A CIP-based numerical simulation of freak wave impact on a floating body

Ocean Engineering 87 (2014) 50–63

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

A CIP-based numerical simulation of freak wave impact on a floating body Xizeng Zhao a,b,c,n, Zhouteng Ye a, Yingnan Fu a, Feifeng Cao d a

Ocean College, Zhejiang University, Hangzhou 310058, China State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Hangzhou 310012, China c State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing Hydraulic Research Institute, China d Institute of Harbor-Channel and Coastal Engineering, Department of Civil Engineering, Zhejiang University of Technology, Hangzhou 310014, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 26 August 2013 Accepted 21 May 2014

The purpose of this work is to develop advanced numerical tools for modeling freak waves impact on a floating body undergoing large amplitude motions. An improved model governed by the Navier–Stokes equations with free surface boundary conditions is presented for nonlinear wave-body interactions, in which a more accurate Volume of Fluid (VOF)-type scheme, the Tangent of hyperbola for interface capturing/Slope weighting (THINC/SW) is adopted for interface capturing. The model is solved by a Constrained Interpolation Profile (CIP)-based high order finite-difference method on a fixed Cartesian grid system. A focusing wave theory is used for freak wave generation. Newly designed physical experiments in a two-dimensional glass-wall wave tank are performed for benchmark validation. Fairly good agreements are obtained from the qualitative and quantitative comparisons between numerical results and laboratory data regarding to distorted free surfaces and large amplitude body motions. Some discrepancies are found for the predicted peak pressure. The effects of grid resolution on body motions and impact pressure are performed for error analysis. The comparison of the numerical results and measured data reveals that the proposed model is capable of reproducing the nonlinear dynamics of the floating body for applications. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Freak wave THINC scheme Wave impact CIP method VOF method Floating body

1. Introduction Floating structures have been widely built and utilized in coastal and offshore engineering, such as floating breakwaters, offshore terminals, drilling platforms, wave energy devices and so on. With operations in the oil, gas field and other energy industry moving to deeper water, floating structures are more likely to be exposed to harsh environmental conditions like freak waves. Freak waves are relatively large and spontaneous ocean surface waves that occur far out at sea, and are a threat even to large ships and ocean liners. Therefore, accurate evaluation of such impact forces and corresponding structure responses is important for the purpose of structure safety and disaster prevention. Usually, the water impacts are characterized by nonlinear phenomena, distorted free surfaces and large amplitude structure responses for floating bodies, and their analyses are very complex. Analytical methods are only available for the simple cases such as the linear problems, while laboratory tests are limited by the high costs and the technical limitations of the experimental facilities. As a result, n Corresponding author at: Ocean College, Zhejiang University, Hangzhou 310058, China. Tel.: þ 86 571 88208891; fax: þ 86 571 88208891. E-mail address: [email protected] (X. Zhao).

http://dx.doi.org/10.1016/j.oceaneng.2014.05.009 0029-8018/& 2014 Elsevier Ltd. All rights reserved.

there is an increasing interest in numerically simulating such nonlinear problems. Over the years, many papers have been dedicated to nonlinear wave-structure interactions and numerous conclusions have been made based on physical experiments and numerical simulations. Cox and Ortega (2002) performed a small-scale laboratory experiment to quantify a transient wave overtopping a horizontal deck and a fixed deck, with the attention focused on the free surface and velocity measurements. Ariyarathne et al. (2012) studied the green water impact pressure due to plunging breaking waves impinging on a simplified, three-dimensional model structure in the laboratory and to relate the impact pressure with the measured velocity as well as void fraction on the deck. Meanwhile, extensive studies on wave-structure interactions have been presented using different numerical methods. For instance, Koo and Kim (2004, 2007) employed a potential theory-based model to investigate the nonlinear interactions between water waves and various surface piercing bodies. Bai and Eatock Taylor (2009) used the higher-order boundary element method to investigate the interaction between fully nonlinear water waves in a wave tank and fixed or floating structures. He and Kashiwagi (2012) proposed a finite element and boundary element coupled method to simulate the interaction of water waves with a vertical elastic plate. Yamasaki et al. (2005), for

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

instance, proposed a finite difference method to study the violent water impact on fixed and moving bodies. Hu et al. (2006) proposed a constrained interpolation profile (CIP) scheme (Yabe et al., 2001) to simulate the extremely nonlinear phenomena such as water slamming on the deck of an advancing ship in waves. Sueyoshi et al. (2008) used a particle-based model to compute the nonlinear motions of a floating body influenced by the water on deck. Hu and Kashiwagi (2009) applied a CIP-based method to investigate the nonlinear wave-body interaction problems involving violent free surface motions. Peng et al. (2013) adopted a viscous-flow based method to study the interactions of water waves with submerged floating breakwaters. Some success has been achieved, but further investigation is needed. In most of the studies listed above, regular wave was usually used as the input wave conditions instead of irregular wave which can consider the effect of bandwidth response. Obviously, regular waves only give information on a single frequency at a time while a large bandwidth of frequencies is required for real sea conditions. Therefore, the focused wave packet is preferred. It has the advantage of low time consuming with a controlled focusing both in time and space, and also high resolution of the bandwidth effect with a range of frequency spectrum input in a very short time with no reflection occurring. The research interests on freak waves have been widely motivated and promoted since the stories of monstrous waves were told by sailors (Draper, 1965). Recently, Nikolkina and Didenkulova (2011) collected the evidence of rogue wave events all over the world during the past five years (2006–2010). It is found that the waves occur not only in deep and shallow zones of the world ocean seas, but also near the coast, where they are manifested as either sudden flooding of the coast or high splashes over steep banks or seawalls. Investigation on the formation of very large water waves has been studied extensively in the past several decades ever since Longuet-Higgins (1952) first investigated the statistics of extreme waves in a narrow-banded random wave field. Numerous studies have shown that the freak wave occurrence may be related to wave energy focusing including a number of factors: wave–wave interactions, wave–current interactions, bathymetry, wind effect, selffocusing instabilities, directional effects, etc. More details on these different mechanisms of freak wave formation were reviewed by Kharif and Pelinovsky (2003) and Dysthe et al. (2008). The wave focusing approach is one of the most powerful methods to generate freak waves with a controlled focusing both in time and space. It was firstly proposed by Davis and Zarnick (1964), then for example applied in different studies by Baldock et al. (1996), Ryu et al. (2007), Ning et al. (2008, 2009), Huang and Lin (2012) and Zhao et al. (2009; 2010). However, considerable attention is mainly paid on the features of wave profile and seldom on related wavestructure interactions. Liu et al. (2010) and Li et al. (2012) investigated the 2-D phase focusing wave and 3-D multi-directional focused wave run-up on a fixed bottom-founded vertical cylinder in an experimental flume. Zang et al. (2010) reported on the interaction of steep waves, both non-breaking and breaking, hitting a fixed bottom-founded vertical circular cylinder in a physical wave flume. Westphalen et al. (2012) dealt with the generation and behavior of focused wave groups and the corresponding forces on fixed horizontal and vertical cylinders in a numerical wave tank. Paulsen et al. (2013) studied the wave impact from phase focused waves on a fixed vertical cylinder by means of laboratory experiments and numerical simulations. Bunnik et al. (2008) used a VOFbased model to predict extreme wave loads on fixed offshore structures due to focused wave groups. Most of the studies listed above only dealt with the waves and fixed structures interactions, with few studies involving a movable structure. Hu et al. (2011) using an in-house CFD flow code studied the wave loading on a wave energy converter (WEC) device in heave motion. Rudman and Cleary (2013) applied the Smoothed Particle Hydrodynamics (SPH)

