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24th Symposium on Naval Hydrodynamics Futuoka, JAPAN, 8-13 July 2002 Prediction of Slam Loads on Wedge Section using Computational Fluid Dynamics (CFD) Techniques Reddy D.N., Scallion T., Chengi Kuo (Universities of Strathclyde and Glasgow, UK) Abstract Numerical prediction of slam loads on a wedge shaped ship section during water entry is carried out using Computational fluid Dynamics (CFD) techniques by employing Finite Volume Method (FVM) for discretization of the flow equations and Volume Of Fluids scheme (VOF) for free-surface capturing. Water impact of a wedge section with 30 degrees of deadrise angle is numerically simulated while taking into account the key factors affecting evaluation of slam loads. History of impact velocity, pressures at distinct locations and the impact force on whole wedge section during water impact are predicted. The main conclusions drawn are that close correlations with the existing experimental data are obtained, and the effects of impact velocity variations, domain size and three dimensionality are significant on the numerical results for slam loads. ~ Introduction Ships operating in open seas undergo severe motions with increasing speeds and experience slam loads on the hull due to continuous buffeting from the waves, leading to loss of ship control and speed, discomfort to crew and passengers, increased wet decks and structural damage. Considering the several adverse effects that slamming loads can cause to the ship, it is essential to estimate slam loads on ship sections at the design stage itself in order to ensure safety, economical design and operation of the ships. Sections at forward are more prone to slamming impacts, and generally have either U shapes or V shapes. Many theoretical (Wagner (1931), Payne (1981), Greenhow(1987), Wilson (1989), Zhao and Faltinsen (1993, 1996), Arai et al. (1989, 1994), Sames et al. (1999), Muzaferija et al.(2000)) and experimental studies (Chuang (1967), Campbell et al. (1980), Zhao et al. (1996)) on prediction of slam loads during water entry of solid bodies have been reported, to name a few, since the pioneering analytical work by von Karman (1929). Comprehensive review on the subject has also been reported by SNAME (1993). Critical review indicates that model experiments produced closer correlations with the ship measurements compared to theoretical methods. However, model experiments are prohibitively costly and time consuming in addition to the difficulties involved in generalizing the specific results for their universal usage. At the same time, theoretical methods have not reached a stage to reliably predict slam loads on any ship plying in a seaway due to the highly complex nature and the associated difficulties in modeling the slamming phenomena apart from the high computational requirements. Even though theoretical methods cannot fully replace the need for experiments, they certainly lead to efficient methods of evaluating different cost effective options available at design stage. With no analytical solutions existing to the complicated conserved flow equations for simulating slamming of the ship sections considering all the influences, the best solution options have traditionally been through the use of numerical methods. Numerical methods of varying vigor and sophistication have been adopted in analyzing water entry problems and mostly consider largely simplified governing equations and boundary conditions based on potential theory and sometimes simplified further through different approaches of reformulation. Their application was largely limited to simple forms like two-dimensional regular and ship shaped sections or three-dimensional regular shapes and resulted in obtaining deeper insight into the slamming problem rather than producing results within engineering accuracy suitable for their practical application. Improvements in computational capabilities have encouraged employment of more efficient numerical methods based on Boundary Element (BE) and Computational Fluid Dynamics (CFD) techniques, but they still required some form of approximation for non-linear free surface boundary. Zhao and Faltinsen (1993, 1996) used boundary element techniques to predict slam loads on wedge and ship sections accounting for splash effects and variations in impact velocity. Their numerical results for ship section compared favorably with the experiments but over predicted for the case of wedge section, and shown that the effect of three- dimensionality is significant and also that the numerical results were sensitive to the length of the introduced jet part. With the emergence of several numerical solution techniques to free surface evolution and improved computational capabilities, CFD methods based on Finite Volume (FV) approach seem to be gaining ground. However, accuracy of the numerical results during water entry of solid bodies

. . . prlman. y requires proper modeling of the physical slamming phenomena in addition to careful consideration of the different key factors affecting them. Application of CFD techniques in Slammine Research To obtain better correlation of the numerical results with the experiments, Arai et al.~1989, 1994), Sames et al.61999), Muzaferija et al..~2000), to name a few, employed CFD techniques to predict slamming loads on arbitrary shaped two-dimensional sections. Arai et al., used finite difference method for discretization of the Euler equations and Volume of Fluids (VOF) method for free surface evolution in order to obtain slam loads on 2D sections assuming constant entry velocity. He did obtain favourable trends but the pressures did not correlate well with those from the drop tests probably due to neglect of variations in impact velocities. Sames et al. (1999) and Muzaferija et al. (2000) used Finite Volume Method (FVM) for discretization of the Navier- Stokes equations and High Resolution Interface Capturing (HRIC) scheme for free surface to predict slam loads on 2D sections by considering varying re- entry velocities. Sames et al. concluded that prescribed vertical velocity histories significantly affected the determination of realistic pressure levels. Muzaferija et al. (2000) obtained slam loads on wedge section considering three-dimensionality with numerical results matching reasonably well with the experiments of Zhao et al. (1996), however under the assumption that these experiments were conducted in restricted waters. The present study considers numerical prediction of slam loads consisting of total impact force and pressures during water entry on a V-shape similar to the case of a wedge dropped onto calm water surface. The present numerical prediction method using CFD techniques accounts for the key factors affecting the accuracy of these slam loads using CFD techniques to achieve better correlations with experiments. 