chapter 3 fem on nonlinear free-surface flow

Numerical Models in Fluid-Structure Interaction

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Chapter 3 FEM on nonlinear free-surface flow K.J. Bai1 & J.W. Kim2 1Seoul National University, Korea 2American Bureau of Shipping, USA 3.1 Introduction Recently there has been a growing need for the treatment of the nonlinear water- wave problems. In treating the nonlinearity, there are basically two different approaches: One is based on the long-wave approximations or a shallow-water theory that gives an approximate formulation including the effect of the nonlinearity. The other is to treat the Laplace equation defined in the fluid domain with the exact nonlinear free-surface boundary conditions and the exact body-boundary condition if a body is present. The former is to treat a simplified nonlinear formulation defined only in a horizontal free-surface plane, which is one dimension less than the original fluid domain.

The free-surface flow problems have been of interest to many naval architects and ocean and coastal engineers for a long time. The generation and evolution of the water waves and their interaction with the man-made structures are the main concerns in the problem. The theoretical investigations on the topics have been usually made in the scope of the potential theory by assuming the fluid is inviscid and incompressible. The most distinctive feature of this problem is the

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presence of the free surface. This is one of the typical free-boundary problems where the part of the boundary should be obtained as a part of the solution. Due to this complexity the theoretical investigation on the problem has been restricted until recently. In the past, the linearized problem has been mainly treated. Sometimes, the nonlinear effect is considered by treating the second-order problem obtained in a systematic perturbation expansion. Another approach, where the leading nonlinearity is included, is to use approximate theoretical models such as the well known KdV and Boussinesq equations based on the long-wave approximation or shallow-water theory. Recently, the application of the Green–Naghdi equation is also proved to be an efficient method for treating nonlinear free-surface flow problems (Ertekin et al., 1986).

In this chapter we will present the finite-element method applied to the nonlinear water-wave problems, i.e., in ship motions, wave resistance, lifting problems, and initial-value problems. The present numerical scheme can be used for the validation of the existing approximate theories, i.e., the KdV and the Boussinesq equations or Green–Naghdi equation as well as for a better prediction for more realistic physical models.

In Section 3.2, we will describe the finite-element method applied to a two- dimensional hydrofoil problem. In this numerical scheme we introduced a buffer zone where the forced damping is introduced to reduce the wave elevation in matching the nonlinear numerical solution with the far-field linear solution. However, in three dimensions the nonlinear wave amplitude will reduce as it propagates to infinity. Thus the introduction of the fictitious damping term is not essential. In Section 3.3, we treat an initial-value problem for a physical model of an axi-symmetric container filled with water freely falling into a flat solid surface. In this problem we included the surface tension. In the numerical procedures, the finite-element subdivisions are made such that the free-surface elevation can be described as the multi-valued functions, since the surface tension as well as the gravity is important in the flow. In Section 3.4, the generation of solitons in a shallow-water towing-tank near the critical speed as well as a numerical towing-tank simulation for arbitrary tank conditions is discussed. In Section 3.5 a preliminary result of a sloshing problem by the same finite-element method used in Section 3.4 is discussed. In Section 3.6 a nonlinear diffraction problem is discussed. In Section 3.7, a brief concluding remark is given.



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3.2 Nonlinear steady waves due to a two-dimensional hydrofoil An application is described of the localized finite-element method to a steady nonlinear free-surface flow past a submerged two-dimensional hydrofoil at an arbitrary angle of attack. The earlier investigations with the linear free-surface boundary condition in Bai (1978) have shown some disagreement between the computed results and the experimental measurements for the cases of shallow submergence. The computational method of solution is the localized finite-element method based on the classical Hamilton’s principle. In the present study, a notable step is introduced in the matching procedure between the fully nonlinear and the linear sub-domains. Additional numerical results of wave resistance, lift force, and circulation strength are presented. Details can be found in Bai and Han (1994). It should be noted that in the earlier treatment of the linear hydrofoil problems, the velocity-potential formulation was used. However, in the following treatment of the nonlinear problem, a stream-function formulation is used since it has a distinct advantage in the numerical treatment. 3.2.1 Mathematical formulation We consider here a steady uniform flow past a fixed two-dimensional hydrofoil submerged in a fluid. The coordinate system is right-handed and rectangular as shown in Fig. 3.1. The y-axis is directed opposite to the force of gravity, and the x-axis coincides with the undisturbed free surface. The unit normal vector n

r is

always directed outward from the fluid domain. We neglect surface tension and assume that the fluid is inviscid, incompressible, and that the motions are irrotational. Here we assume the water depth, H, is constant.

The steady two-dimensional flow is described by a total stream function

),(),( yxUyyx ψ+=Ψ , (3.1)

where U is the incoming uniform flow velocity in the upstream and ),( yxψ is the perturbation stream function. The total stream function Ψ of the incoming uniform flow is set to 0 on the undisturbed free surface and UH− on the bottom. The perturbation stream function ψ must satisfy



86

Figure 3.1: Sketch of boundary configurations.

0),(2 =∇ yxψ (3.2)

in the fluid domain D. The kinematic and dynamic boundary conditions on the free surface FS are as follows:

( )

=∇⋅∇++

=+−=

021

0

ψψηψ

ψηηψ

gU

UorU

y

xx on )(, xySF η= , (3.3a,b)

where )(xy η= and g are the free-surface elevation and the gravitational acceleration, respectively. The boundary condition on the hydrofoil surface 0S can be

Cyx =Ψ ),( on 0S , (3.4)

where the constant C will be determined as a part of the solution. To determine the constant C, we require at the trailing edge denoted by TE an additional condition, i.e., the Kutta condition, which states that the pressure on the upper and lower surfaces is continuous. Therefore, the tangential velocities on the upper and lower surfaces have the same magnitude at the trailing edge and bounded, that is,

x

y

TEc

DS B

S R1 S R2

S F

U

H

0

S 0



87

( ) ( )−+ +∂∂

=+∂∂

ψψ Uyn

Uyn

at TE, (3.5)

where n∂∂ / is the normal derivative and +ψ and −ψ are the perturbation stream function ψ on the upper and lower surfaces, respectively. The boundary condition on the bottom, BS , i.e., Hy −= is

0=ψ on BS . (3.6)

x

y

TEc

SB

SR1

SR2

SF

U

H

0

S0

J 1D 1.1

J 2 J 3

D N DB D1.2

x1 x2 x3

Figure 3.2: Subdomains in nonlinear calculation.

As the radiation condition, we require that no disturbances exist far upstream, that is,

0→∇ψ as −∞→x , (3.7)

and that the perturbed flow velocity is bounded in the far downstream, i.e.,

ψ∇ is bounded as +∞→x . (3.8)

We hereby assume that the solution of eqns (3.2~3.8) can be determined uniquely. 3.2.2 Variational principle A mathematical formulation of the nonlinear free-surface flow phenomena of an incompressible ideal fluid can be described by variational principles based on the



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classical Hamilton’s principle as given in Kim and Bai (1990). Therefore, the formulation given in the previous section can be replaced by an equivalent variational functional formulation as follows. Now the nonlinear problem defined by eqns (3.2–3.8) is replaced by an equivalent Hamilton’s principle for the Lagrangian. The Lagrangian L is defined as

L = T – V + W, (3.9)

where T, V, and W are the kinetic energy, the potential energy, and the work done by external pressure, respectively. Then the Lagrangian L for the above nonlinear boundary-value problem can be defined as

dxgdxdyLFSD ∫∫∫ −∇⋅∇= 2

221],[ ηψψηψ , (3.10)

with the kinematic constraints eqns (3.3a), (3.4–3.6). FS denotes the projection

of the free surface on the y=0 plane. Equation (3.10) can also be written in the following form by using the kinematic constraint of eqn (3.3a),

∫∫ ∫ −∇⋅∇=−

− FS

Ux

Hdxx

UgdxdyL ),(

221],[ 2

2

),(ηψψψηψ

ηψ. (3.11)

In steady nonlinear free-surface problems, the numerical treatment of the radiation condition (3.7) and (3.8) is one of the most difficult obstacles. In linear problems, the radiation condition has been effectively treated by the localized finite-element method, where the numerical solutions in the computational domain were matched to the complete set of the analytic solutions known in truncated sub-domains (Bai and Yeung 1974, Bai 1977, 1978). As an extension of the application of this method to nonlinear problems, a modified variational form is developed in which the degree of the free-surface nonlinearity is artificially reduced to that of the linear problem, by introducing a nonlinear-to-linear transition buffer subdomain between the fully nonlinear computational subdomain and the truncated linear infinite subdomain. This modified variational form was successfully applied to a two-dimensional wave-resistance problem for a nonlifting body in the stream-function formulation in Bai et al (1990).



