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Modeling of Water Flow with Free Surface

V International Conference on Computational Methods in Marine Engineering MARINE 2013

B. Brinkmann and P. Wriggers (Eds)

MODELLING OF WATER FLOW WITH FREE SURFACE MONIKA WARMOWSKA

Polski Rejestr Statków S.A. Al. Gen. Józefa Hallera 126

80-416 Gdask, Poland e-mail: m.warmowska@prs.pl, www.prs.pl

Key words: Free Surface, Harmonic wave, Sloshing, Shallow Water Problem, BEM

Abstract. Simulation of ship motion in irregular waves, modelling of sloshing inside ship tanks or flooded holds and flow of trapped water on fishing vessel deck requires special numerical methods to solve problems describing these phenomena. Accuracy of modelling of the deformed free surface in these phenomena significantly affects the solutions. The paper presents the assumptions, algorithms and results obtained in modelling the free surface and shows results of several years of research performed in Polski Rejestr Statków to enhance ship safety. The studies contribute to better awareness of ship behaviour, capsizing phenomenon, failure of ship structure and other undesired events.

1 INTRODUCTION Modelling a vessel’s motion in waves requires the following models of water flow:

− model of water flow around the ship, − model of water flow inside ship’s tank or damaged holds, − model of trapped water flowing on vessel deck.

Generally the velocity field, the potential of the velocity (if the flow is potential), the pressure field, the free surface position or the changing water domain (e.g. the amount of water mass on fishing vessel) need to be determined in solving the ship hydrodynamics problems in order to simulate vessel motion in waves and motion of liquids in its tanks. Modelling of these phenomenon requires various assumptions to enable the use numerical method. For example, some hydrodynamics problems are solved neglecting the elevation of diffracted waves in determining the hydrodynamic forces acting on the vessel (linear models).

Boundary Element Methods [3], Volume of Fluid technique [1], Finite Elements Method [4], Shallow water method [2] and others [6] are used to solve the hydrodynamics problems.

It is important to choose a model as simple as possible but without loosing sight of the essence of water behaviour. The proper determination of hydrodynamic forces acting on the moving ship in waves is very important in terms of safety.

The paper focuses on the problem of free surface elevation and its importance in the modelling of some phenomenon in vessel motion in waves and motion of liquid in tanks.

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2 PRESSURE FIELD IN REGULAR WAVE The hydrodynamic forces induced by wave and ship motions result in the ship moving in

waves. The linear model assumes a small wave amplitude. Thus the velocity field around the ship is potential and the irregular wave is assumed to be the superposition of regular waves. Assumption of the small wave amplitude in determination of the velocity flow is possible if the main dimensions of ship are significantly larger than wave height.

Figure 1 Big and small ship moving in a regular wave

In the linear model the velocity potential φ of harmonic wave is defined as follows:

)sin(0 tkxer kz ωω

φ −= ,(1)

where r0 is the amplitude of harmonic wave, k is the wave number, ω is the frequency of harmonic wave oscillations (ω2=kg for a deep water), (x, z) is the position of water particle, t is the time treated as a parameter. Axis OZ has an upward direction. The study refers to three methods. In the first method (linear) the pressure pI is obtained from the linearized Bernoulli equation as follows:

kzI eggzpp ζρρ +−= 0 , (2)

where p0 is the pressure on the free surface, ζ is the elevation of the wave surface above a point on undisturbed surface (x,0) defined as follows:

( ).cos0 tkxr ωζ −= (3)

In the second method the pressure pII depends on the position of a wave crest or a wave trough and is approximated by the formula:

( )ζζρρ −+−= zkII eggzpp 0 . (4)

In the third method (presented in report [12]) the wave is described as a cycling particles around their average position:

),sin()( 0000 tkxerxtx kz ω−−= (5)

),cos()( 0000 tkxerztz kz ω−+=

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where (x0, z0) is the average position of a water’s particle. For harmonic wave in deep water it is the orbital motion. In this method the position of free surface is defined by equation (5) for z0 equal to zero. If the average position (x0, z0) of liquid particle is given the value of pressurepIII is obtained from the following formulae:

( )( )00 2200000 15.0 kzkz

III ekrezgpp −+−+−= ζζρ . (6)

where the elevation of the wave surface ζ0 is defined as:

( ).cos 000 tkxr ωζ −= (7)

If the wave amplitude r0 is small the pressure may be approximated by the following formulae:

( )00000

kzIV ezgpp ζζρ −+−≈ . (8)

Figure 2 Orbital motion – position of surfaces for liquid particles with fixed z0 and snapshot time t

In Figure 2, black lines show the surfaces created by cycling liquid particles for fixed z0 (a yellow line is the still water level for particles with the vertical average position z0 equalszero and a blue line for z0 equal to −r0), a green line is the free surface determined by equation (3), red cycles are tracks of liquid particles determined by equations (5).

