MODELLING OF WATER FLOW ON SMALL VESSEL S DECK

Monika Warmowska, Jan Jankowski, Polski Rejestr Statków S.A., al. gen. Józefa Hallera 126, Poland, Gdańsk, 80-416 MODELLING OF WATER FLOW ON SMALL VESSEL S DECK Summary Green water moving on deck of small vessels affects their motion and can contribute to the vessel capsizing. The phenomenon is nonlinear and its description using mathematical differential problems remains to be difficult. This paper presents a numerical model, based on shallow water flow, describing water flow on moving ship s deck. Up to now simplified methods have been applied to determine ship motion in irregular waves. The shallow water method is developed to include it into ship motion equations. The paper presents verification of this method and the results of the simulation of water moving on small vessel s deck. Key words: shallow water problem, water on deck, vessel capsizing, vessel motion

M. Warmowska, J. Jankowski Modelling of Water Flow on Small Vessel s Deck 1. Introduction The amount of water on deck changes continuously during simulation and may lead to capsizing of the vessel. To model this phenomenon we have to take into account boundary conditions such as: velocity field of water motion around the ship, pressure caused by water incoming on submerged deck, motion of ship, mass and acceleration of water jumping over the bulwark, mass of water inflow and outflow trough the openings of bulwark. These problems are well described in references. Dilligham (1981) presents a formulae describing water flow through openings in bulwark and over bulwark. The mass of flowing water depends on the size and shape of the openings, the height of sea wave and the water elevation above the opening. The numerical solution of 2D problem is given by Dillingham (1981). The equation used is valid for a ship at rest. The author applies the random choice method for solving hyperbolic equations. The three-dimensional flow is described by Dillingham and Falzarano (1986), who transform equations to a coordinate system coupled to the ship s centre of gravity. Panatazapoulas (1988) presents a 3D equation of shallow water motion on deck of a ship moving in waves with yow equal to zero. His method is further developed by Huang and Hsiung (1997), who apply the flux differential splitting method to solve the non-linear threedimensional problem describing water flow on deck. The forces and moments of water moving on deck are added to equations of ship motion. The changing water mass and flow of water across the deck and bulwark contribute to the forces and moments acting on the ship. Detailed description of this problem is presented by Belenky (2002). Jankowski & Laskowski (2006) applied a simplified approach used in Ro-Ro ferries damage stability calculations in their computer programs enabling simulation of ship motion in irregular waves (Jankowski, 2007), as the first phase of modeling the water-on-deck effects. They added to the model used in Ro-Ro ferries additional pressure acting on the deck caused by the change of water volume on the deck (Buchner, 2002). This program was used to determine a ship s motion in irregular waves. As the next step the problem of shallow water flow was used to model the water motion on deck, which includes horizontal relative velocity of water (Zienkiewicz, 2005). The added pressure of the dynamic motion of water was obtained using this method. 2. Ship motion in irregular waves obtained using simplified method The simulation of vessel motions in waves is based on numerical solutions of non-linear equations of motion (non-linear model). The hydrodynamic forces and moments defining the equations are determined in each time step, Fig. 1. The forces acting on the vessel can be split into: Froude-Krylov forces, diffraction and radiation forces, rudder forces, non-linear damping and forces induced by water on deck. The forces caused by water on deck are obtained using a simplified model used in Ro-Ro ferries (Jankowski & Laskowski, 2006). The example of simulation of ship motion is presented in Figure 1.

Modelling of Water Flow on Small Vessel s Deck M. Warmowska, J. Jankowski Fig. 1 Time history of ship heave ζ, pitch θ, surface elevation, and roll ϕ The position of water trapped on deck is determined by the horizontal plane and the actual position of the vessel deck in the given time instant. The dynamics of water caused by the motion of water particles in relation to the deck is neglected. The forces and moments caused by water-on-deck are obtained by integrating the hydrostatic pressure determined by water horizontal plane above the deck in the vessel s actual position. Additionally, the vessel s acceleration and the changing heights of the horizontal plane above the deck have been added to the model to better replicate the phenomenon (Jankowski & Laskowski, 2006). The pressure p A in point A on the deck is equal to: where h changing distance of the horizontal plane from the point A in the inertial system, v A, a A velocity and acceleration of the deck point A, determined in the inertial system. The pressure field enables us to determine the force F=(F 1,F 2,F 3 ) and movement M=(F 4,F 5,F 6 ) generated by moving water on deck: ( 1) ( 2) where S the wetted part of surface of the deck and bulwark, R the position vector of points belonging to S, n normal vector.

