DSpace at VNU: Verification of a VOF-based two-phase flow model for wave breaking and wave-structure interactions

Ocean Engineering 33 2006 1565–1588Verification of a VOF-based two-phase flow model for wave breaking and wave–structure interactions Phung Dang Hieua, , Katsutoshi Tanimotob a Department

Ocean Engineering 33 (2006) 1565–1588

Verification of a VOF-based two-phase

flow model for wave breaking and

wave–structure interactions

Phung Dang Hieua, , Katsutoshi Tanimotob

a

Department of Oceanography, Hanoi University of Science, VNU, 334-Nguyen Trai Str.,

Thanhxuan, Hanoi, Vietnam

b

Graduate School of Science and Engineering, Saitama University, 255-Shimo-Okubo, Sakura-ku,

Saitamashi, Saitama-ken 338-8570, Japan Received 25 May 2005; accepted 5 October 2005 Available online 19 January 2006

Abstract

The objective of the present study is to develop a volume of fluid (VOF)-based two-phaseflow model and to discuss the applicability of the model to the simulation of wave–structureinteractions First, an overview of the development of VOF-type models for applications in thefield of coastal engineering is presented The numerical VOF-based two-phase flow model hasbeen developed and applied to the simulations of wave interactions with a submergedbreakwater as well as of wave breaking on a slope Numerical results are then compared withlaboratory experimental data in order to verify the applicability of the numerical model to thesimulations of complex interactions of waves and permeable coastal structures, including theeffects of wave breaking It is concluded that the two-phase flow model with the aid of theadvanced VOF technique can provide with acceptably accurate numerical results on the route

to practical purposes

r2005 Elsevier Ltd All rights reserved

Keywords: Numerical simulation; Two-phase model; Wave breaking; Submerged breakwater; Porous breakwater

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter r 2005 Elsevier Ltd All rights reserved.

doi:10.1016/j.oceaneng.2005.10.013

Corresponding author Fax: +84 4 8584945.

E-mail addresses: hieupd@vnu.edu.vn (P.D Hieu), tanimoto@post.saitama-u.ac.jp (K Tanimoto).

1 Introduction

Along with physical experiments, numerical simulations are useful tools fordesigning coastal structures as well as for understanding natural hydrodynamicprocesses in the field of coastal engineering Among the numerical models, thevolume of fluid (VOF)-type model has been attracted as a potential tool to constructpractical numerical wave channels in the past decade The VOF-type model cansimulate flows including the shape and evolution of the free surface, thus thecomplex free surface boundary can be efficiently simulated

Since the appearance of the VOF method (Hirt and Nichols, 1981) with theSOLA-VOF code (Nichols et al., 1980), the VOF-based model has been applied tosolve various problems in the fields of casting, coating, dynamics of drops, thin film,oil spilling, spray deposition, melting process in metallurgical vessels, shiphydrodynamics, etc In the field of coastal engineering, the VOF-type models arenot yet widely applied but steadily developed

Austin and Schlueter (1982) presented the first application of the SOLA-VOFmodel in the field of coastal engineering Their study was to predict flows in a porousarmor layer of a rectangular block breakwater.Lemos (1992) incorporated a k2eturbulence model in the SOLA-VOF code that allowed a limited description of theturbulence Van der Meer et al (1992) developed a VOF-based model, namelySKYLLA model, with the incorporation of the FLAIR algorithm (a second-orderaccurate VOF method;Ashgriz and Poo, 1991) After that, the SKYLLA model wasfurther developed by Petit et al (1992) and by Van Gent et al (1994) for thesimulation of wave action on and in a porous structure.Iwata et al (1996) used amodified SOLA-VOF model for numerical comparison with experimental data ofbreaking and post-breaking wave deformation due to submerged impermeablestructures

Lin and Liu (1998)incorporated a k2e turbulence model in the SOLA-VOF code,and then studied turbulence generated by breaking waves on a 1

35 sloping bottom.Numerical results were compared with experimental data byTing and Kirby (1994).However, the wave height distribution was not shown for the comparison.Zhao andTanimoto (1998) incorporated a sub-grid scale Smangorinski turbulence model inthe SOLA-VOF code, and applied to study the deformation of breaking waves on asubmerged impermeable reef Their study provided with information of turbulenceeddy viscosity distribution during the waves passing over the shallow reef.Kawasaki(1999)included a non-reflective wave maker source in the SOLA-VOF code to studythe deformation of breaking waves over a rectangle submerged obstacle Numericalresults were then compared with experimental data.Bradford (2000)used a state ofthe art commercial VOF-based software to simulate the surf zone dynamics and thenumerical results were compared with the experimental data (Ting and Kirby, 1994).The numerical results of Bradford underestimated the wave crests near the breakingpoint and inside the surf zone Bradford also found that the numerical resultscalculated using a first-order accurate scheme and using a second-order accuratescheme for the convective terms were not significantly different.Zhao et al (2000)

used a VOF-based model to simulate breaking waves with the condition similar to

that of the experiment byTing and Kirby (1994) The results of Zhao et al showedbetter agreement with the experimental data compared with the results ofBradford(2000) However, the calculated wave crest was overestimated at the breakinglocation, as well as inside the surf zone.

