An Improved Volume-of-Fluid Method for Wave Impact Problems

Transcription

1 An Improved Volume-of-Fluid Metod for Wave Impact Problems K.M. Teresa Kleefsman, Geert Fekken, Artur E.P. Veldman Dept. of Matematics and Computing Science, University of Groningen Groningen, te Neterlands Bogdan Iwanowski FORCE Tecnoloy Norway AS Oslo, Norway ABSTRACT In tis paper, a calculation metod for wave impact problems is presented. Tis metod is based on te Navier-Stokes equations and uses an improved volume-of-fluid (VOF) metod for te displacement of te free surface. Results are sown for a dambreak simulation for wic experimental results are available for comparison. Also drop tests ave been simulated wit wedges and circular cylinders. Te results are very promising for te furter development of te metod. KEYWORDS: Wave loading; moving body; numerical simulation; Cartesian grid; volume-of-fluid. INTRODUCTION Tere is a great need for calculation metods for local penomena of wave impact loading and loading from green water on te deck of a sip. Most of te existing metods are not capable of predicting te local loads. Part of te Joint Industry Project SafeFLOW, initiated in January 2001, is devoted to te development of a metod wic is able to predict local wave impact on floaters operating in te offsore industry. Te development of te metod, called ComFLOW, as started in 1995 wit te simulation of liquid-filled spacecraft, tat are tumbling in space [Gerrits, 2001]. In tis application were te surface tension is te driving force, a good andling of te free surface is crucial. Te metod as also been applied to blood flow troug (elastic) arteries, were no free surface was present [Loots, 2003]. A rater new application was found in te maritime world were slosing inside anti-roll tanks as been simulated. In 1999 a pilot study of te simulation of green water on te deck of an FPSO as been performed, to investigate if te metod is capable of capturing te local flow details on te deck [Bucner, 1999]. Because of promising results, it was decided to move on in tis direction in te SafeFLOW project. Te simulation of fluid flow in ComFLOW is based on te Navier-Stokes equations for an incompressible, viscous fluid. Te equations are discretised using te finite volume metod. For te displacement of te free surface te VOF metod as been used adapted wit a local eigt function wic is essential for a good simulation of te free surface flow. To simulate wave impact in a robust and accurate way, a good coice of te discrete boundary conditions at te free surface turns out to be very important. In tis paper, te model used in ComFLOW is described. Results are sown of simulations for a dambreak wit a box in te flow. Tis simulation can be seen as a model of te water flow on te deck of a sip due to green water. Also results ave been sown of water entry of two-dimensional objects. Comparison wit teory and experiments is available. GOVERNING EQUATIONS Flow of a omogeneous, incompressible, viscous fluid is described by te continuity equation and te Navier-Stokes equations. Te continuity equation describes conservation of mass and te Navier-Stokes equations describe conservation of momentum. In conservative form, tey are given by u nds = 0, (1) V u t dv + uu T nds = 1 (pn µ u n)ds + F dv. (2) ρ Here, is te boundary of volume V, u = (u, v, w) is te velocity vector in te tree coordinate directions, n is te normal of volume V, ρ denotes te density, p is te pressure, is te gradient operator. Furter µ denotes te dynamic viscosity and F = (F x, F y, F z ) is an external body force, for example gravity. In te case tat moving rigid bodies are present in te domain V, te above equations still old, wit te additional condition tat te fluid velocity at te boundary of te object is equal to te object velocity. V Paper No JSC-365 Kleefsman Page: 1 of 8

