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

ARTICLE IN PRESS Ocean Engineering 33 (2006) 1565–1588 www.elsevier.com/locate/oceaneng Verification of a VOF-based two-phase flow model for wave breaking and wave–structure interactions Phung Dang Hieua,Ã, 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 October 2005 Available online 19 January 2006 Abstract The objective of the present study is to develop a volume of fluid (VOF)-based two-phase flow model and to discuss the applicability of the model to the simulation of wave–structure interactions First, an overview of the development of VOF-type models for applications in the field of coastal engineering is presented The numerical VOF-based two-phase flow model has been developed and applied to the simulations of wave interactions with a submerged breakwater as well as of wave breaking on a slope Numerical results are then compared with laboratory experimental data in order to verify the applicability of the numerical model to the simulations of complex interactions of waves and permeable coastal structures, including the effects of wave breaking It is concluded that the two-phase flow model with the aid of the advanced VOF technique can provide with acceptably accurate numerical results on the route to practical purposes r 2005 Elsevier Ltd All rights reserved Keywords: Numerical simulation; Two-phase model; Wave breaking; Submerged breakwater; Porous breakwater ÃCorresponding author Fax: +84 8584945 E-mail addresses: hieupd@vnu.edu.vn (P.D Hieu), tanimoto@post.saitama-u.ac.jp (K Tanimoto) 0029-8018/$ - see front matter r 2005 Elsevier Ltd All rights reserved doi:10.1016/j.oceaneng.2005.10.013 ARTICLE IN PRESS 1566 P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 Introduction Along with physical experiments, numerical simulations are useful tools for designing coastal structures as well as for understanding natural hydrodynamic processes in the field of coastal engineering Among the numerical models, the volume of fluid (VOF)-type model has been attracted as a potential tool to construct practical numerical wave channels in the past decade The VOF-type model can simulate flows including the shape and evolution of the free surface, thus the complex free surface boundary can be efficiently simulated Since the appearance of the VOF method (Hirt and Nichols, 1981) with the SOLA-VOF code (Nichols et al., 1980), the VOF-based model has been applied to solve various problems in the fields of casting, coating, dynamics of drops, thin film, oil spilling, spray deposition, melting process in metallurgical vessels, ship hydrodynamics, etc In the field of coastal engineering, the VOF-type models are not yet widely applied but steadily developed Austin and Schlueter (1982) presented the first application of the SOLA-VOF model in the field of coastal engineering Their study was to predict flows in a porous armor layer of a rectangular block breakwater Lemos (1992) incorporated a k2e turbulence model in the SOLA-VOF code that allowed a limited description of the turbulence Van der Meer et al (1992) developed a VOF-based model, namely SKYLLA model, with the incorporation of the FLAIR algorithm (a second-order accurate VOF method; Ashgriz and Poo, 1991) After that, the SKYLLA model was further developed by Petit et al (1992) and by Van Gent et al (1994) for the simulation of wave action on and in a porous structure Iwata et al (1996) used a modified SOLA-VOF model for numerical comparison with experimental data of breaking and post-breaking wave deformation due to submerged impermeable structures Lin and Liu (1998) incorporated a k2e turbulence model in the SOLA-VOF code, and then studied turbulence generated by breaking waves on a 35 sloping bottom Numerical results were compared with experimental data by Ting and Kirby (1994) However, the wave height distribution was not shown for the comparison Zhao and Tanimoto (1998) incorporated a sub-grid scale Smangorinski turbulence model in the SOLA-VOF code, and applied to study the deformation of breaking waves on a submerged impermeable reef Their study provided with information of turbulence eddy 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 study the deformation of breaking waves over a rectangle submerged obstacle Numerical results were then compared with experimental data Bradford (2000) used a state of the art commercial VOF-based software to simulate the surf zone dynamics and the numerical results were compared with the experimental data (Ting and Kirby, 1994) The numerical results of Bradford underestimated the wave crests near the breaking point and inside the surf zone Bradford also found that the numerical results calculated using a first-order accurate scheme and using a second-order accurate scheme 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 ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1567 that of the experiment by Ting and Kirby (1994) The results of Zhao et al showed better agreement with the experimental data compared with the results of Bradford (2000) However, the calculated wave crest was overestimated at the breaking location, as well as inside the surf zone Isobe (2001) simulated wave overtopping a vertical wall using a modified SOLAVOF-based model Hur and Mizutani (2003) used a VOF-based model to simulate the interaction of waves and a permeable submerged breakwater and to estimate the wave force acting on it Hus et al (2002) included the volume averaged equations for porous flows derived by Van Gent (1995) into the VOF-based model proposed by Lin and Liu (1998), to study wave motions and turbulence flows in front of a composite breakwater Comparisons of the numerical results and laboratory data showed a good agreement Shen et al (2004) used a VOF version of the SOLA-VOF code with a two-equation k2e model to simulate the propagation of non-breaking waves over a submerged bar Their simulated results showed a reasonable agreement with experimental data by Ohyama et al (1995) Zhao et al (2004) introduced a multi-scale turbulence model into the SOLA-VOF code and studied the turbulence in breaking waves on a sloping bottom Their numerical results showed better agreements with experimental data (Ting and Kirby, 1994, 1995, 1996) compared with other results by Bradford (2000) and Lin and Liu (1998), those were simulated using the k2e model However, the discrepancy between simulated and measured results was still significant, roughly about 20% of wave height at the breaking point and more inside the surf zone for the spilling breaker Recently, the Port and Harbor Research Institute (PHRI), Japan (2001) developed a numerical wave channel based on the VOF method (Hirt and Nichols, 1981), namely CADMAS-SURF, which can simulate the wave and structure interaction in a wave channel including the wave overtopping process with acceptable accuracy Some studies based on different methods other than the VOF method for the simulation of wave movements may be shortly mentioned Sakakiyama and Kajima (1992) proposed a modified Navier–Stokes equation extended to porous media and applied the marker and cell method (Hallow and Welch, 1965) to study non-linear waves interacting with permeable breakwaters In their study, the interactions of waves and the rubble-mound breakwater and caisson breakwater covered with armors units were numerically simulated Watanabe et al (1999) proposed a density function method and developed a numerical model using the Constraint Interpolation Profile (CIP) method to study the wave overtopping and vortex generated by overtopping behind a vertical wall Their results presented a good agreement with experimental data by Goda et al (1967) Gotoh et al (1999) studied the wave breaking 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 breakwater with wave overtopping using a three-dimensional large eddy simulation (LES) model proposed by Watanabe and Saeki (1999) It should be pointed out that the studies for the simulation of wave motions mentioned above are based on single-phase flow models, in which the effects of air movement above the free water surface are ignored (only the liquid flow is ARTICLE IN PRESS 1568 P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 simulated and the gas dynamics is neglected) The VOF method is used to track the movement of the free surface, and the density difference between liquid and gas is also neglected As a result, for the case of wave breaking, the trapped air bubbles 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 a linear approximation Velocity components at the surface need estimating either by a linear approximation or by using the continuity equation These approximations may result in missing information about those quantities, and become a source of errors On the simulation of breaking waves, Christensen et al (2002) pointed out that since the mixture of air and water in the roller region, on average, has a smaller density than that of the water, the turbulence produced in the roller would have difficulties in penetrating the underlying fluid Therefore, a large part of the production and dissipation takes place in the roller before it is diffused downward, which explains the overestimation on the turbulence in the surf zone by numerical models so far Moreover, the de-entrainment of air bubbles from the water after the wave breaking may release some wave energies to the air and may contribute significantly to the wave energy dissipation process Therefore, developing a model, which can account for the interaction between air and water, is essential to reduce the shortage of the single-phase model Recently, some numerical studies based on the VOF-based two-phase flow model for the simulation of water wave motions have been reported Hieu and Tanimoto (2002) developed a VOF-based two-phase flow model to study wave transmission over a submerged obstacle Karim et al (2003) developed a VOF-based two-phase flow model for wave interactions with porous structures and studied the hydraulic performance of a rectangle porous structure against non-breaking waves Their numerical results surely showed good agreements with experimental data Hieu et al (2004) simulated breaking waves in a surf zone using a VOF-based two-phase flow model Their numerical results were compared with experimental data provided by Ting and Kirby (1994) for the spilling breaker on a sloping bottom Their results agreed well with the experimental data However, the wave motion in porous media and the 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) is developed for the simulation of wave and porous structure interactions Instead of using an absorbing wave maker method (Zhao and Tanimoto, 1998) in the previous model (Hieu et al., 2004), the internal wave generation source method (Ohyama and Nadaoka, 1991) is used in this study The model is then verified against laboratory experimental data for wave breaking on a sloping bottom and wave breaking over a submerged porous breakwater, in order to give an overall look on the applicability