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The entire fluid domain D is subdivided into four subdomains as shown in Fig. 3.2. We denote the two linear infinite subdomains by 1LD in the upstream and by 2LD in the downstream, respectively. The solutions in these linear subdomains can be represented by the known complete set of analytic solutions in the same way as in linear problems. The nonlinear subdomain and the nonlinear-to-linear transition buffer subdomain are denoted by ND and DB, respectively. The juncture boundary between the upstream linear subdomain and the nonlinear subdomain is denoted by 1J . The juncture boundary between the downstream linear subdomain and the transition buffer subdomain is denoted by

3J . The two intersection points, i.e., the x-coordinates of 1J and J3, are denoted by 1xx = and 3xx = , respectively. The juncture boundary between the nonlinear subdomain and the transition buffer subdomain is denoted by 2J with its x -coordinate 2xx = .

In the nonlinear-to-linear transition buffer subdomain BD , a modification has been made on the Lagrangian and the free-surface boundary conditions in the following way. By introducing a locally linearizing parameter, )(xε , in the transition buffer subdomain BD , the Lagrangian L given in eqn (3.11) can be rewritten as

∫∫∫ −∇⋅∇=−

− FS

Uxx

H

x

xdxx

UgdxdyL ),(

221],[ 2

2

),()(3

1

ηψψψηψηψε

, (3.12)

with the constraints of eqns (3.4–3.6). In the fully nonlinear subdomain, ND , )(xε equals 1 and in the two linear subdomains, 1LD and 2LD , )(xε equals 0,

whereas the value of )(xε is gradually reduced from 1 to 0 in the nonlinear-to-linear transition buffer subdomain in the downstream. There is no need to introduce such a nonlinear-to-linear transition buffer region in the upstream, since the upstream free-surface elevation decays very rapidly. It should be noted that the generations of upstream solutions are absent in the range of the Froude numbers treated here. One can find the application of the present method to this problem in Bai et al. (1989) and Choi et al. (1990).

The new Lagrangian defined in eqn (3.12) is equivalent to the original nonlinear formulation given in eqns (3.2), and (3.4–3.8). However, the following modified free-surface boundary conditions, which replace eqns (3.3a) and (3.3b), are obtained by the new Lagrangian,

0=+ψηU on )()( xxy ηε= (3.13a)



90

02

)( 2 =∇++ ψεηψ xgU y on )()( xxy ηε= , (3.13b)

where 1)(0 ≤≤ xε . It should be noted that the nonlinear problem reduces to a linearized free-

surface problem locally in the neighborhoods of the two juncture boundaries, i.e., 1J and 3J , where the two complete sets of analytic solutions known in 1LD and

2LD are valid. As shown in Kim and Bai (1990), it is easy to show that the Lagrangian formulation given in eqn (3.12) is identical to the bilinear functional formulation in linear problems, and the free-surface conditions given in eqns (3.13a) and (3.13b) can be combined to obtain the linearized free-surface condition. Accordingly, the matching procedure is the same as in the localized finite-element method of Bai (1975, 1977, 1978). However, in the nonlinear subdomain ND , the modified Lagrangian defined in eqn (3.12) recovers the original Lagrangian given in eqn (3.11). For finite-element approximations, the Lagrangian L in eqn (3.12) with the constraints of eqns (3.4–3.6) goes into an operational form in the following way. The m-dimensional subspace of the admissible function space is chosen and let ),....,2,1( miNi = be the basis for the m -dimensional subspace. Then the solution is assumed to be

∑=

=m

iii yxNyx

1

),(),( ψψ , (3.14)

where iψ are coefficients to be determined in eqn (3.14) on the free surface,

∑=

=Fm

kkk xMx

1

)()( ψψ ; ),....,1(),()( Fyikk mkyxNxM == =η , (3.15)

where Fm is the number of nodes on the free surface, ki is the node number of the basis function iN that correspond to the k -th node on the free surface. By substituting eqns (3.14) and (3.15), the Lagrangian L in eqn (3.12) finally reduces to

lklkjiji TUgKL ψψψψ 2

*

221

−= , (3.16)



91

where ∫∫ ∇⋅∇=D

jiij dxdyNNK ; ∫=FS

lKkl dsMMT .

From Hamilton’s principle, eqn (3.16) must satisfy the following equations,

=∂

∂

=∂∂

0

0

*

*

k

iL

L

ψ

ψ . (3.17)

The final nonlinear matrix equation in eqn (3.17) for the computational domains, i.e., ND and BD , can be solved by an iterative procedure of Newton's method. In the procedure of imposing the appropriate upstream and downstream radiation conditions, asymmetric sets of test and trial functions are taken from and the bilinear functional. The detail procedures in the treatments of the radiation condition and the Kutta condition can be found in Bai and Han (1994).

After the stream function has been obtained, the pressure can be computed by Bernoulli’s equation

)(2

22yxyUP ψψρψρ +−−= , (3.18)

where the hydrostatic pressure has been omitted. Then the nondimensional

pressure coefficient PC is defined as

2

2

2 121 UU

PCPψ

ρ

∇−== , (3.19)

and PC is unity at the stagnation point.

3.2.3 Numerical results Computations are made for the flow around a hydrofoil with chord length c in water of finite depth H. The depth submergence h is measured from the undisturbed free surface, y = 0, to the trailing edge as shown in Fig. 3.3. Throughout the computations, we choose the 12% thick symmetric Joukowski hydrofoil at 5-deg angle of attack, which was previously investigated in the



92

experiments by Parkin et al. (1956) and also computed by Bai (1978) with the linearized free surface condition. The value of cH / is taken to be 6 in the computations. The major parameters introduced are nondimensionalized such as Froude number / , / , / ,cF U gc H c h c= where U is the incoming uniform flow velocity. In presenting the pressure distributions on the hydrofoil and the wave profiles on the free surface, the coordinates x (shown in Fig. 3.3) and the free-surface elevation are nondimensionalized by the chord length c such as

ccx /,/ , and c/ , respectively.

x

y

TE

c

U

H

0

⌧

Figure 3.3: Schematic diagram of a hydrofoil. The computations are made for the case when a hydrofoil is submerged near

to the free surface to find the effect of the nonlinearity in the free-surface condition. The pressure distributions on the hydrofoil are computed and compared with experimental results by Parkin et al. (1956) as well as with the linear results of Bai (1978). The wave profiles on the free surface are also computed and compared with the linear results. The results of the wave profiles are shown not only in the fully nonlinear subdomain but also in the nonlinear-to-linear transition buffer subdomain in order to show the entire computed numerical results by the present numerical scheme. However, the computed profiles in the neighborhood of the nonlinear-to-linear transition buffer subdomain should be discarded since we introduced an artificial linearization for a smooth matching in the region.



93

The computations are made for several values of Froude numbers and the depths of submergence. Figure 3.4 shows the result for a Froude number of 0.989 with a depth of submergence, ch = 0.2. The pressure distributions on the hydrofoil are shown in Fig. 3.4(a); there is remarkable agreement with the experimental results compared with the linear result. In Fig. 3.4(b) the nonlinear and the linear wave profiles are also compared. According to Parkin et al., the wave profile at this relatively high Froude number has a smooth shape. In the numerical computations, the convergence of the iteration process is fairly good.

Figure 3.4(a): Pressure distribution ( 2.0/,6/,989.0 === chcHFc ).

3.3 Axi-symmetric transient problem In this section the time-dependent motion of a fluid in a vertical circular cylindrical container with a free surface subject to an impact is discussed. The free surface abruptly rises to a very high level just after this impact. Therefore it seems to be necessary to apply the nonlinear free-surface condition for the problem. This problem is originally investigated by Milgram (1969), who performed a series of experiments and linear analysis for a vertical impact due to



94

the collision between a free-falling container filled with a fluid and the ground. The fluid is assumed to be inviscid and incompressible with its motion being irrotational. However, the surface tension is included since it plays an essential role in the unique restoring force during the free fall of the container. The free-surface shape at the moment of impact is assumed to be in a statically equilibrium state. Now the equilibrium free-surface elevation is used as the initial condition for the computation of the fluid motion after the impact. Throughout the computations, the hydrodynamic forces acting on the container wall is neglected for the motion of the container itself. In other words, the motion of the container itself is prescribed throughout the computation.