2 Boundary conditions At te solid walls of te computational domain and at te objects inside te domain, a no-slip boundary condition is used. Tis condition is described by u = 0 for fixed boundaries, and u = u b for moving objects wit u b te object velocity. Some of te domain boundaries may let fluid flow in or out of te domain. Especially, wen performing wave simulations, an inflow boundary is needed were te incoming wave is prescribed and at te opposite boundary a non-reflecting outflow condition sould be used. In our metod, te wave on te inflow boundary can be prescribed as a regular linear wave or a regular 5t order Stokes wave. Also a superposition of linear components can be used wic results in an irregular wave. At te outflow boundary, a Sommerfeld condition is very appropriate in cases were a regular wave is used. In te case of an irregular wave or a muc deformed regular wave (e.g. due to te presence of an object in te flow) a damping zone is added at te end of te domain. Free surface If te position of te free surface is given by s(x, t) = 0, te displacement of te free surface is described using te following equation Ds Dt = s + (u )s = 0. (3) t At te free surface, boundary conditions are necessary for te pressure and te velocities. Continuity of normal and tangential stresses leads to te equations p + 2µ u n n = p 0 + 2γH (4) ( un µ t + u ) t = 0. (5) n Here, u n is te normal component of te velocity, p 0 is te atmosperic pressure, γ is te surface tension and 2H denotes te total curvature. NUMERICAL MODEL To solve te Navier-Stokes equations numerically, te computational domain is covered wit a fixed Cartesian grid. Te variables are staggered, wic means tat te velocities are defined at cell faces, wereas te pressure is defined in cell centers. Te body geometry is piecewise linear and cuts troug te fixed rectangular grid. Volume apertures (F b ) and edge apertures (A x, A y, and A z ) are used to indicate for eac cell wic part of te cell and cell face respectively is open for fluid and wic part is blocked by solid geometry. To track te free surface, te volumeof-fluid function F s is used, wic is 0 if no fluid is present in te cell, 1 if te cell is completely filled wit fluid and between 0 and 1 if te cell is partly filled wit fluid. Te Navier-Stokes equations are solved in every cell containing fluid. Cell labeling is introduced to distinguis between cells of different caracters. First te cells wic are completely blocked by geometry are called B(oundary) cells. Tese cells ave volume aperture F b =0. Ten te cells wic are empty, but ave te possibility of letting fluid flow in are labeled E(mpty). Te adjacent cells, containing fluid, are S(urface) cells. Te remaining cells are labeled as F(luid) cells. Note tat tese cells do not ave to be completely filled wit fluid. In Figure 1 an example of te labeling is given. E E E E E E E S B B S S F F B F F F F F F F F F F Figure 1: Cell labeling: dark grey denotes solid body, ligt grey is liquid Discretisation of te continuity equation Te continuity and Navier-Stokes equations are discretised using te finite volume metod. Te natural form of te equations wen using te finite volume metod is te conservative formulation as given in Eq. (1) and (2). In tis paper, te discretisation is explained in two dimensions. In most situations, tis can be extended to tree dimensions in a straigtforward manner. In A x wδy A y nδx F b δxδy l A y sδx A x eδy Figure 2: Conservation cell for te continuity equation Figure 2 a computational cell is sown, wic is cut by te body geometry. Wen applying conservation of mass in tis cell, te discretisation results in u w u e A x eδy + v n A y nδx u w A x wδy v s A y sδx + u b (A x e A x w)δy + v b (A y n A y s)δx = 0, (6) were te notation is explained in Figure 2. Discretisation of te Navier-Stokes equations For te discretisation of te Navier-Stokes equations, control volumes are defined containing velocities wic are defined on cell faces. In te case of uncut cells, te control volume of a velocity simply consists of te rigt alf of te cell left of te velocity and te left alf of te cell rigt of te velocity. In case of cut cells te procedure to define control volumes as been explained in detail in [Gerrits, 2001]. Te time derivative in te Navier-Stokes equations is discretised in space using te midpoint rule. Tis results in V u t dv v b v n u b v s u e. = u c t F b c δx c δy. (7) Paper No JSC-365 Kleefsman Page: 2 of 8

3 Here, u c is te central velocity around wic te control volume is placed and Fc b δx c δy is te volume of te control volume. Te convective term is discretised directly from te boundary integral wic is given by uu nds. (8) Note, tat tis integral contains two different velocities: te scalar velocity u is advected wit te velocity vector u. Tis integral is evaluated along all boundaries of te control volume by multiplying te scalar velocity u wit te mass flux troug te boundary u nds. Finally, te convective term discretisation results in a matrix tat is skew symmetric, wic is also a property of te continuous convective operator [Verstappen, 2003]. Te diffusive term, wic for te Navier-Stokes equation in