2 Numerical Method 2.1 Mathematical Model Conservation equations of mass and momentum (Governing Equations) and the boundary conditions as given below are considered to simulate slamming phenomena on an arbitrary impacting section. Governing Equations: Continuity equation: I pv.n dS =0= div v =0 S Momentum equations: Ipv ~7Q+ Ipvi.n AS= J(~/1 grad v-p] .n dS+ J pg dQ dtQ 5 5( J n Approximations: Governing equations neglect the surface forces due to surface tension forces as they are very small compared to the impact forces (Weber number Wend) and do not affect the slamming phenomena, but the same due to viscosity are considered. Similarly, the body forces due to centrifugal, coriolis and electro-magnetic forces do not exist or affect the slamming phenomena and hence are neglected, whereas body force due to gravity (as Fn2 _ 1) is considered. Further, the velocities of the fluid are expected to be small compared to the velocity of sound in water (Mach number Mw < 0.3), and it is fairly reasonable to consider the flow to be incompressible. Boundary conditions: To reduce the computational requirements, the solution domain is defined considering only half the impacting section, since the ship and the hence the transverse sections are traditionally symmetrical about their central vertical plane. Additionally, the body is considered to impact vertically in the same direction that gravitational force acts due to which the fluid flow can also be assumed to be symmetrical about this plane. Bach ( S B ~ Outlet ~ So ~ ~~ | Air | Free Surface ( SF) tI! fl ' I I ' t Figure 1: Schematic diagram for computational domain of Wedge section Inlet (S.): Slamming impact of the body section on to the water surface is simulated by keeping the body stationary and letting the water at the inlet boundary (SI) to move at the velocity of impact vats. At the inlet boundary condition, mass flow rate is fixed irrespective of the internal pressure as An = yin, where n is unit normal vector along the boundary, and vin is the instantaneous velocity of impact. Body Boundary (SRL With the body being stationary and impermeable, fluid velocity at the body boundary is equal to that of the wall itself, , which essentially means that flow exists v = V ._77 only in tangential direction to the boundary surface vt 2

and the velocity perpendicular to the body surface Vn is zero. Additionally, the tangential velocity of the fluid at the wall is equal to that of the wall itself, which follows from the fact that the viscous fluids stick to solid (no-slip condition), i.e., the tangential stresses Ant exist at the body boundary but the normal viscous stresses are zero. Free Surface (SFL TO determine the instantaneous shape of the free surface and the forces exerted on the fluids in contact, both kinematic and dynamic conditions need to be fulfilled on the free surface. Kinematic condition: No convective mass transfer through the free surface; i.e., the fluid velocity component normal to the free surface is equal to the free surface velocity. Dynamic condition: Forces acting on fluids in contact at the free surface are in equilibrium. In the absence of surface tension, this means that the stresses T on both sides of the free surface are equal. In the absence of wind, shear stresses are generally neglected for ship flows. With the further assumption that viscous effects on the free surface boundary are negligible, in which case normal stresses are neglected, the pressures on either side of the free surface are equal, which are simply atmospheric. Walls (Sw~The vertical boundary at far end of the computational domain is considered to be wall where no-slip boundary condition similar to that of body boundary is applied i.e., the normal velocity across the boundary is defined to be zero. Also normal viscous stress is zero at the wall as in the case of impermeable wall with no-slip condition. Svmmetrv plane (Set The vertical boundary at the left end of the computational domain Ss is considered to be symmetry plane wall as this plane coincides with the symmetry plane of the body section. Similar to the case of a wall, here also the velocities normal to the plane vn are zero. At the same time, the tangential shear viscous stresses on the symmetry boundary are zero, but the normal stresses Inn exist. Outlet (SOL Outlet is open to atmosphere and there can be fluid flow across the outlet to maintain the atmospheric pressure. At the outlet boundary, fixed pressure outlet boundary condition is considered with the mass flux adjusted to satisfy continuity equation with the direction of mass flow determined by the pressure inside the outlet boundary is more than the atmospheric pressure or not. The former produces local outlet flow, whereas the latter produces local inlet flow. SOUTH3 SOUTH2 SOUTH1 Bod' 2.2 Numerical Solution Computational Grid Multi-block structured grid with common matching interfaces has been generated for the solution domain considering its simplicity and the ability to maintain conservativeness and reduction in numerical diffusion. Rectilinear grid is chosen with x-axis along the length of the section, y and z-axes along section's breadth and depth respectively as shown in Figure 2. YL . ~ L~ it'' ~ _Y , rUTFT ;_ Figure 2: Sub-division of the Computational Domain The sizes of computational domain (YL, ZL), grid and its distribution are problem dependent, and are chosen to be sufficiently large enough to obtain numerical results independent of these factors. Boundaries of the computational domain along the horizontal (z=0) and vertical axes (y=0), designated as 'Low' and 'South', are divided into two segments (Lowl and Low2) and three segments (South!, South2 and South3) respectively. The other two opposite boundaries are divided in a similar way with the number of cells on opposite sides being same to generate structured grid. With the section's shape remaining constant along x-axis, grid size along x- axis is kept at one cell or CV. Grid points along the segment containing the body boundary are distributed uniformly along the body girth mainly to (i) Ensure that the cell corner points lie on the body boundary for its better representation and (ii) Achieve smaller size of cells where the body curvature is steep within the limits of acceptable cell aspect ratios of 10. Finite Approximations Convection is then approximated using Hybrid Differencing Scheme (HDS), a combination of 2nd order Central Differencing Scheme (CDS) and the first order Upwind Differencing Scheme (UDS) to ensure higher stability and better accuracy by considering the transport property of the convective term in a better manner. The diffusive term is discretized employing 2nd order CDS. The main point to be noted is that, interpolation scheme of order 3