Figure 3.4(b): Wave profiles on free surface

( 0.989, / 6, / 0.2,buffer 15 19cF H c h c= = = = − ).

In the present computations we introduced an automatic mesh generation at some appropriate time to trace the free surface that changes very much. We also introduced splines along which the nodes at the free surface moves up and down. By this method we can represent a multi-valued surface elevation caused by the surface-tension force.



95

3.3.1 Physical model and assumptions

v = 0v = - g.

h

t = t 0 t = 0 t > 0

2g hv = - v = 0

Figure 3.5: Perfect plastic collision of the container.

If a circular container filled with water or some other fluid falls down on the ground as in Fig. 3.5, the fluid begins to move abruptly due to the impact and accordingly the free surface starts its motion. For a mathematical model of this physical problem we introduce the following assumptions. The fluid is assumed to be inviscid and incompressible and the motion of the fluid is assumed to be irrotational. A perfect inelastic collision between the container and the ground is also assumed, i.e., the container is stationary after the collision. If the container begins to fall at the height h with its initial velocity zero, its speed just before the collision will be gh2 , where g is the gravitational acceleration. We assume that the shape of a free surface before the collision is in equilibrium with the surface-tension force for simplicity, since the domain is axisymmetric, it is convenient to use the cylindrical coordinate system. The body-fixed coordinate system Orz is defined in Fig. 3.6. The z-axis points upward opposite to the gravity force at the center of the container and the r-axis is attached to the bottom of it. The radius of the container is R, and the contact angle between the solid wall and the free surface is 0θ , which is assumed to be constant. Because the solution will be independent of θ , the numerical calculation may be performed on the half-plane ( 0>r ) of the vertical cross section D of the container. The unit normal vector n

r points outward on the boundary and τ

r

denotes the unit tangential vector.



96

D

0

z

rR

S W

S F

n

0

Figure 3.6: Definition sketch.

3.3.2 Mathematical model The velocity field of an inviscid and incompressible fluid can be described by a scalar field if the motion of the fluid is irrotational. The scalar function, which is the velocity potential function φ in this case, must satisfy the Laplace equation at all points in the domain D ,

02 =∇ φ in domain D . (3.20)

Because the normal velocities on the solid wall, WS , vanish, we can obtain the

following condition,

0=∂∂

nφ on solid wall, WS . (3.21)

The kinematic condition of the free surface is as follows,

nVn=

∂∂φ on free surface FS , (3.22)



97

where nV is the normal velocity of the free surface. The dynamic condition on the free surface can be obtained from the Bernoulli equation.

( )

+−=++∇⋅∇+

∂∂

21

111)(21

RRPztvg

t a σρ

φφφ& on free surface, FS . (3.23)

Here, aP is an atmospheric pressure on the surface, ρ is the density of the fluid and σ is the surface-tension coefficient. 1R and 2R are principle radii of curvatures of the free surface. If we suppose that the fluid is in equilibrium just before the impact, the velocity potential at the free surface may arbitrarily be set to zero. The velocity potential at the free surface just after the impact can be obtained by integrating eqn (3.23) over the infinitesimal time interval during the impact, i.e.,

∫+

=−

+ −=0

00 )( zdttvt

&φ zgh2−= on FS , (3.24)

because the physical quantities, other than the vertical velocity of the container, are continuous at t=0. 3.3.3 Method of solution The initial- and boundary-value problem given in eqns (3.20) through (3.24) is a free-boundary problem. For a given time t , the shape of the fluid domain as well as the velocity potential given in the domain should be obtained. In this problem the free-surface elevation and the velocity potential on it are the canonical variable for the time evolution (Mi1es, 1977). The surface elevation and the velocity potential on it are evolved in time by the free-surface conditions, eqns (3.22) and (3.23). Then the remaining boundary conditions and the governing equation constitute a boundary-value problem from which the velocity potential in the fluid domain can be obtained. For a numerical computation, as a result, it is important to specify how to represent the free surface and to trace the velocity potential on it. In this section two methods representing the free surface are presented. One is for a single-valued free surface and the other is for a multi-valued free surface. Since the first is a special case of the second, we present the general case of the mu1ti-valued free surface. For a given time 0 ,t t= the position of the free surface can be represented by a curve 0( , ) ( ),r z X s=

r where s is

a parameter such as nodal number. For an interval of time, say ),( 10 tt , we



98

specify the direction of the surface evolution as )(ser

. Then the position vector of the free surface at time t is given as

),(),( tsXzrr

= )(),()(0 setshsX +=r

, 10 ttt << , (3.25)

so that the velocity of the free surface can be represented by a scalar function ),( tsh . We also define the velocity potential on the free surface as

( )ttsXts ),,(),( φφ = . (3.26)

Then the evolution equations of the two canonical variable ),( tsh and ),( tsφ

are given as

neht ∂

∂=

φ1 (3.27.a)

gzGRR

Pt

a −+

++−=

∂∂

2111

21ρσ

ρφ , (3.27.b)

where

neenrr⋅= , ττ

rr⋅= ee (3.28.a)

( )2

2 2

∂∂

−∂∂

+=τφ

τφ

τehehG tnt . (3.28.b)

In the case of the multi-valued free surface, the direction vector )(ser

is taken as the unit normal vector at 0tt = .

For a single-valued tee surface )(ser

is taken simply as the unit upward vector. The velocity potential in the fluid domain is determined from a boundary -value problem,

02 =∇ φ in domain D (3.29.a)

φφ = on free surface FS (3.29.b)



99

0=∂∂

nφ on solid wall WS . (3.29.c)

3.3.3.1 Weak formulation The boundary value problem given in eqns (3.29a,b,c) can be equivalently formulated by the following variational problem.

Find a function Hzr ∈),(φ such that

∫∫ =∇⋅∇D

drdzr 0φψ (3.30.a)

for all test function ( , )r z Hψ ∈ , with constraints

φφ = and 0=ψ on FS . (3.30.b)

Here, H is the tiral and test function space whose elements have square integrable functions and their derivatives in D. The normal derivative of the velocity potential on the free surface can be calculated by the right-hand side of eqn (3.31) by substituting the weak solution for eqn (3.30),

∫ ∫∫ ∇⋅∇=∂∂

FS D

drdzrdlrn

φψφ

ψ (3.31)

for all Hzr ∈),(ψ . It should be noted that the present boundary-value problem might also be

replaced by an equivalent form by a well-known classical quadratic variational functional that is very similar to the present weak formulation. 3.3.3.2 Discretization The foregoing variational problem is approximated by the finite-element method. The fluid domain is approximated by a polygon with a finite number of vortices. On the free surface, the direction vectors ( )e sr are defined piece-wise as shown in Fig. 3.7. Then the domain is discretized by triangular meshes. For a single-valued free surface, the meshes can be simply generated as described in Bai et al. (1989). In the case of a multi-valued free surface, an automatic mesh-generation scheme is developed here. This scheme will be described in the next section.



100

0

ni- 1 n iei- 1

e i

e i+1

i

i- 1

Xi+1B

XiB

Xi- 1B

Figure 3.7: Discretization of the free surface.

To solve the evolution equations, (3.27.a) and (3.27.b), it is required to evaluate

nφ∂∂

from the given values of the velocity potential on the free surface. The

normal derivative of the velocity potential can be obtained from the following procedures. For a given fluid domain at a fixed time, t , the approximate solution may be expressed as a linear combination of the finite-element basis functions, i.e.,

∑=

=NN

jjj zrNzr

1

),(),( φφ , (3.32)

where NN is the number of nodes of the finite elements and { }Nj NjzrN ,,1),,( L= is a piece-wise bilinear basis function whose nodal values are unity at node j and zero at the other nodes. If we make the numbering of the nodes such that the first FN nodes are on the free surface, the test functions are given as

NFii NNizrNzr ,,1),,(),( K+==ψ . (3.33)

Substituting eqns (3.32) and (3.33) into the variational problem, eqns (3.30.a,b), we obtain the following algebraic equations.



