A CIP-Based Method for Numerical Simulations of Violent Free-surface Flowa - Hu

7/27/2019 A CIP-Based Method for Numerical Simulations of Violent Free-surface Flowa - Hu

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J Mar Sci Technol (2004) 9:143157

DOI 10.1007/s00773-004-0180-z

Original articles

A CIP-based method for numerical simulations of violentfree-surface flows

ChanghongHuand MasashiKashiwagi

Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga 816-8580, Japan

major difficulty in computing such wavebody interac-tion problems arises from their extremely complicatedhydrodynamic phenomena. For example, it is necessary

to treat highly distorted or broken free surfaces, and toconsider the effect of the compressibility of water or theelasticity of the body for the case of impacting water.Conventional numerical analysis methods, such as theboundary element method for potential flows or theCFD method using curvilinear grids adapted to botha free surface and a body surface, are not applicableto extremely nonlinear problems, although some effortsare seen to take into account wave breaking by suitablemodeling (see Muscari and Di Mascio1and Olivieri etal.2). Recently there were some papers in which freesurface was captured as a part of solution, thus beingcapable of computing much more complex free surfaces

than conventional surface-fitted methods. Park et al.3

used the MAC method to compute wave-current-bodyinteractions. Andrillon and Alessandrini4used the VOFmethod to compute bow breaking waves. But it is stilltrue that complex free surface phenomena still remainas a challenge to CFD. A new numerical method wasneeded. This new method should not only be able tohandle complicated phenomena associated with wavebody interactions, but should also be a relatively simplescheme which can perform three-dimensional simula-tions in an acceptable spatial and temporal resolution atreasonable cost.

Here, we propose a new CFD simulation approachfor extremely nonlinear free-surface problems. Thisapproach is a finite-difference method based on theconstrained interpolation profile (CIP) algorithm.5,6Thekey points of the CIP method can be summarized as(1) a compact upwind scheme with subcell resolutionfor the advection calculation, and (2) a pressure-basedalgorithm that can treat liquid, gas, and solid phases,irrespective of whether the flow is compressible or in-compressible, by solving one set of governing equations.The method for hydrodynamic problems using the

Abstract A CFD model is proposed for numerical simula-tions of extremely nonlinear free-surface flows such as waveimpact phenomena and violent wavebody interactions. Theconstrained interpolation profile (CIP) method is adopted as

the base scheme for the model. The wavebody interaction istreated as a multiphase problem, which has liquid (water), gas(air), and solid (wave-maker and floating body) phases. Theflow is represented by one set of governing equations, whichare solved numerically on a nonuniform, staggered Cartesiangrid by a finite-difference method. The free surface as well asthe body boundary are immersed in the computation domainand captured by different methods. In this article, the pro-posed numerical model is first described. Then to validatethe accuracy and demonstrate the capability, several two-dimensional numerical simulations are presented, andcompared with experiments and with computations by othernumerical methods. The numerical results show that the

present computation model is both robust and accurate forviolent free-surface flows.

Key words CIP method Violent free-surface flow Multi-phase computation

1 Introduction

In rough seas, ships or ocean structures may experiencehighly nonlinear phenomena such as slamming, wateron the deck, wave impact by green water, or capsizing

due to large-amplitude waves. Such extremely non-linear wavebody interactions sometimes cause local-ized structural damage, and a quantitatively preciseestimation method for wave loads is therefore neces-sary. To date, experiments have been the most practicalway to study these problems, and numerical simulationshave been restricted to a few very simple cases. The

Address correspondence to:C. Hu ([email protected])Received: October 27, 2003 / Accepted: May 11, 2004


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144 C. Hu and M. Kashiwagi: CIP method for violent free-surface flows

pressure-based algorithm is called the CIP combinedand unified procedure (CCUP) method.

The CIP method is a Eulerian approach on a regular,stationary Cartesian grid with multiphase computation.This has the multiple capabilities which we need,e.g., handling the complicated free-surface geometryand treating violently moving floating bodies. Further-

more, since no remeshing calculation is required, thecomputation time can usually be shortened for time-dependent problems. The Eulerian approach was firstproposed in the marker and cell (MAC) method7 nearly40years ago. Since then, this approach has been devel-oped into a number of methods with different interface/front-capturing techniques, such as the volume of fluid(VOF) method8and the level set method.9The motiva-tion for applying the CIP method as the base scheme ofour model originates from the two key features thathave just been described. In particular, the unified pro-cedure coupled with the CIP scheme (CCUP), which is

applied in our CFD model, has the advantage of sim-plicity of coding because only one set of equationsis used for the whole flow field, including the regioninside solid bodies. Although extra computation timeis required for the region inside solid bodies, Xiao10hasshown by a couple of 3-D computations that this unifiedprocedure is relatively easy to implement numericallywith parallelization techniques which take advantageof the rapid progress in the computational power ofmodern computers.