51

method to simulate the impact of a highly non-linear breaking rogue wave on a moored semi-submersible tension leg platform. They considered the effect of wave impact angle and mooring line pre-tension on the subsequent motion of the platform. However, benchmark problem was not presented. Zhao and Hu (2012) studied nonlinear interactions between extreme waves and floating body using an enhanced CIP-based model. They paid attention to the two degrees of freedom (2-DOF) body motions, and computations were compared with experimental results with good agreement obtained when the body surge motion is fixed. In this regard, the CIP-based model is capable of solving such complex problems. To model the interaction between freak waves and freely floating body, several issues have to be resolved. One is the nonlinear distorted free surface related with freak waves, as nonlinearity is one of its main characteristic features. Here, it is overcome by adopting an accurate VOF-type free surface/interface capturing method. Among the available strategies to numerically reproduce an interface, the VOF method is one of the most popular in water-surface capturing, which was first introduced by Hirt and Nichols (1981). The advantages of the VOF method are its mass conservation and easy to implement. Many improved VOF-type schemes have been proposed, such as PLIC–VOF (Youngs, 1982), THINC (Xiao et al., 2005), THINC/WLIC (Yokoi, 2007) (WLIC: Weighed line interface calculation) and THINC/SW scheme (Xiao et al., 2011). In this study, the more accurate THINC/SW scheme is combined with the numerical model to capture the free surface. Another is large amplitude body motions due to distorted free surface. To deal with it, the model is built under the Cartesian grid system to avoid the grid updating. Another is the coupling of the multi-phase interactions including water, air and solid body. To do so, the CIP/CCUP (Constraint interpolation profile/CIP combined, unified procedure) (Yabe and Wang, 1991) is combined with the Cartesian grid system in this research where the multi-phase problem is solved in one set of equations. The Immersed boundary method (Peskin, 1972) is also adopted to treat the fluid–body interactions. The weak coupling of different phases makes its implementation to three dimensions directly. The objectives of the present study are to extend the CIP-based method for simulating freak wave and floating body interactions. An enhanced version of the CIP-based model is proposed to study the interaction of freak waters with a floating structure by introducing the more accurate THINC/SW scheme into the freesurface flow solver. 2-DOF body motions have been presented in our previous paper (Zhao and Hu, 2012). Based on the enhanced model, freak waves nonlinear interacting with a floating body is simulated with considerable attention paid to 3-DOF body motions. Newly designed laboratory experiments are carried out for benchmark validation with numerical simulation. The model will then be used to investigate some physical phenomenon that is not measured experimentally, such as velocity field and pressure contour in the vicinity of the floating body. The rest of the paper is organized as follows. Section 2 gives the CIP-based numerical model. The flow solver and the coupling of fluid-body are briefly introduced. This model combines a more accurate free surface/interface capturing method, the THINC/SW scheme. Experimental set-up for the measurements of both wave elevations and body motion is presented in Section 3. Section 4 displays the numerical results and its comparison with experimental data. A summary and conclusions are given in the final section.

2. A CIP-based numerical model To simulate water waves interacting with a floating body, the equations of motion of the rigid body are solved together with the

52

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

Navier–Stokes equations. A general approach is used to solve the problem in the following way: the CIP method, a high-order difference method, is adopted as the base flow solver in the whole computation domain. The hydrodynamic forces acting on the body are then evaluated by integrating the pressure over the body surface. Using these forces as the input data, the body motion equations can be solved first. Then the body motion such as accelerations, velocities and displacements (translations and rotations) are obtained by integrating in time. The position of the body is also updated and the fluid flow is computed again for the new time. By iterating this procedure until the set time, the body trajectory is reconstructed. More details can be found in the early works (Hu and Kashiwagi, 2004; Hu and Kashiwagi, 2009; Zhao and Hu, 2012) for reference. For the sake of clarity, the key elements of the numerical model are given below. 2.1. Governing equations The wave-body interactions are computed in a two-dimensional viscous-flow based wave tank. Fig. 1 shows the schematic of the wave tank. A stationary Cartesian coordinate system fixed with respect to the earth is chosen with the x-axis on the still free surface and z-axis pointing upward. A piston-type wavemaker is mounted at the left end and a damping zone and increasing grid cells are used at the downstream vertical boundary in order to damp the wave reflection (Hu and Kashiwagi, 2004). A floating body is placed at x¼7.0 m away from the wavemaker. The governing equations for the two dimensional incompressible viscous fluid flow are the Navier– Stokes equations, i.e. the mass and momentum equations, which read in vector form: ! ∇U u ¼ 0