of the VOF-based two-phase flow model to the simulation of wave breaking as well as wave interactions with porous structures Discussions on the effects of porosity of the submerged breakwater to the wave reflection, transmission and dissipation are also given ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1569 Model description 2.1 Governing equations The governing equations accounting for interactions between waves and porous structures are applied for the developments of a numerical wave tank Based on the Navier–Stokes equations, Sakakiyama and Kajima (1992) developed a set of equations for simulation of the unsteady turbulent flows in porous media, where the resistance of the porous media is modeled by the drag force and inertia force Van Gent (1995) developed a different set of equations for porous media, in which van Gent modeled the resistance force using Forchheimer law and inertia term The model of van Gent contains three parameters, and then it requires three empirical coefficients, which are essentially estimated by hydraulic experiments In the model proposed by Sakakiyama and Kajima (1992), there are two empirical coefficients one for inertia force and the other one for drag force In this study, the incompressible fluid is assumed then the set of the modified Navier–equations proposed by Sakakiyama and Kajima (1992) are used as the governing equations for the twophase flow model of water and air: Continuity equation: qgx u qgz w ỵ ẳ qgv qx qz Modified Navier– Stokes equations (momentum in x and z directions):    qu qlx uu qlz wu gv qp q qu ỵ ẳ ỵ g ne lv ỵ qt qx qz qx r qx qx x    q qu qw g ne ỵ þ À Dx u À Rx þ qu , qz z qz qx    qw qlx uw qlz ww gv qp q qw qu ỵ ỵ ẳ þ g ne þ lv qt qx qz qx qz r qz qx x    q qw gz ne ỵ Dz w Rz gv g ỵ qw , qz qz (1) 2ị 3ị where t is the time, x and z the horizontal and vertical coordinates, u and w the horizontal and vertical velocity components; r the density of the fluid; p the pressure; ne the kinematic viscosity (summation of molecular kinematic viscosity and eddy kinematic 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 wave generation; qu, qw the momentum source in x and z direction (the resultants from the convective terms and viscous terms in the momentum equations due to the source of mass in the continuity equation); Dx, Dz the coefficient of energy damping in the x and z direction, respectively; and Rx, Rz the drag/resistance force exerted by porous media ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1570 lv, lx and lz are defined from gv, gx and gz, respectively, using following relationships: lv ẳ gv ỵ gv ịC M , lx ẳ gx ỵ gx ịC M , lz ẳ gz ỵ ð1 À gz ÞC M , here CM is the inertia coefficient The resistance force Rx and Rz are described by the following equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CD ð1 À gx ịu u2 ỵ w2 , Rx ẳ Dx Rz ẳ p CD gz ịw u2 ỵ w2 , Dz ð4Þ (5) (6) here Dx and Dz are the horizontal and vertical mesh sizes in porous media and CD is the drag coefficient The source of mass has the form as follows:  qs at the source location: q¼ (7) others: The momentum source in x and z direction (here we neglect the momentum source contributed by the viscous terms) is, respectively, given as qu ¼ uq, (8) qw ¼ wq (9) 2.2 Free surface boundary The governing equations, which are applied for the simulation domain with both presences of air and water, need special considerations for the boundary between the air and the water The fluid is assumed incompressible then the density is constant in the air zone and in the water zone To distinguish the two zones by an equation, the VOF method (Hirt and Nichols, 1981) is used in this study The VOF method introduces a VOF function F to define the fluid region The physical meaning of the F function is the fractional volume of a cell occupied by the water In particular, a unit value of F corresponds to a cell full of water, while a zero value indicates that the cell contains no water Cells with F value between zero and unity must then contain the free surface The algorithm for tracking the interface consists of two steps In the first step, the interface is approximated by a linear line segment at each cell (Youngs, 1982), which has the value of fractional function between zero and unity In the second step, the interface in each cell is tracked by solving an advection equation of the fractional function F to get the evolution of the fractional function in time The two-dimensional advection equation for the ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1571 fractional function is written as qgv F qugx F qwgz F þ þ ¼ qF , (10) qt qx qz 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 special algorithms other than the conventional finite-difference schemes for solving the advection Eq (10) are considered There are a number of such algorithms as the donor–acceptor (Hirt and Nichols, 1981), FLAIR (Ashgriz and Poo, 1991) and Youngs method (Youngs, 1982), etc In this study, we follow the new and simple PLIC method proposed by Hieu (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 physical quantities 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 whole computation domain with the presence of both air and water, the model needs equations accounting for the variation of density as well as of viscosity If we consider incompressible, immiscible fluids, no phase change between fluids, then the variable density and viscosity can be expressed using the fraction VOF function (Puckett et al., 1997; Renardy et al., 2001) as