Isobe (2001)simulated wave overtopping a vertical wall using a modified VOF-based model.Hur and Mizutani (2003)used a VOF-based model to simulatethe interaction of waves and a permeable submerged breakwater and to estimate thewave force acting on it.Hus et al (2002)included the volume averaged equations forporous flows derived byVan Gent (1995) into the VOF-based model proposed by

SOLA-Lin and Liu (1998), to study wave motions and turbulence flows in front of acomposite breakwater Comparisons of the numerical results and laboratory datashowed a good agreement.Shen et al (2004)used a VOF version of the SOLA-VOFcode with a two-equation k2e model to simulate the propagation of non-breakingwaves over a submerged bar Their simulated results showed a reasonable agreementwith experimental data by Ohyama et al (1995) Zhao et al (2004) introduced amulti-scale turbulence model into the SOLA-VOF code and studied the turbulence inbreaking waves on a sloping bottom Their numerical results showed betteragreements with experimental data (Ting and Kirby, 1994, 1995, 1996) comparedwith other results byBradford (2000)andLin and Liu (1998), those were simulatedusing the k2e model However, the discrepancy between simulated and measuredresults was still significant, roughly about 20% of wave height at the breaking pointand more inside the surf zone for the spilling breaker

Recently, the Port and Harbor Research Institute (PHRI), Japan (2001) loped a numerical wave channel based on the VOF method (Hirt and Nichols, 1981),namely CADMAS-SURF, which can simulate the wave and structure inter-action in a wave channel including the wave overtopping process with acceptableaccuracy

deve-Some studies based on different methods other than the VOF method for thesimulation of wave movements may be shortly mentioned.Sakakiyama and Kajima(1992)proposed a modified Navier–Stokes equation extended to porous media andapplied the marker and cell method (Hallow and Welch, 1965) to study non-linearwaves interacting with permeable breakwaters In their study, the interactions ofwaves and the rubble-mound breakwater and caisson breakwater covered witharmors units were numerically simulated.Watanabe et al (1999)proposed a densityfunction method and developed a numerical model using the Constraint Interpola-tion Profile (CIP) method to study the wave overtopping and vortex generated byovertopping behind a vertical wall Their results presented a good agreement withexperimental data by Goda et al (1967) Gotoh et al (1999) studied the wavebreaking and overtopping at an upright seawall using the particle method.Kato et

al (2002)studied the eddy structure around the head of a vertical wall breakwaterwith wave overtopping using a three-dimensional large eddy simulation (LES) modelproposed byWatanabe and Saeki (1999)

It should be pointed out that the studies for the simulation of wave motionsmentioned above are based on single-phase flow models, in which the effects ofair movement above the free water surface are ignored (only the liquid flow is

simulated and the gas dynamics is neglected) The VOF method is used to trackthe movement of the free surface, and the density difference between liquid andgas is also neglected As a result, for the case of wave breaking, the trapped airbubbles inside the water and the splashed water in the air are not fully treated.Also, at the free surface, unknown physical quantities need to be extrapolated.For example, pressure at a cell nearby the free surface is extrapolated by alinear approximation Velocity components at the surface need estimating either

by a linear approximation or by using the continuity equation These tions may result in missing information about those quantities, and become a source

approxima-of errors

On the simulation of breaking waves,Christensen et al (2002) pointed out thatsince the mixture of air and water in the roller region, on average, has a smallerdensity than that of the water, the turbulence produced in the roller would havedifficulties in penetrating the underlying fluid Therefore, a large part of theproduction and dissipation takes place in the roller before it is diffused down-ward, which explains the overestimation on the turbulence in the surf zone bynumerical models so far Moreover, the de-entrainment of air bubbles from thewater after the wave breaking may release some wave energies to the air andmay contribute significantly to the wave energy dissipation process Therefore,developing a model, which can account for the interaction between air and water, isessential to reduce the shortage of the single-phase model Recently, some numericalstudies based on the VOF-based two-phase flow model for the simulation ofwater wave motions have been reported Hieu and Tanimoto (2002) developed aVOF-based two-phase flow model to study wave transmission over a submergedobstacle Karim et al (2003) developed a VOF-based two-phase flow model forwave interactions with porous structures and studied the hydraulic performance

