A two-phase flow model coupling with volume of fluid and immersedboundary methods for free surface and moving structure problems

Cheng Zhang a,b, Wei Zhang a,c, Nansheng Lin a,d, Youhong Tang e,n, Chengbi Zhao a,d,nn,Jian Gu a,d, Wei Lin a,d, Xiaoming Chen f, Ang Qiu a,d

a Department of Naval Architecture and Ocean Engineering, School of Civil Engineering and Transportation, South China University of Technology,Guangzhou 510641, Chinab Shanghai Rules & Research Institute, China Classification Society, Shanghai 200135, Chinac Guangdong Maritime Safety Administration, Guangzhou 510230, Chinad Naval Architecture and Ocean Engineering Technology, R&D Centre of Guangdong Province, Guangzhou 510641, Chinae Centre for Maritime Engineering, Control and Imaging, School of Computer Science, Engineering and Mathematics, Flinders University, Adelaide 5042,Australiaf The 602 Research Institute of CSIC, Beijing 100024, China

a r t i c l e i n f o

Article history:Received 11 November 2012Accepted 19 September 2013Available online 31 October 2013

Keywords:Nonlinear free surfaceMoving structureTwo-phase flow modelFixed Cartesian gridImmersed boundary methodVolume of fluid method

a b s t r a c t

A numerical model is developed to solve increasing ocean engineering problems involving complexand/or moving rigid structures and nonlinear free surface action with considering air movement effects.The model is based on the two-phase flow model of incompressible viscous immiscible fluids containingvarious interfaces, and employs a coupled immersed boundary (IB) and volume of fluid (VOF) methods.To solve the governing equations, a two-step projection method is employed and the finite differencemethod on a staggered and fixed Cartesian grid is used throughout the computation. The bi-conjugategradient stabilized technique is applied to solve the pressure Poisson equation. In particular, theadvection term is discretized in a composite difference scheme to enhance the stability of the algorithm.The direct forcing IB method is utilized to deal with no-slip boundary condition, while the VOF method,which employs a piecewise line interface calculation technique and a Lagrange method to reconstructand update the interface respectively, is used to track distorted and broken free surfaces. The results ofthis study demonstrate the accuracy and capability of the two-phase model to simulate a moving body infree surface flows while also considering air movement effects.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Efficient and accurate computation of incompressible two-phaseflow problems has enormous value in many scientific and industrialapplications, including marine and coastal fluid-structure interactions,sloshing in tanks, wave loading and run-up (Sussman et al., 2007).Increasingly, applications such as those above require a computa-tional fluid dynamics (CFD) tool which can handle complex andmoving structures, nonlinear free surface, and air motion, tounderstand their natural hydrodynamic processes. Developing

such a tool is thus not only an attractive research topic; it is alsoa difficult task for engineers. The difficulties encountered includeeffective simulation of the motion of various structures with lowcomputational cost, accurate treatment of the nonlinear freesurface boundary, and correct description of the “jump condition”between air and water.

In general, grid methods employed in incompressible two-phaseflow models can be roughly categorized into fixed grid methods andmoving grid methods. In the early stage of CFD studies, Navier–Stokes (N–S) equations were crudely discretized using fixed Cartesiangrid methods for simulations involving simple geometries. Thesemethods are easy to understand and implement when there is ashortage of computing resources. However, for many engineeringproblems necessitate consideration of complex structures and freesurfaces those do not coincide with the fixed grids. As a result, thesemethods have been almost abandoned and replaced by moving gridmethods. Moving grid methods, such as body-fitted grid methodsand interface tracking methods can exactly fit the solid boundary andthe free surface. They are widely used to represent solid surfaces or

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Ocean Engineering

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.oceaneng.2013.09.010

n Corresponding author at: Centre for Maritime Engineering, Control andImaging, School of Computer Science, Engineering and Mathematics, FlindersUniversity, Adelaide 5042, Australia. Tel.: þ61 08 82012138.

nn Corresponding author at: Department of Naval Architecture and OceanEngineering, School of Civil Engineering and Transportation, South China Universityof Technology, Guangzhou 510641, China. Tel.: þ86 20 87111030x3512.

E-mail addresses: youhong.tang@flinders.edu.au (Y. Tang),tccbzhao@scut.edu.cn (C. Zhao).

Ocean Engineering 74 (2013) 107–124

air–water interfaces. Many accurate two-phase models based onthese methods have been developed to simulate ocean engineeringproblems. These methods are based on either the single-block meshsystem or the multi-block mesh system. In the single-block meshsystem, unstructured grids are often used to fit surfaces. When asurface moves, a mesh deformation or a grid re-generation isrequired at every time step. Fitting complex boundary conditionsthrough the single-block mesh technique has proved to be quitedifficult; to the extent that breakdown can occur when a topologicalchange appears. Moreover, using moving grid methods on the single-block mesh system to deal with moving boundaries is relativelytime-consuming and not always successful, due to the deformation,regeneration, and mapping of the grids. To overcome these dis-advantages, the multi-block mesh system was introduced (Beneket al., 1985). In this system, several blocks of structured or unstruc-tured grids are generated in the vicinity of the moving surface, andonly the body-conforming grids follow the body motion. The numberof blocks depends on the complexity of the surface shape, anddifferent blocks obtain their boundary data by extrapolation fromneighboring ones. This method was adopted by Chen and Liu (1999)for studying the flow induced by a ship in harbor, and by Carrica et al.(2007) for studying the wave diffraction problem for a surface ship.Since the scheme exploits the best grid resolution to discretize theflow around the surface, it is much more computationally efficient.When the surface is very complex, however, dozens or evenhundreds of blocks must be generated, entailing a significant increasein computational cost. After comprehensively analyzing the advan-tages and disadvantages of fixed grid and moving grid methods,some researchers in recent decades have again turned to fixed gridmethods and have proposed some ingenious technologies. Consider-ing the scope of this work, only numerical schemes based on thefixed Cartesian grid are discussed here.