The current model with the CIP method is quitedifferent from many other CFD methods for wavebody interaction problems in marine engineering. For

example, a numerical wave tank (NWT) problem istreated as a multiphase problem in our model, whichincludes liquid (water), gas (air), and solid (wave-makerand floating body) phases. The motion of all materials isnumerically solved by one set of hydrodynamic equa-tions in a fixed Cartesian grid. Then the free surface andthe body boundary are treated as two types of phaseinterface: the interface between a liquid and a gas, andthe interface between a solid and a fluid, respectively.Is the CIP method capable of quantitative prediction ofthe solution to extreme wavebody interaction prob-lems? To answer this question is the main purpose of

this article. Some new implementation details for appli-cations of the CIP method will be described, and thefocus will be placed on investigating several features ofthe CIP method that can be important for the problemsof interest, but which have not been studied before.These features include the influence of a term of com-pressible effect in the Poisson equation for incompress-ible computations, and the precision in the calculationof hydrodynamic forces acting on a floating body.

This article is organized as follows. Section 2 intro-duces the governing equations and numerical imple-

mentation of the method. The CIP scheme and theCCUP method are briefly described. The interface cap-turing method, the calculation method for hydrody-namic forces acting on a floating body, and the methodfor constructing a numerical wave tank are outlined. InSect. 3, the current numerical method is applied to threefree-surface problems: propagation and reflection of a

solitary wave, a dam-breaking test in a rectangular tank,and a forced oscillation test with a wedge-type float in atwo-dimensional numerical wave tank. In each compu-tation, a grid refinement test is first carried out, and thena validation test compares this with theoretical or mea-sured results. Some special features associated with theCIP method are discussed using the numerical results.The article ends with some conclusions.

2 CIP-based finite-difference method

2.1 Governing equationsAssuming that there is no temperature variation in theproblem, we start from the following equations for com-pressible fluid:

+

= -

r rr

tu

x

u

xi

i

i

i

(1)

+

=

+u

tu

u

x xfi j

i

j

ij

j

i

1

r

s(2)

where r is the density, ui (i = 1, 2, 3) is the velocitycomponent, and sijis the total stress. For a Newtonian

fluid, the total stress can be written as sij=-pdij+2mSij-2mdijSkk/3, where Sij=(ui/xj+uj/xi)/2 and dijdenotesKroneckers delta. The second term on the right-handside of Eq. 2 represents the body force, such as thegravity force, etc. As there is no temperature variation,the equation of state (EOS) of the problem can bewritten asp=f(r). Theoretically, after the density rissolved by Eq. 1, the pressurepcan be determined. Forincompressible or nearly incompressible flow, however,

as the sound speed C ps = rbecomes very large, asmall numerical noise from the density will produce alarge pressure variation. This means that it is numeri-

cally difficult to obtain an accurate calculation of pres-sure using the EOS. However, this difficulty can beovercome by calculating the pressure independently.Applying the EOS to Eq. 1, the pressure equation canbe obtained as

+

= -

p

tu

p

xC

u

xi

i

i

i

r S2 (3)

Equations 13 are the governing equations for hydrody-namic problems, and will be solved numerically by afractional step method.


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145C. Hu and M. Kashiwagi: CIP method for violent free-surface flows

2.2 The fractional step approach

By applying the fractional step approach, the numericalsolution of the governing equations can be divided intothe following three steps.

1. Advection phase:

r* r r -+

=

n

in

n

itu xD 0 (4)

u u

tu

u

xi i

n

jn i

n

j

* -+

=D

0 (5)

p p

tu

p

x

n

in

n

i

*-+

=D

0 (6)

2. Nonadvection phase (i):

u u

t xS S fi i

j

ij ij kk i

** ** *

-=

-

+

D2 1

3

md

r*(7)

3. Nonadvection phase (ii):

r rr

nin

it

u

x

+ +-= -

1 1**

D(8)

u u

t

p

xin

in

i

+ +-= -

1 11***D r

(9)

p p

tC

u

x

nin

i

+ +-= -

12

1**

Dr s (10)

The advection phase computation of Eqs. 46 isconducted by the CIP method. The nonadvection phaseis divided into a state-related part to reflect the fluidcompressibility, denoted by nonadvection phase (ii),and the remaining part denoted by nonadvection phase(i), which includes a viscous term and a source term.The fractional steps in the present numerical methodare arranged in the order advection phase, nonadvec-tion phase (i), and nonadvection phase (ii). Xiao10hasshown that this procedure yields first-order accuracy intime-integration of the governing equations.

2.3 CIP method

The basic idea of the CIP method is that for advectioncomputation of a variablec, not only the transportationequation ofc, but also the transportation equation of itsspatial gradient, ji=c/xi, are used. Here,crepresentseach of r, ui, andpin Eqs. 1, 2, and 3, respectively. Thetransportation equation of ccan be written as

+

=c c

tu

xHi

i

(11)

By differentiating Eq. 11 with respect to the spatialcoordinates, we obtain the transportation equation ofji.

+

=

-

j jji j

i

j i

j

j

itu

x

H

x

u

x(12)

As shown in the previous subsection, computation ofEq. 12 can also be divided into two steps, an advectionphase and a nonadvection phase. The nonadvectionphase calculation will be included in nonadvectionphase (i). For the advection calculation of Eqs. 11 and12, the following semi-Lagrangian procedures are used.

c c* x x u( ) = -( ))

n tD (13)

j ji in t* x x u( ) = -( )

)

D (14)

where)

cn is an interpolation approximation to cn, and

)

jin =

)

cn /xi. For each computational cell, the interpo-lation function

)

cn can be constructed using a cubic

polynomial in which the coefficients of the polynomialare determined from the continuity condition imposed

oncnand)

jin at the grid points. For a multidimensional

case, several forms have been developed for the cubicpolynomial,5 and details of the two-dimensional CIPscheme that is used in the present numerical model aredescribed in Appendix A.

