Validation of a CIP-based tank for numerical simulation of free surface flows

Transcription

1 Acta Mech. Sin. (2011) 27(6): DOI /s RESEARCH PAPER Validation of a CIP-based tank for numerical simulation of free surface flows Xi-Zeng Zhao Received: 24 August 2010 / Revised: 4 March 2011 / Accepted: 14 March 2011 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011 Abstract A constrained interpolation profile CIP-based numerical tank is developed to simulate violent free surface flows. The numerical simulation is performed by the CIP-based Cartesian grid method, which is described in the present paper. The tangent of hyperbola for interface capturing (THINC) scheme is applied for capturing complex free surfaces. The new model is capable of simulating a flow with violently varied free surface. A series of computations are conducted to assess the developed algorithm and its versatility. These tests include the collapse of water column with and without an obstacle, sloshing in a fixed tank, the generation of regular waves in a tank, the generation of extreme waves in a tank. Excellent agreements are obtained when numerical results are compared with available analytical, experimental, and other numerical results. Keywords CIP method THINC scheme Free surface flow Wave breaking Wave tank 1 Introduction To date, the wave tank has been widely applied by scientists in their studies on water waves and wave-related phenomena. The tank is also a valuable and often irreplaceable source of data and information for engineers resolving practical coastal and offshore engineering problems. It is desirable to have analytic and/or computational tools to perform The project was supported by the Fundamental Research Funds for the Central Universities. X.-Z. Zhao ( ) Colloge of Engineering, Ocean University of China, Qingdao, China RIAM, Kyushu University, Fukuoka, Japan xizengzhao@gmail.com analyses in the early stage of design to reduce the number of different design alternatives that require wave tank testing. Development of efficient wave tank is thus necessary for the seakeeping research. The numerical simulation of free surface flows remains a challenging problem in fluid mechanics, especially for violent flow. It involves many other important processes such as wave breaking and overturning. Accurate numerical simulations will enhance the understanding of the motion of violent free surface flow. The numerical simulation can provide detailed information on the hydrodynamics of a free surface flow, which is not easily obtained by physical experiments or theoretical studies. Therefore, the numerical simulation of a free surface flow with violently varied free surface is not only an attractive research topic but also a difficult task for coastal engineers. Recently with the development of computer technology and computational fluid dynamics (CFD) methods, it is desirable to directly solve the Navier Stokes equation for the simulation of many problems, including the wave breaking and overtopping processes. The major difficulty in the numerical simulation of violent flow is that the topology of free surface may be largely distorted or broken up, which makes it impossible to apply conventional numerical methods such as potential flow solver by boundary element method (BEM) [1], High order spectral method [2 4], finite element method [5 7]. Several challenging works have been reported by using the finite difference method based on solving the Navier Stokes equations in which the free surface is tracked by volume of fluid (VOF) method [8,9], by the particle method [10] and the smooth particles hydrodynamics method [11]. In this paper, we use a CIP-based Cartesian grid method, which is recently named as computation method for extremely nonlinear hydrodynamic (CMEN). This numerical model for predicting hydrodynamics loads associated with strongly nonlinear ship-wave interactions has been developed over a period of years in the Research Institute for Applied Mechanics