Accurate simulation of a moving body in free surface flows on thefixed Cartesian grid is a hot topic in CFD studies. Commonly knownmethods are the cut-cell method and the immersed boundary (IB)method. Like the IB method, the cut-cell method is cost-saving andthe grids are easily generated compared to moving grid methods.Moreover, structures of almost arbitrary shape can be dealt with.However, a major problem of the cell-cut method is that it can createinfinitesimal mesh units that cause serious errors. The IB methoddoes not have this problem, because it directly imposes the velocitiesof boundary conditions on fluid grids rather than “cutting” them.Since Peskin (1972) originally proposed the IB method, manyresearchers have devoted themselves to improving its algorithm.These approaches can be roughly classified into two categories: non-direct forcing methods and direct forcing methods. The original IBmethod, which employs feedback forcing to represent an elasticsurface of the heart valve, falls into the first category. In this method,the interface is smeared and has a finite thickness. Goldstein et al.(1993) and Saiki and Biringen (1996) modified this method andutilized it to represent a solid body, but it has been shown thatfeedback forcing methods are not suitable for rigid boundary condi-tions due to the spurious currents near the interface and restrictedtime steps associated with numerical stability. To overcome theseproblems, a direct forcing method for rigid boundaries was presentedby Fadlun et al. (2000) who evaluated the momentum forcing in N–Sequations. That method was further developed to second-orderaccuracy, for example in the ghost-cell method (Tseng and Ferziger,2003) and the embedded-boundary method (Yang and Balaras,2006). Due to the large amount of special treatments required bylinear or bilinear interpolation, however, implementation of thesemethods is a little tricky when they are applied to 3D problems,especially when very complex and moving boundaries are involved.In fact, some simple direct forcing methods can deal with movingbodies in ocean engineering applications accurately enough if thegrid is finely arranged. In this study, a simplified IB method is

employed to represent solid–fluid interfaces, and it is demonstratedthat it can accurately handle a solid body with complex movements.

Another important topic in developing two-phase flow modelson the fixed Cartesian grid is to find a robust approach to track freesurfaces. Harlow and Welch (1965) developed the first successfulnumerical model that directly solves incompressible N–S equa-tions on such a grid. In the model, a marker-and-cell (MAC)method is proposed to track the position of the free surface.However, the marker information is in general not located at theplace where the velocity is defined, so movements of thesemarkers must be extrapolated from the nearby velocity, whichcan lead to large accumulated errors. This method is thereforeunsuitable for cases with high deformation of the free surface.Moreover, application is very difficult in situations where airvolume is mixed with water. By considering the volume of fluid(VOF), Hirt and Nichols (1981) proposed a different way to trackthe fluid interface. They developed a scheme for advection of theVOF fraction function, called the donor–acceptor method, whichwas effective for tracking fluid interfaces but had relatively lowfidelity. Subsequently, researchers have made significant advancesin developing other schemes to obtain better advection algorithmsfor the VOF function with less diffusion, and with more accuratereconstruction of the interface (Floryan and Rasmussen, 1989;Hyman, 1984; Lin and Liu, 1999; Raad, 1995). Among those schemes,a VOF method with a piecewise linear algorithm proposed byYoungs (1982) is the most widely used. Also employing thepiecewise linear approach, others who have succeeded with theirdevelopments include Ashgriz and Poo (1991); Gueyffier et al.(1999); Harvie and Fletcher (2001); Puckett et al. (1997); Riderand Kothe (1998); Scardovelli and Zaleski (2003), and so on.Because of the efficiency and robustness of the VOF method, it isemployed in the present study.

In recent years, some numerical models combining the IB methodand the VOF method have been developed by farsighted researchers.Sint Annaland van et al. (2006) employed a combined method tosimulate the interaction of a falling droplet and a solid particle in agaseous atmosphere. Results of simulation demonstrated that thiscombined IB–VOF method could deal with free surface flow pro-blems in complex geometries with substantial change in an interfacetopology. Sint Annaland van et al. (2006) and Shen and Chan (2008)applied the combined method to simulate macro-scale interactionsof free-surface waves and submerged solid bodies in 2D. Thesimulation validated the model's capability of handling submergedirregular solid boundaries in a Cartesian coordinate system andaccurately capturing free-surface evolution in the presence of non-linear effects. Furthermore, Shen and Chan (2010) showed themodel's potential as a tool for investigating the formation oftsunamis caused by earthquake through simulations of the deforma-tion of waves by a vertically moving submerged object.

It should be pointed out that many IB–VOF models for thesimulation of free surface problems with rigid structures are basedon one-phase flow models, in which the effects of air movementabove the free surface are ignored (only the liquid flow is simulatedand the gas dynamics are neglected). This neglect leads to twosignificant problems. First, the IB method imposes the boundaryconditions only on fluids, be they water or air. Therefore, utilizationof a one-phase flow model means that the IB method is able tosimulate only submerged structures. This is a major limitation forocean engineering simulations. Second, the density differencebetween liquid and gas is neglected in one-phase flow models. As aresult, trapped air bubbles within the water and splashed water in theair are not fully treated. Besides, unknown physical quantities need tobe extrapolated at the free surface, which can result in missinginformation about those quantities and can become a further sourceof errors. To avoid these problems, it is preferable to construct a two-phase flow model rather than a one-phase flow model in this study.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124108

In this research, the application of simulations is extended toinvestigate the various movements of a cylinder, a dam breaking, andthe sloshing of liquid in a tank, phenomena that are often encoun-tered in the fields of naval architecture, ocean engineering and waterconservation engineering. The present incompressible two-phaseflow model is discretized on a fixed Cartesian grid with a coupledIB–VOF method, and a two-step projection method is utilized tosolve N–S equations. Section 2 describes the details of the numericalmodel. In Section 3, several cases are simulated with the model. First,a cylinder with translation and rotation is tested to demonstrate thecapacity and accuracy of the model for complex and movingstructures. Second, a dam breaking over a dry bed with or withoutan obstacle is simulated. Unlike the simulations of other researchers,a three-phase interface appears in the present simulation, incorpor-ating the overlap of free surface and solid boundary, which is asignificant improvement on many other IB–VOF models. Third, waterentry and exit of a cylinder is performed to further illustrate theapplicability of the model in treating the wave interaction withmoving structures. Last, using the two-phase model, fluid sloshing ina horizontally excited rectangular tank under different excitationfrequencies with or without a baffle is simulated, fromwhich certainphenomena can be observed for engineering application. Finally,some important conclusions are drawn in Section 4.