ð1Þ

! ∂u 1 μ ! ! ! ! ð2Þ þ ð u U∇Þ u ¼  ∇p þ ∇2 u þ f : ρ ρ ∂t ! where u and t are the velocity vector and time, respectively; ρ is the ! liquid density, μ is the viscosity, f is the external force, including gravitational force. The fluid–body interaction is treated as a multi-phase problem that includes water, air and solid body in the model. A fixed Cartesian grid that covers the whole computation domain is used. A volume fraction field ϕ m (m ¼ 1, 2, and 3 indicate water, air and solid, respectively) is adopted to represent and track the interface. The total volume function for the water and body is solved by using the following advection equation. ∂ϕ13 ! þ u U ∇ϕ13 ¼ 0 ∂t

ð3Þ

where ϕ13 ¼ ϕ1 þ ϕ3. The density and viscosity of the solid phase are assumed to be the same as those of a liquid phase to ensure numerical stability. This set of our CFD code is different from most other existing CFD models. The volume function for the solid body

ϕ3 is determined by a Lagrangian method (Hu and Kashiwagi, 2004). The position of the water is calculated by ϕ1 ¼ ϕ13  ϕ3, where the position of the liquid and solid phase ϕ13 is captured by a free surface/interface capturing method see (Section 2.4). The volume function for air ϕ2 is then determined by ϕ2 ¼1.0  ϕ1  ϕ3. After all volume functions have been calculated, the physical property λ, the density ρ and viscosity μ are calculated as follows:

3

λ ¼ ∑ ϕm λ m

ð4Þ

m¼1

2.2. Fluid flow solver For the fluid flow solver, a fractional step procedure is used to solve the governing equations. Applying the fractional step approach, the numerical solution of the governing equations is divided into three calculation steps. A staggered grid is used for the spatial discretization. Firstly, the advection phase calculation is performed by the CIP method (Yabe et al., 2001). To apply CIP scheme for the advection calculation, Eq. (2) is differentiated with respect to the spatial coordinate. It results in

 ! ∂ð∂i u Þ ! 1 μ ! ! ! ! ! þ ð u U ∇Þð∂i u Þ ¼  ∂i u U∇ u  ∂i ∇p þ ∂i ∇2 u þ F ∂t ρ ρ ð5Þ where∂i ¼ ∂=∂xi (i¼ 1, 2). Only the left hand side of Eqs. (2) and (5) is considered and the following equations are solved. ! ∂u ! ! þ ð u U ∇Þ u ¼ 0 ∂t   ! ∂ ∂i u ∂t

ð6aÞ

! ! þ ð u U∇Þð∂i u Þ ¼ 0

ð6bÞ

solutions to these two advection equations by CIP method are as follows: ! !n u ¼ Xðx  u ΔtÞ

ð7aÞ

∂X ! ! ð∂i u Þn ðxÞ ¼ ðx  u ΔtÞ ∂xi

ð7bÞ

where X is the cubic polynomial to approximate the spatial profile ! of u in an upwind cell. The superscript ‘n’ denotes the intermediate time level after the advection step. The main advantage of the CIP scheme is its higher accuracy using less grids comparing with the commonly used upwind schemes. Most of details on this scheme can be found in Yabe et al. (2001).

z U

x 0.4m

Wave generator

Floating body Water 7.0m

Damping domain 5.0m

14.5m Fig. 1. Schematic of the numerical wave tank.

1.4m

Air

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

Secondly, all terms in the right hand side of Eqs. (2) and (5) except for those related to pressure are taken into calculation. ! ∂ u μ 2! ! þ ∇ uþF ∂t ρ

ð8aÞ


 ! ∂ð∂i u Þ μ ! ! ! ! ¼  ∂i u U∇ u þ ∂i ∇2 u þ F ∂t ρ

ð8bÞ

It is called the non-advection step (I). For this step, the following explicit schemes are used for the time marching. !nn !n u u μ !n ! ¼ ∇2 u þ F ρ Δt

ð9aÞ


 ! ! ð∂i u Þnn  ð∂i u Þn μ !n ! !n !n ¼  ∂ i u U ∇ u þ ∂ i ∇2 u þ F ρ Δt

ð9bÞ

denotes the intermediate time level after where the superscript the non-advection step (I). A central difference method is used for the right hand side terms of Eq. (9). Thirdly, the velocity–pressure coupling is considered in the non-advection step (II) as follows: ! ∂u 1 ¼  ∇p ρ ∂t

ð10aÞ


 ! ∂ð∂i u Þ 1 ¼  ∂i ∇p ∂t ρ

ð10bÞ

By taking divergence of Eq. (10a), an equation for pressure can !n þ 1 ¼ 0, the Poisson equation for the be obtained. Assuming∇ U u pressure becomes
 1 1 !nn ∇ U ∇pn þ 1 ¼ ∇ U u ð11Þ ρ Δt Here the superscript ‘n þ 1’ denotes the new time step. The successive over-relaxation method (SOR) is used to solve Eq. (11). Once the pressure is computed from the Poisson equation, the velocity field is then updated by the following operations. !n þ 1 !nn Δt n þ 1 ¼ u  ∇p u

ð12aÞ

ρ




    1 ! nþ1 ! nn ¼ ∂i u  Δt∂i ∇pn þ 1 ∂i u

ρ

tracked by a Lagrange method. By integrating the pressure on the body surface, the hydrodynamic forces acting on the body are first calculated. With Newton's Law, the body motion accelerations, velocities are calculated as follows. The motion of a rigid body is determined by specifying a translational motion to its mass center and a rotational motion to its mass center or a support point. In the present study, the motion is assumed to be in two dimensions. The equations of motion of the mass center of a rigid body is given by m