2 878 X.-Z. Zhao (RIAM) at Kyushu University. The constrained interpolation profile (CIP) algorithm [12] is adopted as the base scheme to obtain a robust flow solver for the Cartesian grid approach. Two- and three-dimensional development of relevant CFD code has been presented in previous papers [13 16]. An improved interface-capturing scheme, the THINC scheme [17], is adopted to avoid drawbacks involved in the original CIP- CLS3 [18] scheme. It should be pointed out that in the problem involving violent free surface flow the CIP-based models are not yet widely applied. The aim of this paper is to present the validation of proposed model for the simulation of motions involving violently varied free surface. This paper is organized as follows. In Sect. 2, the governing equations of the problem and the numerical methods are described. The CIP scheme is briefly described. Then the THINC method is outlined. Section 3 describes the results of our computations. All our calculations are in the context of two dimensions. Comparisons with theoretical solutions and experiments are also presented for discussion. The article closes with some general conclusions drawn based on the present work and some considerations for possible future work. 2 Numerical model description 2.1 Mathematical formulation of the model Two-dimensional incompressible Navier Stokes and continuity equations are employed. The governing equations are expressed as u i t u i + u j = 1 p + F i + 1 (2µS i j ), (1) x j ρ x i ρ x j u i x i = 0, (2) where u i, i = 1, 2 are the velocity components along coordinate axes x i ; t is time; p is hydrodynamic pressure; S i j = ( u i / x j + u j / x i )/2; ρ and µ are the water density and viscosity, respectively. The second term on the right-hand of Eq. (1) is the external force, including the gravitational force. The pressure-velocity coupling is treated in a non-advection step calculation, in which the following Poisson equation is solved ( 1 p n+1 ) = 1 u i. (3) x i ρ x i t x i Equation (3) is assumed to be valid for liquid, gas and solid phase. Solution of it gives the pressure distribution in the whole computation domain. We note that the pressure distribution obtained inside the solid body is a fictitious one, which satisfies the divergence free condition of the velocity field. In this treatment, the boundary condition for pressure at the interfaces is not necessary, and a fast solver or parallel computing technique can easily be applied. This is a very important feature, because the calculation of Eq. (3) is generally the most time-consuming part of computation for this kind of numerical simulation. The two fluids are immiscible, named as Fluids 1 and 2. Density and viscosity are constant in each phase but may be discontinuous at the interface. We use a volume fraction field C to represent and track the interface which is transported by the velocity field of u i C t + u i C = 0. (4) This equation allows for the calculation of density and viscosity. In fact, the average values of density and viscosity are interpolated by the formulas of ρ = Cρ 1 + (1 C)ρ 2, (5) µ = Cµ 1 + (1 C)µ 2. (6) In the simulation, except where otherwise noted, the liquid phase is treated as water and the gas phase is treated as air. Their density and viscosity are ρ 1 = 10 3 kg/m 3, µ 1 = 10 3 kg/(s m), and ρ 2 = 1 kg/m 3, µ 2 = 10 5 kg/(s m), respectively. 2.2 CIP method Time evaluation of Eq. (1) is performed by a fractional step method by which the calculation of the equation is divided into an advection step and two non-advection steps, ie., at each time step, the advection is solved first, then the diffusion, at last the pressure is solved by Eq. (3). In the advection step, the calculation is done by the CIP scheme. The basic idea of the CIP method is that for advection computation of a variable f, not only the advection equation of f, but also the advection equation of its spatial gradient g i = f / x i are calculated by a cubic interpolation method. Therefore, the following equations are used for the advection calculation. f t + u f i = 0, (7) x i g i t + u i g i x i = u i x i g i. (8) Figure 1 shows the principle of the CIP method. When the velocity is constant, the solution of advection equation gives a simple translational motion of wave with a velocity. The initial profile (dot line in Fig. 1b) moves like a dashed line in a continuous representation. At this time, the solution at grid points is denoted by circles and is the same as the exact solution. However, if we eliminate the dashed line as in Fig. 1a, then the profile information inside the grid cell has been lost and it is hard to imagine the original profile and it is natural to imagine a profile like that shown by solid line in Fig. 1c. Thus, numerical diffusion arises when we construct the profile by linear interpolation even with the exact solution as shown in Fig. 1c. The CIP scheme, which is illustrated in Fig. 1d, shows a different way from the con-