2. Numerical model

2.1. Mathematical formula

In this study, incompressible and immiscible fluids are consid-ered. The segregated-fluid approach is chosen for the two-phasemodel, which means that the behavior of each fluid is described byan independent set of equations, while it is assumed that no phasechanging and no slipping between the fluids occur at the interfaceof the fluids. Surface tensions at the interface are ignored. Forincompressible conditions, the governing equations are the massconservation equation and the N–S momentum conservation equa-tion, written as

∂uj

∂xj¼ 0 ð1Þ

∂ui

∂tþuj

∂ðuiÞ∂xj

¼ � 1ρ∂p∂xi

þ 1ρ∂τij∂xj

þaiþ f i ð2Þ

where subscript i¼ 1;2 denotes the 2D geometrical descriptionsand Cartesian tensor notation is used. uj, p and xj are the velocities,pressure and spatial coordinates respectively. ai represents theexternal acceleration, f i represents the external body force and τijis the viscous term given by,

τij ¼ μ∂ui

∂xjþ ∂uj

∂xi

� �ð3Þ

where ρ, μ are the density and viscosity respectively appropriate forthe phase occupying a particular spatial location at a given instanceof time. For immiscible fluids, density and viscosity are constantalong particle paths and they are advected by fluid velocity; there-fore, they satisfy

∂ρ∂t

þuj∂ρ∂xj

¼ 0 ð4Þ

∂μ∂t

þuj∂μ∂xj

¼ 0 ð5Þ

Although the computational cost has been proved to be lowerwith the use of a non-staggered grid mesh than with a staggeredgrid mesh for the same case (Chen et al., in press), non-staggered

grids have a disadvantage in accuracy of simulating complex flows,which can lead to oscillation of pressure. Moreover, the advectionof the volume of fluid in the interface-cell is also more accurateand convenient with a staggered grid. Hence, a staggered gridmesh is applied in this study, as shown in Fig. 1. The velocitycomponents, external acceleration and external forces are definedat the cell faces and the scalar quantities are evaluated at the cellcenter.

2.2. Solution algorithm

For the simulations of the engineering cases in Section 3, i.e.the various movements of a cylinder, the dam breaking and thesloshing liquid in a tank, the finite difference method on a fixedCartesian grid is applied to the governing equations throughoutthe computation. In this study, the two-step projection method isused. The governing equations are discretized on a staggered gridusing first order time, marching with an implicit Crank–Nicolsonscheme (Le et al., 2008) for the viscous terms and the Adams–Bashforth scheme (Kim and Moin, 1985) for the advection terms.

During the whole computational process, the intermediatevelocity is computed first, and then a pressure is obtained bysolving the Poisson equation derived from enforcing the continuityconstraint, and lastly the final velocity is updated by simplealgebraic operations. The detailed solution algorithm of the two-step projection method applied in this study includes the followingmain steps.

For a given time step n, pn, uin, ui

n�1, μn, ρn, ain, f in are known.

For the next time step, an intermediate velocity field uin, which

does not satisfy the continuity equation, is derived from theHelmholtz equations with implicit values

uin�ui

n

Δt¼ � uj

∂ui

∂xj

� �nþð1=2Þ

� 1ρn

∂pn

∂xiþ 1

2ρn

∂∂xj

μn ∂uin

∂xjþ ∂uj

n

∂xi

� �þμn ∂ui

n

∂xjþ ∂uj

n

∂xi

� �� �

þainþ f i

n ð6Þwhere

uj∂ui

∂xj

� �nþð1=2Þ¼ 3

2uj

n ∂uin

∂xj

� �� 1

2uj

n�1 ∂uin�1

∂xj

� �ð7Þ

Fig. 1. Staggered grid mesh.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 109

The superscript indicates the time level and Δt is the time-stepsize, which varies according to the velocity. The Helmholtzequations are solved by the Gauss–Seidel method. The secondstep is to solve the Poisson equation to obtain the pressure pnþ1

∂∂xj

1ρn

∂pnþ1

∂xj

� �¼ 1Δt

∂ujn

∂xjþ ∂

∂xj1ρn

∂pn

∂xj

� �ð8Þ

It is noted that the discrete Poisson equation (Eq. (8)) can becombined in a matrix form as ½A�½P� ¼ ½B�, where A is a matrix ofcoefficients, P is a matrix with discrete pressure pnþ1 as itscomponents, and B is a matrix formed by the known right-sidevariables of Eq. (8). In this study, the bi-conjugate gradientstabilized (Bi-CGSTAB) method (van der Vorst, 2003) with apreconditioner of incomplete LU is used.

The difference between the corrected velocity field uinþ1,

which must be divergence-free to satisfy the incompressiblecondition, and the intermediate velocity field ui

n is due to theeffect of the change of pressure at the new time step. Thecorrected velocities are then found as

uinþ1 ¼ ui

n� Δtρn

∂pnþ1

∂xi� Δtρn

∂pn

∂xi

� �ð9Þ

After the velocity field at the new time step has been found, thefree-surface position can be reconstructed through the VOFmethod, and then the distribution of density and viscosity caneach be determined. As well, according to the velocity of the solidboundary, the external force f i

n can be determined by the IBmethod for the next time step. In particular for the non-inertialcoordinate system for fluid sloshing, external acceleration ain canbe obtained according to the movement of the system.

2.3. Interface treatment by the volume of fluid method

To reconstruct the free surface mentioned above, the VOFmethod is used to capture the air–water interface. Unlike in aone-phase model, the free surface in a two-phase flow model isconsidered as a fluid–fluid interface rather than a free-surfaceboundary condition, but the ways to apply the VOF method aresimilar in both types of model.

Defining the volume fraction of the gas phase F as 0, the liquidphase F as 1, the interface as 0oFo1, the advection equation ofthe interface can be described as

∂F∂t

þuj∂F∂xj

¼ 0 ð10Þ

To determine the variable F in every cell at each time step, it isimportant to select carefully two core algorithms of the VOFmethod, namely the interface reconstruction and volume advectionalgorithms, which significantly influence computational accuracyand efficiency.