M. Warmowska, J. Jankowski Modelling of Water Flow on Small Vessel s Deck 3. Modeling of water flow on small vessel s deck using the shallow water method The phenomenon of water flow on small vessel s deck can be divided into the following stages: 1. the inflow of water over upper edge of bulwark, 2. the inflow and outflow of water from deck through openings in bulwark, 3. the flow of sea water over the submerged vessel deck, 4. the dynamic water motion on deck. In this model it is assumed that the volume of water on deck, varying in time, depends on the difference in heights between the wave surface and the following edges: the upper edge of the bulwark, and the lower edges of openings in the bulwark. The velocity field around the ship is not disturbed by the ship and its motion. The field causes the flow of seawater in case the deck is submerged. The dynamic water motion over the deck is directed along the deck. The vertical acceleration a z can be neglected. It is assumed that viscosity forces can be neglected. The motion of water is described using Euler equations. The assumption enables to model the problem of dynamic water motion on deck as the shallow water flow problem. Using such a model the following can be determined: the shape of free surface elevation, the velocity field of water particles over the deck, the pressure, forces and moments acting on the deck caused by water moving over the deck. 3.1. Inflow and outflow of water over the bulwark and through the openings It is assumed that the flow rate of water volume over the bulwark can be calculated as the flow over a weir, whereas the flow through the openings in the bulwark is modeled as a flow through a submerged orifice in a dam. The general formula for the flow rate is: ( 3) where q changing mass of water, c correction coefficient for non-stationary flow, established experimentally, b the width of the orifice or the fragment of bulwark above which the deck is flooded, H vertical distance between the wave profile and the water free surface on the deck at a point considered (positive if the wave exceeds the water level on the deck), d the depth of water at the orifice or the instantaneous elevation of wave profile above the deck edge at the orifice. Formula (3) assumes various forms depending on relative water levels inside and outside the deck hollow and on the position of the opening in the bulwark (Pawłowski, 2004). Formula (3) is applied separately for the upper edge of bulwark and the openings in the bulwark.

Modelling of Water Flow on Small Vessel s Deck M. Warmowska, J. Jankowski 3.2. Deck submerged in water It is assumed that the velocity field around the ship is the velocity of the undisturbed sea wave. In the case of deck submerged in a wave, the water particle velocity over the deck, near ship s bulwark, is calculated taking into account the velocity field being the average of the deck water velocity field and the wave velocity field. 3.3. The shallow water model The shallow water problem is solved in four steps (Warmowska, 2008), determining: 1. the domain Ω occupied by water, 2. the pressure field, 3. the horizontal components u x, u y of velocity u, 4. the vertical component u z of velocity u, 5. the forces and moments generated by moving water over the deck. The motion of the free surface S F is described by the following equations: ( 4) Equations (4) are integrated using the Runge-Kutta method. The nodes of the net determining the free surface moving in time are updated in each time step by interpolating the function describing free surface over nodes (x Ai, y Ai, z Ai ) of constant Euler grid of deck. It is assumed that the vertical acceleration a z can be neglected. The pressure field on deck p A is obtained integrating third Euler equation defining water motion. As a result we obtain:, ( 5) where p a pressure on the free surface S F corresponding to atmospheric pressure, f Z vertical component of the force acting on the water in point (x,y,z A ). Horizontal components of the velocity field in the domain Ω are determined from the two first Euler equations: ( 6) for pressure determined by formula (5), where f x, f y horizontal components of the force acting the water in point (x,y,z A ).

M. Warmowska, J. Jankowski Modelling of Water Flow on Small Vessel s Deck The vertical component u z of the velocity field in the domain Ω is determined from the equation of mass conservation. Additionally, in the shallow water model it is assumed that horizontal velocities u x and u y do not depend on the vertical coordinate z. Basing on this assumption, the equation determining vertical component u z takes the form:. ( 7) The forces and moments are obtained from formulas (2). 4. Verification The model was verified (Warmowska, 2008) testing the following cases of water motion: 1. with constant water mass inside tank moving with constant acceleration, 2. trapped on non moving open deck, 3. with constant water mass inside tank and moving in 3D, 4. trapped on small vessel, moving on irregular wave (Warmowska, 2010), 5. trapped on moving vessel with and without open stern. In case 1, procedures solving the problem of free surface movement were tested. The second case checked the problem of the impact of sea wave motion on water flow over the deck. In case 3, the 3D flow of water inside the closed tank was verified. In case 4, the forces and moments obtained using simplified method (with horizontal, non disturbed free surface of water over deck) and shallow water method were compared. 4.1. Tank moving with constant acceleration In this case the rectangular deck moved with constant acceleration. During the simulation after a few seconds the flat free surface inclined at a constant angle to the deck (Fig. 2). Fig. 2 The form of free surface of water on deck moving with constant acceleration a x =1m/s 2