follows: r ẳ F ịra ỵ F rw , (11) n ẳ F ịna ỵ F nw , (12) where is the air density and rw is the water density and rw are molecular kinetic viscosity of air and water, respectively It is worth mentioning that the kinetic boundary condition is satisfied by the advection Eq (10) and the dynamic boundary condition for the free surface is automatically satisfied with the Navier–Stokes equations 2.4 Turbulence model For the estimation of the small-scale turbulence generated during wave breaking and contribution of sub-grid scale turbulence, a turbulence model similar to LES is incorporated The basic for the LES simulation is a spatial filtering of the Navier–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 represented directly in the simulation using the filtered Navier–Stokes equations In the present study, the governing Navier–Stokes equations are filtered using Smagorinsky scheme (Smagorinsky, 1963) In the Smagorinsky scheme, the momentum exchange by the sub-grid scale turbulence is transported by means of an eddy viscosity term The eddy viscosity (nt ) is determined from the strain rate (Sx,z) of the flow field The ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1572 formula for estimation of the eddy viscosity for a two-dimensional model is given as nt ¼ ðC s DÞ2 ð2S x;z Sx;z Þ1=2 , (13)   qu qw ỵ ẳ , qz qx (14) Sx;z D ẳ Dx Dzị1=2 , (15) where Cs is the model parameter with value in the range of 0:1pC s p0:2, 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, the source 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 The source function is added to the mass conservation equation (continuity equation) in order to generate the desired incident waves, while the damping zone works as an energy 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 i Zi ỵd < 3Tt 2U if tp3T; Dxs Zs ỵd qs ẳ (16) i Zi þd : 2U t43T; Dxs Z þd s where qs is the source function, T is the incident wave period, d is the still water depth, Ui and Zi are the time variations of horizontal velocity and water surface estimated by a third-order Stokes wave theory, t is the time, Dxs is the mesh size in the horizontal direction at the source location and Zs is the real water surface displacement at the source location In Eq (16), the amplitude of the source function is gradually intensified in the duration of three wave periods in order to guarantee a stable 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 is given by hydrostatic pressure The air density is chosen as 1.2 kg/m3 and the fresh water 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 continuity equation ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1573 2.7 Method of solution The governing equations are discretized by a finite-difference scheme on a staggered grid mesh The velocity components are evaluated at cell sides, while scalar quantities are evaluated at the cell center The SMAC method (Simplified Marker and Cell Method) is used to get the time evolution solution of the governing equations The resultant Poisson equation of pressure correction due to the SMAC method is solved using a bi-conjugate gradient method (Van der Vorst, 1992) In the previous 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 the momentum equations However, for a problem with the presence of porous media, the non-conservative CIP scheme falls to give good approximations for the simulation of wave and porous structure interactions in the two-phase flow model To solve the problem, in this study, a high resolution MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws)-type second-order accurate scheme (Nessyahu and Tadmor, 1990) is employed for the convection terms, and the second-order central scheme is used for the viscous terms The computational procedure in detail can be found in Hieu et al (2004) Here the brief explanation is given as follows: (a) (b) (c) (d) (e) (f) Give initial values for all variables Give boundary conditions for all variables Solve explicitly the momentum equations for the predicted velocities Solve the Poisson equation for the pressure corrections Adjust the pressure and velocity Solve the advection equation of VOF function using the PLIC algorithm for tracking 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 Model verification The model described above is validated against two experimental tests The first is wave breaking on a sloping bottom with the experimental conditions by Ting and Kirby (1994, 1996) The second is the simulation of wave interactions with a submerged porous structure 3.1 Wave breaking on a sloping bottom 3.1.1 Test conditions The condition similar to the laboratory experiment by Ting and Kirby (1994) is used for testing the numerical simulation of breaking waves on a sloping bottom In the laboratory experiment, a beach with uniform slope of 35 is connected to a region ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1574 Damping zone Wave generation source Air zone 1.0m z x SWL Water zone dc = 0.40m d 0.38m slope=1/35 5m 5m 17m Fig Sketch of the simulation domain with constant depth d c ¼ 0:40 m Fig shows the schematic view of the numerical wave channel where the coordinate system is chosen so that x ¼ is located at the position with the still water depth d ¼ 0:38 m on the slope, and z ¼ is located at the still water level (SWL) the same as the coordinate system in the laboratory experiment Test waves are regular and the incident wave heights HI and periods T in the constant water depth are 0.125 m and 2.0 s for the spilling breaker, and 0.128 m and 