3 Validation of a CIP-based tank for numerical simulation of free surface flows 879 ventional high-order schemes to reconstruct the profile inside a grid cell. It uses the grid point values and its spatial derivative in two grid points to form a cubic polynomial to approximate the profile [12]. Fig. 1 The principle of CIP method. a Initial profile; b Exact solution after advection; c Linearly interpolated; d Interpolated by CIP method 2.3 Free-surface treatment by THINC scheme We apply the THINC scheme for free surface capturing. The THINC scheme is a VOF type method. The difference is that the VOF method uses a Heaviside step function as the characteristic function, while the THINC scheme uses a smoothed Heaviside function, the hyperbolic tangent function. A 1-D THINC scheme is described in the following. Multi-dimensional computations are performed by a directional splitting method. The one-dimensional advection equation for a density function C can be written in conservation form as follows C + (uc) = C u t x. (9) Equation (9) is descretized by a finite volume method. For a known velocity field u n, integrating Eq. (9) over a computational cell [x i 1/2, x i+1/2 ] and a time interval [t n, t n+1 ] we have C i n+1 = C i n + 1 (g i 1/2 g i+1/2 ) x i + t C i n x (un i+1/2 gn i 1/2 ), (10) i where x i t n+1 = x i+1/2 x i 1/2, t = t n+1 t n, g t±1/2 = t n (uc) i±1/2 dt is the flux across the cell boundary (x = xi+1/2 x i±1/2 ), and C i = 1 C(x, t)dx is the cell-averaged x x i 1/2 density function defined at the cell center (x = x i ). The fluxes are calculated by a semi-lagrangian method. Similar to the CIP method, the profile of C inside an upwind computation cell is approximated by an interpolation function. Instead of using a polynomial in the CIP scheme, the THINC scheme uses a hyperbolic tangent function so as to avoid numerical smearing and oscillation at the interface. Since 0 χ 1, and the variation of χ cross the free surface is step-like, a piecewise modified hyperbolic tangent function is used to approximate the profile inside a computation cell, which is shown as χ x,i = α 2 { 1 + γ tanh [ ( x xi 1/2 β x i )]} δ, (11) where α, β, γ, δ are parameters to be specified. α and γ are parameters used to avoid interface smearing, which are given as follows C i+1, if C i+1 C i 1, α = C i 1, otherwise, (12) 1, if C i+1 C i 1, γ = 1, otherwise. Parameter β is used to control the sharpness of the variation of function. In this paper, we use β = 3.5 which corresponds to one mesh spacing smoothing. Parameter δ is used to determine the middle point of hyperbolic tangent function, and is calculated by solving the following equation 1 x i xi+1/2 x i 1/2 χ i (x)dx = C n i. (13) After χ i (x) is determined, the flux at the cell boundary can be calculated by Eq. (9). In Fig. 2, u i+1/2 > 0 is indicated by the dashed area. After all of the fluxes across the cell boundaries have been computed, the cell-integrated value at the new time step can be obtained by Eq. (10). This cell-integrated value is used to determine the free surface position; therefore, mass conservation is automatically satisfied for the liquid. Fig. 2 Concept of the THINC scheme 3 Numerical results In this section, a series of numerical experiments are conducted to demonstrate the versatility and accuracy of the model. Our main focus is laid on the validation and the numerical accuracy for free surface profile calculation.