As the advection of spatial gradients in each com-putation cell can be solved, and only the information(value and its spatial gradients) at the grid points ofone cell is needed for the interpolation function, theCIP scheme has both a subcell resolution feature and acompact structure. Therefore, for a multiphase com-putation in which there are discontinuities or largegradients for physical quantities at the interfaces, theCIP scheme can maintain sharpness better than otherupwind schemes.

2.4 The nonadvection phase calculation

For nonadvection phase (i), the Euler explicit scheme isused for the time-integration. The physical nature in-

volved in nonadvection phase (ii) is fluid compres-sibility. Since resolving the acoustic wave propagationrequires extremely small time steps and is beyondour interest, the implicit scheme is used for the time-integration. Taking the divergence of Eq. 9 and substi-tuting uin+1/xi using Eq. 10, we obtain the pressureequation

=

-+

+ +

x x

p p

C t t

u

xi

n

i

ni

i

1 11 1

2 2r

r

r*

*

*

**

s D D(15)


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This is a Poisson-type equation for the pressure calcu-lation. The first term on the right-hand side is the contri-bution of compressibility. As Eq. 15 is valid for theliquid, gas, and solid phases, we can obtain the pressurefield in the whole computation domain by solving thisequation. Then the boundary conditions for the pres-sure at the interface between different phases are not

necessary, and a fast solver or parallel computing tech-nique can easily be applied. This is a very importantfeature, because the calculation of Eq. 15 is generallythe most time-consuming part of this type of computa-tion. Another advantage of Eq. 15 is that it provides avery simple and robust way to calculate hydrodynamicforces on a moving body in a fixed Cartesian grid, as willbe described in Sect. 2.6.

For a perfect incompressible fluid, we can assumethat Cs=. Then Eq. 15 takes the form

=

+

x x t

u

xi

n

i

i

i

1 11

r

r

*

**

D

(16)

This is a conventional Poisson equation for incom-pressible flows. We will use Eq. 16 for most of thecomputations shown here because under the assump-tion of an incompressible fluid, instead of solving Eq. 1,the density can be obtained directly by Eq. 19, as shownin the next subsection. A comparison will be made inSect. 3.2 to check the difference between computedresults using Eqs. 15 and 16.

2.5 Interface capturing method

The moving body boundary and the free-surface bound-ary are distinguished by a density function fm, which issolved by the equation

+

=f fm

im

itu

x0 (17)

For the numerical wave tank problem shown in Fig. 1,m = 1, 2, and 3 denote liquid, gas, and solid phases,respectively.

There are two types of interface that need to be cap-tured in the numerical simulation, i.e., the interface be-

tween gas and liquid, the so-called free surface, and theinterface between solid and fluid, such as a floating bodyboundary. As the behaviors of the two types of interfaceare quite different, we will use different capturingmethods for them.

The free surface is determined by solving Eq. 17 withthe CIP method. Like most of the Eulerian methods,

the original sharp phase interface may become a layerwith finite thickness due to the numerical diffusivity.However, owing to the subcell resolution feature of theCIP method, the thickness grows very slowly as thecomputation proceeds. Therefore, in many cases it isconsidered that this degree of interface diffusion is ac-ceptable for computations when the time correspondsto actual experiments.

A continuum surface force (CSF) model11is also in-corporated in the present computation code to approxi-mate the effect of surface tension, in which the surfacetension is considered as a continuous three-dimensional

effect across the interface. The surface tension forceincluded in the body force term fi of Eq. 2 has theexpression

fS = -

s

r

f

ffS 1

1

1 (18)

Here, ssis the surface tension coefficient. By applyingthis model, no extra treatment is needed even when theinterface is topologically distorted.

For the floating body boundary, we only consider therigid body case. Instead of the computation using Eq.17, a direct computation method has been developed todetermine the density function for the solid phase f3.The basic idea of this method is to map the geometryinformation of a moving body to a fixed Cartesian grid.A detailed description of the method for the two-dimensional case is shown in Appendix B. The advan-tage of this method is that the solid body boundaryposition can be obtained accurately without any nu-merical diffusion.

After the density function for all phases is deter-mined, the physical properties for each computation cellcan be calculated by the equation

l f l=

= m mm 13

(19)

where ldenotes the viscosity, sound speed, etc. Underthe incompressible fluid assumption, the density canalso be determined by Eq. 19.

2.6 Hydrodynamic forces on a floating body

The hydrodynamic force acting on a floating body, Fi,can be calculated by integrating the pressure and skin

damping zone

x

x3

1

d

wave

maker

floating body

solid phase

gas phase

liquid phase

Fig. 1. Schematic view of a numerical wave tank


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friction along the body surface. For the rigid body case,the formulation is written as

F F F p n A S n Ai i i ik k

A

ik k

A

= + = -( ) +( ) ( ) p v d dd m2 (20)

where Fi(p)represents the force due to the pressure, and

Fi(v) represents the force due to the friction. Adenotes

the surface of the floating body, and niis the i-th compo-nent of the outward unit normal vector.