ð12bÞ

2.3. The coupling of fluid–body interaction To model the body motions, the wave–body interaction is coupled by using the Fractional Area Volume Obstacle Representation (FAVOR) method. The FAVOR was developed initially by Hirt (1993) and shown to be one of the most efficient methods to treat the immersed solid bodies (Xiao, 1999). The effect of a moving solid body on the flow is included by imposing the velocity field of the solid body into the flow at the solid edge. The following equation is introduced to update the local information of the fluid domain covered by the body. ! ! ! U ¼ ϕ3 U b þ ð1  ϕ3 Þ u ð13Þ ! ! where U b is the local velocity of the solid body and u denotes the flow velocity obtained from the fluid flow solver. The advantage of the present treatment of the coupling of fluid and body is its simple implementation to a three-dimensional problem. The solid phase ϕ3 is obtained using a virtual particle method (Hu and ! Kashiwagi, 2009). U b , the local velocity of the solid body, is

dV c ¼F dt

ð14Þ

where m is the mass of a rigid body, Vc is the center of mass velocity and F is the total force on the body. The equation for the angular velocity ω about the center of mass is I

nn

53

dωc ¼M dt

ð15Þ

where I is the moment of inertia and M is the total torque about the center of mass or a support point. The hydrodynamic force acting on the body is calculated by integrating the pressure on the body surface as follows: F ¼ ∬A ð  pδik Þnk dA

ð16Þ

where nk is the k-th component of the outward unit normal vector, and A denotes the surface of the solid body. As a brief summary, the coupling of fluid and solid body is calculated as follows. The velocity and pressure are first solved in the section of flow solver. The resulting hydrodynamics forces on the solid body are then computed. Eqs. (14) and (15) are used to compute the translational and the rotational velocity of the rigid body. The solid phase ϕ3 is updated according to its new position. The effects from the solid body on the fluid are enforced as a boundary condition of Eq. (13) during the time integration. Finally, repeat above calculations until the set time. 2.4. THINC/SW for the free surface treatment In this study, an accurate interface capturing scheme, the THINC/ SW scheme (Xiao et al. 2011) is used to calculate the free surface. The THINC/SW scheme is a VOF-type method. In the THINC/SW method, a variable steepness parameter is adopted instead of the constant steepness parameter that is used in the original THINC scheme. This variable parameter helps to maintain the thickness of the jump transition layer. Also, a one-dimensional THINC scheme is described in the following paragraph. In this procedure, multi-dimensional computations can be performed by a directional splitting method. The one-dimensional advection equation for a density function ϕ is written in conservation form as follows: ∂ϕ ∂u þ ∇ðuϕÞ ¼ ϕ ∂x ∂t

ð17Þ

Eq. (17) is discretized by a finite volume method. For a known velocity un, integrating Eq. (17) over a computational cell [xi-1/2, xi þ 1/2] and a time interval [tn, tn þ 1] results in

ϕi

nþ1

n

¼ ϕi þ

1

Δx i

ðg i  1=2  g i þ 1=2 Þ þ

Δt n n ϕ ðu  un Þ Δxi i i þ 1=2 i  1=2

ð18Þ

R tn þ 1   uϕ t 7 1 dt is tn 2 R x 1=2 ϕðx; tÞdx the flux across the cell boundary (x¼xi7 1/2) andϕ ¼ Δ1x xiiþ1=2

where Δxi ¼ xi þ 1=2  xi  1=2 ,Δt ¼ t n þ 1  t n , g t 7 1 ¼ 2

is the cell-averaged density function defined at the cell center(x¼xi). The fluxes are calculated by a semi-Lagrangian method. Similar to the CIP method, the profile of ϕ 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

54

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

numerical error is defined and estimated by n ex ex Error ¼ ∑i;j ϕi;j  ϕi;j =∑i;j ϕi;j

tangent function in order to avoid numerical smearing and oscillation at the interface. Since 0r χ r1, and the variation of χ across the free surface is step-like, a piecewise modified hyperbolic tangent function is used to approximate the profile inside a computation cell, which is displayed as follows:


  x  xi  1=2 α χ x;i ¼ 1 þ γ tanh β δ 2 Δxi

n Here ϕex i, j is the exact solution of ϕi, j. The result of numerical error is summarized in Table 1. In addition, the shape distortion after one rotation is evaluated as shown in Fig. 3. The dotted contour line indicates the exact shape and the solid contour line shows the computational solution. It can be seen from these figures and the table that a finer grid produces better shape retention, and the numerical error of the THINC/SW scheme is lower than that of the original THINC scheme.

ð19Þ

where α, γ, δ, β are parameters to be specified. Parameters α and γ are used to avoid interface smearing, which are given by

α¼

8 <ϕ

iþ1

: ϕi  1

if ϕi þ 1 Z ϕi  1 otherwise

( ; γ¼

1

if ϕi þ 1 Z ϕi  1

1

otherwise

ð23Þ

ð20Þ 3. Laboratory experiments

parameter δ is used to determine the middle point of the hyperbolic tangent function, and is calculated by solving the following equation: 1 Δx i

Z

xi þ 1=2

xi  1=2

n

χ i ðxÞdx ¼ ϕi

Details of the experiments can be found in Zhao and Hu (2012). However, for the completeness of this study, a brief summary of wave generation and wave-body interaction is as follows. The laboratory experiments have been performed in a twodimensional glass-wall wave flume at the Research Institute for Applied Mechanics (RIAM) of Kyushu University, Japan. It was performed by the first author when he was working as a research fellow at RIAM, Kyushu University. The wave flume is 18 m long, 0.30 m wide, and filled with tap water to a depth of 0.4 m. A wedge-type wavemaker is located at one end of the flume to generate target wave trains. Another wedge-type absorbing wavemaker is placed at the opposite end to help damp incident waves. The experimental set-up is displayed in Fig. 4. A photo of the floating body is shown in Fig. 5(a), while Fig. 5(b) shows a photograph of the carriage and guide rail. A simple box-shaped geometry with a deckhouse is used. The floating body is 0.5 m long, 0.29 m wide and 0.123 m high. The width of model (0.29 m) is nearly as same as the width of wave flume (0.3 m). Then, the 3D experimental test can be approximately regarded as a 2D case. The main model geometrical and hydrostatic parameters are summarized in Table 2. This model of a box-type floating body is connected with a heaving rod through a rotational joint. The rotational joint is placed initially at the still free surface. The heaving rod is set in and moves smoothly between the slider mechanisms installed in a carriage on the guide rails. The body is