In the present model, a second-order piecewise linear interfacecalculation (PLIC) method of the VOF method is used to recon-struct the interface and to determine the VOF fluxes. The linearinterface reconstruction is achieved from knowledge of both thenormal vector of the interface and the intercept of the interfaceline on the meshes. The modified Youngs' least square method(Rider and Kothe, 1998), which is second-order accurate, isemployed to estimate the normal vector of the interface. In termsof the free-surface advection algorithm, a Lagrangian advectionmethod according to Gueyffier et al. (1999) is used after recon-struction of the interface in each cell at each time step. Afterimplementing the above algorithms, the fluid fraction variable Fcan be updated to the next time step. Then the local density and

viscosity can be calculated from the fluid fraction variable F. It isnoted that the update process of the VOF function is basically theupdate of density, and therefore the transport of density with theinterpolation of velocity may result in air bubble entrainment ordroplets after reconstruction of the interface. No special treat-ments are needed in this two-phase flow model when dealingwith flow with broken free surfaces.

2.4. Treatment of a moving structure by the immersed boundarymethod

To determine the external force f in for description of rigid

structures with the IB method, a simplified direct forcing IBmethod developed by us (Chen et al., in press) is used to apply ano-slip boundary condition on the solid wall boundaries. Com-pared to other IB methods, the present method is very suitable forengineering applications because of its simple procedures and costsavings. Moreover, its accuracy can be enhanced by arranging afine grid near the solid–fluid interface to meet the computationalrequirements of the applications.

With the present method, the whole computational domain isoccupied by incompressible fluids, but the solid body is identifiedby a volume of solid (VOS) function η, and the velocity field in thesolid body is prescribed. As shown in Fig. 2, the function ηcoinciding with the location of velocity vectors equals 0 and 1 forfluid and solid, respectively. After the solid body is identified,the terms f i

n are calculated by momentum equation. DiscretizingEq. (2) in explicit scheme and then rearranging it, we obtain

f in ¼ unþ1

i �uni

Δt�rhs ð11Þ

where rhs includes the viscosity terms, the pressure terms, theadvection terms and the external acceleration terms. To prescribethe velocity of the solid body with Usinþ1 at the point where η¼ 1,the forcing term can be described as

f in ¼ ηi

Usinþ1�uni

Δt�rhs

� �ð12Þ

Since the external force terms f in are obtained and substituted

into Eq. (6), the velocity field in the solid body is imposed, andthen the effects of structures can be represented. Besides thissimple calculation of forcing terms, there is no other specialtreatment with the present IB method in the entire computationalprocess. Therefore, for a grid, the computational time for cases of astationary and a moving body is almost the same, due to the fixedgrid arrangement that essentially reduces computational cost andmemory requirements. Furthermore, rather than employing theintegration method over the interface, simple addition of theexternal force is used to calculate the forces acting on the solidbody even if it is moving (Shen et al., 2009). These advantages are

Fig. 2. Illustration of solid and fluid points with η¼ 1 and 0.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124110

significant for engineering applications with complex and movingstructures.

2.5. Non-inertial coordinate system for fluid sloshing

Because forced sloshing is studied, the moving frame needs tobe considered. In this study, a non-inertial coordinate system asshown in Fig. 3 is used, which translates and rotates relative to theinertial coordinate system. It is noted that the origin is located atthe left wall of the rectangular tank and on the still water surface.Thus, a non-inertial coordinate system can be used to representthe general roll (displayed by θ) or pitch of the tank. In the non-inertial coordinate system, the external acceleration ani in the N–Sequations includes the gravitational acceleration, the translationaland rotational inertia forces, the vector form of which is trans-ferred as follows (Lin, 2008):

a¼ g� du0

dt� dw

dt� r�2w � dr

dt�w � ðw � rÞ ð13Þ

where u0 and w are the translational and rotational velocityvectors respectively in the non-inertial coordinate, and g¼�g sin θ i

!�g cos θ j!

. Further, r is the position vector of thepoint of interest relative to the rotational origin. The second termof the right-hand side is the translational inertia, and the third,fourth and fifth terms, which are due to the rotational motion,represent angular acceleration, Coriolis force and centrifugal force,respectively.

Numerical simulations of the fluid sloshing in a horizontallyexcited 2D rectangular tank are presented using this two-phaseflow model, so the external force can be simplified as

f′¼ g� du0

dt; ð14Þ

and then the external acceleration ani can be obtained as the formof the gravitational acceleration and the horizontal acceleration ofthe tank.

3. Numerical results and discussion

3.1. Various movements of a cylinder in fluid

To demonstrate the superiority of the present IB method formoving and complex structures, two cases of a cylinder with

translational and rotational motions are carried out. Compared tothe majority of other methods for dealing with a moving cylinder,the IB method is easy to use in simulating these two cases becauseonly the external forcing terms in the fluid need to be predicted,which means that the grid is constant in the whole computationalprocess. Furthermore, we also demonstrate that the ease of thissimplified algorithm does not sacrifice accuracy of the results.It should be noted that all previous numerical data cited in thissection were obtained by moving grid methods, and comparisonswith these data can better demonstrate the high accuracy of thepresent IB method.

The drag and lift coefficients used in this section are defined as

Cd ¼ � Fx0:5ρU1D

; ð15Þ

Cl ¼ � Fy0:5ρU1D

; ð16Þ

where Fx and Fy are the forces acting on the solid body in the twodirections; these forces are the same as the corresponding variantsin Shen et al. (2009); U1 is the velocity scale which is defineddifferently in the two cases; D is the diameter of the cylinder.

3.1.1. Translational motion of a cylinder in fluidThe first case is tested here to validate the present IB method

for structures with translation. The case of an in-line oscillatingcylinder, the diameter D of which is 1, is simulated in a fluiddomain Ω¼ ½�15;15� � ½�7:5;7:5�, the sketch of which is shownin Fig. 4, and the equations of the motion are

xcðtÞ ¼ �A sin ð2πf tÞ; ð17Þ

ucðtÞ ¼ �U1 cos ð2πf tÞ; ð18Þ

where U1 is the maximum velocity of the oscillation, A is theamplitude and f is the frequency. Two characteristic parameters ofthe Reynolds number, Re¼U1D=ν, and the Keulergan–Carpenternumber, KC ¼ U1f =D, are set to 100 and 5 respectively as thestandard values of comparison with former experimental andnumerical data. A 380�200 grid is employed with the minimumgrid size of δx¼ δy¼ 0:02 near the cylinder.