higher than second makes sense only if the surface integrals are also approximated using higher order follllulae. Higher order schemes may reduce numerical diffusion but give unstable solutions. For discretizing the pressure term, the mid point rule approximation for the surface integrals is used with the pressure term interpolated linearly to the center of cell faces. Source term containing the body forces due to gravity, represented by a volume integral, cannot be replaced by surface integrals and hence this non-conservative term is integrated over the cell volume by a simple second order accurate approximation, which also becomes exact since the fluid density p is considered to be constant. For the present case of simulating slamming phenomena, implicit scheme for transient terms is considered due to its unconditional stability and ability to obtain physically realistic and bounded results and does not put restrictions on the size of the time step through courant condition. Free Surface The shape and position of the free surface keeps changing with time during slamming of the ships and if the cells, which lie on the free surface boundary, are known, implementation of the free surface boundary conditions is straightforward. In general, dynamic condition of the free surface is implemented directly whereas the kinematic condition is used to update the free surface position for each time step in an iterative manner. Volume of Fluid scheme (VOF) is employed for interface capturing, as the same is considered to be computationally more efficient and also treats the overturning of free surface. Here, a scalar transport equation as given below for the volume fraction 'c' of fluid is solved in addition to the usual conservation equations. ,9 JcdQ + |cv.n dS = 0 The values of volume fraction indicate the Guides) present in the cell. where c=0 and 1 indicating the , ~ cells filled completely with air and liquid respectively, and any value between 1 and 0 indicating presence of both air and water. Physical properties of the fluids involved in the computational domain depend water and air, and the volume fraction c, and are obtained through piece wise interpolation. Limiting factors imposed on the volume fraction c to preserve sharpness of the free surface interface are as follows: c=0 if C<cmin=Q4 and c=1 if c2cma,=0.6 Discretization of the convective term in the above mentioned transport equation should neither produce numerical diffusion nor unbounded values of volume fraction, i.e., c should lie between the minimum and maximum value of the neighbor cells. Numerical diffusion produces smearing of the free surface interface with the smearing region increasing with time, ultimately leading to inaccurate results. Numerical diffusion is minimized both by reducing the grid size and using a higher order convection scheme, namely van Leer scheme (1974~. Temporal scheme used for evolution of the free surface being explicit, the Courant condition for every time step l\t is to be satisfied for achieving numerical stability as given below. If t < min ~ Ay A z ~ u v w Pressure correction techniques based on SIMPLEST algorithm method are used for calculating the pressure field in the computational domain. implementation General purpose CFD code 'PHOENICS' is utilised in implementing the present numerical method. Varying time step sizes based on courant condition are considered to optimize on computational time. Variations in impact velocity profiles are incorporated through user-defined code in an external file. Grid data for the solution domain containing the co-ordinates (x, y, z) of all the grid corner points is generated separately and input to the code. Relaxation parameters are set for the different variables like velocities and pressure to promote convergence by 'slowing down ' the changes made to the values of during the solution procedure. Relaxations for pressure and velocities are carried out by the Linear and False time step relaxation methods respectively. For the pressure the value of the linear relaxation parameter is chosen to be between 0.1 and 0.4, whereas for velocities false time-step is set, depending upon the least value of the ratio of characteristic length to velocity in the whole computational domain. Stopping the outer and inner iterations for solving the algebraic equations depends upon the convergence rate and the residual norm. The residual norm, a ratio of the sum of the residuals over the whole computational domain and a chosen reference value for the variable prior to the first inner iteration, is used as reference point for checking the convergence. For obtaining the residual norm, reference value is taken as small fraction (0.1 %) of typical value of the variable. All the computations have been camed out using a Pentium II, 300 MHz, 64 MB RAM machine. 4

3 Incorporation of Deceleration effects (ii) It can be seen from the experimental data (Zhao et. al., 1996) that the impact velocities are not constant through out the slamming period, but vary depending upon the geometrical shape and mass of the section itself, known as 'deceleration effects'. This is mainly due to the changes in the total force history experienced by the section during the drop tests, i.e., resultant of the slamming (both hydrodynamic and hydrostatic) impact force and the weight of the body acting in opposite directions. Impact pressures on the body surface are known to be functions of the section's shape and instantaneous impact velocity. It is therefore imperative for any numerical model to be incorporated to consider variations in impact velocities during slamming. Implementation of the velocity variations during the numerical solution is relatively straight forward if the impact velocity profile is known in priori. However, this may not always be available before hand and the same may need to be estimated during simulation itself. Deceleration effects are therefore implemented in the present numerical method through one of the procedures briefed below. Assigning