5.0 s for the plunging breaker, respectively In the numerical wave channel, the computation domain is discretized using an orthogonal 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 the computation domain together with a sponge damping zone on the left The sloping bottom 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 time profiles of wave quantities Numerical results in terms of wave crest, trough and mean water level are used to compare against the experimental data (Ting and Kirby, 1994) Velocities in a cross-section inside the surf zone are also compared with the experimental data (Ting and Kirby, 1996) 3.1.2 Results and discussions Fig shows the time profile of water surface elevation at the location x ¼ À0:5 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 are almost stable after 30 s from the starting time of computation This gives us a confidence in getting the mean wave quantities from the time profiles after a specific time Fig shows the simulation results compared with experimental data (Ting and Kirby, 1994) for mean wave quantities (wave crests, troughs and mean water levels) In the figure, lines indicate the simulated results and circles indicate the measured data The wave crests and troughs are determined from the mean water level It is ARTICLE IN PRESS free surface elevation (m) P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 0.2 x = -0.5m 0.15 0.1 0.05 -0.05 -0.1 10 15 20 25 (a) free surface elevation (m) 1575 30 35 40 45 50 30 35 40 45 50 time (s) 0.2 x = 6.4m 0.15 0.1 0.05 -0.05 -0.1 10 15 20 25 (b) time (s) Fig Time profiles of water surface elevation present cal.results Expt.Data (Ting &Kirby, 1994) Cal.results (Bradford, 2000) 0.2 0.15 Surface elevation (m) Breaking point Cal results (Zhao etal.,2004) Cal results(CADMAS-SURF) Wave crest 0.1 Mean water level 0.05 -0.05 -0.1 Wave trough -2 10 12 Cross-shore distance x (m) Fig Distributions of wave crests, wave troughs and mean water levels of the spilling breaker case on the slope ARTICLE IN PRESS 1576 P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 seen that the wave crests and troughs are well simulated by the model Especially, at the breaking point, the simulated breaking wave height is very accurate compared with simulated results by Zhao et al (2004) and Bradford (2000) However, at the very near shore area, the present simulated results of wave crests are overestimated The simulated mean water level is also slightly underestimated compared with the experimental data The reasons for those disagreements could be due to the complicated 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 those simulated by Zhao et al (2004), by Bradford (2000) and by using CADMAS-SURF model It should be pointed out that the simulation using CASDMAS-SURF model was carried out with a fine mesh (Dx ¼ 0:02 m, Dz ¼ 0:01 m) and the sloping bottom was treated by the partial cell and wall function technique The Stokes fifth-order wave source was applied for generating the incident waves The CADMAS-SURF model 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-SURF model prove that the incorporation of the treatment of air motions in the present model has contributed significant improvement on the accuracy of the numerical simulations of wave breaking Thus, the effects of the air movement on the wave motion under wave breaking processes are not negligible Comparing the results by the present model with the results by Hieu et al (2004) (see Fig in Hieu et al., 2004), it is seen that the simulated wave crests by the present model are more accurate 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 models accurately simulated the breaking point location and distribution of wave trough The better results obtained in this study may be due the incorporation of the wave generation source method and the MUSCL-TVD scheme in the present model The water surface elevation, horizontal and vertical velocity components in a cross-section located at x ¼ 7:272 m inside the surf zone is considered for the comparisons with experimental data (Ting and Kirby, 1996) Fig shows the variation of phase averaged water surface at the cross-section In the figure, 0.6 h − 0.4 0.2 -0.2 0.2 0.4 0.6 0.8 t /T Fig Comparison of simulated and measured phase averaged surface elevations at x ¼ 7:275 m Line: simulated results; circle: measured data (Ting and Kirby, 1996) ARTICLE IN PRESS 0.5 t/ T 0.5 t/T 0.4 0.2 0.2 0.2 -0.4 0.5 t/T w (m/s) 0.4 -0.2 (b) 0.6 0.4 0.2 -0.2 -0.4 -0.6 0.4 w (m/s) w (m/s) (a) 0.6 0.4 0.2 -0.2 -0.4 -0.6 u (m/s) 0.6 0.4 0.2 -0.2 -0.4 -0.6 u (m/s) u (m/s) P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 -0.2 -0.4 0.5 t/T 1577 0.5 t/T 0.5 t/T -0.2 -0.4 Fig Variation of phase averaged velocity components at location x ¼ 7:275 m: (a) horizontal components and (b) vertical components From left to right, the vertical locations are z ¼ À0:04 m, z ¼ À0:08 m and z ¼ À0:12 m Lines: simulated results; circle: measured data (Ting and Kirby, 1996) ¯ z¯ is the the water surface elevation z is normalized by the local water depth h ẳ d ỵ z; mean water level from the SWL In the numerical simulation, the phase averaged quantities are averaged over five waves on the stable time profiles A good agreement between simulated results and measured data is clearly observed, although some difference is noticed Fig 5(a) and (b), respectively, shows the phase averaged horizontal and vertical velocity components at three depth levels, z ¼ À0:04 m, z ¼ À0:08 m and z ¼ À0:12 m, in the cross-section It