4 880 X.-Z. Zhao 3.1 Collapse of water column Test simulation of the collapse of water column is a useful benchmark providing extreme conditions to assess the numerical stability as well as the capability of the model in treating the free surface problem. The collapse of water column is a classic problem in free surface hydrodynamics, and as its name implies, it is basically the sudden collapse of a rectangular column of fluid onto a horizontal surface. The broken water column problem gets its name due to its use in modeling the sudden failure of a water column. The relatively simple geometry and initial conditions associated with the problem have made it a popular validation case for various surface tracking and capturing schemes. A laboratory experiment is conducted at the Research Institute for Applied Mechanics, Kyushu University. In the experiment, the evolution of the free surface is recorded by a high-speed digital video camera. The experiments are repeated more than 10 times and it is found that the repeatability is good. A variable grid is used for the computation, in which the grid points are concentrated near the floor and the left wall. To show grid convergence of the test case, we use two grids, coarse mesh (205 79) and fine mesh ( ) with its minimum mesh grid ranging from 0.8 mm to 0.4 mm, as shown in Fig. 3. The nondimensional height at the left wall and the leading-edge positions of the collapsing water column versus the nondimensional time are shown in Fig. 4, respectively, and are found in good agreement with the experimental data presented by Martin and Moyce [19]. In Fig. 4, the effect of the time step is also studied, and minor difference is found. It can be seen that the results from the calculations using different meshes and time steps are nearly identical and thus a nearly converged result can be obtained by using the mesh and time step set above. Fig. 3 Schematic view of water column collapse experiment Fig. 4 Comparison between computation and experiment. a Position of leading edge; b Height of water column Figure 5 shows the free surface at different times of the simulation compared with photographs of the experiment and overall good agreement is found, in which the fine grid is used. The free surface is indicated by density function contours, with three lines showing φ = 0.05, 0.5 and 0.95, where the solid line denotes φ = 0.5. The simulation predicts well not only in the first stage, i.e., the development of water flow along the floor after the sudden collapse of water column, but also in the second stage of the flow development, i.e., the flow after impact onto the wall, overturning and breaking of the free surface as well as air entrapment are observed. Another more interesting simulation of a collapsing dam is made with an obstacle fixed at the bottom of tank, as shown in Fig. 6. For this well-known example, experimental data [10] and a wide range of numerical results obtained with other methods are available [20 22]. In the computation, a variable mesh is used, in which the grid points are concentrated near the floor and the obstacle. the grid number is with the minimum mesh spacing of 1 mm. the time step is set to t = 10 4 s. The predicted positions of the interface for t = 0.1 s, 0.2 s, 0.3 s, 0.4 s, and 0.5 s are shown in Fig. 7 together with still photographs taken by Koshizuka [10]. The free surface motion is predicted well for both the first few time steps considered and at later times even when the surface is highly contorted with considerable spray, breakup and overturning. When the thin front has passed the obstacle and reaches the right wall of the tank and a tongue of water is created connecting the left corner of the obstacle with the opposite wall, it encloses a region full of air. A small second front develops close to the top of the

5 Validation of a CIP-based tank for numerical simulation of free surface flows 881 obstacle. After that, the bridge-like water structure collapses. Fig. 5 Development of the collapsing water column. a Computation; b Experiment

6 882 X.-Z. Zhao 3.2 Sloshing in a fixed tank Fig. 6 System of collapsing dam with an obstacle The sloshing of a liquid wave with finite amplitude under the influence of gravity is another classical test case for the free surface flow. The considered situation is the same as used by Suzuki [23]. Initially the depth h of water is 1.0 m in the equilibrium state, and its surface is defined by y(x) = a cos(k 0 x + π/2), (14) where k 0 = 2π/L is the wavenumber and L = 2 m is the wavelength. The wavelength is chosen to be equal to the width of the tank. The domain is descretized with 200 uniform cells in the horizontal direction and 192 variable cells in the vertical direction. Slip and zero-gradient boundary conditions are used for the velocity and pressure at all boundaries, respectively. The initial velocity distribution is set to zero everywhere. Fig. 7 Development of the collapsing water column with an obstacle. a Computation; b Photographs from Koshizuka [10]