In Fig. 5, two velocity profiles u=U1 and v=U1 along the y-axisat four different locations, x¼ �0:6D, 0:0D, 0:6D and 1:2D, forthree different phase-angles, ϕ¼ 180 3 , 210 3 and 330 3 , are com-pared with the experimental data of Dütsch et al. (1998). Fig. 6shows the time history of the drag coefficient, Cd, of the cylinder,together with the numerical results of Dütsch et al. All thesecomparisons show good agreement and demonstrate that thepresent IB method is adequately accurate to deal with translatingstructures in the present model.Fig. 3. The inertial and the non-inertial coordinates.

Fig. 4. Sketch showing a cylinder with translational motion in fluid.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 111

3.1.2. Rotational motion of a cylinder in fluidThe second case is tested to validate the IB method for structures

with rotation. In this case, the flow passes a rotating cylinder, thediameter D of which is 1; a sketch of the computational domainΩ¼ ½�10;30� � ½�10;10� is shown in Fig. 7. Two characteristic

parameters of the Reynolds number, Re¼U1D=ν, and rotation rate,α¼ 0:5Dω=U1, are used to identify the flow, where U1 in this caseis the free stream velocity and ω is the angular velocity of thecylinder. To compare the experimental data of Coutanceau andMenard (1985) and the numerical results of Chen et al. (1993),Re¼ 200 and α¼ 2:07 are considered. A 390�200 grid with theminimum grid size of δx¼ δy¼ 0:02 near the cylinder is utilized.

Fig. 5. Velocity profiles u=U1 and v=U1 along the y-axis at four different x locations for three different phase-angles. Lines are the present results; symbols are theexperimental results: and at x¼ �0:6 D; and at x¼ 0:0D; and at x¼ 0:6D; and at x¼ 1:2D. (a) 1801, (b) 2101 and (c) 3301.

Fig. 6. Time history of the drag coefficient: lines are the present results; symbolsare the experimental results.

Fig. 7. Sketch showing a cylinder with rotational motion in fluid.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124112

Comparisons of the present results of instantaneous stream-lines at different times with other experimental data and numer-ical results are shown in Fig. 8, where good agreement is observed.Furthermore, for quantitative analysis, Fig. 9 displays the timehistory of the lift and drag coefficients with the numerical resultsof Chen et al. (1993), and very good agreement is achieved again.These comparisons demonstrate that the present IB method is arobust method for rotating structures in the present model.

3.2. Dam break over a dry bed

Simulations of a dam break over a dry bed have been investi-gated numerically by various researchers, such as Greaves (2005);Jeong and Yang (1998, 1999) and Ubbink (1997). These researchersfound that simulations carried out by a one-phase model wereinsufficiently accurate, due to their neglect of the air effect and theerror of extrapolation on the free surface boundary condition.Furthermore, a dam break with an obstacle cannot be simulatedusing a one-phase IB–VOF model. The goal of this section is todemonstrate the capacity and accuracy of the present two-phase

model, especially considering the three-phase interface, i.e., theoverlap of free surface and solid boundary.

3.2.1. Dam break without an obstacle in the flowFirst, the problem of a dam break over a dry bed is investigated,

and the computed results are compared with the laboratoryexperimental data of Martin and Moyce (1952). In the experiment,a rectangular column of water in hydrostatic equilibrium isconfined between two vertical walls. In Fig. 10, the dimensionsof the dam are presented and the aspect ratio (height to width) ofthe initial water column is two. At the beginning of the experi-ment, the right vertical wall of the rectangular column of water isinstantaneously removed. When the dam fails, under the down-ward action of gravity, the still water column behind the wallbegins to collapse. This 2D dam break problem is a very usefulbenchmark test. The test provides extreme conditions for asses-sing numerical stability, as well as the capability of the model totreat the free surface problem.

As shown in Fig. 11, the non-dimensional surge front positionsX ¼ x=L versus the non-dimensional time T ¼ t

ffiffiffiffiffiffiffiffiffiffiffi2g=L

pobtained by

Fig. 8. Comparison of the instantaneous streamlines at different times for a rotating cylinder of Re¼ 200 and α¼ 2:07. (a) The present numerical results; (b) theexperimental results of Coutanceau and Menard (1985); and (c) the numerical results of Chen et al. (1993).

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 113

the present two-phase model, which is implemented on 32�32,64�64, and 128�128 uniform grids respectively, compare to thenumerical results of our one-phase model and experimental dataof Martin and Moyce (1952). Fig. 12 shows a similar comparison ofthe non-dimensional height of the water Y ¼ y=2L versus the non-dimensional time T ¼ t

ffiffiffiffiffiffiffiffig=L

pat the left side wall between the

simulations and experiment. The calculated results comparesatisfactorily with the experimental data and are obviously moreaccurate than the results from the one-phase model. Thus, thesuperiority in capability and accuracy of the two-phase model forthe simulation of the free surface problem is demonstrated.

Snapshots of the evolution of the free interface and velocityvectors calculated on a uniform 128�128 grid at different times tare presented in Fig. 13. At t ¼ 0:1 s, the column of water is greatlyaccelerated and moves rapidly along the bottom wall. Later, thewater column impacts against the right-hand wall and climbs upit. Then the water against the right-hand wall begins to fall backunder the influence of gravity. When the surge front falls downfrom the right-hand wall and plunges into the bottom water, air isentrapped by the water, forming air bubbles. The tongue of themoving water impinges on the left wall and again traps airbubbles. The velocity vector field shows that a large vortex isformed in the air in the vicinity of the water surface.

3.2.2. Dam break with an obstacle in the flowFollowing Section 3.2.1, another simulation of a dam break, this

time with an obstacle in the flow, is carried out. The physicalmodel is established according to the simulation of Ubbink (1997),as shown in Fig. 14, and all parameters of the physical model areidentical to those of Ubbink. The computational domain is dis-cretized by a 100�100 uniform grid. Free-slip boundary condi-tions are applied to the left and right boundaries and no-slipboundary conditions are applied to the other boundaries.