is observed from the figures that both horizontal and vertical velocity components are well simulated However, from the surface to the bottom, the accuracy of simulation decreases for the horizontal velocity, whereas an opposite tendency is observed for the vertical velocity components Near the free surface, the numerical simulation seems to overestimate the positive peak of vertical velocity component Fig presents the comparisons of simulated results and experimental data for the plunging breaker case It is also observed that good agreements are also obtained for the plunging breaker The agreements for mean water level are better than those in the spilling breaker case Overall, it is concluded that the model can simulate well the spilling and plunging wave breaking on the sloping bottom 3.2 Wave breaking over a submerged breakwater 3.2.1 Test conditions In this test, laboratory experiments for wave interaction with a submerged porous breakwater have been conducted in a physical wave channel, at the Hydraulic ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1578 Breaking point Present cal results Surface elevation (m) 0.2 Expt Data (Ting & Kirby, 1994) Wave crest 0.1 MWL Wave trough -0.1 -2 10 12 Cross-shore distance x(m) Fig Distributions of wave crests, wave troughs and mean water levels of the plunging breaker case on the slope Laboratory, Saitama University The wave channel is 18 m long, 0.7 m high and 0.4 m wide A submerged breakwater is set at a distance approximately 10.5 m from the wave maker paddle and built by stones with mean diameter Dm ¼ 0:025 m The breakwater is 0.33 m high, 1.16 m wide at the base and has the porosity 0.45 The water depth is kept 0.376 m Fig shows the experimental setup The incident wave height and period are 0.092 m and 1.6 s, respectively Waves are measured at 38 locations in the offshore side, lee side and on the breakwater During the experiment, wave breaking was observed on the top of the breakwater In the numerical simulation, a regular grid mesh is used with grid size Dx ¼ 0:02 m and Dz ¼ 0:01 m The wave source is set at the location similar to the wave paddle in the physical experiment The breakwater is modeled by a set of porous meshes with a drag coefficient CD and an inertia coefficient CM At the end of the numerical wave channel, a damping zone is set to avoid wave reflections Time profiles of water surface elevation at the locations of wave gauges and distribution of wave height around the breakwater are used to compare with the experimental data 3.2.2 Results and discussions In the numerical model, the drag coefficient CD and inertia coefficient CM must be determined in advance There are two methods to determine these coefficients, one is a direct measurement from hydraulic experiments (van Gent, 1995) and the other is to apply numerical simulations using several sets of CD and CM to find best results compared with experimental data (Karim et al., 2003) In the present study, the second method is followed Based on available literatures (Sakakiyama and Kajima, 1992; Mizutani et al., 1996; Karim et al., 2003), here, a range of C D ¼ 0:5, 1.0, 1.5, 2.0, 2.5, 3.5, 4.0 and C M ¼ 0:5, 1.0, 1.5, 2.0, 2.5 are considered for the trial–error investigation Fig shows the influences of CM on the wave height distribution for a fixed value C D ¼ 3:5 It is seen from the figure that the variations of CM have a considerable ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1579 38 capacitance wave gauges G1 G12 G17 G31 G34 G38 Wave damper Incident waves 0.30m SWL 0.33m d=0.376m 0.43m 1.16m x x=0m Fig Sketch of experiment and submerged porous breakwater 1.5 CD=3.5 Measured H/HI (Cm=0.5) (Cm=1.0) 0.5 (Cm=1.5) (Cm=2.0) -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 x/L Fig Influence of inertial coefficient on wave height distribution Lines: simulated results; black triangles: measured data effect to the wave height distribution in the whole region With higher values of CM, the wave reflection is higher on the offshore side On the leeside, the wave height distribution changes in the complicated manners with the change of CM As well known in the literature, the waves on the leeside of the breakwater consist of the transmitted waves (fundamental waves) and higher harmonic waves generated by wave breaking and non-linear shallow effects (Huang and Dong, 1999; Kawasaki, 1999) The changes of CM result in different conditions on delaying the phase of the transmitted waves through the porous breakwater Therefore, the distributions of wave height on the leeside are considerably effected According to the results, the appropriate value for CM can be estimated as about 1.2 Fig shows the comparison between numerical results by the fixed C M ¼ 1:2 and variable CD and the experimental data We can see that the influence of CD on the ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1580 1.5 CM=1.2 Measured (CD=0.5 ) H/HI (CD=1.0 ) (CD=1.5 ) (CD=2.5 ) 0.5 (CD=3.0 ) (CD=4.0 ) -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 x /L Fig Influence of drag coefficient on wave height distribution Lines: simulated results; black triangles: measured data wave reflection is very small but significant on the wave transmission The wave transmission is higher for the smaller value of CD; however, the pattern of wave height distribution is similar The appropriate value for CD is in the range from 1.5 to 2.5 By applying the trial–error process, we found that the inertia and drag coefficients, respectively, 1.2 and 2.5 are best fit to