x- direction is given by 1 µ u ds, (9) ρ n is discretised along all boundaries of te control volume. To ensure stability, te term n u is discretised in cut-cells as if te cells are uncut. Te error introduced tis way is small and as no influence in te convection-dominated simulations. Te discretisation results in a symmetric matrix wic is negative definite. Te pressure term is discretised as a boundary integral, resulting for te Navier-Stokes equation in x-direction in pn x ds =. (p e p w )A x c δy. (10) Here, p e and p w are te pressure in te eastern and western cell respectively, A x c is te edge aperture of te cell face were te central velocity is defined. Te external force is discretised similar to te time derivative, resulting for te x-direction in F x dv =. F xc Fc b δx c δy. (11) V Here, F xc is te force at te location of te central velocity. A detailed explanation of te discretisations described in tis section is given in [Gerrits, 2001]. Temporal discretisation Te continuity and Navier-Stokes equations are discretised in time using te forward Euler metod. Tis first order metod is accurate enoug, because te order of te overall accuracy is already determined by te first order accuracy of te free surface displacement algoritm. Using superscript n for te time level, te temporal discretisation results in Mu n+1 = 0, (12) Ω un+1 u n + C(u n δt )u n = 1 ρ (M T p n+1 µdu n ) + F n. (13) Te continuity equation is discretised at te new time level to ensure a divergence free velocity field. Te spatial discretisation is written in matrix notation were M is te divergence operator, Ω contains cell volumes, C contains te convection coefficients (wic depend on te velocity vector) and D contains diffusive coefficients. Solution metod To solve te system of equations, te equations are rearranged to u n+1 = ũ n + δtω 1 1 ρ M T p n+1, (14) were ũ n = u n δtω 1 (C(u n )u n µ ρ Dun F n ). (15) First, an auxiliary vector field ũ n is calculated using Eq. (15). Next, Eq. (14) is substituted in Eq. (12) wic results in a Poisson equation for te pressure. From tis equation te pressure is solved using te SOR (Successive Over Relaxation) metod were te optimal relaxation parameter is determined during te iterations [Botta, 1985]. Once te pressure field is known, te new velocity field is calculated from ũ n using te pressure gradient. Free surface displacement After te new velocity field as been calculated, te free surface can be displaced. Tis is done using an adapted version of te volume-of-fluid metod first introduced by [Hirt, 1981]. A piecewise constant reconstruction of te free surface is used, were te free surface is displaced by canging te VOF-value in a cell using calculated fluxes troug cell faces. Te original VOF metod as two main drawbacks. Te first is tat flotsam and jetsam can appear, wic are small droplets disconnecting from te free surface. Te oter drawback is te gain or loss of water due to rounding of te VOF function. By combining te VOF metod wit a local eigt function as introduced in [Gerrits, 2001], tese problems do not appear any more. Te resulting metod is strictly mass conservative. FREE SURFACE BOUNDARY CONDITIONS At te free surface, boundary conditions are needed for te pressure and te velocities. Te pressure in surface cells is calculated as an interpolation or extrapolation from te pressure in an adjacent fluid cell and te boundary condition at te free surface. Te velocities in te neigbourood of te free surface can be grouped in different classes (see Figure 3). Te first class contains E E E S FF, FS, SS: momentum equation S S S F S F F F SE: extrapolation EE: tangential free surface condition Figure 3: Different caracters of velocities near te free surface te velocities between two F-cells, between two S-cells and between an S- and F-cell. Tese velocities are determined by solving te momentum equation, so te velocities are called momentum velocities. Te second class consists of te velocities between an S- and an E-cell. Tese velocities are determined using boundary conditions wic will be described below. Te last class consists of velocities between two E-cells wic are sometimes needed to Paper No JSC-365 Kleefsman Page: 3 of 8