ð21Þ

parameter β is used to control the sharpness of the variation of the color function. In the original THINC scheme, a constant β ¼3.5 is usually used which may result in ruffling the interface which aligns nearly in the direction of the velocity. Therefore, a refined THINC scheme, the THINC/SW, by determining adaptively according to the orientation of the interface was proposed by Xiao et al. (2011). In a two-dimensional case, parameters β could be determined by the following equations: (

βx ¼ 2:3jnx j þ 0:01 βy ¼ 2:3 ny þ 0:01

ð22Þ

where n ¼ (nx, ny) is the unit norm vector of the interface. After χi(x) is determined, the flux gi at the cell boundary can be calculated. In Fig. 2, for ui þ 1/2 40 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. (18). This cell-integrated value is used to determine the free surface position. Therefore, mass conservation is automatically satisfied for the liquid or water part. A validation test, known as Zalesak's problem (Zalesak, 1979), is performed with the THINC/SW scheme and the original THINC scheme. This test is one of the most popular scalar advection tests. Also, cases with different grid sizes are carried out for parametric study. A velocity field is given by u ¼(y  0.5, 0.5 x) with Δt ¼2π/ 628. In general, one revolution is completed in 628 time steps. The

Table 1 Numerical errors for Zalesak's test problem. Grid number

100  100

200  200

500  500

THINC THINC/SW

9.11  10  2 5.16  10  2

4.95  10  2 2.58  10  2

2.04  10  2 1.01  10  2

ui

Fi x

0

1/ 2

Fi

1

x

1

i 1

i 1

i

i 2

x

0 up

ui

1/ 2

t

Fig. 2. Concept of the THINC/SW scheme.

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

55

THINC

THINC/SW Fig. 3. Zalesak's problem after one rotation (solid for simulation and dashed for theory): THINC (Left) and THINC/SW (Right).

Unit: m 11.0

7.0 5.1

Floating body

0.1875 Wave gauge Wave maker 0.225

0.2 Wave absorbing 0.25 device f=0.023

Rotational joint

Center of gravity

D=0.1 B=0.5

Carriage (2.13 kg)

h=0.4

Spring (3.82 N/m)

Heaving Rod (0.276kg)

Guide Rail

Free Surface

Bottom of tank

Floating Body (14.5kg)

Fig. 4. Experimental setup.

Fig. 5. (a) Photograph of the floating body (left); (b) photograph of the carriage and guide rail (right).

56

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

free to move in heave and pitch. The surge motion is restrained by a spring, connecting the carriage and the guide rail. The masses of the floating body, heaving rod and carriage are m1 ¼ 14.5 kg, m2 ¼0.276 kg and m3 ¼2.13 kg, respectively. The corresponding masses for heave, pitch and surge motion are (m1 þ m2), (m1) and (m1 þm2 þm3), respectively. The allowed body motions are measured by potentiometers. The prescribed wave parameters are checked with wave probes located along the wave flume. The positions of the wave probes are x ¼3.0 m, 5.1 m, 7.0 m, 8.9 m and 11.0 m away from the wave generator, respectively. The sampling rate for the wave probes in these measurements is 100 Hz, while a high sampling frequency of 1000 Hz is chosen for the body motion and pressure measurements. A pressure gauge is placed on the deckhouse at a height of 0.01 m above the deck to record the water-on-deck impact pressure, as shown in Fig. 6. The experiments are recorded by a high-speed video camera for a qualitative understanding of wave-body interactions. The physical measurements have been utilized to study 2-DOF case by Zhao and Hu (2012). In this study, temporal water elevations and body position measurements are used to investigate 3-DOF case.

4. Results In this section, the CIP-based numerical results and their comparisons with experimental measurements are presented including body motions, free surface elevations and impact pressure. As mentioned before, the results of two-dimensional simulations with 2-DOF body motions have been analyzed previously (Zhao and Hu, 2012). Here the results of two-dimensional simulations with 3-DOF body motions are presented and compared with the experimental data. Corresponding velocity field and pressure domain around the body are also observed based on the numerical data. The discrepancies between experimental and numerical results are discussed. Table 2 Main parameters of the floating body. Item

Value (m)

Length Breadth Draft Gyration radius Center of gravity (from the bottom)

0.5 0.29 0.10 0.1535 0.0796

4.1. Wave generation Freak waves are generated numerically and experimentally by the dispersive focusing method (Baldock et al., 1996). Wave conditions for generation are described as follows. A computational domain of 14.5 m  1.4 m is discretized using a variable grid with the minimum gird 0.03 m  0.005 m. For the focused waves, the component wave frequency ranges from 0.6 to 1.6 with fp ¼0.83 s as the peak frequency (Tp ¼1.2 s is the peak period). The wave frequency range is divided into Nf ¼29 components with equal intervals. The waves are focused at tf ¼ 20.0 s and at xf ¼ 7.0 m away from the wavemaker. The total simulation time is chosen to be 30 s. The amplitudes of the individual wave components ai are calculated based on a JONSWAP spectrum (Zhao and Hu, 2012). First, the time histories of the wave elevations (without the body) along the tank are presented and compared with the experimental data. Focusing wave amplitudes, Af ¼0.03 m and Af ¼0.07 m have been examined, and the input positions are adjusted to focus the wave energy at the target location. For the case of Af ¼0.03 m, the nonlinearity can be neglected since the amplitude is small with the wave steepness ε ¼ka¼ 0.136. It is so-called a linear wave focusing case, here k is the wave number and a is the wave amplitude. While for Af ¼0.07 m with the wave steepness ε ¼0.317, the effect of the wave nonlinearity on the wave focusing cannot be neglected. So, it is called the nonlinear wave focusing case. Comparisons of the free surface elevation along the tank between numerical result and experimental data are displayed in Fig. 7. Notice that the computational result coincides pretty well with the experimental data, and the wave elevation reaches its largest value at the focusing position (x/h¼17.5), with a symmetric distribution of wave profile about the focusing time. Similar observations can be made concerning the behavior of the wave profile at space anti-symmetric about the focusing position. Fig. 8 illustrates details of the simulated wave profile at the focusing positions close to the focusing time, and comparison with the experimental data, where the wave profile is normalized by the focusing amplitude A. The line denotes the simulated result, while the symbol indicates the experimental data. It can be seen that the nonlinear focusing crest at the focusing position becomes higher and narrower, while the adjacent wave troughs become wider and shallower than the linear focusing case. Therefore, it implies that the nonlinearity creates a steeper wave envelope, and this observation is also supported by the experimental data. As a summary, the freak wave generation and evolution along the tank have been captured by the present model in this study. The effect of grid resolution on the wave generation is performed and depicted in Fig. 9 for three different grid resolutions. They are a