the instantaneous impact velocity at every time step from the known velocity profile with respect to impact time. Estimating the instantaneous impact velocity during the numerical simulation itself based on the different instantaneous forces acting on the impacting body. In the first method, impact velocity profile measured during the experiments is fitted into a polynomial as a function of impact time. In the second method, impact velocity of the body is estimated during the simulation at every time step, with the initial impact time being the instant when body touches water surface. The first step involved in calculating instantaneous impact velocities is to estimate the instantaneous acceleration of the body from the resultant vertical force (F) acting on the section. The total resultant vertical force (F) per unit length of the impacting body is the difference between the vertical slamming force (S), which also includes the hydrostatic force, and body weight (mg). ~ = m 2 = ~ - mg Mass of the freely falling body is therefore an additional factor influencing the impact velocity profile with time, consequently the pressures and hence the total slam force. The vertical slamming force S is calculated by integrating the instantaneous cell pressures obtained numerically, over the whole impacting surface by considering the corresponding impact area of each cell in that direction, whereas the weight of the body (mg) remains constant during the whole impact period. From the resulting instantaneous body acceleration, it is straightforward to get the body velocity by multiplying the acceleration with the time step, which is very small in the order of milliseconds used for numerical simulation. 4 Numerical Results To illustrate applicability of the present numerical method to practical ship design problems on slamming, it is necessary to validate the numerical model based on CFD techniques through correlations with the model experiments. This can only be achieved by clear understanding of the different factors influencing both the numerical and experimental results and implementing the same accordingly. These influencing factors broadly fall into two categories as those . . Affecting the numerical accuracy of the results like proper representation of the boundaries, computational domain, grid and cell sizes etc. Representing the actual test condition like body decelerations, position of testing tank wall boundaries with respect to body, three- dimensionality etc. Details of Physical model Experimental results as obtained by Zhao et. al. (1996) for the water impact of the wedge section have been used to validate the present numerical results. The geometrical and experimental details of the wedge section used by Zhao are given in Table 1 below. Geometrical details of Wedge test section Length of the section L [m] Breadth of the section B Eml ., Vertical distance from Keel to knuckles D Eml ~ . Length of measuring section Lm Em] Weight of drop-rig including ballast [kg] Weight of Measuring section [kg] Initial Impact Velocity Via [m/s] Table 1: Wedge section details (Zhao et. al., 1996) 1.000 0.500 0.l4s 0.200 241.0 14.5 ~ 1SO For better validation of the numerical results, direct comparison with more than one parameter obtained during the experiments as listed below is considered. Impact velocity profiles, Vertical impact force on the whole section, and Pressures at certain locations on the surface. 5

4.1 Deceleration Effects Variations in drop velocity of the body over impact period, known as 'deceleration effects', have been incorporated in the numerical model in order to simulate the actual test condition of drop tests. Observation of the experimental data (Zhao et al., 1996) reveals that the total impact period for the Wedge section is around 0.025 seconds with the impact force peaking at 0.0158 seconds. Initial impact velocity for the wedge section is 6.150 m/s and the experimental data shows that the impact velocity do not remain constant during the whole impact period. Impact of wedge section is numerically simulated considering the deceleration effects following both the methods i.e., using (i) known velocity profile from the experiments (Zhao, et. al., 1996) and (ii) estimating the instantaneous impact velocity during simulation itself. In the first method, impact velocity data V(t), measured during the experiments, is fitted into a polynomial as a function of impact time as given below. v(~) = 6.~s+~6.0~*` - 6398.94*r + ~02439.20*~+ 2006636.38*~4, 0.0<~<0.025 sec In the second method, at every time step during simulation, the impact velocity is calculated based on the different instantaneous forces acting on the impacting body as briefed in Section 3. Computational Domain Size Initial computational domain size ratios of Dy =10 and Dz =2 are chosen along y and z- axes for generating the grid over the computational domain. Here, Dy is the ratio of domain width YE to body half breadth (B/2), where as Dz is the ratio of the domain depth above deck or below keel of the body Zig to the body depth D (Figure 2~. Computational domain size along the x-axis is chosen to be one meter. Grid and Cell sizes Since, the shape of the impacting section does not change along its length, grid size along x-axis is kept at one cell or CV, whereas grid sizes along y and z axes are chosen in such a way that the total number of cells in the whole computational domain is kept at optimum level to reduce the computational time but produce grid independent results. Distribution of grid points is carried out using stretching functions so that the cells closer to the body boundary are as small as possible apart from ensuring that the neighboring cells at each segment boundary do not vary much in size to achieve numerical stability during simulation. Details of minimum cell sizes corresponding to a particular grid size chosen initially for the computational domain is given in Table 2. Item | Lowl | Low2 |Southl|South2|South3| CO°maP | Glid Cells 60 30 30 60 30 90 X12O C~etfChlhcineg I 1.000 1.161 1.085 1.000 1.085 din cel I 4.16e-3 4.19e-3 2.57e-3 2.41e-3 2.41 e-3 ~ 406e 3 Table 2: Initial Grid and Cell Sizes for the computational domain Results Convergence of the solutions is characterized by the amount of the sum of residuals during each time step. A preliminary study on the number of sweeps showed that a minimum number of 15-20 sweeps were necessary to achieve convergence. Numerical simulations with different impact velocity profiles have been carried out and the impact forces obtained on the