In this numerical model, the forces can also be calcu-lated easily by integration over the whole computationdomain. Applying Gauss theorem to Eq. 20, we obtain

F F Fp

xV

S

xV

p

x

S

x

i i i

iV

ik

kV

i

ik

k

= + = -

+( )

= -

+( )

( ) ( )

p v

3 3

d d

d d

2

2

m

fm

fW WW W

(21)

where V and W denote the space occupied by thefloating body and the whole computation domain, re-spectively. As the pressure, the velocities, and the den-sity function for all computation cells are obtained bythe procedure described in the previous subsections, wecan directly calculate the hydrodynamic forces on amoving body in the fixed Cartesian grid using Eq. 21.The advantage of using Eq. 21 is that we do not need toknow the exact position and orientation of the bound-ary surface in order to calculate the unit normal vector.However, it is necessary to check the accuracy of theforce computation by Eqs. 20 and 21, and this will bedone through a numerical example in Sect. 3.3.

2.7 Absorbing boundary condition for NWT

In order to perform simulations in a numerical wavetank with a finite computation domain over a long pe-riod of time, a nonreflection boundary condition is re-quired at the downstream boundary (see Fig. 1). In thisstudy, an artificial damping zone is placed at the down-

stream boundary (xs1


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Re = =r

m

r

m

1

1

1

1

dU d gd(23)

where r1 = 103 kgm-3, d = 0.1m, and U gd= is thephase speed. We can then obtain different values of Reby changing m1, as shown in Table 1. Our computationsshow that the physical properties of the gas phase mayalso affect the results. In order to compare these withother relative single-phase results, the density and vis-cosity of the gas were chosen as r2=1kgm-3and m2=0,respectively.

The computation grid shown in Table 1 is determinedby a grid refinement test, in which an inviscid solitarywave is simulated. Such an inviscid case is convenient

for code verification because the solitary wave can beknown exactly by theoretical analysis, and the waveprofile does not change during propagation. In the com-putation, we used the slip condition for the bottomboundary and let m1=0. Figure 2 shows the result of thegrid refinement test. Six types of grid were used. Thecomputed wave heights by grids 4, 5, and 6 are almostthe same, i.e., grid convergence is achieved. It is alsoclear from the figure that the variable grid (grid 6), inwhich the grid points are concentrated near the freesurface and the bottom boundary in the vertical direc-tion and in the region containing the wave crest in the

horizontal direction, gives a converged result althoughthe grid number is much less than the uniform grid (grid5). This also confirms that our variable mesh treatmentis successful, and thus grid 6 was used for the followingcomputations. On the other hand, even for the con-verged results, a slight attenuation of the wave height isfound in the computation. This can be explained by thenumerical diffusion of the present method.

The computations were then carried out to investi-gate the effect of viscosity on the propagation of a soli-tary wave by using the different values of m1shown in

Fig. 2. Grid convergence test for an inviscid solitary wavepropagation computation

10 0 10 20

10 0 10 20

Fig. 3. Solitary wave propagation for different Reynoldsnumbers

Fig. 4. Damping of wave height with distance for differentReynolds numbers. CIP, constrained interpolation profile

Table 1. A no-slip condition was applied on the bottomboundary. Figure 3 shows computed free-surface eleva-tions between t/t0=0 and t/t0=12. The computed freesurface is denoted by the line of f1=0.5. It can be seenthat the amplitude of wave decays due to viscous damp-ing and the decay in amplitude decreases with increas-ing Reynolds number.

In Fig. 4, the calculated attenuation of the waveheight with distance traveled by the wave is comparedwith the theoretical results of Mei.13The result obtainedby the computation, including the surface tension effect,

13


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which is modeled by Eq. 18, is also shown in Fig. 4. Itcan be seen that all computed wave heights are lowerthan the corresponding theoretical results. The mainreason for this is that the theoretical analysis only con-

siders the viscous effect at the bottom of the channel,whereas the present computations are for the viscousflow in the whole fluid region. It can also be seenthat the effect of surface tension slightly increases theattenuation of the wave height, but we note that theeffect of surface tension depends on the spatial scale ofthe problem. Another computation on a solitary wavewith a larger reference depth has shown a much smallerdifference between the results with and without thesurface-tension effect. As the surface-tension effect isnot the most important issue of the present research, theother computations in this paper were performed with-out the surface-tension model.

The second computation on a solitary wave is a reflec-tion of a wave from a vertical wall. The computationgrid shown in Table 1 is determined by a similar checkto the one shown in Fig. 2. The computed time variationof wave run-up at the wall is shown in Fig. 5, and goodcomparison is obtained for the run-up stage with theinviscid theory of Power and Chwang.14The differencein the run-down stage is due to the viscous effect at thevertical wall that reduces the run-down speed in thepresent fully viscous computation. Figure 6 shows acomparison of the computed maximum run-up versusthe incident wave amplitude with some available ex-

perimental and theoretical results. By comparing thesewith the experiment, we can see that the prediction bythe present numerical method is better than that ofother theoretical models.

3.2 Dam-breaking problem

The water flow after the sudden break of a dam iswidely used as a test problem for checking the com-putation accuracy for a largely distorted free surfacebecause of its simple initial and boundary conditions.

The development of water flow along the floor afterthe sudden break of the dam has been a conventionaltarget for numerical studies, but our research interest ison the second stage of the flow development, i.e., theflow after impact onto the vertical wall. In the second

stage, overturning and breaking of the free surfaceas well as air entrapment are observed, and the com-putation of these complicated phenomena is a morechallenging subject.