From Figs. 15 and 16, the performance of evolution of theinterface together with velocity vectors can be compared withUbbink's (1997) simulation and experimental results at the timest ¼ 0:1 s, 0:2 s, 0:3 s, 0:4 s, 0:5 s and 0:6 s respectively. Ubbink'ssimulation was performed with the “compressive interface captur-ing scheme for arbitrary meshes” method and the experiment wascarried out using an open-topped container indicated by thedotted line in the photographs (Ubbink, 1997). From Figs. 15 and16, very satisfactory agreement is achieved with Ubbink's simula-tion and experimental results. In particular, it is noted that thesheet of water behaves remarkably when it impinges against theobstacle and the right wall. As seen in Fig. 15, the leading-edgewater is obstructed by the obstacle and forms a tongue-shapedwave which bounces up from the upper left corner of the obstaclein the direction of the opposite wall at t ¼ 0:2 s. Simultaneously,a remarkable eddy establishes, exactly at the peak of the tongue-shaped wave, that can also be seen clearly in Ubbink's simulation.When t ¼ 0:3 s, the sheet of water keeps advancing in the direction

Fig. 9. Time history of the (a) lift and (b) drag coefficients for a rotating cylinder ofRe¼ 200 and α¼ 2:07.

Fig. 10. Dimensions of the computation domain of the dam.

Fig. 11. Surge front position. Fig. 12. Height of water column.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124114

of the right wall. Then at t ¼ 0:4 s, the tongue-shaped water sheetimpinges against the right wall, trapping air beneath it. Conse-quently, the fall of the water under the influence of gravity will beresisted by the trapped air but cannot be stopped. Ultimately, thewater strikes the bottom wall and rushes toward the obstacleagain. At the same time, a secondary tongue-shaped mass of wateris generated, falling from the upper right corner of the obstacle.These phenomena can also be seen in both simulation results andthe snapshot at t ¼ 0:5 s. When t ¼ 0:6 s, the secondary tongue-shaped water mass keeps growing and then impinges with thesheet of water rushing towards the obstacle along the bottomwall.As a result, the water to the right of the obstacle becomesturbulent and contains a high amount of air. From the wholeevolution of the interface and features of the flow distribution, theconclusion can be drawn that the results simulated in this studyagree well with Ubbink's simulation results and experimentalobservations.

To further validate the model quantitatively, another similarsimulation is performed, referring to the work of Cruchaga et al.(2009). As Fig. 17 shows, the evolution of a dimensionless interfacevertical position (y=L) at the left wall is recorded for comparisonwith the numerical simulation and experimental results of Cru-chaga et al. In Fig. 17, the numerical results of the current studyfollow a similar trend to the simulation and experimental averageof the results of Cruchaga et al. In particular, the curves before the

first trough show excellent agreement with the simulation. Mean-while, the curve of the present study performs more unsteadilythan the other two curves. The reason for the difference is the usein the current study of the integral height of the interface insteadof the actual height of the continuous free surface, considering thepossible turbulence of small drops above the surface.

3.3. Water exit and entry of a horizontal circular cylinder

The problem of water exit and entry of a cylinder is significantin various practical applications. Such complicated physical pro-cesses include breakup of free surface, wave–body interaction,wave–vortex interaction, etc. Investigating these processes isuseful for understanding the impacts of wave energy convertersunder extreme waves. An early and typical study of water exit andentry of a horizontal circular cylinder was presented by Greenhowand Lin (1983), who conducted experiments to show the consider-able difference of free surface deformation between the entry andexit processes. This problem has subsequently been studied withdifferent numerical methods by many researchers, includingGreenhow and Moyo (1997); Lin (2007) and Yang and Stern(2009). In this study, this typical problem is revisited based onthe present IB–VOF viscous two-phase flow model, for furthervalidation.

The uniform computational domain and principal parametersfor the both water exit and entry problems are shown in Fig. 18.The domain is 40 m� 24 m, the water depth is h¼ 20 m, and theradius of this horizontal cylinder is R¼ 1 m. In these simulations,the gravity is set to be g¼ 1:0 m=s2 and the cylinder is given aconstant upward or downward velocity V from T ¼ Vt=d¼ 0 tothe end of the simulation. The dynamic viscosity and density ofwater are 1� 10�3 kg=ms and 1� 103 kg=m3, while those of airare 1:8� 10�5 kg=ms and 1 kg=m3, respectively. A non-uniformgrid 400�300 is adopted and a minimum grid size of Δx¼Δy¼ 0:05 m, which is the same as that in Lin's (2007) work, isused to cover the path of the cylinder and near the air–waterinterface.

3.3.1. Water exit of a cylinderFor the water exit problem, the cylinder is initially placed under

a calm water surface and the distance of its center to the air–waterinterface is d¼ 1:25 m. The cylinder is then given a constantimpulsive upward velocity V ¼ 0:39 m=s. Thus the two dimension-less parameters are ε¼ R=jdj ¼ 0:8 and Fr ¼ V=

ffiffiffiffiffiffiffiffigjdj

p¼ 0:35.

Fig. 19 shows snapshots of the interactions between thecylinder and the two-phase flow. Fig. 19(a) shows the air–waterinterface position in the initial calculation. According to Fig. 19(b),two shear layers develop along the left and right sides of the

Fig. 13. Dam break interface evolution and velocity vectors for grid 128�128.

Fig. 14. Dimensions of the computation domain of the damwith an obstacle on thebottom wall.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 115

cylinder as the cylinder moves upward. Later, two vortices shedfrom these two shear layers interact with the air–water interfaceand form two dipoles beneath the interface. As shown in Fig. 19(h)–(j), waves are generated in the exit process and they arebroken due to strongly negative pressures arising on the cylindersurface. Water droplets can also be captured in this simulation,as shown in these figures. In short, the simulation results andobservations are close to the large-eddy simulation of two-phaseturbulent flow model (Yang and Stern, 2009).