7 Validation of a CIP-based tank for numerical simulation of free surface flows 883 Fig. 7 Development of the collapsing water column with an obstacle. a Computation; b Photographs from Koshizuka [10] (continued) In order to validate the proposed scheme, timedependent free-surface motions at the tank centre for different wave steepness parameters a/h are compared with the corresponding first-order and the first plus second-order analytical solutions [24 26]. Numerical results, for steepness a/h = 0.01, are shown in Fig. 8 and are compared with the linear solutions. Figure 8a shows the time histories of the wave elevation at the centre of the tank, whereas Fig. 8b shows the corresponding amplitude wave spectrum. The application of the present approach to waves of small steepness enables us to verify the accuracy of the present numerical scheme because the outcome of the model should be basically the same as the linear solution. With small steepness, there is good agreement between the numerical results and the linear solutions. Fig. 8 Comparison between linear solution: a = 0.01h. a Position of interface at the center of the tank; b The corresponding spectrum For the case of small steepness, the nonlinearity effect is of secondary importance. The free surface at the centre of the tank can be predicted by the linear wave theory. Figure 9 shows plots of the position of the interface at the centre of the tank and the corresponding amplitude wave spectrum for the case of larger amplitude, a/h = 0.1. Numerical results are compared with the corresponding analytical solution given by the first order and second order potential theory. With the input steepness of a/h = 0.1, the nonlinearity effects are no longer negligible and cause some modification in wave elevation and wave spectrum, as presented in Figs. 9a and 9c. The wave elevation can not be predicted well by the linear solution. Second-order harmonic waves can be found within the numerical results (see Fig.10c). There is evidently a need to introduce a more advanced theoretical solution to improve the prediction. The second order solution is applied, and it is found that it is very close to the nonlinear numerical solution in the initial few cycles, it diverges from the analytical solution at later times, as shown in Fig.9b. Also, it can be observed (Figs. 9b and 9d) that the phase-shift grows in time between the second order analytical solution and the numerical model. Figure 9d shows the associated spectra. It can be observed that small additional frequencies due to non-linear mode to mode interaction are present for the large initial amplitude cases (Fig. 9d). We note that they are responsible for the large disturbances in the free surface elevation. Owing to the absence of terms above the second order, the analytical solutions are incapable of modelling fully the behaviour of steep (and therefore highly nonlinear) free-surface waves. The behavior of the numerical solution mentioned above has also been observed in simulations using other numerical methods [6, 24 27].

8 884 X.-Z. Zhao Fig. 9 Position of interface at the center of the tank and the corresponding spectrum: a = 0.1h. a, c Comparison with linear solution; b, d Comparison with second order solution Several snapshots of the results after t 0 = 7.125T (phase lag causes the t 0 not equal to NT, N = 1, 2, 3, ) are shown in Fig. 10, where both the location of the water surface and the velocity vectors are plotted. T is the theoretical period of sloshing of the first mode, and can be obtained by the dispersion relationship. The maximum values of the velocity (V max ) on each figure are calculated to show the scale of the velocity. After a quarter of the period the potential energy of the system has been transferred to kinetic energy and the velocities reach their maxima. After a half period all the kinetic energy has been transferred back into potential energy with the velocity almost back to zero. Fig. 10 Plots of wave position and velocity vectors for one period of sloshing after t 0 = 7.125T

9 Validation of a CIP-based tank for numerical simulation of free surface flows 885 Fig. 10 Plots of wave position and velocity vectors for one period of sloshing after t 0 = 7.125T (continued) 3.3 Generation of regular wave in a tank In the following part, the generation of regular waves is investigated. Three wavelengths are chosen to validate the ability of the numerical model in simulating the water waves with different wave wavelengths. The computed free surface elevations are compared with linear wave solutions and the second order Stokes solutions at x/h = The linear analysis is supported within the frame of linear wave-maker theory [28]. The numerical results are compared with the Stokes solution to evaluate the effects due to the interaction of wave components in the wave train [2]. Case studies are presented in two dimensions with a constant water depth of h = 0.4 m. Computation conditions are presented in Table 1, where h is the water depth, A is the wave amplitude, T is the wave period, L is the wave length and ε is the wave steepness. The computation is carried out in a two-dimensional wave tank (14.1 m long, 0.8 m height). On the downstream boundary, a damping zone is used [13]. Two meshes are used, the coarse grid (420 65) and the fine grid ( ) with a minimum grid spacing of 5 mm and 3 mm around the still free surface and the wavemaker boundary. The time step is set to t/t = In Fig. 11, the simulated free surface elevation for test Case 1 and Case 2 against time at x/h = 12.5 are compared on the two grids, from which it can be seen that the results from the calculations using the coarse grid (420 65) and the fine grid ( ) are nearly identical and thus a nearly converged result can be obtained by using the coarse grid set with meshes. The coarse grid is used for the following computations. Table 1 Wave conditions for regular wave Case 1 Case 2 Case 3 h/m 0.4 A/cm T/s L/m ε/ka Fig. 11 Wave elevation at x/h = 12.5 using two meshes Figure 12 presents a comparison of the free surface elevation for Case 1 at x/h = 12.5 between the simulation and the linear wave solution; while Fig. 13 displays its comparison with the second order Stokes solution. It can be noticed for Case 1 that the simulation results agree well with the theoretical solutions as there is minor difference between the linear solution and the second order Stokes solution, where nonlinearity effects can be neglected. However, for Case 2,