whole wedge section are shown in Figure 3. p )11 lag I: Shim ~ lag Vetci~ Vs ~ picttm e 6 .4 ED ~ 5.6 =52 A ~ 4.8 he 4A 4D . ~ ~ Experiments ...... constantVeL I ~ ,j l Vay~g Vex Sodom ill Vay~g Vet (Ca'cchted) . O.000 ODO5 OD1O OD15 ~ pacts e i: Seconds 0.020 OD25 3 )11 IDG! :Shaa ~9 brceYs ~~6ctt~ e | a Expedients - constantVel I 12000 . . I _0ar~n 0~1 l~n~mn~ ~l, ..i~0anrnn Sol Ira~,llAm~l l 10000 ~ 8000 z 6000 ~ 4000 an = ~ 2000 ~ Ol i~"6 . ~ I OD20 OD25 j . - ., ~ . a_ ~ ~,"~ - _~ . - ., U . - _ \w "_~_ - , ~ ODOO OD05 OD1O 0.015 hpactto e r: Seconds Figure 3: Decelerations effects on Slamforce Discussion Impact forces obtained during constant velocity impacts show very poor correlation with experimental results as can be expected, with discrepancy being more than 100%. Incorporation of deceleration effects into the numerical model has improved the correlation between numerical and experimental results significantly with the forces obtained based on instantaneous calculated impact velocities have a closer agreement with the experimental results compared to the case of pre- assigned (polynomial) values. Better agreement is 6

noticed during initial stages of impact, but differences are larger at later stages with the peak impact forces from experiments and numerical solutions on the measuring section being 5100 N and 6200 N respectively, even though the time instant at which these peak values occur is nearly same. This also is the time instant at which the flow is nearing the knuckle or separation point. Also, the impact velocity profile estimated during the simulation is not matching well with the experiments. Zhao et. al. (1996) had argued that the differences in peak impact force on the wedge section considering Boundary Element Methods are mainly due to the cross flow or three-dimensional effects. This also explains the differences in impact velocities, which can be attributed to the fact that impacting velocities during simulation are estimated based on the impacting force acting on the total length of the section, which may not be constant along its length. Zhao had further concluded that the discrepancies in impact force would be significant beyond the time instant at which the flow reaches a position where the ratio of body breadth to length crosses 0.25. Since the present section falls in this category, it is considered worthwhile to carry out few additional simulations on three-dimensional sections. Further, Sames et. al. (2000) had concluded that discrepancies can also occur due to the effects of domain (analogous to tank size during experiments). Before undertaking studies on cross flow, investigations have been carried out to obtain solutions independent of all the effects due to different factors affecting numerical solutions like, grid size, computational domain size etc. 4.2 Effect of Grid Size Effect of grid size on the numerical accuracy has been carried out by considering four different grid sizes namely Coarse, Medium, Fine and Very fine Grids, with the number of cells doubled successively for the four cases along y and z axes as given in Table 3 below. The minimum size of cells obtained along each region for the section is given Table 4 respectively. It can be observed that the cell sizes are the lowest near the body. Computational details for impact of wedge section considering different grid sizes are given in Table 5. GRID SIZE Coarse Med. Fine Very Fine i WEDGE SECTION: GRID DISTRIBUTION Lowl~ow2~outh~South2| South3 15 8 8 15 8 . 30 15 15 30 15 60 30 30 60 30 ~- l20 60 60 120 60 Near body (Whole Domain) 15X15 (23X31) 30X30 (45X60) -60X60 (9OX120) 120X120 (180X240) Table 3: Grid Sizes for computational domain Grid Size Coarse Medium Fine ~ Very Fine . Lowl 3.00 l 47 0.72 0.36 Wedge Section: Minimum Cell Size [mm] Low2 10.79 4.52 2.02 1.19 Southl 10.92 6.3 2.78 1.31 South2 7.00 3.60 1.69 0.85 South3 101.50 5.08 2.71 5.28 Whole Domain 300 1.47 0.72 0.36 Table 4: Min. ell siz es corn to differ rent G. id sizes Grid Size Coarse Medium . Fine Very fine Wedge: Computation time on Pentium II, 300 MHz, 64 Mb RAM machine l No. of CVs: I Near body | (Comp. Dom.) 1 225 ( 713) 900 ( 2700) l 3600 (10800) 14400 (43200) T Min. l I Cell | I Size | I im] | .623e-3 ~.81 le-3 | 12.406e-3 Tl.203e-3 1 No.of Time Steps (nt) 54 112 222 456 IMin. |time step l size dt | |secl e4 1 .25e4 4.70e-S |2.80e-5 ICPU Time I [has: | mind 10 o4 0:15 1:25 Table 5: Computation time for different grid sizes Numerical results consisting different slamming parameters like impact velocity profile, pressures at distinct locations and the impact force on the whole wedge section are presented in Figure 4. Observation of time series of impact velocity profiles vertical impact force and pressures clearly indicate that effect of grid size is not significant and a grid size of 60X60 near body (90X120 in the whole domain) is considered to be sufficient to capture all the flow features for the impact of the wedge section. (A) WEDGE Summing Velocity Vs Impact time 1 6.4 r 1 .~ 6.0 ~ 1 ~s'6t h Grid :15 X 15 r 1 ~ 4.8 ~ - Grid :30 X30 I ~ I I Grid :60 X 60 05 4~4 ~ Grid 120X 120 4.0 1 o.ooo ~ ' - ~ 1 0.005 0.010 0.015 0.020 0.025 1 Impact bme in Seconds | l I (B) WEDGE: Slamming force Vs Impact time 7000 6000 1 ~40001 ~ ~ 1 3000 1 ~ ~ x _ ~ \ in, 2000 -1 _~ 1 =Grid 60 X60 1 E 1000 1 '' 1 Grid: 120 X 120 1 1 1 ~ 1~ 1 I ~ 0-r I 1 0.000 0.005 0.010 0.015 0.020 0.025 I Impact time in Seconds (C) WEDGE: Pressure distribution over Breadth ( at impact time t ~ 0.0202 seconds ) 0.00 0.05 0.10 0.15 0.20 0.25 Section Breads in m Figure 4:Effect of Grid size on Slamming parameters 7