4 solve te momentum equation. Tese are determined using te tangential free surface condition time is 375 s treatment SE vel: mass conservation treatment SE vel: extrapolation model test SE-velocities Te coice for te metod to determine te velocities at te cell faces between surface and empty cells is very important for te robustness and te accuracy of te flow model. Furtermore, tis as an influence on te occurrence of numerical spikes in te pressure signal (see Figure 11 were numerical spikes occur). Te reason for pressure spikes is tat tere is not always conservation of mass in a surface (or empty) cell (depending on te boundary conditions). Wen suc a cell becomes a fluid cell, te pressure reacts wit a spike to restore mass conservation. In te following paragraps two metods will be described for te definition of free surface velocities. Metod 1: conservation of mass in S-cells Te first metod demands conservation of mass in surface cells. Te SE-velocity or velocities at te faces of a surface cell are cosen suc tat conservation of mass is satisfied in tat cell. Te advantage of tis metod is tat no spikes will occur in te pressure signal, because conservation of mass is already satisfied if tis surface cell canges to a fluid cell. Tere are two disadvantages. Te first one is tat tis metod is not always robust. In te case of Figure 4 te SE-velocity will become very large. If tis configuration does not cange during a number of time steps, te metod will diverge. S S E S S E Figure 4: Very large SE velocity wen using metod 1 Te oter disadvantage is tat te metod does not give accurate results in wave simulations. Tis is sown by te dased line of Figure 5. Metod 2: extrapolation from direction of fluid In te second metod, te SE-velocities are calculated as an extrapolation from te direction of te fluid. Using a linear extrapolation gives very accurate results for steep wave simulations (see Figure 5). But linear extrapolation can lead to instability of te metod wen te velocity field is not smoot. In tat case constant extrapolation is a better coice. In practice, a metod as been cosen wic is an engineering mix between te two metods described above. To prevent spikes in te pressure signal, te mass conservation principle as been cosen wen a cell canges label from surface cell to fluid cell. In all oter cases linear or constant extrapolation is used from te direction of te main body of te fluid, wic gives a ig accuracy and a very robust metod. wave elevation (m) x axis (m) Figure 5: Different metods for free surface velocities in a steep wave simulation STABILITY In te case of uncut cells wit fixed objects, te stability of te equation containing te time integration term and te convective term is given by te CFL-restriction δt u 1. Here, is te size of te uncut cell. Wen cut cells are present, tis criterion is not canged. Tis result is not directly straigtforward wen looking at te equation containing te time derivative and te convective term u t = Ω 1 C(u, u b )u (16) were u b is te object velocity. Te matrix Ω is a diagonal matrix containing te volumes of te cells, so tese entries can become arbitrary small for cut cells. So te elements in te matrix Ω 1 can become arbitrary large. But, wen estimating te eigenvalues of te convective matrix using Gerscgorin circles by being order O( Ω u), it can be concluded tat te same stability criterion is needed as for te uncut-cells case. Wen moving objects are present, te story becomes somewat different. Now, te CFL-criterion does not guarantee stability anymore, because te eigenvalues of Ω 1 C(u, u b ) are of order O(Ω 1 u b ) wic means tat tey can become arbitrary large. To cancel te effect of Ω a formulation based on a weigted average of te fluid velocity and te boundary velocity sould be applied in te cells cut by te moving object. To avoid smearing of te interface in cases were it is not necessary to stabilise te convective term, te following discretisation is used u n+1 = Ω n+1 (Ω n+1 + Ω ) 1 (u n + δt(ω n+1 ) 1 ( C n u n )) + (I Ω n+1 (Ω n+1 + Ω ) 1 )u n+1 b (17) were Ω = Ω n+1 Ω n is te difference between Ω s at two different time steps. Te factor Ω n+1 (Ω n+1 + Ω ) 1 is cosen because ten te stabilising term is only used wen te body is moving; note tat it equals unity for fixed objects. Te maximal stabilisation is required wen te object is moving normal to its boundary, wereas no stabilisation is needed wen te object is moving tangential to its boundary (see Figure 6). A detailed explanation of te stability of te convective terms is given in [Fekken, 2004]. Paper No JSC-365 Kleefsman Page: 4 of 8