Table 1 shows the pressures obtained using these four methods for the following wave parameters:

− the amplitude r0 is equal to 2,25 meters, − the period T of wave oscillation equals to 9,5 seconds, − the average position (x0, z0) is situated at a point (0,−2).

The position of oscillating liquid particle (x, z) is calculated using equation (5). The wave elevation ζ in method I and II is determined by equation (3) and the wave elevation ζ0 in method III and method IV – by equation (7).

The differences between method I and method III reaches 3.92Pa (for t equal to zero). The differences between method II and method III is not greater than 2,41Pa. The relative errors will increase if the cycling particle is reaching the wave surface and the errors will decrease if the cycling particle moves away from the wave surface.

The pressure calculations are simpler in the first and second method than in the third method. The method I and II are usually used to solve problems when the main dimensions of

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the vessel exceed significantly the wave amplitude. In the case of fishing vessel motion, when the vessel draught is about 1to 3 meters, the third method, describing the orbital motion of the water particles, should be used [13].

Table 1: Pressure obtained by four methods

t [s] x[m] z[m] ζ[m] ζ0[m] pI[Pa] pII[Pa] pIII[Pa] pIV[Pa] 0.00 0.00 0.06 2.25 2.25 22.10 19.93 17.99 18.18 0.79 1.03 -0.22 2.00 1.95 22.09 20.39 18.25 18.44 1.58 1.78 -0.97 1.28 1.13 22.05 21.37 18.96 19.15 2.38 2.06 -2.00 0.21 0.00 22.01 21.99 19.93 20.11 3.17 1.78 -3.03 -0.97 -1.13 21.96 21.59 20.89 21.083.96 1.03 -3.78 -1.89 -1.95 21.94 20.52 21.60 21.784.75 0.00 -4.06 -2.25 -2.25 21.93 19.93 21.86 22.045.54 -1.03 -3.78 -1.89 -1.95 21.94 20.52 21.60 21.78 6.33 -1.78 -3.03 -0.97 -1.13 21.96 21.59 20.89 21.08 7.13 -2.06 -2.00 0.21 0.00 22.01 21.99 19.93 20.11 7.92 -1.78 -0.97 1.28 1.13 22.05 21.37 18.96 19.15 8.71 -1.03 -0.22 2.00 1.95 22.09 20.39 18.25 18.44 9.50 0.00 0.06 2.25 2.25 22.10 19.93 17.99 18.18

3 SLOSHING INSIDE SHIP TANK Solving of the problem of water motion inside ship’s tanks or holds focuses on determining

of the free surface. A domain occupied by a fluid surrounded by the tank’s walls changes during the simulation.

Figure 3 Sloshing phenomena in partially filled tank

Tank filled over 30% If the tank is filled over 30% the free surface hitting a ceiling causes a dangerous

consequences such as a construction failure. The forces induced by sloshing are taken into account in ship’s motion equations [9].

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If the oscillation of tank meets the natural period of fluid motion in the tank a standing wave is observed. The linear problem can be used only for wave of a small amplitude and when the volume of water does not reach the tank top. An observation of the standing wave velocity shows that the maximum value of velocity appears when moving free surface pierces the calm water plane (blue line in Figure 4). When a maximum elevation is achieved gravity forces generate accelerated wave motion.

Figure 4 Free surface of standing and absolute velocity field

One of the best methods for determining the ship motion equations accounting for sloshing problems and forces generated by liquid motion in partially filled tank (over 30%) is the Boundary Element Method [10]. In this problem the domain occupied by liquid is bounded. The potential flow is assumed. The velocity potential satisfies Laplace problem with Neumann conditions on tank walls Sc and on the free surface SF.

The boundary-value problem determining the velocity potential in the tank has the following form:

Ω∈=∆ P ,0φ , (9)

vnn

⋅=∂∂φ

, CSP ∈ ,

0φφ = , FSP ∈ .