The numerical results of boundary element simulation reportedby Greenhow and Moyo (1997) were found be close to thetheoretical series expansion results derived by Tyvand and Miloh(1995) for certain cases. Fig. 20 shows the comparison between thecurrent results and the boundary element simulation (Greenhow

and Moyo, 1997), further validating the abilities of our model. Verygood agreement can be observed for the two time instancesT ¼ 0:4 and T ¼ 0:6 given here. The present model has no difficultyproceeding further up until the body is completely detached fromthe water surface at T ¼ 4:0, whereas Greenhow and Moyo's modelcan only be carried out up to T ¼ 1:0 when the free surface breaks.

3.3.2. Water entry of a cylinderFor the water impact and entry problem, the parameters and

overall setup are the same as for the water exit problem except thatthe cylinder begins its downward motion from a height of d¼ 1:25mto the calmwater. A larger constant downward velocity V ¼ �1:0 m=s

Fig. 15. Evolution of interface together with velocity vectors.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124116

is applied to create a stronger impact. This renders the dimensionlessparameters ε¼ R=jdj ¼ 0:8 and Fr ¼ V=

ffiffiffiffiffiffiffiffigjdj

p¼ 0:89.

Fig. 21 shows the simulation results of the water impact andentry problem at different times. According to Fig. 21(b) and (c),water is pushed up during the initial impact of the body on thefree surface but its motion follows the curvature of the bodysurface. From Fig. 21(d) to (f) two oblique jets are formed thatplunge to two sides of the water surface as the body moves furtherdownward. The vortices shed from the shear layers along thesurface of the cylinder attach to the air–water interface and followthe interface motion. As the cylinder reaches greater depth asshown in Fig. 21(g)–(l), a large amount of water is pulled down-ward and a large surface depression persists even after the body iscompletely submerged in the water. It can be seen from the figuresthat the air–water interface positions and vortices contours arealso very close to the results of Yang and Stern (2009).

3.4. Fluid sloshing in a horizontally excited rectangular tank

Fluid sloshing in a tank is a typical problem during marinetransportation of liquefied natural gas (LNG) and other types offluid. Faltinsen (1978) proposed a linear analytical solution forfluid sloshing in a horizontally excited 2D rectangular tank andthis solution has been widely used. However, this solution is notavailable for many current investigations that aim to reduce theeffect of the fluid sloshing impact, such as considering baffles intanks. In this section we demonstrate that the present two-phasemodel is an advantageous tool for such investigations and providesome significant phenomena.

3.4.1. Fluid sloshing without a baffleThis section presents the numerical simulation results for fluid

sloshing in a horizontally excited 2D rectangular tank without abaffle, using the two-phase model. These results are compared

Fig. 16. Simulation by Ubbink (left) and observation in experiment (right) (Ubbink, 1997).

Fig. 17. Comparison of dimensionless interface vertical position at the left wallbetween the numerical results of the current study, simulation, and the experi-mental average from Cruchaga et al. (2009).

Fig. 18. Sketch of the computational domain of the water exit problem.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 117

with the linear analytical solution developed by Faltinsen (1978).

η¼ 1g

∑1

n ¼ 1sin ðknðx�L=2ÞÞ cos hðknhÞ½�Anωn sin ðωntÞ

�Cnω sin ðωtÞ�� 1gAωðx�L=2Þ sin ðωtÞ ð19Þ

ωn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigkn tan hðknhÞ

pð20Þ

Fig. 19. Water exit problem: the air–water interface position (solid black line) and vortices contours (�10oωo10 with intervals of 0.1). (a) T¼0.0, (b) T¼0.2, (c) T¼0.4,(d) T¼0.6, (e) T¼0.8, (f) T¼1.0, (g) T¼1.5, (h) T¼2.0, (i) T¼2.5, (j) T¼3.0, (k) T¼3.5, (l)T¼4.0.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124118

Fig. 20. Comparison of the air–water interface profiles: “–” present simulation; “o” boundary element simulation (Greenhow and Moyo 1997). (a) T=0.4 and (a) T=0.6.

Fig. 21. Water impact and entry problem: the air-water interface position (solid black line) and vortices contours (�20oωo20 with intervals of 0.2). (a) T¼0.0, (b) T¼0.2,(c) T¼0.4, (d) T¼0.6, (e) T¼0.8, (f) T¼1.0, (g) T¼1.5, (h) T¼2.0, (i) T¼2.5, (j) T¼3.0, (k) T¼3.5, (l) T¼4.0.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 119

kn ¼2n�1

Lπ ð21Þ

Cn ¼ωDn

ω2n�ω2 π ð22Þ

An ¼ �Cn�Dn

ωð23Þ

Dn ¼ 4ωAð�1Þn�1

L cos hðknhÞ1

k2nð24Þ

where L is the tank length, h is the filled water depth, and n is themode number.

In this study, the parameters L and h are set as 0:57 m and0:15 m, as shown in Fig. 22. The lowest natural frequencyω1 ¼ 6:0578 rad=s can be therefore computed by Eq. (20). Thetank is subjected to excitation expressed as a velocity function, i.e.,u¼ �A cos ðωtÞ where A¼ bω is the velocity amplitude withb¼ 0:005 m being the displacement amplitude and the angularfrequency ω being the external excitation frequency.

In the following simulations, we have chosen ω=ω1 to be equalto 1.1, 0.95, 0.9, 0.85 and 0.8. The grid system adopted here has 114uniform horizontal meshes and 104 non-uniform vertical meshes,with the minimum dx¼ 0:0025 m being near the free surface.

The results of the time history of free surface elevation for thedifferent excitation frequencies at x¼ 0:02 m and x¼ 0:55 m areshown in Figs. 23–27. Because the distances between the two

Fig. 22. Dimensions of the computation domain of fluid sloshing in a tank withoutbaffles.

Fig. 23. Comparison of the time history of free surface elevation ω=ω1 ¼ 1:1.

Fig. 24. Comparison of the time history of free surface elevation ω=ω1 ¼ 0:95.

Fig. 25. Comparison of the time history of free surface elevation ω=ω1 ¼ 0:9.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124120

positions and the left and right wall respectively are 0:02 m, theresults of the two positions are symmetric. Comparisons of theanalytical solution and the numerical results of the two-phasemodel show that they are in agreement at the two positions andthe numerical result deviates from the linear analytical solutiondue to the nonlinear effect and the air effect. The linear analyticalsolution always shows a symmetric wave pattern, whereas thenumerical results can present a typical nonlinear wave phenom-enon, i.e. the wave crest becomes sharper and the trough becomesflatter near the resonant frequency. That phenomenon is clearlyseen from these figures.