of a rectangle porous structure against non-breaking waves Their numerical resultssurely showed good agreements with experimental data Hieu et al (2004) simu-lated breaking waves in a surf zone using a VOF-based two-phase flow model.Their numerical results were compared with experimental data provided by Tingand Kirby (1994)for the spilling breaker on a sloping bottom Their results agreedwell with the experimental data However, the wave motion in porous media andthe non-reflective wave source method were not included in the model by Hieu

et al (2004)

In this paper, a VOF-based two-phase flow model by Hieu et al (2004) isdeveloped for the simulation of wave and porous structure interactions Instead ofusing an absorbing wave maker method (Zhao and Tanimoto, 1998) in the previousmodel (Hieu et al., 2004), the internal wave generation source method (Ohyama andNadaoka, 1991) is used in this study The model is then verified against laboratoryexperimental data for wave breaking on a sloping bottom and wave breaking over asubmerged porous breakwater, in order to give an overall look on the applicability ofthe VOF-based two-phase flow model to the simulation of wave breaking as well aswave interactions with porous structures Discussions on the effects of porosity ofthe submerged breakwater to the wave reflection, transmission and dissipation arealso given

2 Model description

2.1 Governing equations

The governing equations accounting for interactions between waves and porousstructures are applied for the developments of a numerical wave tank Based on theNavier–Stokes equations, Sakakiyama and Kajima (1992) developed a set ofequations for simulation of the unsteady turbulent flows in porous media, where theresistance of the porous media is modeled by the drag force and inertia force.VanGent (1995)developed a different set of equations for porous media, in which vanGent modeled the resistance force using Forchheimer law and inertia term Themodel of van Gent contains three parameters, and then it requires three empiricalcoefficients, which are essentially estimated by hydraulic experiments In the modelproposed bySakakiyama and Kajima (1992), there are two empirical coefficients onefor inertia force and the other one for drag force In this study, the incompressiblefluid is assumed then the set of the modified Navier–equations proposed by

Sakakiyama and Kajima (1992) are used as the governing equations for the phase flow model of water and air:

Dzw
Rz
gvg þ qw, ð3Þwhere t is the time, x and z the horizontal and vertical coordinates, u and w thehorizontal and vertical velocity components; r the density of the fluid; p the pressure;

ne the kinematic viscosity (summation of molecular kinematic viscosity and eddykinematic viscosity); g the gravitational acceleration; gv the porosity; gx and gz

the areal porosities in the x and z projections; q the source of mass for wavegeneration; qu, qwthe momentum source in x and z direction (the resultants from theconvective terms and viscous terms in the momentum equations due to the source ofmass in the continuity equation); Dx, Dzthe coefficient of energy damping in the xand z direction, respectively; and Rx, Rzthe drag/resistance force exerted by porousmedia

lv, lx and lz are defined from gv, gx and gz, respectively, using followingrelationships:

lv¼gvþ ð1
gvÞCM,

lx¼gxþ ð1
gxÞCM,

here CMis the inertia coefficient

The resistance force Rxand Rzare described by the following equations:

The source of mass has the form as follows:

q ¼ qs at the source location:

2.2 Free surface boundary

The governing equations, which are applied for the simulation domain with bothpresences of air and water, need special considerations for the boundary between theair and the water The fluid is assumed incompressible then the density is constant inthe air zone and in the water zone To distinguish the two zones by an equation, theVOF method (Hirt and Nichols, 1981) is used in this study

The VOF method introduces a VOF function F to define the fluid region Thephysical meaning of the F function is the fractional volume of a cell occupied by thewater In particular, a unit value of F corresponds to a cell full of water, while a zerovalue indicates that the cell contains no water Cells with F value between zero andunity must then contain the free surface The algorithm for tracking the interfaceconsists of two steps In the first step, the interface is approximated by a linear linesegment at each cell (Youngs, 1982), which has the value of fractional functionbetween zero and unity In the second step, the interface in each cell is tracked bysolving an advection equation of the fractional function F to get the evolution of thefractional function in time The two-dimensional advection equation for the

fractional function is written as

where qF is the source of F due to the wave maker source method

The sharp gradient of VOF function F needs to be conserved at the free surface,otherwise the interface between air and water will lose its definition; the specialalgorithms other than the conventional finite-difference schemes for solving theadvection Eq (10) are considered There are a number of such algorithms as thedonor–acceptor (Hirt and Nichols, 1981), FLAIR (Ashgriz and Poo, 1991) andYoungs method (Youngs, 1982), etc In this study, we follow the new and simplePLIC method proposed byHieu (2004)