The value of potential φ0 on free surface SF, occurring in boundary condition determining the shift of the free surface, is determined by (e.g. (12))

Figure 5 shows the method of surface shift in each time step of a simulation, [10]. There are different methods to determine a new surface of a liquid’s domain [10]. In the

first method points describing the free surface are shifted for fixed x. The value of the wave elevation is describe as follows:

xnt

2

1

∂∂+

∂∂−=

∂∂ ξφξ .

(10)

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(x,y)

α

(n1,n2)

∂φ∂x

∂φ∂y

( , )

(− n1,− n2 )∂φ∂n

∂φ∂n

(x+ ∆ t,y+ ∆ t )∂φ∂x

∂φ∂y

(x,y+( − tgα) ∆t)∂φ∂x

∂φ∂y

(x,y − ∆t)∂φ∂y

∂∂

∂∂

zx

φφ ,

∆

∂∂− t

zzx

φ,

( )zx,

( )21,nn

∂∂−

∂∂− 21, n

nn

nφφ

∆

∂∂−

∂∂+ ttg

xzzx αφφ,

∆

∂∂+∆

∂∂+ t

zzt

xx

φφ ,

Figure 5 Methods of a free surface shift

When the inclination of the free surface is not too big the derivation of the elevation ζwith respect to horizontal direction x is negligible and the second method may be applied:

.nt ∂

∂−=∂∂ φξ (11)

In the third method the free surface is moved according to formulae:

xdt

dx

∂∂= φ

, zdt

dz

∂∂= φ

.(12)

Each of the methods mentioned above imply a different formula determining the derivatives of velocity potential φ with respect to time t. In the last method the velocity potential is obtained from the differential equation defined as follows:

gzt

−∇=∂∂ 2

21 φφ

.(13)

This method requires the setting of a new grid defining the domain boundaries in each time step of the simulation.

Low liquid level in a tank or in a damaged hold When the level of liquid in a hold (e.g. in a damaged condition) is below 30% of the hold’s

height other methods are needed to solve the problem of sloshing. In this case a bore wave is observed (Figure 3). This problem is modelled with assumptions that vertical velocity is small thus vertical acceleration is negligible. It implies the assumption that horizontal velocities u1 and u2 do not depend on the vertical coordinate z. The dynamic water motion over the deck is directed along the hold.

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One of the methods suitable for this kind of phenomenon is the method used for a shallow water problem. An algorithm of a solution is developed, which determines:

− the domain Ω occupied by water, − the pressure field, − the horizontal components u1 , u2 of velocity u, − the vertical component u3 of velocity u, − the forces and moments generated by moving water in the hold.

In this method the free surface shift is determined by the following equations:

1udt

dx = , 2udt

dy = , 3udt

dz = ,(14)

where velocity (u1, u2, u3) describes the motion of water particles in relation to the bottom/deck of the hold. The pressure p in water over the bottom/deck is obtained as follows:

( ) ( )dssyxfpzyxpz

zyxhz

a

bb

+

+=),,(

3 ,,,, ρ ,(15)

where h(t,x,y,zb) is the distance between bottom/deck in the point (x,y,zb) and the point (x,y,zb+h(t,x,y,zb)) belongs to the free surface and (fx, fy, fz) is the external force (gravity and that generated by accelerating vessels).

The earlier assumptions, Euler equations of liquid motion and the principle of mass conservation result in the following formula determining the vertical velocity u3:

( )bzzqyxy

uyx

x

uzyxu −

+∂∂−

∂∂−= ),(),(),,( 21

3 ,(16)

where q represents the change of water’s mass on the hold bottom/deck due to the water flow in and out of the hole due to damage.

Shallow water method describes the problem of liquid motion in three-dimensional space but the solution is determined by the conditions on the liquid boundaries including the free surface (similarly to BEM). Figure 6 presents results of sloshing inside a tank (e.g. hold). Motion of the tank induces the motion of water.

Figure 6 Free surface and velocity field obtained using shallow water method

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4 WATER MOTION OVER THE FISHING VESSEL The shallow water problem can be also used to model the flow of seawater on the vessel

deck [11], [13]. The amount of the water mass trapped on deck keeps changing during the simulation as showed in Figure 7. This phenomenon significantly affects the flow of water on the vessel deck and therefore it is included to the shallow water problem approximating the flow on the deck (16). The mass of water entering the vessel deck in time is accounted for.

Figure 7 Flow over a bulwark

Other phenomena which significantly affect the flow of water on the vessel deck is showed in Figure 8. In this phenomenon water on the deck is connected with seawater and there is a boundary between these two domains. The velocity field of water around the ship affects the velocity field of water on the deck. It is taken into account as a boundary condition in differential equation (16).