5 te simulation are sown togeter wit images of te video of te experiment (at te same moments in time). Te smaller pictures ub ub ub ub Figure 6: Left: boundary moving normal to itself: maximal stabilisation is required; rigt: boundary moving tangential to itself: no stabilisation is required From te diffusive term, also a stability criterion follows wit a restriction on te time step. In te case of uncut cells, tis cri2 terion is given by δt 2ν, were ν denotes kinematic viscosity. Because te diffusive term is discretised as if all cells were uncut ( staircase approac), te above criterion is also valid in our model. Figure 8: Snapsots of a dambreak simulation wit a box in te flow compared wit experiment at time 0.4 and 0.56 seconds DAMBREAK SIMULATION At te Maritime Researc Institute Neterlands (MARIN), experiments ave been performed for breaking dam flows. Tese experiments can be seen as a simple model of green water flow on te deck of a sip. Te dambreak is a very popular validation case, because te set up is easy, no special in- or outflow conditions are needed. A large tank of 3.22 by 1 by 1 meter is used wit an open roof (scale 1:15). Te rigt part of te tank is first closed by a door. Beind te door, 0.55 meter of water is waiting to flow into te tank wen te door is opened. At a certain moment, a weigt is released wic opens te door. In te tank, a box is placed wic is a scale model of a container on te deck of a sip. inside te snapsots sow te water in te reservoir. Tere is a very large agreement between te simulation and experiment. Te moment in time wen te water is first itting te box is te same. Te sape of te free surface, bending a bit forwards in te second picture, is seen in bot experiment and simulation. In te simulation, te free surface as some ripples, and tese are due to te computational grid. Figure 7: Measurement positions for water eigts and pressures in te dambreak experiment Figure 9: Horizontal wave probe along te side of te tank During te experiment, measurements ave been performed of water eigts, pressures and forces. In Figure 7, te positions of te measured water eigts and pressures are sown. In te tank, four vertical wave probes ave been used. One in te reservoir and te oter tree in te tank. Te box was covered by eigt pressure pick-ups, four on te front of te box and four on te top. Te forces on te box were also measured. To determine te velocity of te water wen entering te tank, a orizonal wave probe is used near te side wall of te tank. As starting configuration of te simulation wit ComFLOW, te water in te rigt part of te domain is at rest. Wen te simulation is started, te water starts to flow into te empty tank due to gravity. In Figure 8 two snapsots of te early stages of In Figure 9 te time istory of te orizontal wave probe is sown, compared wit te simulation. Te first stage of te simulation compares very well wit te experiment, te velocity wit wic te water is flowing into te tank is predicted very well. By te time te water its te box, tere is a difference between te experiment and te simulation. In te experiment, te water is slowed down, te wave probe is totally covered by water only after 2 seconds. In te simulation tis is te case after 0.5 seconds. Wen looking at te movie of simulation and experiment, tis difference is not present, suggesting an inconsistency in te wave probe measurements. Te flow reaces te end wall at te same time in simulation and experiment. So te appearing difference in Figure 9 is not a real difference between simulation and Paper No JSC-365 Kleefsman Page: 5 of 8

6 experiment. In Figure 10, te water eigt at two locations is sown: in te reservoir, and in te tank just in front of te box. Te agreement in bot pictures is very good until te water as returned from te back wall (after about 1.8 seconds). After tat some differences occur, but te global beaviour is still te same. Wen te water as returned from te wall, te water eigt at probe H2 is largest. Te water flows back to te reservoir, were it turns over again at about 4 seconds. Te moment tat tis second wave meets te wave probe at H2 again is almost exactly te same in simulation and experiment. coming back from te wall is overturning. Tis difference is a real difference wic cannot be explained properly at te moment. Several spikes appear in te pressure signals, wic are spikes in all te graps at te same moment. Tese spikes occur, because some water enters an empty cell wic is completely surrounded by cells wit fluid. Ten, tis empty cell canges to a fluid cell in one time step witout being a surface cell in between. Tis cange in label results in a pressure spike over te wole pressure field. Te results of te dambreak simulation are in