10 886 X.-Z. Zhao Fig. 12 Free surface elevation at x/h = 12.5 for Case 1. a Comparison with linear solution; b Comparison with second order Stokes solution Fig. 13 Free surface elevation at x/h = 12.5 for Case 2. a Comparison with linear solution; b Comparison with second order Stokes solution the second order Stokes solution predicts well the simulation result. Figure 14 shows the simulation results of Case 3, long wave, and their comparison with the theoretical solutions. There are great discrepancies between the simulation results and the theoretical solutions. In Fig. 14a, the discrepancy between the linear solutions and the computations arises mainly from the wave-wave nonlinear interactions. While in Fig. 14b, the disagreement is mainly caused by the linear wavemaker theory that generates undesired free waves for relatively long waves [29]. Figure 15 shows the simulation results using second order wavemaker theory. It can be found that the simulated free surface elevation is consistent with that given by the second order Stokes solution. More details can be found in Refs. [3] and [29]. Fig. 14 Free surface elevation at x/h = 12.5 for Case 3. a Comparison with linear solution; b Comparison with second order Stokes solution 3.4 Generation of extreme wave in a tank In this part, generations of extreme wave is simulated to verify the applicability of the present model to wave trains. The numerical simulations reproduce the physical experiments described by Zhao [30]. The experimental investigations are carried out in a 50 m long, 3.0 m wide and 1m deep wave flume at the State Key Laboratory of Coastal and Offshore