2.3 Equations of density and viscosity for two-phase flow model

In order to minimize the effects of the inaccurate interpolation for some physicalquantities at the free surface in the single-phase model, in this study a finite air zone

is included in the computation domain above the free surface To solve the wholecomputation domain with the presence of both air and water, the model needsequations accounting for the variation of density as well as of viscosity If weconsider incompressible, immiscible fluids, no phase change between fluids, then thevariable density and viscosity can be expressed using the fraction VOF function(Puckett et al., 1997;Renardy et al., 2001) as follows:

2.4 Turbulence model

For the estimation of the small-scale turbulence generated during wave breakingand contribution of sub-grid scale turbulence, a turbulence model similar to LES isincorporated The basic for the LES simulation is a spatial filtering of theNavier–Stokes equations The length scale of the filtering depends on the grid size,which means that for a finer grid, a larger part of the turbulent motion is representeddirectly in the simulation using the filtered Navier–Stokes equations In the presentstudy, the governing Navier–Stokes equations are filtered using Smagorinsky scheme(Smagorinsky, 1963) In the Smagorinsky scheme, the momentum exchange bythe sub-grid scale turbulence is transported by means of an eddy viscosity term Theeddy viscosity (n) is determined from the strain rate (S ) of the flow field The

formula for estimation of the eddy viscosity for a two-dimensional model is given as

where Csis the model parameter with value in the range of 0:1pCsp0:2, and Dx and

Dz are, respectively, the grid sizes in x and z direction

2.5 Wave maker source method

In order to minimize the reflection of waves at the wave maker boundary, thesource wave maker method (Ohyama and Nadaoka, 1991) is employed in this study.The method consists of two parts, the source function and the damping zone Thesource function is added to the mass conservation equation (continuity equation) inorder to generate the desired incident waves, while the damping zone works as anenergy dissipation zone by adding a resistance force proportional to the flow velocity

to the momentum equations The equation for the source function is written as

is gradually intensified in the duration of three wave periods in order to guarantee astable regular wave train (Brorsen and Larsen, 1987)

2.6 Initial and boundary conditions

At the initial time, the still water condition is assumed in the computation domain.The velocity is set equal to zero for whole computational region, and the pressure isgiven by hydrostatic pressure The air density is chosen as 1.2 kg/m3and the freshwater density is 998.2 kg/m3

For the boundary between fluid and solid body, the no-slip condition is adopted

in this study At the top boundary, where the computation domain is connected

to the open air above, the continuative conditions are applied for velocity.These conditions mean that the velocity components fully satisfy the continuityequation

2.7 Method of solution

The governing equations are discretized by a finite-difference scheme on astaggered grid mesh The velocity components are evaluated at cell sides, while scalarquantities are evaluated at the cell center The SMAC method (Simplified Markerand Cell Method) is used to get the time evolution solution of the governingequations The resultant Poisson equation of pressure correction due to the SMACmethod is solved using a bi-conjugate gradient method (Van der Vorst, 1992) In theprevious model proposed by Hieu et al (2004), the non-conservative CIP scheme(Yabe and Aoki, 1991) was used for the approximation of the convective terms in themomentum equations However, for a problem with the presence of porous media,the non-conservative CIP scheme falls to give good approximations for thesimulation of wave and porous structure interactions in the two-phase flow model

To solve the problem, in this study, a high resolution MUSCL (MonotonicUpstream-centered Scheme for Conservation Laws)-type second-order accuratescheme (Nessyahu and Tadmor, 1990) is employed for the convection terms, and thesecond-order central scheme is used for the viscous terms The computationalprocedure in detail can be found inHieu et al (2004) Here the brief explanation isgiven as follows:

(a) Give initial values for all variables

(b) Give boundary conditions for all variables

(c) Solve explicitly the momentum equations for the predicted velocities

(d) Solve the Poisson equation for the pressure corrections

(e) Adjust the pressure and velocity

(f) Solve the advection equation of VOF function using the PLIC algorithm fortracking the free surface

(g) Calculate the new density and kinetic viscosity based on the VOF values.(h) Calculate the turbulence eddy viscosity

(i) Return to step (b) and repeat for next time step until the end of specified time

3 Model verification

The model described above is validated against two experimental tests The first iswave breaking on a sloping bottom with the experimental conditions by Ting andKirby (1994, 1996) The second is the simulation of wave interactions with asubmerged porous structure