3.2.1 Experiment

A laboratory experiment was conducted at theResearch Institute for Applied Mechanics, KyushuUniversity. The geometry of the experimental set-up isshown in Fig. 7. As this is a small-scale tank, the viscouseffect at the tank walls might be an important factor.The water in the tank is colored, and the temperature of

the water is adjusted to be the same as that of thesurrounding air in order to reduce the temperatureeffect on the pressure sensor. A pressure sensor withdiameter of 0.8 cm is installed on the right-hand verticalwall at point A in Fig. 7. In the experiment, the pressureis measured and the development of the free surface isrecorded by a high-speed digital video camera.

The experiment was repeated eight times. All pres-sure data measured at point A are shown in Fig. 8 bycircular symbols, and the mean value is shown by a solidline. Although a certain scatter is seen in the measured

Fig. 5. Time variation of run-up at the wall Fig. 6. Maximum run-up versus incident wave amplitude

1cm

50cm68cm

12cm

12cmA

x

yz

Fig. 7. Schematic view of the dam-breaking experiment

s

A

14

s

A

14

15

16


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data, the repeatability of the experiment seems good,and the measurements are considered to be reliable forthe purpose of validation. The mean value is used for acomparison with the numerical simulation.

3.2.2 Grid refinement test

The computation conditions were set as in the experi-ment. A variable grid was used for the computation, inwhich the grid points are concentrated near the floorand the right-hand wall. To perform a grid refinementtest, we use three grids, 1, 2, and 3, as indicated in Fig. 9.

The minimum grid spacing varied from 0.8mm to0.2mm.

Figure 9 shows a comparison of the time series of thepressure at point A, for which the computations wereperformed using three different grids. The pressure fieldwas solved by the incompressible Eq. 16. Good generalagreement was found for all the computations. How-ever, for computations with grids 2 and 3, the position ofthe first peak and its value agree well with the experi-ment, whereas with grid 1, the coarse grid, a delay forthe first peak was observed and the peak value was

lower than the measured value. In the result with thefine grid (grid 3), a strong oscillation at the first peakand oscillations around the second peak were alsofound. The reason for this is not very clear at present.

From Fig. 9, it can also be seen that the prediction ofthe pressure in the latter stage (t>0.75s) is poor for allthree grids. In the experiment, the flow at this stage isthree-dimensional and severely broken. Therefore, weconsider that this disagreement is due partly to the 2-Dcomputation, and partly to the smearing of the com-puted free surface in this stage.

Figure 10 shows free-surface profiles near the cornerat t=0.7 s obtained using different grids. The computedfree-surface profile with grid 1 is quite different fromthe other two results. Since the difference betweenthe results with grids 2 and 3 is small for both the pres-sure and the surface profile, it was considered that grid2 is sufficient, and that was used for the followingcomputations.

Figures 11 and 12 are the computed time variation ofthe free surface and the pressure distribution, respec-tively. Grid 2 is used for the computation. In Fig. 11, thecomplicated phenomena of the second stage of flowdevelopment can be seen; the flow hits the vertical wall,overturns, and hits the free surface again with a break-

ing of the free surface as well as air entrapment. In Fig.12, the contour ranges fromp-patm=100Pa to 1500Paat increments of 100 Pa. Here, patm means the initialpressure of the atmosphere. It was found that a verylarge spatial variation appears near the lower part of thevertical wall at the time of water impact.

3.2.3 Effect of compressibility

The CIP method was originally developed for com-pressible flows. Later improvements made it computa-tionally applicable to incompressible problems as well.

Fig. 8. Pressure measured in the experiment shows the scatterof all measured data. Solid line, mean value; circular symbols,the measured data in all tests

Fig. 9. Time variation of the pressure at point A. Computa-tions using different grids. The pressure field was solved usingEq. 16

Fig. 10. Free-surface profile at t= 0.7s. Computations usingdifferent grids

grid 3

grid 1

x (m)

z(m)

0.1 1.11

0

0.05

0.1

0.15

0.2

grid 2


9/15


By using Eq. 15, both compressible and incompressibleflows can be solved by the same scheme. However, un-der the assumption of an incompressible fluid, we tendto use Eq. 16 instead of Eq. 15 to calculate the pressurefield in order to reduce the computation time. Then anatural question is how different the results of theimpact pressure are by using Eq. 15 or 16.

Figure 13 shows a comparison of the computed pres-sures at point A obtained with Eqs. 15 and 16. The soundspeed used for the computation with Eq. 15 was Cs=330m/s for gas and Cs=1400m/s for liquid. Differenceswere found in both the position of the first peak and the

peak value. A better prediction was obtained with Eq.16, which is the Poisson equation for incompressibleflow. On the other hand, the computation with Eq. 16,the Poisson equation with a compressible term, givesthe result without oscillation. We may conclude that thecompressible term in the Poisson equation stabilizes thepressure calculation for surface-impact problems. Sinceusing Eq. 15 requires extra effort in calculating the den-sity, the computations shown here were carried out us-ing Eq. 16, the incompressible version of the Poissonequation, when there is no other explanation.

t = 0.0 sec

t = 0.1 sec

t = 0.4 sec

t = 0.5 sec

t = 0.6 sec

t = 0.7 sec

t = 0.8 sec

t = 0.9 sec

t = 0.2 sec

t = 0.3 sec

Fig. 11. Time-series of computed free-surface development

Fig. 13. Time variation of the pressure at point A. Pressurecomputed using different types of Poisson equation