101

φφ = , FNi ,,1 K= (3.34.a)

∑=

=NN

jjijK

1

0φ , NF NNi ,1+= , (3.34.b)

where

∫∫ ∇⋅∇=D

jiij drdzrNNK ,

and { }FNi ,,1, K=φ is the nodal values of φ , obtained from the previous time step. Then we approximate the normal derivative of the velocity potential on the free surface as

∑=

=∂∂ FN

jjj zrN

n 1

),(ϕφ on FS . (3.35)

Then the variational equation, (3.31), can be written as

∑=

=FN

jFjij NiM

1

,,1, Kϕ ; ∫=FS

Jiij drdzrNNM . (3.36)

The two principal radii of curvature, which are required to evaluate the surface-tension terms given in eqn (3.8.b), are approximated as follows. The first curvature 1/1 R at the i-th node of the discrete free surface can be approximated as

)ˆˆ(211

11−

−−

+×=

ii

ii llsign

Rττ , (3.37)

where il is the length of the i-th element at the discrete free surface and iτ̂ is the unit tangential vector of the i-th free surface element. The sign convention of the curvature is defined in Fig. 3.8.



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+

z

-

z

Figure 3.8: Sign convention of the curvature.

The second curvature 2/1 R can be expressed by using the r-component of

the unit normal vector.

rn

Rr−=

2

1. (3.38)

3.3.4 Automatic mesh generation

3.3.4.1 Dynamic circular linked list Data read from a file or generated from a program at run time can be stored in dynamically allocated memory to form a data structure like a circular ring as in Fig. 3.9. This data structure has many advantages to the commonly used static array. It is easy to insert or delete a data item to or from this data structure. Even if we do not know the number of the data item, we will not have any difficulty to form this data structure. These features are applied to represent the boundary of the discrete domain. If we make a domain discrete, the boundary of this domain may be represented by a polygon. The vertices of this polygon are stored in the dynamically linked circular list.



103

Next Prior

DataItem

Next Prior

DataItem

Next Prior

DataItem

Prior Next

DataItem

Prior Next

DataItem

Prior Next

DataItem

Figure 3.9: Data structure for mesh generation.

3.3.4.2 Finite element mesh generation algorithm Figure 3.10 is a polygon that constitutes a discrete domain. If we connect two vertices of that polygon or introduce a new point in the domain, we can make a triangle as in Fig. 3.11. Repetitions of this procedure may fill the domain with triangles.

Figure 3.10: Discrete domain before triangulation.



104

Figure 3.11: Triangulation of the fluid domain.

3.3.5 Numerical results and discussions Throughout the present computations, the density of the fluid is 998 (kg/m3), the surface-tension coefficient is 0.0728 (N/m), the gravitational acceleration 9.81 (m/s2) and the contact angle between the free surface and the wall is 30°. All variables are nondimensionalized and all results are illustrated in dimensionless values. The length is nondimensionalized by the radius R , time by gR / , the velocity potential by gRR . Figure 3.12a-d show the results when the radius is 0.5 (m), the height is 0.05 (m) and single-valued elevation is assumed. Figure 3.13a-c show the results of the multi-valued surface-elevation model when the radius is 0.5 (m) and the falling height is 0.1 (m). At the center, the surface elevation is curved by the effect of surface tension. This case was also tested by using the free-surface elevation by a single-valued interpolation function and the results showed some numerical error as the slope of the wave elevation becomes very large. This error is presumably due to the fact that the true solution is a multi-valued function for the wave elevation in this case.

The computed results showed an abrupt rise of the surface elevation at the center that was observed in the experiments by Milgram (1969). Especially when either the radius of the container is small or the falling height is large, the free surface becomes multi-valued due to the effect of the surface tension.

If the impact is very large, we can observe that the fluid is divided into many drops in the experiment. The method used here cannot simulate this extreme case.



105

(a) t=0.0, 0.1,···, 1.0 (b) t=1.0, 0.3, 1.9

(c) t=1.9, 2.5, 2.8 (d) t=2.8, 3.0, 3.2, 3.4

Figure 3.12: Free-surface elevation.



106

(a) t=0.0, 0.2,…, 1.0 (b) t=1.0, 1.3, 2.0

(c) t=2.0, 2.4

Figure 3.13: Free-surface elevation.



107

3.4 Numerical towing-tank simulation In this section we describe a finite-element method applied to a nonlinear free-surface flow problem for a ship moving in three dimensions. The physical model is taken to simulate the towing-tank experimental conditions. The exact nonlinear free-surface flow problem formulated by an initial/boundary value problem is replaced by an equivalent weak formulation. This problem has been considered by Bai et al. (1989, 2002).

The model is assumed to have vertical wall sides and stretches from the free surface to the bottom for simplicity. Specifically, the model has a wedge-shaped bow and stern with a parallel middle body. For this model, the well-known generation of the solitons in the upstream and smooth waves in the downstream is simulated simply by taking the depth of water to be shallow. Also treated is a transom-stern ship model. This method can treat arbitrary water depth and practical ship geometries. Thus the present method is not restricted to the shallow-water problem or a special ship model geometry stretched from the free surface to the bottom, even though the computations are made for a simple model in shallow water in this section. 3.4.1 Mathematical formulation To make our formulation a scale-independent form, we nondimensionalized all physical variables by ghhh /,, 3ρ for length, mass and time, respectively. After nondimensionalization, the governing equation and the boundary conditions can be written as

0t),x(2 =∇r

φ in D (3.39)

xhn nF−=φ on oS (3.40)

nz

xht nF φζζ 1

+−= on FS (3.41)

pF xht −−∇−−= ζφφφ 2

21 on FS (3.42)



108

0=nφ on z = -1 (3.43)

0=nφ on By ±= , (3.44)

where gh

UhF = is the depth Froude number, 221/1 yxzn ζζ ++= is defined

on the free surface and all variables are redefined as nondimensionalized ones. 3.4.2 Variational formulation We introduce a variational formulation that is equivalent to the above problem. First, we define the variational functional, J and the Lagrangian L as

∫=*

0

tLdtJ (3.45)

dVdSdSLDSS

tFF

22

21

21

∫∫∫∫∫∫∫ ∇−−= φζφζ , (3.46)

where FS is the projection of FS on Oxy plane and *t is the final time. Taking

the variations on J first with respect to ζ , we can obtain ζδ J as

∫ ∫∫

∇−−=

*

0

2

21t

St dSdtJ

F

δζφζδζφδζδ ζ

[ ] [ ] dStS ttF

0* == −= ∫∫ φδζφδζ (3.47)

∫ ∫∫

+∇+−

*

0

2

21t

St dSdt

F

δζζφφ .

Next the variations on J with respect toφ , φδ J can be obtained as

∫ ∫∫∫∫∫

∇⋅∇−=

*

0

t

DSt dVdSdtJ

F

δφφδφζδ φ (3.48)



109

∫ ∫∫ ∫∫∫

∇+

−=

*

0

21t

S Dn

zt dVdS

ndt

F

δφφδφφζ .

Here φζ δδδ JJJ += .

Equation (3.47) means that the stationary condition on J for the variation with respect to ζ recovers the dynamic free-surface condition in each time step and that the wave elevation at t=0, *t should be specified as the constraints. Equation (3.48) shows that the stationary condition on J for the variation of φ recovers the kinematic condition on FS and the governing equation.

The above variational form was previously given by Miles (1977) and is slightly different from that given by Luke (1967). In the present variational formulation the wave elevation ζ is assumed to be known at t=0, *t , whereas Luke assumed the potential φ to be known at both initial and final times additionally. The present functional has an advantage over the original Luke’s variational functional in treating the nonlinear free-surface boundary conditions. 3.4.3 Finite element discretization In the original initial-boundary value problem, the admissible solution should be twice continuously differentiable. But in the above variational method, it is sufficient that the admissible trial functions have the square integrable properties of the function φ and its first derivative in space. This enables us to seek an approximate solution in a wider class in the variational method.