Figure 8 Deck submerged in water

The pressure obtained from formulae (15) determines the values of forces acting on the ship. The results obtained using shallow water method were compared to those using the simplified method [5], [8]. The simplified model is used in Ro-Ro ferries damage stability calculations where the volume of water on deck depends only on the distance between the

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wave surface and deck water surface and does not depend on the velocity field on the deck. Figure 9 presents force Fd3 (vertical component) generated by water on deck, increasing the draught, and Figure 10 the rolling moment Fd4, responsible for vessel capsizing.

220 240 260 280t [s]

-600-400-200

0200Fd3 [kN] shallow water method

simplified method

Figure 9 Time history of vertical force Fd3 (increasing the draught)

220 240 260 280t [s]

-800-400

0400800

Fd4 [kNm] shallow water method

simplified method

Figure 10 Time history of rolling moment Fd4

In the shallow water model the mass of water does not change as rapidly as in the simplified method used for calculating the mass of water in the Ro-Ro vessels in damage conditions. The rolling moment matches well for both methods.

5 CONCLUSIONS Different numerical methods are used to solve problems describing ship motion in waves

affected also by forces generated by sloshing of liquids in partially filled tanks, by trapped and moving water on the fishing deck etc. This paper focuses on the problem of free surface elevation as this phenomenon significantly impacts the determination the velocity field, and thus the pressure field and forces acting on the vessel.

For smaller vessels, for which the wave amplitudes are much bigger in relation to the vessel dimensions than in the case of big ships, more accurate methods determining the free surface should be used (equations (5) and (7)).

The potential flow can be used to approximate the flow in the ship tank (sloshing phenomenon) filled more than in 30%. This paper presents the Boundary Element Method used to solve the problem. The free surface is determined from equations (12) and the value of potential on the free surface from equation (13). When the level of liquid in a hold is below 30% the method of shallow water problem can be used to approximate the sloshing in the hold. The shallow water problem can be also used to model the flow of seawater on the vessel deck. This phenomenon is significantly affected by water flowing on and off of the deck, therefore, it is included in the method approximating the flow on the deck.

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REFERENCES

[1] Hirt, C.W. and Nichols, B.D. „Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries”, Journal of Computational Physics, 1981.

[2] Zienkiewicz O.C. and Taylor, R.C. The finite element method, 4th Edition, Vol. 1, McGraw Hill, 1989.

[3] Faltisen, O.M. and Timokha A.N ”Adaptive Multimodal Approach to Nonlinear Sloshing in a Rectangular Tank”, Journal of Fluid Mechanics, Vol. 432, 2001.

[4] Wu, G.X. and Ma, Q.W. and Taylor, R.Eatock „Numerical simulation of sloshing waves in 3D tank based on finite element method”, Applied Ocean Research, Vol. 20, No. 6, 1998.

[5] Belenky V., Luit D., Weems K., Shin Y.-S., „Nonlinear ship roll simulation with water-on-deck”, 6th International Ship Stability Workshop, Webb Institute, NewYork, 2002.

[6] Bertram, V. „Practical Ship hydrodynamics”, Butterworth-Heinemann, 2002. [7] Warmowska, M., „Numerical simulation of 2d sloshing in a rotating rectangular tank”,

Marine Technology Transactions, Vol. 15, Special issue – XXI SNO, 2004. [8] Jankowski J., Laskowski A., „Capsizing of small vessel due to waves and water trapped

on deck”, Proceedings of the 9th International Conference STAB 2006, Brasil, 2006. [9] Warmowska, M. and Jankowski, J. „Ship motion affected by moving liquid cargo in

tanks”, Systems − Journal of Transdisciplinary Systems Science, Vol. 11/1, Wrocław, 2006.

[10] Warmowska, M. „Numerical simulation of liquid motion in partly filled tank”, Opuscula Mathematica”, Vol. 26/3, Kraków, 2006.

[11] Warmowska, M. „Problem of water flow on deck”, Archives of civil and mechanical engineering, Vol. VII, No. 4, Wrocław, 2007.

[12] Warmowska, M. „Fala biegnca”, Raport Techniczny Nr 50, Polski Rejestr Statków, 2010.

[13] Warmowska, M. and Jankowski, J. „Safety of fishing vessels”, Symposium Safe Shipping on the Baltic Sea, Gdask Polski Rejestr Statków, 2011.