4.3 Influence of Domain boundaries The main aim of these investigations is to identify the minimum domain size based on the impacting body dimensions, which yield domain independent numerical solutions. Domain sizes in both the directions, i.e., along breadth (YL) and depth (ZL) of the impacting body, have been varied systematically to study its effects on the numerical results. Details of the systematic variation of the domain size ratios DB (= YL/(B/2)) and DO (= ZL / D) as considered are given in Table 6. Actual dimensions of the computational domain sizes will be dependent on the physical size of the section. Simulations are carried out considering independent variation of the domain size ratios (DB, DD) to have a better understanding of their effect on the slam loads. Wedge section: Domain size along y and Taxis YL 1m1 DB ~ Y. /(B/2) ZI Iml , 0.75 3 0.290 1.25 5 0.580 2.50 10 0.870 3.75 15 1.160- - DD=ZI /D 2 4 . 6 8 Table 6:Domain sizes along y and z-axes Numerical results containing (i) impact velocity variations and (ii) impact force on the section are presented in Figure 5 and Figure 6 for the domain variations along breadth and depth respectively. 3)! eDGE:Sims ~'Vetcg`Vs Attire 6.0 5.6 ~ i.: .R 4.8 D ~ g .- 4.0 6.4 - ~ pacts e ~ Seconds ¢~11 EDGI:S~m s'b~ceVs &;acttill6 ~ pact ~ e :~ S econds Figure 5: Effect of Domain size along section 's breadth (Y-axis) '}' IDO1'Sbs ~ it'Tetcl~g ~ pectin ~ 6A . . . . _- ^,.L ~ .~ ~ ~ {D. ODOO 0~05 0.010 OD15 0~20 OD2S ~P3 t~e~SeCO:6S {till IDGI:Sh. ~ ii' Arc ~ pecitic ~ 6000 solo logo ; - ~ 3000 ,, 2000 .~, 1000 Van I _~ I=' -' - ~ 2L~ - 6 ZL~ - 8 9' O D20 O D2S . O TOO O DOS O [10 O [15 ~ PaCt~ e ~ S8CO3dS Figure 6: Effect of Domain size along section 's depth (z-axis) These results (Figure 5 and Figure 6) clearly indicate that, domain independent results are obtained when the domain ratios DB and DD are greater than 10 and 4 respectively. All further simulations on these sections are thus carried out keeping these limitations in mind to obtain domain independent results. 4.4 Effect of Viscosity and Gravity Viscosity effects on the water entry problems are analysed by considering the coefficient of viscosity of water as l .O l e-6 m2/sec. Gravitational forces were introduced into the numerical model in the form of body force acting vertically down along the negative z-axis. Simulations including gravity, gravity and viscosity together were carried out as shown in Figure 7. Effects due to viscosity and gravity are found to be minimal probably due to low fluid velocities and size of the section. Greenhow (1989) also concluded that gravitational forces are important only when impact time t > V/2g, where g is acceleration due to gravity. 11 cDa ~ · a ~e ~ ing Sbxc- V. ~ proceed - ~ ~n anon 500 400 B 300 .. 2 0 0 ~ 100 ; ~ O 0 . Jr N 0 V is c 08 icy, N 0 C ra vibes j A_ | w id] V ~sc08ily o ~/ W ith G savvy . / . W ith C m viny a n d V is c o ~ ity | l o.ooo o.oos o.olo o.o.s ~ ^~~ ~ pretty · ~ Abscond. 0.020 0.025 Figure 7: Effect of Gravity and Viscosity on impact force of Wedge section 8

It can be observed that numerically obtained value for peak impact force on the measuring length of wedge section (0.2m), which is independent of grid and domain sizes is around 5600 N. whereas the same from the experiments is around 5100N. Also the impact velocities from the numerical solution are lower than those obtained from the experiments especially at later stages of impact, indicating that the numerically estimated impact forces are higher, which is also the case. To understand this discrepancy and the effects of cross flow on the impact forces, further studies are carried out considering the three dimensional wedge section. 4.5 Effect of Cross flow To study the effects of Cross Flow along the sections length (x-axis) on the slamming force during drop tests, impact of three-dimensional wedge section as used during the experiments (Zhao et al. (1996~) is considered. For this purpose, the computational domain for the wedge section is extended along the longitudinal direction (A ~ beyond the section's length (L/2) as shown in Figure 8. In this regard, domain size ratio Do along x-direction is defined as the ratio of domain width XL. to body half Length (L/2~. Few parametric studies are carried out to identify the minimum domain and grid size ODIN along x-axis, which yield domain and grid independent numerical solutions. ~ z ~ i L/2 7.L D ..... .... Body ZL X Vin /~ ~ ~ ~ - Figure 8: Extent of Computational domain along section 's length (x-axis) Grid Generation The Wedge section used in drop tests has constant sectional shape along its length with the measuring section located exactly at the center. The impacting section is symmetrical about both x-y and y-z planes. By exploiting this double symmetry of the impacting section, the computational domain is discretized considering only one quarter of the section to reduce the total number of CVs in the whole computational domain leading to reduced computation time. Grid and Domain sizes alone length Grid and domain sizes along the x-direction (length) are varied systematically considering the grid size of 40X40 (near body) along y and z-axes due to limitations in computational facility. It was also observed that results considering this grid size are not significantly different from using grid of size 60X60. Details of the systematic variations of grid and domain sizes (D~) along x-axis are given in Table 7 and Table 8 along with the actual dimensions of the domain, which naturally varies with the length of the impacting body (L/2 = 0.5 m). 3D Wedge section: 3D Grid size along x-axis Grid Grid Grid in Min.Cell Size Near body whole Domain Size iml Coarse SX40X40 1 SX60X80 3.608e-3 Fine 1 OX40X40 20X60X80 3.608e-3 Table 7: Grid sizes along length 3D Wedge section: 3D Domain size along x-axis Details of Grid and Domain sizes along y and z axes Grid Size (Near Body) 5X40X40 Grid Size (Whole Domain) I OX60X80 . Ye (DB) 2.500m(10) Zip (DD) 0.870m (8) . Case | XL DL 1 ~ 0.75 1.5 2 ~ 1.50 3 3 1 2.50 5 4 ~ 5.00 10 Table 8: Computational domain sizes along length Computational time Details of computational time required for undertaking the simulations on Wedge section using 3D grid are given in Table 9. Grid Size Coarse Fine No. of CVs: Near body (WholeDomain) SX40X40 (lSX60X80) 10X40X40 (20X60X80) 3D Wedge section: Computation time on Pentium II, 300 MHz, 64 Mb RAM machine . Min. Cell Size lml 3.608e-3 3.608e-3 No. of Time Steps (nt) 152 152 l Min.time step size dt |sec 8.9e-S 8.9e-S CPU time h ml 10:17 14:25 Table 9: Details of computation time for different 3D grid sizes of Wedge section Slamming impact forces obtained for different grid and domain sizes along x-axis are shown in Figure 9 and Figure 10 respectively. Figure 9: Time series of slammingforces for different grid sizes along x-axis 9