As the first step in the numerical procedure, the fluid domain is discretized into a finite number of finite elements. In this study, the finite elements are generated such that projections of x and y coordinates on the horizontal plane are fixed but the other coordinate, i.e. the z-axis, is allowed to move vertically in time. This restriction makes the regridding and computating considerably simpler. But, it is not always necessary to impose this restriction in general. The trial basis is denoted by { } NiiN ,...,1= and ζ is approximated by the span of the restrictions of { } NiiN ,...,1= on FS , which is also continuous and piecewise differentiable on FS .

);,,()(),,,( ζφφ zyxNttzyx ii= (3.49)



110

),()(),,( yxMttyx kkζζ = , (3.50)

where

ζζ

==

zik zyxNyxMk

);,,(),( , FNk ,...,1= , (3.51)

and FN is the number of nodal points on FS and ki is the nodal number of the basis function iN of which the node coincides with that of the free-surface node k. Summation conventions for the repeated indices are used here. It should be noted that the basis function,{ } NiiN ,...,1= , is dependent on the free-surface shape, z= ),,( tyxζ , but its restriction on FS is a function of (x,y) and independent of ζ . The special property of { }

FNkkM ,...,1= results from the finite-element subdivision employed here. Once the trial function is approximated by using the above basis function, the Lagrangian, L, for these trial solutions is obtained as

jkjkjijijkji PKTLk

ζζφφζφ21

21

−−= & (3.52)

∫∫=FS

jkkj dSMMT (3.53a)

∫∫=FS

jkkj dSMMP (3.53b)

∫∫∫ ∇⋅∇=D

jiij dVNNK . (3.53c)

The tensors, kjK , kjP are the kinetic and potential energy tensors and kjT is a tensor obtained from the free-surface integral, which can be interpreted as a tensor related to the transfer rate between these two energies. It is of interest to note that in eqn (3.53), kjkj PT = . The stationary condition on ∫= LdtJ gives the following Euler–Lagrange equation.

jjijkj kKT φζ =& , (3.54)

jkjjk

ijiikj P

KT

jζφ

ζφφ −

∂

∂−=

21& , for k=1,…, FN , (3.55)

0=jijK φ , kii ≠ . (3.56)



111

Here, eqns (3.52) and (3.53) are the nonlinear ordinary differential equation for { }

Fk Nkik ,...,1,

=φζ and eqn (3.54) is the algebraic equations for { }

kiii ≠φ , which is the constraint for the above two equations. It can be easily shown that the solution of the above discretized problem satisfies the conservation of mass and total energy, i.e.

0,

=

∑

jkjkjP

dtd ζ , (3.57)

0,,

=

+∑∑

jkjkjk

jijiji PK

dtd ζζφφ . (3.58)

This property of the conservations is independent of the tensor ijT if it satisfies

∑∑ =k

kjk

kj PT . (3.59)

It should also be pointed out that the direct use of eqn (3.55) leads to some difficulty in the computations. This difficulty arises from the first term in the right-hand side that is the derivative of the kinetic energy tensor with respect to the wave elevation. We have avoided this difficulty by utilizing the fact that eqn (3.55) is equivalent to the condition of vanishing of the right-hand side in eqn (3.47). In eqn (3.47), ζδ can be regarded as test functions on FS . Then eqn (3.55) can be given as

∫∫∫∫ ∇−−=FF

k Sj

Stzjijk dSMdSMT 2

21 φζφφ& kjkP ζ− . (3.60)

And in eqn (3.48), φδ can also be regarded as test functions on FS . Therefore

eqn (3.54) can be given as

∫∫=F

k Sznjijk dSnMT /φζ& . (3.61)

Consequently, the Laplace equation in eqn (3.56) is no more than a constraint. In order to make the first derivative terms of φ written as terms on the free surface, we utilized the same relations as Zakharov (1978). Then our system becomes the Hamiltonian,



112

tztt ttyxyxtzyx ζφζφφ += )),,,(,,(),,,( , (3.62a)

xzxx ttyxyxtzyx ζφζφφ += )),,,(,,(),,,( , (3.62b)

yzyy ttyxyxtzyx ζφζφφ += )),,,(,,(),,,( . (3.62c)

Using the above relation, we can rewrite eqn (3.60) as

∫∫

∇⋅∇−

∇⋅∇+=

Fk S

ssssz

nzjijk dS

nnMT φφζφ

φφ

22

21&

kjkP ζ− , (3.63)

where

∂∂

∂∂

=∇yxs , and 221/1 yxzn ζζ ++= .

If the integrals in eqns (3.55) and (3.63) and the integrals in eqns (3.54) and (3.61) are evaluated exactly, they are equivalent. However, in the present computation these integrals are calculated by integral quadrature rules. Therefore, the conservation of energy may not be satisfied exactly due to the error caused by the numerical integration. This test result is given in the next section. 3.4.3.1 Numerical instability In nearly all time-dependent nonlinear free-surface flow calculations, the numerical instability can be observed after a sufficiently long time. This instability is similar to a saw-toothed shape alternatively above and below a smooth curve. Longuet-Higgins and Cokelet (1976) discussed that the growth may be partly physical, being similar to the growth of short gravity waves by horizontal compression of the crests of longer waves. In reality, these instabilities are partly damped by viscosity, which we have neglected.

In order to remove the instabilities, Bai et al. (1989) used the lumping and upwinding schemes in the convective term occurring in the case of using the moving coordinate on the free-surface boundary conditions and maintained the numerical stability in the integration with respect to time. But they experienced irregular free-surface shapes in the downstream. There was an additional



113

difficulty in choosing an appropriate value of the upwinding factor for a general nonlinear free-surface problem because the factor is determined from the linear dispersion relation.

As an alternative, a method of smoothing out this instability was introduced by Longuett-Higgins and Cokelet(1976). They were able to remove the instabilities effectively by using the 5-point Chebychev filtering method. In the present chapter, we adopted this smoothing scheme and obtained good results. We used the following filtering function:

kxxn

nk xaxaxaxaaxxf =+++++== |)()( 33

2210 L

kxxn

nk xbxbxbxbb =

−−+++++−+ |)()1( 1

13

3210 L , k=j, j±1,…, j±n,.

(3.64) where the coefficients can be obtained from 2n+1 points. We used the 5-point filtering scheme as follows:

( )21022

22102 ++++ ++++= jjjj xbbxaxaaf δδδ (3.65-a)

( )11012

21101 ++++ +−++= jjjj xbbxaxaaf δδδ (3.65-b)

00 baf j += (3.65-c)

( )11012

21101 −−−− +−++= jjjj xbbxaxaaf δδδ (3.65-d)

( )21022

22102 −−−− ++++= jjjj xbbxaxaaf δδδ (3.65-e)

)( jj xff = , jkjkj xxx −= ++δ (k=-2,-1,0,1,2) . (3.65-f)

Then the filtered value at the j-th point, jf̂ , can be obtained from the above

equations as



114

( ) ( )( )

( ) ( )( )

1 1 1 10

1 1

2 2 2 21 1

2 2

ˆ2

/ /.

2

j j j j j jj

j j

j j j j j jj j

j j

x f f x f ff a

x x

f f x f f xx x

x x

δ δ

δ δ

δ δδ δ

δ δ

+ + − +

+ −

+ + − −+ −

+ −

+ − += =

−

− − −+

−

(3.66)

If a constant grid spacing is taken, i.e., hxxx jjj ≡−= ++ 11δ , then eqn (3.66)

reduces to that used in Longuet-Higgins and Cokelet (1976) as

( )2112 4104161ˆ

++−− −+++−= jjjjjj ffffff . (3.67)

3.4.3.2 Time integration on the free surface Once we discretize the computational domain into a number of finite elements and perform integrations in terms of three space variables, we obtain a set of ordinary differential equations in matrix form given as follows,

jz

nkjjkjhjkj n

TCFT φζζ ζ +−=& , (3.68)

jkjjikijjkjhikj PCCFTj

ζζφφφ φ −+−= ),(~21& , (3.69)

ijij fK −=φ , (3.70)

where the coefficients are defined as

∫∫ ∂∂=

FSjxkkj dSMMCζ , (3.71a)

∫∫ ∂∂=

FSjxkkj dSNMCφ , (3.71b)

∫∫

∇⋅∇+=

FSss

z

nzkjikij n

nMC2

2),(~ζφ

φζφ dSss

∇⋅∇− φφ , (3.71c)



115

∫∫=0S

xihi dSnNFf . (3.71d)

It should be pointed out that eqn (3.70) can be interpreted as a constraint to eqns (3.68) and (3.69). Equation (3.44) is obtained from the boundary-value problem with an essential condition (Dirichlet type) on the free surface and a natural condition (Neumann type) on the body surface. In the solution procedure, the constraint, i.e., eqn (3.44) is first solved by the GMRES method. Then the matrix, kjT , is inverted by the same method. Here, the other matrices, which are dependent on the free-surface shape, are treated as known values from the previous time step. The final form in eqns (3.68) and (3.69) is solved by the fourth-order Runge-Kutta method with minimum truncation error. 3.4.4 Results and discussions

0 20 40 60

-0.5

0

0.5

1

1.5Energy Conservation

PEKEPE+KE

Figure 3.14: Numerical test of energy conservation,

(K : kinetic energy; P: potential energy; E=K+P: total energy). The numerical results in this section are taken from Bai et al. (2002). The computations are made to test the conservation of energy in the case of two-

K.E

., P

.E., T

.E.