11 Validation of a CIP-based tank for numerical simulation of free surface flows 887 Engineering, Dalian University of Technology, China and the water depth in the experiment is set to 0.4 m. The waves are generated by a piston wavemaker and wave reflections are absorbed by a 4 m foam layer placed at the downstream end of the flume. Wave gauges (WG) are used to measure the surface elevation along the flume. In the study by Zhao [30], different cases are investigated with different input amplitudes, different peak frequencies and frequency widths. In the physical experiment the focusing area is set from 13.8 m to 17.0 m downstream of the wavemaker, where 21 wave gauges are installed. In the computation, the focal domain is set at 5 m away from the wavemaker. Figure 16 presents the schematic view of the numerical model. 1.6 s with d f = 0.6. The focusing point is taken to be where the surface elevation is a maximum and the surface elevation in the following figures has been shifted in time to coincide with one another. Table 2 Wave conditions for extreme waves Case 4 2 Case 5 6 Case 6 2 Case 7 6 Case 8 2 Case 9 6 A/cm f c ( f min, f max ) 0.83 (0.68, 0.98) (0.53, 1.13) (0.38, 1.28) Fig. 15 Free surface elevation at x/h = 12.5 for Case 3 using second order wavemaker theory Fig. 16 Schematic view of the numerical model Case studies are presented in a two dimensional context with a constant water depth of h = 0.4 m. The computation domain is the same as used in the part of regular wave. The time step is set to t/t c = , where T c is the peak period corresponding to the peak frequency f c. The coarse grid shown in Sect. 3.1 is used for the following computations. Wave conditions run in this section are given in Table 2. Other computation conditions are listed as follows number of wave components: N f = 29, focusing position and time: x p = 5.0 m, t p = 10 s, total time integration performed up to t = 20 s. Several cases are conducted with different frequency widths d f = 0.3, 0.6, 0.9 with fixed peak frequency f c and two focusing amplitudes A, where A represents the linear sum of the component wave amplitude (A = N f a i ). Moreover, different peak periods T p are considered from 1.0 s to Figure 17 considers the free surface elevations at the focal point resulting from three frequency widths d f = 0.3, d f = 0.6 and d f = 0.9, respectively. Computations are carried out for two input wave amplitudes (A = 2 cm, 6 cm), in which coarse mesh 1 are used. The measured time series of the free surface elevation are also shown in Fig. 17 for comparison, which shows that the calculations agree well with the experimental data. Considering the effect of the frequency width, it can be noticed that the difference between the maximum focused wave height and the sideband-wave height increases with increasing frequency width. For the case of d f = 0.3, the difference between the maximum amplitude and the sideband-wave amplitude is small. The generated wave is somewhat like regular wave, rather than extreme wave. For the case of d f = 0.9, a wide range of frequency is covered. The maximum is much larger than the sideband-wave amplitude. The effect of the frequency shape on the wave-body is unclear. Therefore in practical application, the wave spectrum should be carefully selected. To further validate the numerical model, the simulations of extreme wave group with different peak frequencies f c are compared with physical results. The free surface elevations at the focal point resulting from three different peak frequencies f c are displayed in Fig. 18, in which two input focusing amplitudes are considered with a fixed frequency width, d f = 0.6. The numerical conditions are listed in Table 3. For the case of T p = 1.0 s, A = 6 cm, the main difference Table 3 Wave conditions for extreme waves Case 11 2 Case 12 6 Case 13 4 Case 14 8 Case 15 4 Case 16 8 A/cm f c ( f min, f max ) 1.0 (0.70, 1.30) 0.71 (0.41, 1.01) 0.63 (0.33, 0.93)

12 888 X.-Z. Zhao Fig. 17 Comparisons between physical data and computations for different frequency widths of the focusing amplitude between the experimental results and the computation arises from the fact that the generated wave is too steep. For this case, the generated wave is close to the breaking point. From Fig. 18, we can notice that the numerical results agree well with physical data, except for some difference concerning the trough, where the wave trough in the experiment is deeper than that of computations. From the figures shown above, it can be noticed that the present numerical results agree well with the experimental data. To summarize, we can claim that the numerical model can well predict the generation and transformation of specified waves. 4 Conclusions In this study, a two-dimensional numerical tank has been developed for simulating violent free surface flow based on the THINC scheme and the CIP method. The present method is validated through comparisons with theoretical solutions and experimental data. The present model is firstly used to simulate water column collapse with and without obstacle, where complex phenomena like wave breaking and overturning can be predicted, and the simulation results agree well with experiment data. The present model is also used to simulate fluid sloshing in a fixed tank. The numerical results compare well with established analytical results. The model is then used to simulate the generation of regular waves by input incident velocity, where a free surface is present. Regular waves with different wave lengths are simulated and the simulation results agree well with analytical solutions. Finally, as a demonstration, the model is further tested by simulating the generation of extreme waves. The numerical results are found in very good agreement with experimental data. All the important physical phenomena, e.g., wave breaking with and without obstacle, sloshing in a fixed tank, regular wave generation, and extreme waves, etc., have been reasonably captured in the present tank. From these results, one can conclude that the present method has enough accuracy in calculating problems with complex free surfaces. Further investigations on the applicability of the present model for simulating water-floating body interaction are necessary.

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