91 ~DO~s8~ ~ ~g db=:e Ve ~ DOtto 7000 6000 4000 ~ 3000 . f ._~.~XL~ ~ 10 `_ D 2000- Jo - XL~- 5 me_ ~ j 10 0 0 7 _XL /L 1 .5 O o.ooo 0.005 0.010 0.015 0.020 0.025 ~ ~ate" · ~ s-cood. Figure 10: Time series of slammingforces for different domain sizes along x-axis Results from Figure 9 indicate that the effect of grid size along x-axis is not significant, understandably due to the fact that the section shape does not change along the section length. Results from Figure l O indicate that a domain size ratio of Do= 5 along section's length is enough to obtain domain independent results. The slamming impact forces obtained on each transverse strip along the length are presented in Figure 11. This figure clearly shows the effect of cross flow, with the total impact force on the section reducing as we move from center to the edge (section 1 to section 5~. In other words, as the immersed sections breadth to length ratio increases, considering the length to be equal section's position along length from the edge, the effect of cross flow becomes more significant in reducing the impact force on the whole section. Wedge :Slamming Force Vs Impacttime 3000 - .......... .. ~ 2 ~ 0 0 ~ # \ E Jr S action 1 E 1000 · ~ Section 2 of Jo Section 3 SOO ~ Section 4 O ~ ~ S action 5 0 0.005 0.01 0.015 0.02 0.025 Im pact tim ~ in secede Figure 11: Slammingforces on different transverse sections of Wedge section - Section I is at center, Section 5 is nearest to the edge along section length edge compared to the sections at the center, and the same can be observed in Figure 13 considering the pressure distribution at a particular time instant. Further, this cross flow observed initially at the edges slowly spreads towards the center with the increasing impact time as the flow is nearing knuckles upwards. Free surface elevations obtained for the section 1 at center are shown in Figure 14. .. . . Figure 12: Flow distribution along the length of Wedge section at t=0.292 sec . . Figure 13: Pressure distribution along the length of Wedge section at t=0.292 see it' /~L · 0 '49 see ~1 TIC Is /~ - ~ - 0.110 sec - 0.044 see Figure 14: Free Surface elevations at different time instants during impact of Wedge section Comparison of 2D and 3D results Comparison of the results obtained for the Wedge section considering two-dimensional and three- dimensional grid is shown in Figure 15. Flow patterns along the impacting wedge section are shown in Figure 12. It can be observed that, the fluid initially flowing upwards along positive z-direction tends to move away from the center to the edges of The maximum impact force on the measuring section the section longitudinally (along x-axis), clearly reduces from 5600 ~N/0.2m] in 2D case to indicating the 'Cross flow'. This cross flow reduces 5200tN/0.2m] in 3D case. This comparison shows the vertical component of the impacting velocity interesting features about the effects of cross flow or thereby reducing the local impact pressures on the three-dimensionality as summarized below. section. In other words, reduction in the impact force on the section occurs due to the cross pow of the fluid from center to the edges of the section's length. This reduction is greater for the sections nearer to the · Numerically calculated impact velocity profile matches very closely with the experimentally measured values 10

. . Vertical Impact force on the measuring section and also the whole section in 3D case is less than those obtained from the 2D case, clearly indicating the effect of cross flow and three dimensionality. Vertical Impact forces obtained from 2D and 3D cases match well during initial stages of impact but differ during later stages of impact with the 3D case predicting smaller magnitudes indicating occurrence of cross flow. . . . O.000 o.oos 0.010 0.015 ~ p.ct~ ~ ~ atonal 7000 6000 ~ A A ~ VVV ' _ 4000 0 3000 2000 # 1000 o O .00 0.020 0.025 ~ it\ O 2D C ad: 40X40 1 3D C ndt lOXdOX40 | 0.005 0.010 0.015 0.020 0.025 Figure 15: Effect of Three-dimensionality on slamming parameters of Wedge section' Comparison with the Experiments and other Theoretical data Results obtained using the present improved numerical method consists of (i) impact velocities during drop tests, (ii) pressure distributions along the boundary of the impacting body and (iii) Slamming impact force on the measuring section and (iv) free surface evolution at different time instants during the impact. All the results except impact velocity profiles are compared with the Boundary Element Method (Zhao et al., 1996) and the CFD method (Muzaferija et al., 2000~. A simultaneous comparison of all the results is made with the experimental results also as obtained by Zhao. Both Zhao and Muzaferija use the experimentally determined impact velocity profiles for their numerical simulation whereas the present CFD method estimates the same during simulation itself. Zhao et al. carried out simulations using two dimensional grid and corrected the impact forces on Wedge section based on the Meyerhoff (1970) results to account for the cross flow and three dimensionality. Muzaferija et al. carried out initial simulations using 2D grid and accounted for cross flow by undertaking further simulations using 3D grid. The present numerical calculations also consider both 2D and 3D grids for simulating the water impact of Wedge section. Comparison of results on Wedge section from all the methods is shown from Figure 16 to Figure 19. Figure 16 and Figure 17 show comparison of vertical impact velocity and forces on the section respectively. Pressure distributions on the surface and at locations of the section and different time instants are shown in Figure 18 and Figure 19 respectively. W ZDC Z s 8 ~e ~ iz~g V.bc~ty Ve be pecttl. 6 .: 6.~ . 5.6 :^ S.2 - 4.8 - .4 .o . 0.000 Exp. (Zhao etaL, 199 _ C ~ D (P resent) 0.005 0.010 0.015 ~ pacts e n 5 econds 0.020 0.025 Figure 16: Comparison of impact velocities on Wedge section | 11 ~DC ·'81~ - ~ ins lb=- Va ~ pectt~ - 7000 .._ +' - '- ~` 6000. ..~-'~ ~ solo ~~' ~ \`, ~ 4000 ~~ .~; 3 0 0 0 . `~ ~ _ Hi. ,11~ 2000 ~ `` Exp. (Zhao stat, 1996) ·~ BEM (Zhao etaL, 1996) 1000 . - _ {' CFD ~ uza~ ctaL ,2000 0, ~ ~ CFD (Present) 0.000 o.oos u.ulu u.u;~ . b~pacttile h Seconde - .020 0.02s Figure 17: Comparison of slammingforces on Wedge section obtainedform different methods Comparison of slamming forces on wedge 30 section indicates that the results obtained by the present numerical method correlate well with the experiments Figure 17. Zhao (1996) obtained better correlation by correcting the impact forces for cross flow and three-dimensionality. Muzaferija et al. obtained still better correlations for the peak slamming force with slight deviations with the experimental values are observed during the initial and final stages of impact. It may be however be remarked here that he achieved these results by considering a particular situation where the domain boundaries are restricted with domain size ratio (Dx) along length being 0.25. The results obtained using the present method did not encounter this situation. This can probably be due to the fact that he had concluded that domain independent results are achievable with domain size ratios of DO = 3 and DB = 10. But it was observed during the present studies 11