Furthermore, according to Eq. (20), the wave history is mainlymodulated by the natural frequencies ωn(n¼ 1;2⋯) and theexcitation frequency ω. For the former, the lowest naturalfrequencyω1 is dominant and others make a far smaller contribu-tion. Therefore, the wave time history is actually dominated by theexcitation frequencies ω and ω1.

3.4.2. Fluid sloshing with a vertical baffleFluid sloshing in a horizontally excited 2D rectangular tank

with a vertical baffle is also simulated using this two-phase model.The dimensions of the computational domain of the fluid sloshingin a tank with a vertical baffle are shown in Fig. 28.

We have chosen ω=ω1 to be equal to 0:95. The results of thetime history of free surface elevation for different vertical baffleheights Hb at x¼ 0:02 m and x¼ 0:55 m are shown in Fig. 29. It isclearly evident from the figure that as the baffle height Hb

increases, the sloshing motion of the fluid becomes slightlyweaker due to the blocking effects of the baffle on the fluidconvection. At the same time, the behavior of the free surfacebecomes more stable. Fig. 30 presents snapshots of the liquidmotion including the free surface deformation and the velocity fordifferent vertical baffle heights.

In order to discuss the detailed mechanism of the evolution ofvortices in the baffled tank, interaction processes of the evolutionof vortices with different vertical baffle heights are depicted in

Fig. 31(a)–(c). It is evident from the figures that as the baffle heightHb increases, more and larger vortices appear in the baffled tankdue to the hydrodynamic damping of the baffle, including theblocking effects and the viscosity of the baffle walls. We nextdiscuss the detailed evolution of vortices in the baffled tank withHb ¼ 0:12 m.

First, a vortex is generated near the baffle tip due to a separatedshear layer. This vortex is destroyed by the wave sloshing from rightto left. As time progresses, a larger vortex generated to the right ofbaffle tip becomes apparent. Then a strong vertical jet occurs alongthe right wall of the baffle, intercepting the link between the vortexand the baffle, and subsequently vortex shedding occurs. Thevortices continue shedding from the baffle tip and then interactwith the sloshing flow, resulting in the formation of a snaky flow.The head of the snaky flow impinges on the free surface andpartitions the flow near the free surface into two parts. Theseparated free surface stream, interacting with the snaky flow,creates other vortices. The direction of the snaky flow varies withtime on account of the interactions among vortices and sloshingflow. As the baffle height increases, it slightly suppresses the fluidsloshing because of the hydrodynamic damping of the baffle,including the blocking effects and the viscosity of the baffle walls.

Fig. 27. Comparison of the time history of free surface elevation ω=ω1 ¼ 0:8.

Fig. 26. Comparison of the time history of free surface elevation ω=ω1 ¼ 0:85.

Fig. 28. Dimensions of the computation domain of fluid sloshing in a tank with avertical baffle.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124 121

Fig. 29. Time history of free surface elevation for different vertical baffle heights ω=ω1 ¼ 0:95.

Fig. 30. Snapshots of the velocity vector with different baffles ω=ω1 ¼ 0:95.

Fig. 31. Velocity vector near the vertical baffle ω=ω1 ¼ 0:95 and (a) Hb ¼ 0:04 m (b) Hb ¼ 0:08 m and (c) Hb ¼ 0:12 m at t ¼ 1; 2; 3; 4; 5 s respectively.

C. Zhang et al. / Ocean Engineering 74 (2013) 107–124122

4. Conclusion

In this study, an incompressible viscous immiscible two-phaseflow model on a fixed Cartesian grid is developed. This work isbased on a PLIC–VOF method for treatment of the air–waterinterfaces and an IB method for treatment of the moving struc-tures. Detailed numerical methods for the solution of the presentmodel, such as the two-step projection method for velocity–pressure coupling, Helmholtz equations and Poisson equationsolvers, VOF solver, and immersed boundary treatment, are dis-cussed. Furthermore, as forced sloshing problems are studied,a non-inertial coordinate system is introduced.

Several cases with different purposes are used to comprehen-sively validate the two-phase flow model. The present IB methodis first tested to simulate a cylinder with various movements.The results demonstrate the capacity and accuracy of this methodin the model for a moving and complex solid–fluid interface,which is important for applications in ocean engineering. Then, adam break over a dry bed with or without an obstacle is analyzedby the two-phase model. This case shows that there are nolimitations to simulating submerged structure problems in thepresent IB–VOF model. Moreover, consideration of air effectsmeans that the simulations are closer to the realistic situationthan is possible with one-phase models, and this naturally leads toan increase in accuracy. To further illustrate the applicability of themodel in wave–structure interaction problems, the cases of waterentry and exit of a cylinder are presented. The air–water interfaceprofiles are compared with other numerical results and very goodagreement is obtained. Some intricate wave–structure, vortex–structure and wave–vortex interactions are observed.

After some validations are presented, fluid sloshing in a hor-izontally excited rectangular tank under different excitation fre-quencies is numerically investigated. First, the results of caseswithout baffles are shown to be similar to the linear analyticalsolutions. Then, to reduce the wave height near the left and rightwalls, baffles of different heights are set on the tank bottom. It is ofinterest to investigate to what extent the height of the baffleinfluences the free surface elevation and vortices in the water.From analyses and comparisons of the results of simulations, it isobvious that with the increase in baffle height, the waves near thewalls become calmer and the vortices near the baffle become largerand more frequent.

The results of this study show that the present two-phase flowmodel with the IB method and the VOF method is a potential toolfor simulation of ocean engineering applications with nonlinearfree surfaces and structures of arbitrary shape and motion.The fixed Cartesian grid system makes the model simple, accurateand stable. Current study is focused on further extending themodel to three dimensions, and investigating some more complexengineering problems.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (51009069) and the Fundamental ResearchFunds for the Central Universities – South China University ofTechnology, Ministry of Education of China (2012ZZ0097).

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