good agreement wit te experiment. Te global beaviour of te fluid is te same and te impact peak of te pressure agrees well, especially at te lowest part of te box. WATER ENTRY AND EXIT Figure 10: Vertical water eigts in te reservoir H4 (left) and te tank H2 (rigt) In tis section, validation results of water entry and exit of twodimensional objects are presented. Te simulations ave been carried out in two dimensions. Te objects are moving according to a constant prescribed motion. Figure 12 presents free surface profiles for te entry of two wedges. Te wedges ave deadrise angles of 30 and 45 degrees, respectively. Te simulation results ave been compared wit potograps of experiments by Greenow [Greenow, 1983]. Te Figure 11: Pressure time istories of P1 (upper left), P3 (upper rigt), P5 (lower left), and P7 (lower rigt) Te moment te wave its te box is perfectly captured by te simulation as can be seen from Figure 11. Here te pressure at point P1 and P3 at te front of te box and at te top of te box, P5 and P7 (see Figure 7), are sown. Te magnitude of te impact pressure is te same for simulation and experiment at pressure point P1 (te lowest on te box), but is underpredicted by te simulation at point P3. Te moment te return wave its te box again (at about 4.7 seconds) is again visible in te graps. In te bottom graps of Figure 11, were te time istory of pressure transducers at te top op te box are sown, a clear difference occurs between simulation and experiment. After about 1.3 seconds, tere is a wiggle in te simulation wit a duration of 0.5 seconds, wic is not present in te experiment. Before tis point, te water its te top of te box wen te wave Figure 12: Snapsots of wedge entry wit deadrise angles 30 degrees (up) and 45 degrees (down) simulations ave been performed on a grid of 300x280 computational cells. Te visual comparison between te experiments and te simulations is very good. Te jets wic are formed aside of te wedge are created by ComFLOW to a certain extent, but te small details of te jets and te spray are not reproduced, mainly due to te limited number of grid cells. A visual comparison of te free surface development of cylinder entry is presented in Figure 13. Te simulation as been performed on a grid of 400x400 points. Te same can be concluded for te cylinder as for te wedges: te details of te spray are not reproduced by ComFLOW. Te total ydrodynamic force on te cylinder during te first stage of te impact as been calculated and compared wit ex- Paper No JSC-365 Kleefsman Page: 6 of 8

7 Figure 13: Snapsots of te water entry of a circular cylinder perimental results of Campbell and Weynberg [Campbell, 1980], also reported in [Battistin, 2003]. In Figure 14 te slamming coefficients of te cylinder entry wit different entry velocities ave been plotted versus te nondimensional penetration dept. Te slamming coefficient is given by C s = F/ρRV 2 wit F te total vertical ydrodynamic force, R te radius of te circular cylinder and V te entry velocity. Besides te experimental result of Campbell and Weynberg, also te teory of Von Karman (1929), reported by Faltinsen [Faltinsen, 1990], as been included. Tis teory is based on potential teory. For te very initial stage of te entry of a circular cylinder, te vertical ydrodynamic slamming force can be estimated by F = V ρ π 2 (2V R 2V 2 t), (18) were t denotes time wit t = 0 at te moment of first impact. Te comparison between te experiments of Campbell and Weynberg and te simulations is reasonable. It can be seen tat te initial impact is a bit underpredicted by ComFLOW for all entry velocities. Te initial impact is more in agreement wit te teory of Von Karman. In a later stage, te results are in good agreement wit te experiments of Campbell and Weynberg. Te results of ComFLOW are almost similar for different entry velocities, wic confirms near perfect scaling wit V 2. To study te convergence of te metod under grid refinement, te circular cylinder entry simulations ave been run wit different grids also. Te results are presented in Figure 15. It can be seen tat te coarseness of te grid as a very large influence on te formation of te jets aside te cylinder. A very fine grid is needed to capture te jets. However, te formation of te jets does not ave a large influence on te total ydrodynamic force, because of te almost zero pressures inside te jets [Battistin, 2003]. In Figure 16, snapsots of simulations of cylinder exit are compared wit potograps of experiments [Greenow, 1983]. Te Froude