that the domain size along depth is very sensitive and the same need to be increased further to 4 or 6 to achieve domain independent results. This fact could also be verified considering the 2D results, where the peak slamming force from the present method is 5600 (N/0.2m] (Figure 15), whereas the same obtained by Muzaferija is around 6000 tN/0.2m]. Incidentally, this value of 6000 tN/0.2m] is similar to that obtained for a domain ratio of 2 along depth (Figure 6~. Basically, as the section is flatter, the domain ratio has to be larger. It is thus believed that, domain size restriction could perhaps be the reason for his results to be slightly deviating from the experimental results. UL}ll EDGE:Pr.~or'VaD't;Deci~ isp~ct~. {Platdep~ ~ OD125s ) 8 6 =~ l 1~ 43 0 2 O_ ~ O ~ me ~ O ~ ~ Exp.(2hao,19961 ~ 8E11 2hao,19961 A _ ~ CFD (S3:es,20001 CFD presents .000 0.005 OD10 OD15 h pactec e ~ S econds ) ~ I D G I: P r Acute Va:i~tiDe ~ ~ pact IP2etde~tt 0.0375s ) OD20 OD25 . 1 ~ E P lZb2o~l996) ~ BEE 12hao,19961 .- . ~ ~ CFD ISa~es,2000)CFD $resent) 0.00 OD1 OD1 OD2 ~ pacts e ~ Seconds O D2 0.03 (C )'1 IDS I: Pn8BSU~ Vln~t60 v its ~ pacttiB 6 ( P 3 atd.pti r 0.0625 11t ) Exp.~2hao,1996) ~ DEt' (2hao,1996) - ~ | ~ CFD iSa~es,20001 - CFD present 0.020 0.02S once o.oos OD10 OD1S pretty e ~ Seconds ) ~ ~ D G I: P rhesus V8D1UDB v is ~ pact tie 6 ( P. atlepth ~ 0.0875 ~ ) 8 6 a ODOO O.OOS OD10 OD1S ~pactts e ~ Seconds 10 · 8 E:p.i2hao,1996) ~ Bell 12hao,l996 CFD iSa~es,20001 CFD Resents O D20 0.02S ) ~ IDS I: P 28SU~ VasatiDe v ith ~ p8Ctti' (P5 ~td6pth a 01318 ~ ) ~ E:p.¢hao,1996~ ~ BE11 12hao,1996~ 6. · CFD (Sa~es,20001 CFD Resend- ~ . 2. . O- O D 00 O DOS O D10 0 D1S ~ pacts e ~ Seconds 0.020 0 D2S Figure 18: Comparison of Slam pressures histories at different locations on the surface of Wedge section obtainedirom different methods (A) WEDGE: Pressure distribution along She surface ~ at Impact time ~ ~ 0.00435 "coeds ) 6 {¢ 5; 3~} 4 ~ oc 3 c 2 O- . _ , . 0.000 0.200 0.400 0.600 Nondim~nabnal Depth d S - Sian _ _ ~ Exp. (Zhao et al.,1996) · BEM (Zhao et al., 1996) CFD (Muzaferija et al., 2000 CFD (Present) o.eoo Boon (B) WEDGE: Pressure dIstrlbutIon along the surface ( at Impact time ~ ~ 0.0158 seconds ) Exp. (Zhao et al.,1996) · BEM (Zhao et al., 1996) CFD (Muzafenja et al., 2000 ) CF D (present) 7~ K e 7 6 ·r^ s t.84 lo 2 1 O 0.00~) 0200 0.400 o.eoo NandImendonal Depth d Season 0.800 1.000 6 4 83- 3 's2 O.XO 0200 (C) WEDGE: Pressure distribution along the surface ( at Impact time ~ ~ 0.0202 seconds ) I ~ Exp. (Zhao et al.,1996) · BEM (Zhao et al, 1996) CFD (Muzafenja et al., 20X) CFD (Present) 1 ~ 0.400 o.eoo 0.800 1.000 No~lm~i~ Depth°'Sectbn Figure 19: Comparison of Slam pressures along the surface of Wedge section obtainedfrom different methods at different time instants 5 Conclusions Numerical solution method using CFD techniques considers slamming of two dimensional wedge section produce reasonably accurate results for different slamming parameters suitable for design purposes. Finite volume discretization techniques considering simple rectilinear grid deals with the slamming phenomena quite effectively. Application of volume of fluids method has captured the free surface evolution including breaking and overturning effectively. The present numerical method also considers the key factors affecting the slamming of wedge section including varying impact velocities, domain size and cross flow. Based on the present studies the following conclusions are drawn. There are two main conclusions: Firstly, close correlations of the present numerical results with the experiments have been obtained considering the histories of both impact velocity and slam loads for the water entry of wedge section during drop tests. Secondly, Effect of impact velocity variations and computational domain size, cross flow and three- dimensionality on the numerical accuracy of the results is significant. 12

Acknowledgements This work is a part of PhD study, which has been financially supported by Lloyds Register of Shipping (LRS), London, and contribution of LRS is gratefully acknowledged. Special thanks are due to Dr. Susan E. Rutherford and Dr. Paul C. Westlake of LRS for their constant support throughout the project. We also thank Prof. Brian Baxter for his invaluable comments and suggestions. The opinions expressed herein are those of the authors. References Arai, M. and Matsunaga, K (1989J: A Numerical Study of Water Entry of Two-dimensional Ship- Shaped Bodies. PRODS, 75/1-8. Arai, M., Cheng, L. Y. and Inoue, Y. (1994~: A computing Method for the Analysis of Water Impact of Arbitrary Shaped Bodies, Journal of the society of Naval Architects of Japan, Vol. 176 233-240. Campbell, I.M.C. and Weynberg, P.A. (19809: Measurement of Parameters Affecting Slamming. Wolfson Unit for Marine Technology, Report no. 440. Greenhow, M. (19877: Wedge Entry into initially Calm Water. Applied Ocean Research Vol 9, No.4, 214-223. Hirt, C.W. and Nichols, B.D. (1981J: Volume of Fluid (VOF) method for the Dynamics of Free Boundaries. Journal of Computational Physics, Vol. 39, 210-225. Meyerhoff; W.K (19709: Added Masses of Thin Rectangular Plates Calculated from Potential Theory, SNA ME. 100-1 1 1. Muzaferija, S., Peric, M., Sames, P. C. and Schellin, T.E.,(20009: A Two-Fluid Navier-Stokes Solver to Simulate Water Entry. Proc. Of 22nd Symposium on Naval Hydrodynamics, 638-65 1. Sames, P. C., Schellin, T.E., Muzaferija, S. and Peric, M. (19999: Application of a Two Fluid Finite Volume Method to Ship Slamming. Journal of OfFshore Mechanics and Arcitic Engineering (OMAE),Vol.121,no. 1. SNAME (19939: Notes on Ship Slamming. Technical and Research Bulletin 2-30. van Leer, B. (19749: Towards the Ultimate Conservative Difference Scheme. IV. A New Approach to Numerical Convection. Journal of Computational Physics 23 276- , , 299. Con Karman, T. (19299: The Impact on Sea plane Floats during landing, N.A. C.A. T.N. No. 321. Wagner, H. (19329: Uber Stoss- und Gleitvorgange an der Oberfache von Flussigkeiten, Z.A.M.M, 12~4), 193-215 (in German). Wilson, S.K (19899: The Mathematics of Ship Slamming, D.Phil Thesis, Oxford University. Zhao, R. and Faltinsen, O. (1993~: Water Entry of Two Dimensional Bodies. Journal of Fluid Mechanics. Vol. 246, 593-612. Zhao, R., Faltinsen, O. and Aarsnes, J.V. (19969: Water Entry of Arbitrary Two Dimensional Sections With and Without Flow Seperation. Proc. of 21St Symposium on NavalHYdrodYnamics, 118-138. 13