Pressure sensor

0.01m

Fig. 6. Details of the pressure transducer.

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

Exp Cal

Af=0.03m

Exp Cal

1.0 x/h=7.6

0.5

Af=0.07m x/h=7.6

0.5 f

f

1.0

57

0.0

0.0

-0.5 1.0

-0.5 1.0

x/h=12.7

0.5 f

f

0.0

0.0

-0.5 1.0

-0.5 1.0

x/h=17.5

x/h=17.5

0.5

0.5 f

f

0.0

0.0

-0.5 1.0

-0.5 1.0

x/h=22.2

0.5

x/h=22.2

0.5 f

f

x/h=12.7

0.5

0.0

0.0

-0.5 1.0

-0.5 1.0

x/h=27.4

x/h=27.4

0.5 f

f

0.5

0.0

0.0

-0.5

-0.5 0

5

10

15

20

25

t/Tp

0

5

10

15

20

25

t/Tp

Fig. 7. Comparison of the focused wave profile between computation and experiment: Af ¼0.03 m (Left) and 0.07 m (Right).

Fig. 8. The enlarged view of the focused wave profile for different focusing amplitudes.

coarse grid Nx  Ny ¼ 520  43 with the minimum gird 0.03 m  0.015 m, a middle grid Nx  Ny ¼736  56 with the minimum gird 0.02 m  0.01 m, and a fine grid Nx  Ny ¼ 1180  90 with the minimum gird 0.01 m  0.005 m. The focusing amplitude Af ¼0.07 m is chosen. Also, a variable grid is adopted for the simulation, in which the grid points are concentrated near the free surface and the left-hand wave generation boundary. The minimum grid size is ranged from 0.005 m to 0.03 m. The computed wave profiles by three grids are almost the same, i.e., grid convergence is achieved. 4.2. Wave–body interactions In this subsection, freak wave impact on a 2-D floating body is considered with the nonlinear focused waves. Considerable

Fig. 9. Simulated wave elevations at the focal position. Three different grid sizes are considered (fine grid, middle grid and coarse grid).

attention is paid to the body motions, free surface profiles, and velocity field and pressure domain around the body. The pressure time histories upon the deckhouse are also presented for water on deck phenomena. Also, the numerical results are compared with experimental data for validation. Computations are carried out by using the numerical tank with the floating body placed at x¼ 7.0 m away from the wave maker as shown in Fig. 1. However, several simplifications are made in order to save computation time. First, waves are generated by prescribing an inflow velocity similar to a piston-like wavemaker instead of simulating the wavemaker motion. Second, a shorter numerical tank (14.5 m) than the physical wave tank (18.0 m) is used, and a damping zone is adopted other than simulation of the absorbing wavemaker in the experiment. In the computation, a stationary Cartesian grid is employed with a grid number of 648(x)  248(y)

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X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

Af=0.07m

Exp Cal

0.4 0.0

5

10

15

20

Af=0.07m

Exp Cal

Heave

2-DOF

-0.4 0

the end of the water impact according to the heave motion and pitch motion. At that moment, the flow structure around the body is extremely complex including the water–air interaction, bubbles, turbulence and structure vibration and so on. It is very hard to obtain the exact hydrodynamics force acting on the body in such complex flow field. The impulsive impact pressures on the deckhouse are shown at the bottom of Fig. 10. Notice that two peaks phenomenon of the pressure variation can be predicted well by the present model. One is caused by the first high-speed water impact along the deck, while another is caused by water fall along the deckhouse; the second peak is larger than the first peak. In another word, the water fall is more dangerous for structure safety. Meanwhile, more DOF body motions appear to decrease the impact pressure, where the peak pressure for 2-DOF is larger than that of 3-DOF. Although the exact impact force is predicted less accurately here, there might be several possible reasons for this problem. First could be the three-dimensional effect, since the present simulations are in two dimensions. Second could be the grid resolution selected. Therefore, the effects of grid resolution on water on deck impact pressure and body motions are considered and checked below. The results using different grid sizes for the wave–body interactions are compared with measured data as shown in Figs. 11 and 12. Three different grid sizes are used: coarse grid (totally 618  246 grids), middle grid (totally 748  324 grids) and fine grid (totally 1138  554 grids). The “coarse” grid case corresponds to a simulation

(yc-yc0) /Af

(yc-yc0) /Af

and a minimum grid spacing of Δx¼Δy¼ 0.003 m around the body and the free surface. The grid system is the “coarse” grid, which will be pointed out below. The time step is dynamically determined to satisfy the stability criterion of a Courant–Friedrichs– Lewy condition (CFL ¼ 0.1) with the total simulation time up to 30.0 s. Fig. 10 illustrates an extensive comparison of the simulated and measured body motions, free surface profiles before the impact pressure found on the deckhouse due to water on deck. Fig. 10 (a) shows the 2-DOF body motions with the surge motion fixed in column (a), while Fig. 10(b) displays the 3-DOF body motions with all modes free in column (b). It can be seen that the present numerical results are in good agreement with the experimental data. It appears that the 2-DOF results are better than the 3DOF as the flow structure in the 3DOF is more complicated than that of 2DOF during the wave-body interactions. Comparing column (a) with column (b), little difference can be found for the heave motion, pitch motion and free surface elevation before the body. By checking carefully, the maximum heave motion in positive direction for the case of 3-DOF is little larger than that of 2-DOF. Meanwhile, for the case of 3-DOF, the numerical model predicts the peak clockwise pitch motion less accurately. However, the trend of the pitch motion has been captured by the present model. For the surge motion, discrepancy between the numerical results and the experimental data can be found from its maximum surge motion. The surge motion reaches its maximum value almost at