number, defined by F r = V 2 Rg, is taken Te simulation as been performed on a grid of 300x280 points. Te visual comparison between simulation and experiment is very Figure 14: Slamming coefficient of te entry of a circular cylinder compared to te experiments of Campbell & Weynberg and te teory of Von Karman Figure 15: Effect of grid refinement on te jets formed during water entry of a circular cylinder: from left to rigt 100x100, 200x200 and 400x400 grid cells good. Te free surface development is well resolved by Com- FLOW, also te very tin layer of fluid around te cylinder is present. CONCLUSIONS In tis paper, a numerical metod is described for te prediction of local impact loads on floating structures. Te metod is based on a cut-cell approac on a fixed Cartesian grid, and is stable even wen very small cells appear. For te displacement of te free surface, an improved VOF metod is used, wic results in full mass conservation. Te determination of te free surface boundary conditions ave sown to ave a large influence on te accuracy of wave simulations. To improve tis furter, te metod will be extended to a two-pase fluid flow model, in wic tese boundary conditions will not be needed any more. Te metod as been validated using two kinds of simulations. Te first simulation is a dambreak simulation wit a box in te flow. Time series of pressure and water eigt ave been compared wit measurements from MARIN. Te results are satisfying, especially wen considering tat te flow is very violent. Te global beaviour of te fluid is very muc te same. Te second kind of validation consists of water entry and exit of wedges and cylinders. Te development of te free surface as been com- Paper No JSC-365 Kleefsman Page: 7 of 8

8 REFERENCES Figure 16: Snapsots of cylinder exit at t=0.9 seconds and t=1.06 seconds, wit exit velocity V=1 m/s pared wit experimental results, were a good comparison as been found. Also te jets appearing at te sides of te wedges and cylinders are resolved by ComFLOW, altoug te details of te spray are not reproduced. Using a finer grid gives a large improvement for te reproduction of te jets. Te ydrodynamic forces during te impact of a cylinder entering te water ave been compared wit experiments and teory, wic sow a rater good agreement. Te next step in te validation of te metod will be to simulate green water due to large waves wic flow over te deck of a floating vessel. In te coming years, te metod will be extended wit a coupling to an outer domain were waves are generated using a muc ceaper diffraction code. In tis way, te Com- FLOW domain can be limited to te close surroundings of te places of impact. Battistin, D., Iafrati, A., Hydrodynamic loads during water entry of two-dimensional and axisymmetric bodies, J. Fluids Struc., 17, , Botta, E.F.F., Ellenbroek, M.H.M., A Modified SOR Metod for te Poisson Equation in Unsteady Free-Surface Flow Calculations, J. Comp. Pys., 60, , Fekken, G., Bucner, B., Veldman, A.E.P., Simulation of greenwater loading using te Navier-Stokes equations, J. Piquet, editor, Proc. 7t Int. Conf. on Numerical Sip Hydrodynamics, , Nantes, Campbell, I.M.C., Weynberg, P.A., Measurement of parameters affecting slamming, Report No. 440, Wolfson Unit of Marine Tecnology, Tec. Rep. Centre No. TO-R-8042, Soutampton, Faltinsen, O.M., Sea loads on sips and offsore structures, Cambridge University Press, Cambridge, Fekken, G., Numerical Simulation of Free-Surface Flow wit Moving Rigid Bodies, PD tesis, University of Groningen, Marc 2004, URL: ttp:// Greenow, M., Lin, W.-M. Nonlinear free surface effects: experiments and teory, Rep. No 83-19, Department of Ocean Engineering, MIT, Cambridge, Mass, Gerrits, J., Dynamics of Liquid-Filled Spacecraft, PD tesis, University of Groningen, December 2001, URL: ttp:// Hirt, C.R., Nicols, B.D. Volume of fluid (VOF) metod for te dynamics of free boundaries, J. Comp. Pys., 39, , Loots, G.E., Hillen, B., Veldman, A.E.P., Te role of emodynamics in te development of te outflow tract of te eart, J. Eng. Mat., 45, , Verstappen, R.W.C.P., Veldman, A.E.P., Symmetry-preserving discretization of turbulent flow, J. Comp. Pys, 187, , ACKNOWLEDGMENT Te project is supported by te European Community under te FP5 GROWTH program; te autors are solely responsible for te present paper and it does not represent te opinion of te European Community. It is also supported by 26 parties from te industry (oil companies, sipyards, engineering companies, regulating bodies). Paper No JSC-365 Kleefsman Page: 8 of 8