116

dimensional free oscillations and the test result is given in Fig. 3.14. In the computations the length of the tank is taken to be 40 with the water depth being 1. The length of the half-cosine hump is 5 and its height 0.5. The final time is 60 with the time step of 0.05. Figure 3.14 shows that the total energy is conserved fairly well throughout the computations.

The first computed model is shown in Fig. 3.15. The model is a vertical wall-sided wedge-shaped bow and stern extended from the free surface to the bottom of the tank. The length of the wedge ship, the length of the parallel middle body, and the beam are denoted by L , mL , and 2b. The tank has a width of 2B and a mean water depth of h. The ship is assumed to move with a constant velocity of –U along the centerline of the tank.

2b

y = B

y = - B

Lm

L

x

y

Figure 3.15: Sketch of ship model and tank in horizontal plane view. The z-axis

is against the gravity.

In presenting our computed results, all the physical quantities are

nondimensionalized by ghhh /,, 3ρ for length, mass and time, respectively, as mentioned previously.

Wedge model : L = 8, b = 0.4 , Lm = 4 Computation domain : x = (-30,30), y=(-4,4) Finite-element meshes : 150×10×1 elements Mesh sizes : ∆x = 0.4,∆y = 0.4, ∆t = 0.1

t



117

Figures 3.16 (a) and (b) show the results of wave profiles as time increases with the step of 20 s in a nondimensionalized form for 9.0=hF . Figures 3.17(a) and (b) show the results of wave profiles for 0.1=hF .

-10

1

Z

-30

0

30

X

-4

0

4

Y

(a) After 80 s

-10

1

Z

-30

0

30

X

-4

0

4

Y

(b) After 100 s

Figure 3.16: Computed free-surface elevation for 9.0=hF .



118

-1

01

Z

-30

0

30

X

-4

0

4

Y

(a) After 80 s

-10

1

Z-30

0

30

X

-4

0

4

Y

(b) After 100 s

Figure 3.17: Computed free-surface elevation for 0.1=hF .



119

0 20 40 60 80 100

t

0

0.2

0.4

0.6

0.8

Rw

0 20 40 60 80 100

Rw(Wave Resistace)Present Fn = 0.9Present Fn = 1.0Present Fn = 1.1Kim&Bai(1989) Fn = 0.9Kim&Bai(1989) Fn = 1.0

0 20 40 60 80 100

Figure 3.18: Comparisons of the wave resistance for different Froude numbers (B=4,b=0.4).

Figure 3.18 shows the comparisons of the computed results of wave

resistance for 9.0=hF and 0.1=hF by Bai et al. (1989) and Bai et al (2002) where substantial modifications are made for a transom-stern ship model. The comparison is good in the wave resistance as shown in Fig. 3.18. It can be interpreted that the previously computed wave resistance of Bai et al. (1989) was not contaminated by the irregular wave profiles in the downstream region. The second computational model with a transom stern is shown in Fig. 3.19. This model also has a vertical-wall-sided wedge-shaped bow and a parallel middle body extending from the free surface to the bottom of the tank. The length of the wedge, the length of the parallel middle body, and the beam are denoted by L , mL , and 2b, respectively. The tank geometry is the same as the first model.

Transom model: L = 30, Lm = 25, B=30 Computation domain: x = (-60,60), y=(-30,30) Mesh sizes: ∆x =0.5,∆y =0.5,∆t = 0.01



120

2b

y = B

y = - B

Lm

L

x

y

Figure 3.19: Sketch of transom-stern ship model and tank in horizontal plane view. The z-axis is against the gravity.

-30

-15

0

15

30

Y

-60 -30 0 30 60

X

-1

0

1

Z

-60

-30

0

30

60

X

-30

0

30

Y

Figure 3.20: Contour plot and wave profile at nF =1.8.



121

For the transom-stern ship model many cases are computed for various values of Froude numbers. More details on the numerical condition for the dry bottom can be found in Bai et al. (2002). Figure 3.20 shows the wave contour and wave profiles at 8.1=nF for the case of 2b=15, 2B=60, L=30, and Lm = 25.

The figures show approximately triangular shapes of a dry bottom generated behind the transom-stern models, which were also observed in the experiments. This numerical method can be used as an efficient method to treat nonlinear free-surface flow problems. Even though the sample computations are made for rather special and simple cases, i.e., a ship in a shallow-water tank, the present method can accommodate arbitrary water depth (i.e., tank depth), tank width, and ship geometry even for a transom-stern model. This method can also be applied to the sloshing problem, the local flow around the sail of a submarine, and the dam-breaking problem. 3.5. Nonlinear sloshing problem In this section a nonlinear sloshing problem in a LNG tanker is numerically simulated. During excessive sloshing, the sloshing-induced impact load can cause critical damage on the tank structure. Recently, this problem became one of the important issues in FPSO design. A three-dimensional free-surface flow in a tank is formulated in the scope of the potential flow theory. The mathematical formulation of this nonlinear sloshing problem is the same as in the previous section, i.e., in Section 3.4, for the numerical towing-tank simulation, except the Neumann condition on the tank ends is prescribed here. Since all the numerical procedures are the same as in the previous section, we leave out the entire numerical procedures such as variational formulation, finite-element approximation, time integration, etc. Here we only give the numerical results. The hydrodynamic loading on the pillar in the tank is computed and compared with other results. More details and numerical results can be found in Kyoung et al. (2003). 3.5.1 Numerical results Figures 3.21 through 3.23 show the time history of the impact forces. The force acting on the pillar is non-dimensionalized by SU 22/1 ρ . ρ is the density of the fluid. The velocity ,U is the critical speed of the bore, which is gh in linear



122

shallow-water theory. S is the projected area, namely hD. The maximum values of non-dimensionalized impact loads lies between 1.0 and 1.4 for the present computed cases. The wave loads increase as the water depth and the height of the bore increase in the cases of the ratio of water depth to the length of the tank being relatively low. Figure 3.22 shows the wave profiles at the instant of the maximum impact load. Impact loads on the pillar located in the middle of the tank have maximum value when the wave crest hits the pillar.

0 12.978 25.956 38.934 51.912Time(sec)

-1.5

-1

-0.5

0

0.5

1

1.5

Figure 3.21: Impact force on the pillar versus time: h= 4 m.

-2

-1

0

1

2

Z

-20

-10

0

10

20

X

-5

0

5

Y

Figure 3.22: Wave profile for 4m depth at t=37.084 s.

SU

Fx

2

21 ρ



123

The wave amplitude computed here is roughly proportional to the water depth of the tank. As concluding remarks, a series of numerical computations are made for the impact load on the pillar in the middle of the shallow filling tank. A very steep bore is observed and it makes the maximum impact load on the pillar. The maximum wave load occurs when the wave crest hits the pillar and is proportional to the water depth (filling depth) squared, approximately.

1 2 3 4Water Depth (Meter)

0

50

100

150

200

250

300

Fx

Fitted ForceOriginal DataFitted Curve

Figure 3.23: Impact forces versus the water depth.