the experimental data Fig 10 shows the time profiles of water surface elevation at six points in the wave channel Two points are located on the offshore side (x1 ¼ À1:90 m; x12 ¼ À0:60 m), one on the top of front slope (x17 ¼ À0:10 m) and three others on the leeside (x31 ¼ 0:75 m; x34 ¼ 1:44 m; x38 ¼ 2:04 m) of the breakwater It is clearly observed that the numerical results agree well with the experimental data Especially, the numerical model can simulate well the waves on the leeside of the breakwater in the case with the presence of wave breaking As seen from Fig 10(d)–(f), the detachment of the secondary waves is clearly observed Thus, the model is capable of reproducing the features of the evolution and decomposition of the waves The good agreement between numerical results and experimental data gives us a confidence in the ability of the model on the simulation of wave interactions with porous structures The well-simulated results obtained in this study also confirm that the modified Navier–Stokes equations proposed by Sakakiyama and Kajima (1992) are excellent governing equations for the simulation of wave motions in porous media Snapshots of velocity field and free surface position around the submerged porous breakwater are shown in Fig 11(a) at the time of wave trough on the breakwater and Fig 11(b) at the time of wave crest on the breakwater In Fig 11(a), a hydraulic jump is clearly observed in front of the front slope of the breakwater The largest water velocity is observed at the free surface and at the top of front slope The velocity inside the porous breakwater is much smaller than that on the top of the breakwater due to resistant force of the porous media In Fig 11(b), it is seen that the water velocities on the top of the breakwater are very strong and much stronger than those in Fig 11(a) There is a big difference between the velocities on the top ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1581 and inside the breakwater The largest velocity is observed near the top of the leeside slope Strong water velocities at the top of front and leeside slopes may be a key issue for the damage of a breakwater in practice Fig 12 shows the vertical distribution of dynamic pressure at several phases in the cross-section x ¼ 0:15 m at the center of the breakwater In the figure, the phase 0.1 Measured Simulated x = -1.90m (m) 0.05 -0.05 -0.1 14 14.5 15 15.5 (a) 16 16.5 17 0.1 18 Measured Simulated x = -0.60m 0.05 (m) 17.5 time (s) (b) -0.05 -0.1 14 14.5 15 15.5 (b) 16 16.5 0.1 17.5 18 17.5 18 Measured Simulated x = -0.10m 0.05 (m) 17 time (s) -0.05 -0.1 14 14.5 15 15.5 (c) 16 16.5 17 time (s) 0.1 Measured Simulated x = 0.75m (m) 0.05 -0.05 -0.1 14 (d) 14.5 15 15.5 16 16.5 17 time (s) Fig 10 Time profile of water surface elevation at several locations 17.5 18 ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1582 0.1 Measured Simulated x = 1.44m (m) 0.05 -0.05 -0.1 14 14.5 15 15.5 (e) 16 16.5 17 0.1 18 Measured Simulated x = 2.04m 0.05 (m) 17.5 time (s) -0.05 -0.1 14 (f) 14.5 15 15.5 16 16.5 17 17.5 18 time (s) Fig 10 (Continued) interval is 151, i.e the time interval is T=24 The dynamic pressure is also normalized by the quantityrw gH I , and vertical coordinate is normalized by the still water depth d It is seen from the figure that when the phases change from the wave crest to wave trough, the dynamic pressure in the water layer on the top of the breakwater is almost constant in the vertical direction (see Fig 12(a)) Below the constant part of the dynamic pressure in the vertical direction, there is an abrupt change of the dynamic pressure from the outside to inside of the breakwater when the phase is at the wave crest Due to the presence of the breakwater, the vertical distributions of dynamic pressure, when the phases change from the trough to the crest, are quite different from that when the phases change from the crest to the trough As seen in Fig 12, the vertical distributions of dynamic pressure shown in Fig 12(a) and in Fig 12(b) are different It is seen from the Fig 12(b) that the highest negative dynamic pressure gradient in the vertical direction is not observed at the wave trough phase but at the middle phase between the trough and crest This negative dynamic pressure may contribute some effects to the stability of stone on the top of the breakwater Because of the direct simulation using the Navier–Stokes equations, the information on the velocity components, pressure and water surface variations are available It is worth knowing how the porosity of the breakwater effects to the wave reflection, wave transmission and wave dissipation As well known in the literature, when waves pass over an obstacle the wave fission occurs (Lin, 2004; Huang and Dong, 1999) In addition, the wave breaking produces higher harmonic waves, which make the wave profile continuously evolve from location to others Therefore, the conventional definitions of wave reflection and transmission coefficient using the ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1583 1m/s 50 z(cm) 40 30 20 -0.5 0.5 0.5 x(m) (a) 1m/s 50 z(cm) 40 30 20 -0.5 x(m) (b) Fig 11 Snapshots of velocity field around the breakwater: (a) wave trough time and (b) wave crest time ratio of wave heights become inaccurate In this study, we use definitions for the wave reflection and transmission coefficients based on the concept of energy conservation The formulae are as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E offshore À E incident KR ¼ , (17) E incident rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E lee-side KT ¼ , (18) E incident qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K D ¼ À K 2R À K 2T , (19) E ẳ E pot ỵ E kin , (20) ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1584 0.6 Wave trough Wave crest z/d 0.2 -0.2 -0.6 -1 -0.8 -0.4 pd / (a) 0.4 0.8 wgHi 0.6 Wave trough Wave crest z/d 0.2 -0.2 -0.6 -1 -0.8 -0.4 pd / (b) 0.4 0.8 wgHi Fig 12 Vertical distributions of dynamic pressure at x ¼ 0:15 m with the time interval T=24: (a) from wave crest to wave trough and (b) from wave trough to wave crest (L: incident wave length; pd: dynamic pressure) Big arrows in the figure show the direction of phase change E pot ¼ E kin ¼ T T Z Z T z rw gz dz, dt Z Z T dt (21) z rw u ỵ w2 ị dz, h (22) where KR is the reflection coefficient, KT is the transmission coefficient and KD is the dissipation coefficient; z is the water surface displacement from the SWL E indicates the wave energy Investigations of the influence of porosity of the breakwater on the wave reflection, wave transmission and wave dissipation when waves are passing the submerged breakwater were carried out with the porosity variation from to The incident wave energy was estimated for the case without the breakwater The case with zero value of porosity means that the breakwater is solid while the unit value ARTICLE IN PRESS P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 1585 1.2 KR, KT, KD KT KD 0.8 0.6 0.4 0.2 KR 0.2 0.4 0.6 Porosity 0.8 Fig 13 Influences of porosity to the reflection, transmission and dissipation coefficients means there is no breakwater Fig 13 shows the variations of the reflection, transmission and dissipation coefficients versus the change of porosity It is seen that the reflection coefficient monotonic decreases versus the increasing of porosity The transmission coefficient gradually decreases down to about 0.6 when the porosity increase from zero to 0.55, then it increases with further increasing of porosity The effective porosity about 0.55 in minimizing the transmission coefficient found in this study is very close to that about 0.52 found in the study by Huang et al (2003), in which the investigation on a solitary wave passing a submerged rectangular porous obstacle was conducted Also from the figure, the dissipation coefficient increases to a maximum value of 0.75, then decreases when the porosity changes from zero to unity It is observed that the maximum energy dissipation happens when the porosity is 0.6 With the case of impermeable submerged breakwater (porosity ¼ 0), it is seen from the figure that the energy dissipation (K 2D ) due to the wave breaking and turbulence processes is about of 25% of the incident wave energy It is almost half of the total energy dissipation in the case of maximum performance of the submerged breakwater in the wave energy dissipation This result suggests to us that the energy dissipation during wave breaking processes is the key issue in designing the submerged breakwaters The effective of porosity in respect to the energy dissipation for this submerged breakwater is in the range from 0.5 to 0.65 Conclusions In this study, a VOF-based two-phase flow model has been verified against some laboratory data The model is based on the spatial averaged Navier–Stokes equations extended to porous media and the new VOF method for the treatment of the free surface boundary It is confirmed that the present numerical model can simulate well the wave breaking on a sloping bottom as well as the wave breaking over a submerged breakwater The results of the present study also confirmed that with appropriate values of inertia and drag coefficients, the present numerical model can accurately simulate the complicated interactions of waves and porous structures ARTICLE IN PRESS 1586 P.D Hieu, K Tanimoto / Ocean Engineering 33 (2006) 1565–1588 The velocity and pressure has been numerically investigated for the case of wave interaction with a submerged porous breakwater By applying the model, the distributions of velocity, vertical distribution of the dynamic pressure can be observed The dynamic pressure has the maximum positive pressure near the SWL, and the negative maximum pressure at the trough level below the SWL Investigations on the influence of porosity to the wave energy dissipation show that there is an effective range of porosity, which gives the maximum wave energy dissipation For the present submerged breakwater with the fixed values of inertia and drag coefficients, the effective value of porosity is about 0.6 However, the inertia and drag coefficients might be variable with the variation of the porosity of the submerged breakwater in the real Thus, more investigations with variable inertia and drag coefficients together with the variations of porosity could be necessary in the future The results of the present study also reveal that the energy dissipation due to wave breaking is significant portion in the total energy dissipation Therefore, it should be considered as the key issue in designing an effective breakwater The results of this study confirmed that the two-phase flow model with the aid of VOF method is very useful tool for the investigation of fluid dynamics including the non-linear interactions of waves and porous structures Further investigations on the applicability of the present model for simulating wave overtopping coastal structures are necessary Acknowledgments A part of this work has been done under the financial 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