Fig. 14. Time variation of the pressure at point A. Computa-tions using different velocity boundary conditions at the tankwalls. BC, boundary condition

t = 0.9 sec

t = 0.8 sec

t = 0.7 sec

t = 0.6 sec

t = 0.5 sec

t = 0.4 sec

t = 0.3 sec

t = 0.2 sec

t = 0.1 sec

t = 0.0 sec

Fig. 12. Time-series of computed pressure distribution


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3.2.4 Effect of wall boundary condition

For a small-scale problem such as the current dam-breaking experiment, the viscous effect on the wall mustbe important. A numerical simulation of this problemwould be affected by using a slip or no-slip boundarycondition at the wall. The results shown in Sect. 3.2.2and 3.2.3 were obtained using the no-slip condition, andwe shall illustrate that the slip condition is not good forthis example.

Figure 14 shows the time variation of the pressure atpoint A obtained by using different wall boundary con-ditions. The computation with a no-slip boundary con-dition gives a much better result than that with a slip

boundary condition. The reason can be found in Figs.15, 16, in which the velocity fields and pressure fields,respectively, are shown. The vortex near the corner isclearly seen in the no-slip result, but not in the slipresult. Correspondingly, the pressure fields are quitedifferent in the two computations.

3.3 Hydrodynamic force on a heaving wedge

The third example tested is the computation of wavebody interactions. It is well known that a successful

x (m)

z(m)

1.14 1.15 1.16 1.17 1.18

0

0.01

0.02

0.03

0.04

no slip wall BC

x (m)

z(m)

1.14 1.15 1.16 1.17 1.18

0

0.01

0.02

0.03

0.04

slip wall BC

Fig. 15. Velocity fields at t=0.4s. Computations using differ-ent velocity boundary conditions at the tank walls

100

200

200

300

400

400

500

500

600

600

700

800

900

x (m)

z(m

)

1.14 1.15 1.16 1.17 1.18

0

0.01

0.02

0.03

0.04

slip wall BC

100

200

200

200

300

300

300

400

40

0

400

500

500

500

500

600

600

x (m)

z(m)

1.14 1.15 1.16 1.17 1.18

0

0.01

0.02

0.03

0.04

no slip wall BC

Fig. 16. Pressure fields at t=0.4s. Computations using differ-ent velocity boundary conditions at the tank walls. The unit ofthe contour labels is Pa.

a

b=2.5a

0.5a

a x

z

Fig. 17. Schematic view of the wedge for the forced heavingcomputation

prediction of a floating body in waves requires an accu-rate calculation of the hydrodynamic forces. Here, wewill check the numerical accuracy of the two force cal-culation methods described in Sect. 2.6.

3.3.1 Three-phase computation

The computation example is a forced oscillation inheave with a wedge-type float, as shown in Fig. 17.


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For this problem, a laboratory experiment was con-ducted by Yamashita,17and a numerical simulation bythe potential theory was performed by Kashiwagi.18

The characteristic length for the float is a= 0.0792m,and the other computation details are shown in Table 2.The wavenumber is defined as Ka=w2a/g, where w is

the frequency of the forced oscillation. We considerthat this problem is a good example of a three-phasecomputation to check the present numerical model,because the computation domain includes a liquidphase (water), a gas phase (air), and a solid phase(float).

A grid convergence test was carried out first. Figure18 shows a comparison of the time variation of thecomputed vertical force acting on the wedge using fourtypes of variable grid. The wavenumber is Ka = 0.4.The vertical force is divided into a pressure-related

force Fz(p) and a friction-related force Fz

(v), which are

obtained by the volume integration method (Eq. 21).For the pressure related force, computations using grids2, 3, and 4 resulted in a converged time variation. Onthe other hand, the friction-related force increaseswith increasing grid density. However, as the pressure-related force is 1000 times larger than the friction-related force by the present computation, grid 3, whichis considered to be sufficient for an accurate calculationof the pressure-related force, is used for the followingcomputations.

In order to show the concept of our numerical model,a snapshot of the computed flow field (velocity and

pressure) near the wedge is shown in Fig. 19. The com-puted free surface is represented by the line f1= 0.5.The thin viscous boundary layer near the boundary ofthe wedge is unclear owing to the boundary-embeddingtreatment. It should be pointed out that using thepresent grid, the boundary layer cannot be fully re-solved, i.e., the viscous effect is only approximately con-sidered by the numerical model. We will check whethersuch an approximation of the boundary layer affects thecalculation accuracy of the hydrodynamic force on thebody in the following section.