3.6 Nonlinear diffraction problem In this section, the diffraction of highly nonlinear Stokes waves by a vertical circular cylinder is numerically simulated in the time domain. The Stokes waves, input at the numerical wave maker, are obtained numerically from the two-dimensional steady solution of the finite-element method. A new matching scheme is developed to match the two-dimensional wave at the far field and the three-dimensional diffracted wave in the vicinity of the cylinder. As a numerical example, the diffraction of Stokes waves with various steepnesses by a circular cylinder is tested. The wave elevation and run-up on the cylinder are calculated and compared with the available theoretical results. More applications to multiple cylinders can be found in Kim et al. (2003).



124

3.6.1 Mathematical formulation We consider the free-surface flow of an inviscid and incompressible fluid, in water of uniform depth, h. The coordinate system is chosen such that the z-axis is directed against gravity and the Oxy-plane coincides with the still-water level. The location of the free surface is denoted by ( )tyxz ,,ζ= and the bottom as

hz −= , where h is a constant. The fluid domain is denoted by D, and some boundaries of the fluid domain, i.e., the free surface, sea bottom and vertical cylinder, are denoted by SF, SB and S0, respectively. The fluid domain is unbounded in the horizontal plane. However, the computational domain is desired to be as small as possible without losing numerical accuracy.

D

x

y

SR

SD

SfR

SR

SR

SR

(a) Plan view

x

y

z

SF

SBZ=- h

Z = (x,y,t)ζ

SD

(b) Perspective view

Figure 3.24: Definition sketches of computational domain.



125

With this in mind we truncate the original infinite-fluid domain and we introduce an appropriate new numerical radiation boundary, SR, which will be discussed further later when we discuss a numerical example. Figure 3.24 shows a computational domain and sub-domains, which will be described later.

Hereafter, we nondimensionalize all the physical variables such as the mass density of the fluid, ,ρ gravitational acceleration, g, etc. The characteristic length, l, is taken as unity. The characteristic length may be the water depth, h, or the wavelength, ,λ depending on the particular problem, and this will be indicated accordingly. The governing equation for the velocity potential, ( )tzyx ,,,φ , in the fluid domain and the kinematic boundary condition on the

bottom and cylinder surface can be written as

02 =∇ φ in D. (3.72)

0=∂∂

zφ on z = -h (3.73)

0=∂∂

nφ on S0 , (3.74)

where n is the outward vector normal to the cylinder surface. On the free surface, Sf, the kinematic and dynamic conditions can be written as

yyxxzt ∂∂

∂∂

−∂∂

∂∂

−∂∂

=∂∂ ζφζφφζ on Sf , (3.75)

( ) 021

0 =−+∇⋅∇+∂∂ Rg

tζφφφ on Sf , (3.76)

where R0 is the Bernoulli constant. The above problem can be completed by specifying an initial condition and a radiation condition at infinity on the horizontal plane. We assume that the free-surface flow is caused by an incoming Stokes wave whose surface elevation is wζ and velocity potential is wφ . Then, at infinity, the following radiation condition should be satisfied:



126

∞→

+=

+= ras

rO

rO ww

1,1 ζζφφ , (3.77)

where 22 yxr +≡ . It should be noted that a different radiation condition should be used if the fluid domain is partially confined, such as in a wave tank. Since we cannot solve the problem numerically in an infinite-fluid domain, the radiation condition at infinity is replaced by numerical matching conditions at the finite truncated boundary, SR , i.e.,

, onw w RSφ φ ζ ζ= = , (3.78)

which is valid only when the minimum radial distance, r, from the cylinder to the radiation boundary, SR, is sufficiently large because the far-field propagating wave decays as ( )rO /1 . To accelerate the decay and to minimize the required distance of the radiation boundary, an artificial damping term is added in the free-surface conditions, eqns (3.75) and (3.76), in a transition buffer domain, SfR:

( )( )wyxyyxxzt

ζζµζφζφφζ−−

∂∂

∂∂

−∂∂

∂∂

−∂∂

=∂∂ , on SfR (3.79)

( ) ( )( )wyxRgt

φφµζφφφ ˆˆ,

21

0 −−−−∇⋅∇−=∂∂

on SfR, (3.80)

where ( )tyx ,,,ˆ ζφφ ≡ and ( )tyx www ,,,ˆ ζφφ ≡ . The damping function ( )yx,µ to be defined later varies from zero at the inner boundary of SfR to a constant value at the outer area of SfR. The far-field wave solution wζ can be any nonlinear transient wave elevation that will be specified as a basic flow input while ( )wζζ − is the diffracted wave elevation that radiates to infinity and decays. In the present application, the Stokes wave is adopted as nonlinear progressive waves of permanent form. The Stokes-wave solution has also been calculated here by the two-dimensional version of the present finite-element method.



127

3.6.2 Finite-element method Following Bai and Kim (1995), the initial-boundary value problem, given by eqns (3.72) through (3.80), is solved by a finite-element method based on Hamilton’s principle. The additional damping terms in the free-surface conditions are treated as a pressure and mass flux distribution on the free surface.

Since the fluid domain, and accordingly the surface elevation, changes in time, it is more convenient to transform the depth dependence, i.e., the z-coordinate by a new variable γ as

( ) ( ).,,,,,,, * tyxtzyxhhz

γφ=φ+ζ+

=γ (3.81)

In the transformed coordinate, the water depth is unity for all time. Then we can represent the trial solution as

( ) ( ) ( ) ( )∑∑= =

γφ=γφM

mm

N

iimi fyxNttyx

e

1 1

* ,,,, , (3.82)

where ( ) ( ) ( ) ( ){ }γγγγ Mffff ,...,,, 321 is the set of interpolation functions (also called the basis function, shape function, pyramid function, etc.) in the vertical direction, and ( ) ( ) ( ){ }yxNyxNyxN

eN ,,...,,,, 21 are the set of interpolation functions defined on the horizontal plane that is the projection of the computation domain. Here M and Ne denote the number of interpolation functions in the vertical direction and the horizontal plane, respectively. The free- surface elevation is also represented by

( ) ( ) ( )∑=

ζ=ζeN

iii yxNttyx

1

,,, . (3.83)

As shown by Bai & Kim (1995), the even-order polynomial function in the transformed depth coordinate provides an optimal interpolation when the water depth is uniform. Thus we use this interpolation function with a slight modification that is the introduction of the even-order polynomials by the Chebychev polynomials for better numerical stability, as follows:

( ){ }[ ] 1,cos12cos121;1 1

1 >γ−−== − mmff m . (3.84)



128

For the interpolation function in the xy-plane, a three-node triangular element is used. In the present computations, the triangular prism elements are used. Here one should note that the so-called h-version is adopted in the horizontal plane while the p-version in the finite-element method is used in the depth direction. Once the spatial discretization has been made by the number of finite-elements, the Laplace equation, (3.72), reduces to a system of algebraic equations for the velocity potential. An iterative scheme (Jacobi conjugate gradient method) is used to solve the system of equations. The matrix involved in the algebraic equations is highly sparse. Only the non-zero entries of the sparse matrix are stored. The free-surface conditions given by eqns (3.79) and (3.80) lead to time-evolution equations for the surface elevation and the velocity potential on the free surface. They are first-order differential equations and can be integrated, for example, by the Runge–Kutta (4th-order) method.

3.6.3 Diffraction by a vertical circular cylinder The run-up of the Stokes wave on a circular cylinder of radius a, is simulated here. The computational domain is rectangular, with a length of 24a and half-width of 8a, as shown in Fig 3.25. The projected area of the fluid domain on the xy-plane is subdivided into 8987 triangular elements and 4670 nodes as shown in Fig. 3.25. For the vertical interpolation the fourth-order polynomial, or M = 3 has been used in eqn (3.82).

x

y

-10 -5 0 5 100

2

4

6

8

Figure 3.25: Finite-element mesh. Coordinates are nondimensionalized by the cylinder radius, a.



129

The damping function is assumed as

( ) ( ) ( ) ( ) ( ){ }yxyxyx yxyx µµ−µ+µµ=µ 0, , (3.85)

where

( )

( )( )

( )( )

≤<−−π

≤<

≤<−−π

=µ

.,2

cos

;,0

;,2

cos

4343

42

32

2112

12

xxxxxxx

xxx

xxxxxxx

xx ( ) ( )

( )

≤<−−π

≤<=µ .,

2cos

;0,0

2121

221

yyyyyyy

yyyy

(3.86)

Specifically, the parameters in the above equation are taken as

1 2 3 4

1 2 0

14 , 10 , 6 , 10 ;4 , 8 ; 1.

x a x a x a x ay a y a µ= − = − = == = =

For the input Stokes wave, the following parameters have been used:

1.0, / 0.1,0.2,0.3,0.4,0.5.kh ka H h= = = The ratio between the wavelength, water

depth and the radius of the cylinder were taken from Ferrant (1998) for comparison purposes. For the input Stokes wave, the following parameters have

been used: 1.0, / 0.1,0.2,0.3,0.4,0.5.kh ka H h= = = The ratio between the

wavelength, water depth and the radius of the cylinder were taken from Ferrant (1998) for comparison purposes.

