eq. (15) directly to deduce the same result as Eq.(19). In the frequency domain, two separated equations (Eqs. (13) and (14) and one combined equation (Eq. (15)) predict the same boundary condition for the wave problems.

Conclusions of this letter are drawn here. A combination of the kinematic and dynamic free-surface boundary conditions for water wave problems in the time domain may leads to spurious initial conditions for a real physical concern. The initial conditions should impose on the terms, which involve the derivative with respect to the time, in the original equations for the kinematic and dynamic boundary conditions before they are mathematically combined with higher-order derivatives, for example,Eqs. (5) and (12). For the boundary-value problems in the frequency domain, the separated and the combined boundary conditions predict the same results since no initial conditions are required. The judgement for surface waves in a single fluid also holds interfacial waves between two layers of fluid[12-13].