Table 2. Computation condition for the forced oscillation test

Wave makerGeometry a=0.0792mMotion z(t) =Zsin(2pt/T)Amplitude Z/a=0.6Wave number Ka=w2a/g=0.2, 0.4, 0.6, 0.8, 1.0Water depth d/a=7.6

Grid and time-step

Grid number 460 (horizontal) 165 (vertical)Min grid spacing Dx/a=0.02, Dz/a=0.02Time-step Dt/T=5 10-4

Fig. 18. Comparison of the time variation of the hydrody-namic forces acting on the wedge for computations using dif-ferent grids. Ka=0.4

40

40

8080

12120

16160

200200

240240

velocity

pressure

Fig. 19. Computed flow field at t/T=10 near the wedge. Theunit of the contour labels is Pa


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The experiments were carried out for three differentoscillation amplitudes, i.e., Z/a=0.2, 0.4, and 0.6, andthe nonlinear computations by BEM were for Z/a=0.2,0.6, and a linear one for the case of infinite water depth.The present computation was conducted for the largestamplitude of Z/a = 0.6 only. Fourier analysis of thecomputed results is carried out in the same way as thatused in the experiments. As the first-order hydrody-

namic forces, we show the heave-added mass (A33) andthe damping coefficient (B33), and as the second-orderhydrodynamic forces, we show the amplitude (F2) andthe phase lead (d2) of the time-varying force with thesecond harmonic of the oscillation frequency and thetime-averaged steady force (F0). The definition of thesequantities can be found in the reference paper.18Boththe first-order (Fig. 20) and second-order (Fig. 21)hydrodynamic forces obtained by the present numericalmodel compare well with the experiments and the non-linear BEM results. However, we should note thatthe predicted damping force coefficient B33 is slightly

lower than in the experiment. This is due partly to theincorrect prediction of the skin friction because of theinsufficient resolution of the boundary layer.

3.3.3 Two force calculation methods

As described in Sect. 2.6, the hydrodynamic forcesacting on a floating body can be obtained by two meth-ods: the surface-integration method represented byEq. 20 and the volume-integration method representedby Eq. 21. By the surface-integration method, we meanintegration along the body surface, and by the volume-

Fig. 20. First-order hydrodynamic forces on a wedge

Fig. 21. Second-order hydrodynamic forces on a wedge

3.3.2 Hydrodynamic forces

Figures 20 and 21 show comparisons of the hydrody-namic force coefficients acting on the wedge-type floatwith a half-beam over draft ratio (a/b) equal to 0.4.

Fig. 22. Comparison of the time variation of the hydrody-namic forces acting on the wedge for computations using dif-ferent force calculation methods. Ka=0.4

BEM (Kashiwagi18

)Exp. (Yamashita17

) present CIP

Z = 0.6 a

Z = 0.2 a

Z = 0.4 a

Z = 0.6 a

Z = 0.2 a

Z = 0.6 a

-F0

/2

gZ2

F2

/2

gZ2

2(

deg.

)

Ka

BEM (Kashiwagi17

) Exp. (Yamashita18

)present CIP

Z = 0.6 a

Z = 0.2 a

Z = 0.6 a

Z = 0.2 a

Z = 0.4 a

Z = 0.6 a

Linear

Ka


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integration method, we mean integration over thewhole computation domain.

Figure 22 shows the time variation of the hydro-dynamic forces acting on the wedge obtained by thetwo force calculation methods. The wave-number isKa = 0.4. Only a slight difference was found for Fz(p),

whereas Fz(v)obtained by the surface-integration method

is larger than that obtained by the volume-integrationmethod. As discussed in relation to Fig. 18, grid con-vergence cannot be achieved for the friction forcecalculation. Therefore, this discrepancy between thetwo methods is mainly due to the inadequate gridresolution.

The dependence of the difference in integrationmethod on the computed hydrodynamic forces is shownby Figs. 23 and 24. The volume-integration methodshows a better performance for the added-mass predic-tion, while the surface-integration method gives a betterprediction of the damping coefficient. For the second-order hydrodynamic forces, the two methods give thesame results except for the steady component F0

(2).

4 Conclusions

We have introduced a new numerical approach forextremely nonlinear free-surface problems in the fieldof marine engineering. The wavebody interaction istreated as a multiphase problem in which the freesurface as well as the body boundary is immersed inthe computation domain. One set of governing equa-tions is used to represent the motion of all materials,and the governing equations are solved numerically ona nonuniform, staggered Cartesian grid by a CIP-basedfinite-difference method.

Several 2-D numerical simulations were carried out.The numerical results provide an assessment of the pro-posed model and a partial answer to the question: Is

the CIP method capable of quantitative prediction ofextreme wavebody interaction problems? In particu-lar, what we most want to know, but have not investi-gated before, is how accurate the CIP calculation is forthe hydrodynamic force acting on the body. Therefore,as well as proposing some new developments for theimplementation of the CIP method, the originalityof this paper lies rather in the quantitative assessment ofthe precision of the hydrodynamic force calculation.The main findings by the numerical results presentedare given below.

For the solitary wave propagation computation, grid

convergence can be achieved, but damping of the waveheight by the numerical diffusion is still found. Thedependence of the damping on the Reynolds numbercompares well with the theoretical results. The resultof the dam-breaking computation shows that the com-puted peak value of the impact pressure and the mea-surements are in good agreement, and are not verysensitive to grid density if a sufficiently large grid num-ber is used. As numerical predictions of the impact pres-sure are still very difficult in this type of computation,the present CIP method is shown to be quite satisfac-

a

a

Ka

Fig. 23. First-order hydrodynamic force coefficients on thewedge obtained by different force calculation methods

Fig. 24. Second-order hydrodynamic force coefficients on awedge obtained by different force calculation methods


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tory. The last example we presented, the forced heavingoscillation test with a wedge-type float in a 2-D numeri-cal wave tank, is another challenging subject for a Car-tesian grid approach. This is a three-phase computation,and a new method was developed to calculate themotion of a solid body. The accuracy of the two force-integration methods, the surface-integration method

and the volume-integration method, were also investi-gated. We found that the relatively simple method, thevolume-integration method, is nearly as accurate as thesurface-integration method. The precision of hydrody-namic force predictions using the volume-integrationmethod is comparable to that using the nonlinear BEMwhen compared with the experiments. The resultsimply that for a wavebody interaction problem, if thepressure-related hydrodynamic force acting on the bodyis dominant, a quantitatively correct prediction can beachieved by the current CIP-based method.