0.4

Heave

3-DOF

0.0 -0.4 0

25

5

10

15

20

Pitch Pitch (deg)

Pitch (deg)

Pitch 10 0 -10 0

5

10

15

20

5

10

15

0

5

10

15

20

10

0 0

5

10

15

20 Elevation at x/h=12.7

0

5

10

15

20

Pressure

1

17.0 t/Tp

25

Pressure Pressure (kpa)

2

16.5

25

0 -1

25

25

t/Tp

3 Pressure (kpa)

20 Surge

t/Tp

0 16.0

15

1

0 -1

5

1

-1

25

η/Af

η/Af

20 Elevation at x/h=12.7

1

-10

2

0 0

0

0

Surge

1

-1

10

25

(xc-xc0) /Af

(xc-xc0) /Af

2

25

17.5

18.0

1

0 16.0

16.5

17.0 t/Tp

Fig. 10. Body motions due to focused waves: Tp ¼ 1.2 s, Af ¼0.07 m: 2-DOF (Left) and 3-DOF (Right).

17.5

18.0

Exp Fine grid Middle grid Coarse grid

0.4 0.0

Af=0.07m

2-DOF

Heave

(yc-yc0) /Af

(yc-yc0) /Af

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

-0.4

59

Af=0.07m

Exp Fine grid Middle grid Coarse grid

0.4 0.0

3-DOF

-0.4

Pitch Pitch (deg)

Pitch (deg)

Pitch 10 0

10 0 -10

-10

1 0

0 -1 1

η/Af

η/Af

1

Elevation at x/h=12.7

1

0

0

5

10

15

20

Surge

2

(xc-xc0) /Af

(xc-xc0) /Af

Surge

-1

Heave

Elevation at x/h=12.7

0

-1

25

0

5

10

15

20

25

t/Tp

t/Tp Fig. 11. Effect of the grid resolution on the body motions: 2-DOF (Left) and 3-DOF (Right).

Exp Fine grid Middle grid Coarse grid

1.5

Exp 1) grid Fine Middle grid Coarse grid

3-DOF

Pressure(kpa)

Pressure(kpa)

2-DOF

1.0 0.5

1.5 1.0 0.5 0.0

0.0 16.6

16.7

16.8

16.9

17.0

17.1

t/Tp

16.8

16.9

17.0

17.1

17.2

t/Tp

Fig. 12. Effect of the grid resolution on the impact pressure due to green water: 2-DOF (Left) and 3-DOF (Right). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

with the minimum grid of 0.003 m, the results of which are the same as those already shown in Fig. 10. For the middle grid and fine grid cases, the minimum grid of 0.002 m and 0.001 m are chosen, respectively. In Fig. 11, the effect of the grid resolution on the body motions is depicted and the computed results are almost the same for both 2-DOF and 3-DOF cases. This implies that global quantities such as the wave-induced body motions are not so sensitive to the grid resolution. The numerical results have already converged with the coarse resolution. The effect of the grid resolution on the impact pressure is depicted in Fig. 12. It can be seen that the numerical prediction of the first peak pressure with fine resolution is higher than that with coarse resolution, and agrees well with observed results. While for the second peak pressure, little improvement can be noticed. This indicates that local quantities like the first peak pressure are obviously improved with an increase in the grid resolution. But the second peak pressure is not so sensitive to the grid resolution. Further study is warranted for improvement in the accuracy of the coupling of wave–body interactions.

Figs. 13 and 14 show a qualitative comparison of free surface and body position between computation and experiment for 2-DOF and 3-DOF cases, respectively. Meanwhile, the numerical simulations of the pressure field and the velocity vector around the body at different moments are presented, which may give some insight in the internal flow structure during water impact. In order to obtain clear flow structure, the “middle” grid system is chosen for the computations. The results reveal the freak wave collides with the offshore structure and the pressure and velocity abruptly changes around the structure during water impact. For the water on deck phenomena, little difference can be found no matter for the fixed body (Greco et al., 2005) or for the freely floating body in this study. The process of the freak wave impact on the floating body can be divided into several stages. Firstly, the water front approaching the body is steepened near the intersection with the deck (Fig. 13, t¼ 19.9 s and Fig. 14, t ¼20.0 s). The results reveal that before the wave approaches the body, the flow structure is less complicated with not vortex and simple

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X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

t=19.9s

Surface and velocity

t=20.1s 1m/s

t=19.9s

1m/s

t=20.1s

Pressure domain

4000 3200 2400 1600 800 0

4000 3200 2400 1600 800 0

4000 3200 2400 1600 800 0

t=20.7s 1m/s

1m/s

t=20.3s

4000 3200 2400 1600 800 0

t=20.5s

t=20.5s

t=20.3s

t=20.9s 1m/s

t=20.7s

4000 3200 2400 1600 800 0

1m/s

t=20.9s

4000 3200 2400 1600 800 0

Fig. 13. Qualitative comparison of the water surface elevations, flow velocity fields, pressure domain and body positions at different phases between computation and experiment with 2-DOF body motions.

pressure contour. As the water climbs the deck, a steep slope is generated at the front edge of the deck and the body shows clockwise pitch response accordingly (Fig. 13, t¼ 20.1 s and Fig. 14, t¼20.2 s). At this moment, flow separation occurs and a vortex is generated and develops into a circular shape, which can be seen in both the velocity field and pressure domain around the left side of the body. Then, the wave front collides with the deckhouse and the first peak impact pressure appears. Around that moment, the pitch motion reaches its first peak value. Because of the vertical wall, the wave front is deviated upward and a vertical jet is

deflected. It rises vertically up the wall and then slows down due to gravity effects (Fig. 13, t¼ 20.3 s and Fig. 14, t ¼20.4 s). Looking at the flow structure in the figures, the velocity below the body of the left half of domain changes its direction to horizon. While for the right half of domain, the velocity below the body changes its direction to the top right corner. Totally, the body suffers from an anticlockwise torque, so it makes the body with an anticlockwise rotation. Finally, the surface fluid motion is converted into a water run-down subjected to the gravity force, and then overturns to be a re-entry against the underlying free surface.