A quasi-steady state for the wave elevation has been reached after 5~6 periods in all cases of the simulated wave heights. Numerical simulation is performed for up to t/(h/g)1/2 = 100, which is equal to about 15 periods.

The comparison of the maximum run-up with the results of Ferrant (1998) is given in Table 3.1. Good agreement can be seen. Figure 3.26 shows the snapshots of the free-surface elevation for H/h = 0.5. Significant run-up and a steep rooster-tail can be observed at the weather and lee sides, respectively. The radiating waves generated by the cylinder are mostly higher-harmonic waves that have wavelengths and periods shorter than the incoming waves. These higher- harmonic waves interact with the incoming Stokes waves and generate the



130

instability of very short waves. In the present numerical scheme, the free-surface elevation is assumed to be single valued and no breaking waves are allowed. Presumably, the short-wave instability observed in the numerical result can be related to the spilling breaker observed in nature and experiments. Table 3.1: Maximum run-up compared with numerical results of Ferrant (1998).

H/h 0.1 0.2 0.3 0.4 0.5

F.E.M. 1.90 2.12 2.38 2.68 2.99 Ferrant (1998)

1.87 2.06 2.38 N/A N/A

Figure 3.26: Snapshots of surface elevation around a cylinder. 3.7 Closing remarks In this chapter the applications of the finite-element method to various nonlinear free-surface flow problems are described. The finite-element method has been proven to be a useful method of solution for highly nonlinear free-surface flow problems.



131

Acknowledgments This work was supported by a Grant (No.R01-2000-000-00321-0) from the Korea Science & Engineering Foundation, the Korea Research Foundation Grant. (KRF-2002-005-D00033), Research Institute of Marine Systems Engineering, College of Engineering, Seoul National University. We thank Dr. Jo-Hyun Kyoung for his assistance during the preparation of this manuscript. References [1] Bai, K.J., A Localized Finite-Element Method for Steady, Two-Dimensional

Free-Surface Flow Problems. Proc. First International Conference on Numerical Ship Hydrodynamics, Edited by Schot and Salvesen, David W. Taylor Naval Ship R&D Center, Bethesda, Maryland, U.S.A., 1975.

[2] Bai, K.J., A localized finite-element method for steady three-dimensional free-surface flow problems. Proc. 2nd Int. Conference on Numerical Ship Hydrodynamics, University of California, Berkeley, pp. 78-87, 1977.

[3] Bai, K.J., A localized finite-element method for two-dimensional steady potential flows with a free surface. Journal of Ship Research, 22(4), pp. 216-230, 1978.

[4] Bai, K.J., Wave Resistance in a Restricted Water by the Localized Finite Element Method. Proc. Workshop on Ship Wave Resistance Computations, David W. Taylor Naval R&D Center, Bethesda, Maryland, 1979.

[5] Bai, K.J., A localized Finite-Element Method for Three Dimensional Ship Motion Problems. The Third International Conference on Numerical Ship Hydrodynamics, Paris, France, June 1981.

[6] Bai, K.J., Kim, J.W. & Kim, Y.H., Numerical Computations for a Nonlinear Free Surface Flow Problem. Proc. 5th Int. Conf. on Numerical Ship Hydrodynamics, 24-28, Hiroshima, Japan, pp. 403-418, 1989.

[7] Bai, K.J. & Kim, J.W., A Localized Finite Element Method for Nonlinear Water Wave Problems. International Series of Numerical Mathematics, 106, pp. 67-74, 1989.

[8] Bai, K.J., Park, J.K. & Kim, J.W., Nonlinear Waves in a Cylindrical



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Container Following Vertical Impact. The Second Japan-Korea Joint Workshop on Ship and Marine Hydrodynamics, June 28-30, 1993, Osaka, Japan, 1993.

[9] Bai, K.J., Kim, J.W. & Lee, H.K., A Localized Finite Element method for nonlinear free-surface wave problems. Proceedings of the 19th Symposium on Naval Hydrodynamics, Hague, Washington, D.C., pp. 113-139, 1994.

[10] Bai, K.J. & Han, J.H., A Localized Finite-Element Method for the Nonlinear Steady Waves due to a Two-Dimensional Hydrofoil. Journal of Ship Research, 38(1), pp. 42-51, 1994.

[11] Bai, K.J., Kyoung, J.H. & Kim, J.W., A Numerical Computations for a Nonlinear Free-Surface Problem in Shallow Water. Journal of Offshore Mechanics and Arctic Engineering, 125(1), pp. 33-40, 2003.

[12] Choi, H.S., Bai, K.J., Kim, J.W. & Cho, I.H., Nonlinear Free Surface Waves Due to a Ship Moving Near the Critical Speed in a Shallow Water. Proc. 18th Symposium on Naval Hydrodynamics, Office of Naval Research, Aug. 1990, Ann Arbor, Mich. U.S.A., 1990.

[13] Ferrant, P., Run-up on a Cylinder due to Waves and Current: Potential Flow Solution with Fully Nonlinear Boundary Conditions. Proc. 8th ISOPE Conf., Montreal, Canada, 3, pp. 332-339, 1998.

[14] Ertekin, R.C., Webster, W.C. & Wehausen, J.V., Waves Caused by a Moving Disturbance in a Shallow Channel of Finite Width. Journal of Fluid Mech. 169, pp. 275-292, 1986.

[15] Kim, J.W. & Bai, K.J. A note on Hamilton’s principle for a free surface flow problem. Journal of Society of Naval Arch., Korea, 27(3), pp. 19-30, 1990.

[16] Kim, J.W. & Bai, K.J., A Finite-element Method for Two-Dimensional Water-Wave Problems. International Journal for Numerical Methods in Fluids. 30(1), pp. 105-121, 1999.

[17] Kim, J. W., Kyoung, J. H., Ertekin, R. C. & Bai, K. J. A Finite-Element Computation of Wave-Structure Interaction by Steep Stokes Waves. Submitted to Journal of Waterway, Port, Coastal, and Ocean Engineering, 2003.



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[18] Kyoung, J.H., Kim, J.W. & Bai, K.J., A Finite Element Method for the Nonlinear Sloshing Problem with a Bottom-Mounted Vertical Cylinder. OMAE, Cancun, Mexico, June 8-13, 2003.

[19] Longuet-Higgins,M.S. & Cokelet,E.D., The Deformation of Steep Surface Waves on Water, I. A Numerical Method of Computation. Proc. R. Soc. London Ser. A 350, pp. 1-26, 1976.

[20] Luke, J.C., A variational principle for a fluid with a free surface. Journal of Fluid. Mech. 27, pp. 395-397, 1967.

[21] Miles, J.W., On Hamilton’s Principle for Surface Waves. Journal of Fluid Mech., 83, pp. 153-158, 1977.

[22] Milgram, J.H., The Motion of a Fluid in a Cylindrical Container with a Free Surface Following Vertical Impact. Journal of F1uid Mech., 37(3), pp. 435-448, 1969.

[23] Parkin, B.R., Perry, B. & Wu, T.Y., Pressure distribution on a hydrofoil running near the water surface. Journal of Applied Physics, 27, pp. 232-240, 1956. (Also in California Institute of Technology, Hydrodynamics Laboratory Report No. 47-2, 1955).

[24] Swan, C., Taylor, P.H. & van Langen, H., Observation of wave-structure interaction for a multi-legged concrete platform. Applied Ocean Research, 19, pp. 309-327, 1997.

[25] Zakharov, V.E., Stability of Periodic Waves of Finite Amplitude on the Free Surface of a Deep Fluid. Journal of Appl. Mech. Tech. Phys., 9, 190, 1968.


chapter 3 fem on nonlinear free-surface flow

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