3.1 Wave breaking on a sloping bottom

3.1.1 Test conditions

The condition similar to the laboratory experiment byTing and Kirby (1994)isused for testing the numerical simulation of breaking waves on a sloping bottom Inthe laboratory experiment, a beach with uniform slope of 1 is connected to a region

with constant depth dc¼0:40 m.Fig 1shows the schematic view of the numericalwave channel where the coordinate system is chosen so that x ¼ 0 is located at theposition with the still water depth d ¼ 0:38 m on the slope, and z ¼ 0 is located at thestill water level (SWL) the same as the coordinate system in the laboratoryexperiment Test waves are regular and the incident wave heights HIand periods T inthe constant water depth are 0.125 m and 2.0 s for the spilling breaker, and 0.128 mand 5.0 s for the plunging breaker, respectively.

In the numerical wave channel, the computation domain is discretized using anorthogonal uniform grid with mesh size Dx ¼ 0:02 m in the horizontal direction, and

Dz ¼ 0:01 m in the vertical direction A wave generation source is set inside thecomputation domain together with a sponge damping zone on the left The slopingbottom is treated using the partial cell technique and wall function (Rodi, 1993)

A computation time of 50 s is set for the simulation in order to get stable timeprofiles of wave quantities Numerical results in terms of wave crest, trough andmean water level are used to compare against the experimental data (Ting andKirby, 1994) Velocities in a cross-section inside the surf zone are also compared withthe experimental data (Ting and Kirby, 1996)

3.1.2 Results and discussions

Fig 2shows the time profile of water surface elevation at the location x ¼
0:5 m

on the horizontal bottom and x ¼ 6:4 m nearby the breaking point From the results,

it is confirmed that the time profiles of water surface elevation at both locations arealmost stable after 30 s from the starting time of computation This gives us aconfidence in getting the mean wave quantities from the time profiles after a specifictime

Fig 3shows the simulation results compared with experimental data (Ting andKirby, 1994) for mean wave quantities (wave crests, troughs and mean water levels)

In the figure, lines indicate the simulated results and circles indicate the measureddata The wave crests and troughs are determined from the mean water level It is

d c = 0.40m SWL

Water zone

0

1.0m Damping zone Wave generation source Air zone

Fig 1 Sketch of the simulation domain.

Cal results (Zhao etal.,2004) Cal results(CADMAS-SURF)

Mean water level

seen that the wave crests and troughs are well simulated by the model Especially, atthe breaking point, the simulated breaking wave height is very accurate comparedwith simulated results byZhao et al (2004)andBradford (2000) However, at thevery near shore area, the present simulated results of wave crests are overestimated.The simulated mean water level is also slightly underestimated compared with theexperimental data The reasons for those disagreements could be due to thecomplicated turbulence dissipations in the surf zone, which is inadequately simulated

by the simple Smangorinsky’s turbulence model However, as seen from the figure,the simulated results by the present model are much more accurate than thosesimulated byZhao et al (2004), byBradford (2000)and by using CADMAS-SURFmodel It should be pointed out that the simulation using CASDMAS-SURF modelwas carried out with a fine mesh (Dx ¼ 0:02 m, Dz ¼ 0:01 m) and the sloping bottomwas treated by the partial cell and wall function technique The Stokes fifth-orderwave source was applied for generating the incident waves The CADMAS-SURFmodel is known as one of the most accurate single-phase models for the simulation

of wave motion among the coastal engineer community The better results obtained

by the present model in the comparison with the results by CASDMAS-SURFmodel prove that the incorporation of the treatment of air motions in the presentmodel has contributed significant improvement on the accuracy of the numericalsimulations of wave breaking Thus, the effects of the air movement on the wavemotion under wave breaking processes are not negligible Comparing the results bythe present model with the results byHieu et al (2004) (see Fig 3 in Hieu et al.,

2004), it is seen that the simulated wave crests by the present model are moreaccurate than those by Hieu et al in the area before the breaking point However,inside the surf zone, a similar distribution of wave crests is observed Both modelsaccurately simulated the breaking point location and distribution of wave trough.The better results obtained in this study may be due the incorporation of the wavegeneration source method and the MUSCL-TVD scheme in the present model.The water surface elevation, horizontal and vertical velocity components in across-section located at x ¼ 7:272 m inside the surf zone is considered for thecomparisons with experimental data (Ting and Kirby, 1996) Fig 4 showsthe variation of phase averaged water surface at the cross-section In the figure,

-0.2 0 0.2 0.4 0.6