Modelling of Water Flow on Small Vessel s Deck M. Warmowska, J. Jankowski 4.2. Inflow of wave on non moving deck Fig. 3 The impact of the sea wave on the flow over the deck In this case the deck was on the level of the sea surface. The sea wave was regular and flowed onto the open deck. The mass of water kept changing. The superposition of incoming and reflected wave could be observed (Fig. 3). 4.3. Water motion on oscillating deck The simulation of water motion on deck moving with a harmonic acceleration had also been carried out. The results were compared with results obtained by other authors and in experiments (Huang Z.-J., Hsiung C., 1997). Various frequencies of the deck oscillations were applied. When the frequency was equal to one and half of primary natural frequency, the bore could be clearly observed (Fig. 4). Fig. 4 The oscillating tank The water motion simulation in the tank is described by 3D model. Coupled sway and pitch motion of the tank are presented in Fig. 5. When the frequency was equal to second natural frequency, the free surface was not much disturbed (the Figure on right). In this case the wave over the tank was the superposition of two waves: coming and reflected.

M. Warmowska, J. Jankowski Modelling of Water Flow on Small Vessel s Deck Fig. 5 The motion of tank coupled by sway and pitch The simulation enables visualization of the velocity field changing in time. In the case of progressive wave formation, the largest velocity values are created in the head of the wave (Fig. 6). Fig. 6 The velocity field distribution (3D and projection on the plane OXY) 4.4. Comparison between two models The forces and moments F d caused by water on deck are calculated by two models: the simplified method the model used in Ro-Ro ferries damage stability calculations (presented in chapter 2); the shallow water model (presented in chapter 3). The comparison of forces generated on the vessel deck was calculated with the use of two models (simplified and presented) for the same vessel motion making it possible to compare the forces. Fig. 7 Simulation of the water flow on the moving vessel s deck, 262s

Modelling of Water Flow on Small Vessel s Deck M. Warmowska, J. Jankowski The irregular wave, determined by significant wave height H s =6m and mean period T z =8s, followed the vessel moving with the forward speed u=6m/s. The angle between the vector of forward vessel velocity and the wave vector was 30 degree. Fig. 8 presents force F d3 generated by water on deck, which increases the vessel draught, and Fig. 9 the rolling moment F d4, responsible for vessel capsizing. Fig. 8 Time history of vertical force F d3 (increasing the draught) When the vessel is in a relatively calm wave trough and the free surface on deck is almost a horizontal plane both methods match well, (in period (240s, 248s)). There are some differences in period (248s, 260s). In shallow water model the mass of water does not change as rapidly as in the simplified method Fig. 8. In the simplified method the volume of water on deck depends only on the difference between the wave surface and deck water surface and does not depend on the velocity field on the deck. In period (260s, 265s) the water surface in the simplified method drops immediately, while in the shallow water method the velocity field in the water on deck falls down slower Fig. 7. The rolling moment matches very well for both methods (Fig. 9); this is probably because the following wave is considered for these conditions, which does not generate significant water motion across the deck. Fig. 9 Time history of rolling moment F d4

M. Warmowska, J. Jankowski Modelling of Water Flow on Small Vessel s Deck 4.5. Water flow on deck with or without the open stern For an open stern vessel the mass of seawater inflow on deck and its outflow is faster than the water mass flowing on closed stern vessels. Fig. 10 The vertical forces caused by water on deck In the case of closed stern vessels the water stays longer on the deck. It increases the heave of the vessel and it changes the GM value. A similar situation is observed when the orifices are small or closed. The closed bulwark around small vessels deck may cause vessel capsizing. 5. Conclusions The shallow water model with appropriate models of boundary conditions such as: influence of seawater, water plunging on the deck, motion of the ship, can be applied to simulate the problem of water flow over the ship deck, moving in irregular wave. The computer program, developed in PRS, simulates the flow of water on deck. It takes into account the sea wave undisturbed by ship s presence, the motion of vessel in the wave, the shape of deck and the orifices. The study shows that the velocity of the water flow on deck impacts the forces generated on the deck. In the future, the diffraction velocity field of sea waves around the ship will be added to the boundary conditions of the velocity field of water on deck (Wroniszewski, 2010). REFERENCES [1] V. Belenky, D. Luit, K. Weems, Y.-S. Shin (2002), Nonlinear ship roll simulation with water-on-deck, 6 th International Ship Stability Workshop, Webb Institute, NewYork. [2] B. Buchner (2002), Green water on ship type offshore structures, PhD Thesis, Delft University of Technology. [3] J.T. Dillingham (1981), Motion studies of a vessel with water on deck, Marine Technology, Vol. 18, No. 1. [4] J.T. Dillingham, J.M. Falzarano (1986), Three-dimensional numerical simulation of green water on deck, 3 rd International Conference on the Stability of Ship and Ocean Vehicles, STAB 86, Gdańsk.

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