Appendix A

Equations for the 2-D CIP method

Rewriting Eqs. 11 and 12 in the two-dimensional form,we have

+

+

=c c c

tux

vy

0 (A1)

+

+

=c c cx x xt

ux

vy

0 (A2)

+

+

=

c c cy y y

t u x v y 0 (A3)

The terms on the right-hand side of Eqs. A1A3 arenot shown because they can be included in thenonadvection phase calculation. Considering a gridpoint (i, j), we can find an upwind cell with four gridpoints, (i,j), (iw,j), (i,jw), and (iw,jw), as shown in Fig.25. Here, iw = i-sign(u), jw =j-sign(v). Then a cubic

polynomial can be constructed to approximate the spa-tial distribution of the value c in the upwind cell, asfollows:

X ,x h x x h xh h x

xh h x h

( ) = + + + ++ + + + +

C C C C C

C C C C C

303

212

122

033

202

11 022

10 01 00(A4)

There are ten unknown coefficients in Eq. A4, Cmn,which will be determined by the values of cn, cnx, and c

ny

at grid points (i,j), (iw,j), and (i,jw), and the value ofcn

at the grid point (iw,jw). We obtain

C iw j i j

iw j i j

xn

xn

n n

30 1

132

= ( )+ ( )[ ]{- ( ) - ( )[ ]}

x c c

c c x

, ,

, , (A5)

C iw j i j

iw j i j

xn

xn

n n

20 1

1

2

2

3

= - ( ) + ( )[ ]{

+ ( ) - ( )[ ]}

x c c

c c x

, ,

, , (A6)

C i jw i j

i jw i j

yn

yn

n n

03 1

132

= ( ) + ( )[ ]{- ( ) - ( )[ ]}

h c c

c c h

, ,

, , (A7)

C i jw i j

i jw i j

yn

yn

n n

02 1

12

2

3

= - ( )+ ( )[ ]{+ ( ) - ( )[ ]}

h c c

c c h

, ,

, , (A8)

C iw jw iw j i jw

i j

xn

xn

xn

x

n

21

1 12

= ( ) - ( ) - ( )[

+ ( )] ( )

c c c

c x h

, , ,

,(A9)

C iw jw iw j i jw

i j

yn

yn

yn

yn

12

1 12

= ( ) - ( )- ( )[+ ( )] ( )

c c c

c x h

, , ,

, (A10)

C iw jw iw j i jw

i j C C

n n n

n

11

1 1 21 1 12 1

= ( ) - ( ) - ( )[+ ( )] ( ) - -c c c

c h h

, , ,

, x x (A11)

C i jx

n10 = ( )c , (A12)

C i jy

n01 = ( )c , (A13)

C i jn

00 = ( )c , (A14)where z1=-sign(u)Dx and h1=-sign(n)Dy. Then thevalues after the advection calculation,c*,c*x, andc*y, canbe obtained directly as follows:

c x h* , ,i j( ) = ( )C (A15)

cx h

x x xh m

x i j* ,,

( ) = ( )

= =

C(A16)

Fig. 25. Interpolation concept for the CIP method


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cx h

h x xh m

y i j* ,,

( ) = ( )

= =

C(A17)

where z=-sign(u)Dtand h=-sign(n)Dt.

Appendix B

Calculation method for the density function for a 2-D

rigid body

For a rigid body, since the geometry does not changewith time, a Lagrangian method was developed to cal-culate f3 to obtain the solid-body boundary positionsvery accurately without any numerical diffusion. Forthe two-dimensional case, the main procedure is asdescribed below.

1. The two-dimensional body boundary is approxi-

mated by a series of straight line segments (pk,pk+1),k=1 ~N, as shown in Fig. 26a.2. The coordinates for the end points (xpk, zpk) are cal-

culated by the equations

x x x x z zpk c pk c pk c= + -( ) - -( )0 0 0 0cos sina a (B1)

z z x x z zpk c pk c pk c= + -( ) + -( )0 0 0 0sin cosa a (B2)

where (xc, zc) is the mass center of the floatingbody, a is the roll angle, the superscript 0 denotesthe initial value (xc, zc), and a is calculated in aLagrangian way because the hydrodynamic forces onthe body can be obtained by the method described inSect. 2.6.

3. All of the intersection points (nodes) of line seg-

ments and grid lines are then calculated. For eachcomputation cell, if there are more than two nodes,the cell is considered as a cell which includes thesolid-body boundary, and the area of the solid bodyin this cell is computed to determine f3, as shown inFig. 26b.

References

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13. Mei CC (1983) The applied dynamics of ocean surface waves.Wiley, New York14. Power P, Chwang AT (1984) On reflection of a planar solitary

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Fig. 26. Method for calculating the density function of thesolid phase for the 2-D case. a2-D body represented by theenclosing line segments. b Density function for a boundarycell

a

b

A CIP-Based Method for Numerical Simulations of Violent Free-surface Flowa - Hu

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