X. Zhao et al. / Ocean Engineering 87 (2014) 50–63

t=20.0s

Surface and velocity

t=20.2s 1m/s

t=20.0s

P

1m/s

P

4000 3200 2400 1600 800 0

t=20.8s 1m/s

4000 3200 2400 1600 800 0

P

4000 3200 2400 1600 800 0

t=20.6s

1m/s

t=20.4s

P

4000 3200 2400 1600 800 0

t=20.6s

t=20.4s

t=20.2s

Pressure domain

61

t=21.0s 1m/s

t=20.8s

P

4000 3200 2400 1600 800 0

1m/s

t=21.0s

P

4000 3200 2400 1600 800 0

Fig. 14. Qualitative comparison of the water surface elevations, flow velocity fields, pressure domain and body positions at different phases between computation and experiment with 3-DOF body motions.

A backward plunging wave forms and hits the deck (Fig. 13, t¼ 20.5 s and Fig. 14, t ¼20.6 s). Meanwhile, the second peak pressure occurs. The anticlockwise body rotation compresses the flow structure and the vortex moves clockwise around the body corner for a 2-DOF case (Fig. 13, t ¼20.5 s). While for the 3-DOF case, different flow structures can be found and the vortex departs from the body because of the horizontal body motions (Fig. 14, t¼ 20.6 s). After that, the water on deck strikes the underlying water with air entrapment (Fig. 13, t¼20.7 s and Fig. 14, t¼20.8 s),

where violent nonlinear fluid–structure interactions can be observed. Wave breaking and bubbles around the body can be found in both the laboratory data and the computation results. For the 2-DOF case, the vortex moves upward along the left side of the body besides several small sizes of the vortex are generated. It is almost the same situation for the 3-DOF case. Look closely, horizontal vortex separations occur as the floating body suffers from surge motion for the 3-DOF case. As the waves pass by, the water surface calms down slowly with “foams” and bubbles

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around the body (Fig. 13, t¼20.9 s and Fig. 14, t¼21.0 s). The body oscillates with a small amount of water on deck appeared. For the velocity field, new vortex is generated for both 2-DOF and 3-DOF cases. These comparisons of water surface around the body indicate that the distorted free surface flow happens during the wave impact and is also captured numerically. The velocity and pressure fields all suggest that complex flow structure such as vortex is generated during the wave–body interactions, which can be noticed in most snapshots of Figs. 13 and 14. Accordingly, the floating body encounters a deep heave and a large amplitude pitch motion. Thus, it is one of the most nonlinear forms of water motion including breaking, bubbles, vortex and large amplitude body motions. As compared with the laboratory data, these processes can be modeled in the numerical simulations.

body motions, but the local first peak pressure due to water on deck is improved with finer grids. Comparisons of numerical results and experimental data show that the proposed CIP-based model is capable of reproducing the freak wave–structure interactions, and can provide an effective approach for nonlinear problems where traditional models are not applicable. Although the results reported here are preliminary about a simple body in two dimensions, they clearly show the CIP-based model could be used to deal with distorted free surface impact problems (wave breaking and water–air mixing) and to predict the corresponding body responses (large amplitude body motions). Further studies will be reported that consider the three dimensions and a specific structure such as a breakwater.

Acknowledgments 5. Conclusions A CIP-based multi-phase fluid flow model has been proposed for investigating the freak wave impact on a 2-D floating body. The model governed by the Navier–Stokes equations is solved by a CIP-based high-order finite difference method on a fixed Cartesian grid system, which contribute to a robust flow solver for the governing equations. Fluid–body interaction is treated as a multiphase flow problem with water, air and solid phase solving one set of governing equations. A VOF-type scheme, the THINC/SW scheme is implemented in the model to accurately define the free-surface configuration. In order to validate the numerical results, physical experiments have been performed in a 2-D wave flume. The two-dimensional benchmark problem could be used to validate our/other CFD code for nonlinear wave–body interactions. Waves are generated using a wave focusing theory and its interaction with a box-shaped floating body is studied. High speed camera images of the impact events are recorded, wave profile along the flume and motions of the body are measured, impact pressure on a vertical deckhouse along the body deck is measured. Qualitative and quantitative comparisons between numerical results and laboratory data have been presented. Considerable attention is given to the free surface profile, nonlinear body responses and impact pressure due to water-on-deck numerically and experimentally. The experimental conditions were simple and both the body motions, impact pressure due to water on deck and the free surface elevations at particular positions are measured and can be easily used for comparison by other numerical simulations. Results show that the CIP-based model can be used to reproduce the freak wave nonlinear interacting with the floating body, where the distorted free surface like wave breaking and water–air mixing, and large amplitude body motions have to be dealt with. Fairly good agreement is obtained for the prediction of the free surface profile round the body and the body motions numerically and experimentally. Water on deck is captured numerically with several main features like “dam break” flow, “jet” flow, “run-down” flow, water overturning, wave breaking and water–air mixing and so on. Velocity vector and pressure domain are also displayed around the floating body and strong viscous process such as the vortex generation and disappearance are accurately reproduced using the proposed model. Vortex generation occurs mainly around the corner of the floating body for the 2-DOF case. While for the 3-DOF cases, vortex separations are found to move horizontally because of the surge body motions. The predicted impact pressure due to water on deck shows that the same tendency with laboratory results for both 2-DOF and 3-DOF. However, the peak impact pressure due to water on deck is underpredicted for both 2-DOF and 3-DOF. Effect of the grid resolutions is performed. It is found that no obvious difference could be seen in the

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