Numerical simulation of sloshing in tanks

J.-P.V. Cordonnier

Laboratoire de Mecanique des Fluides, Ecole

Centrale de Nantes, 44072 Nantes, France

Abstract

In this paper, a two-dimensional numerical simulation based on the Marker-and-Cell (MAC) method is used for the study of sloshing phenomena inside tanks ofany shape. The free surface is treated according to a modification of the generalMAC method developed at the University of Tokyo called TUMMAC. Theimpacts on the walls are studied by letting the time step free of reducing to copewith the high velocities appearing during the sloshing. Comparisons withexperiments show a good agreement as far as the wave shape is concerned, thefluid viscosity seeming to be larger than expected since the wave amplitudelooks underestimated.

1 Introduction

The oscillations of fluid inside partially filled tanks, excited by the shipmovements, are of great concern to structural designers, since sloshing loadscan reach considerable values and damage the tank. This is the case in the designof liquid cargo carriers such as LNG tankers. The present study treats thecomplete two-dimensional problem in numerically simulating very fast impactphenomena which can be observed in relatively slow tank movements. The gaspart which is above the liquid inside the tank is not simulated which means thatthe non-liquid part is assumed empty. The TUMMAC method used here is amethod which can be applied for solving a large variety of flow problems withfree-surface since over-turning waves can be generated. External and internalflows can be studied using a rectangular mesh system in the fixed coordinatesystem or in a frame moving with the body. In the case of fluid motions insidetanks, the Navier-Stokes equations are written in the relative coordinate system.Boundary conditions on the walls assure free slip of the liquid in order to avoidsticky behaviour of it on the ceiling. Special care of the time step gives thepossibility of simulating high velocities as soon as they appear when impactoccurs.

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198 Marine, Offshore and Ice Technology

2 Description of the numerical method

2.1 Governing equationsIncompressible fluid motions are managed by the following equations,

pV+(V.V)V + -V/> = vAV + Fdt p

|y.v = o

In these equations, F is the external force field, V is the fluid velocity field, p

the density of the fluid and v its kinematics viscosity.In the case of a two-dimensional flow in the gravity field, the z-direction beingupward vertical, these equations are the Navier-Stokes equations in the fixedcoordinate system,

du du du I dp ( d*u d*u— + u — + w — + —^ = Vl -r-T + -r-rdt dx dz p ox

dw dw dw 1 dp-| --- =

dx dz

where w,w are the velocity components in the directions x , z respectively andg = 9.81 m/s is the gravity acceleration.The tank is excited by random plane motions whose velocity V (f) can be

separated into a translation velocity C(f) and a rotational velocity Q(0 about afixed point M^ (x zj so that, in any point M (*,z), the velocity of the tank is

The Navier-Stokes equations are then written in a coordinate system attached tothe tank by performing on the partial derivatives the following transformation,

4- - >-£--V£.V + QAdt dt *abs. rel.

and if V =V-V represents the fluid velocity relative to the moving coordinatesystem, this equation becomes,

The Navier-Stokes equation shows in the right-end side extra-terms of accele-ration which can be computed at every time steps.A staggered mesh system is used with constant grid spaces Ax,Az in thehorizontal and vertical directions respectively. A finite difference scheme is used

to discretize these equations. A time increment At is chosen at the starting timeand is checked at each time step n in order to satisfy the following condition inany place of the fluid domain:

If this condition is not satisfied, the time increment for the following time step is

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Marine, Offshore and Ice Technology 199

max 1 + maxAt \At}

I Ax Az \

The variables u,w and P are the velocity components and the pressure, respecti-vely. They are evaluated at the positions illustrated in fig.l, where subscripts i,kindicate the cell location.

Figure 1: Locations of finite difference variables in a cell

At each step of the time marching simulation, the velocity components u"* and^n+i an^ estimated for every cells inside the fluid from the variables computed atthe previous time step n by the following equations:

At i n n \U, i, — UL, t 7~\*i* ~ ;-u)

pAx^

/lf/nn nn \

pAzwhere

duw

in which the square brackets represent the finite difference approximation of theterms inside them and where

dt

assuming the rotational displacement of the tank being around a fixed point

(% z ) and the angular position 0 being counted in clockwise direction andwhere

U~L= —

w ,. = —-

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200 Marine, Offshore and Ice Technology

and (Xj Zifr) is the lower left corner of the cell i,k.Any movement can be simulated, harmonic or random displacements may beimposed to the tank.

2.2 Boundary conditionsSince tanks of any shape and containing any inside walls can be studied, apreliminary program is used to discretize the given tank so that any cell in thecomputing domain can be simplified into three kinds: cell containing a boundarysegment, cell without any boundary segment inside the fluid domain and cellwithout boundary segment outside the fluid domain. For the cells containing aboundary segment, various geometrical parameters are computed in order todefine, depending on the segment location inside the cell, the estimated values ofthe velocity components that cannot be ordinary computed. In order to satisfythe boundary condition: a limit velocity V^ is computed from the velocity

components (//,v) in the cell

Uf = r(ru - pw)

where (p,r) are the components of the unit vector n normal to the boundarysegment towards the fluid domain. If we impose free-slip condition on thewalls, the limit velocity is V^. For the no-slip condition, the limit velocity is

zero.For interpolations, the velocity profile is assumed to be decreasing as the squareroot of the distance to the wall so that:

V

Figure 2: Velocity interpolation on the walls

2.3 Free surface conditionsThe free surface is followed in its movement at every time step in computing thevelocity of marker particles. The new free surface line is then approximated by anew set of points (marker particles) that are the intersections between the meshsystem and the displaced free surface (fig. 3).The volume of fluid is then computed and a vertical correction is applied in orderto keep the fluid volume a constant along the numerical simulation.The pressure on the free surface is the atmospheric pressure. A cell flaggingtechnique is used to allow for any shape of the free surface since over-turningphenomena may occur.

2.4 Solving techniquesThe pressure inside the fluid is updated at every time step by an iterative

procedure using an over-relaxation factor F. It is usually taken equal to 1.5,except in special cases where it can be reduced to 1. The iterative procedure is

stopped when convergence is obtained, that is if P"' - P"[ < P mr is satisfied

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Marine, Offshore and Ice Technology 203

the horizontal direction Ox :The tank is build in translucid material. Perfect two dimensional fluid motionshave always been observed during experiments, the tank being 0.15 m thick inthe third direction Oy.Various filling of water have been tested. The tank is completely closed and theair above the free surface is kept at atmospheric pressure.After a starting period depending on the amplitude, harmonic displacements areobtained by an electric motor. The frequency is adjusted by the motor speed ofrotation. Complete record of the tank position is made in order to apply exactlythe same displacements in the numerical simulation . A visual mark is used toacknowledge the starting time of the simulation on the video record.A video-tape recorder gives 50 pictures per second. They are numerised,analysed by an image processor and the fluid domain is deduced at every timestep giving the fluid contour. Both experimental and numerical pictures aresuperimposed on the same one to verify the tank position along the simulation.The comparison is limited to the free surface shape, but fluid velocities shouldalso be compared in order to find the origin of any discrepancy.

4. Results

Among many films, it is difficult to select some pictures which never givedynamical aspects of the phenomenon that have to be presented. Nevertheless, itis possible to present some results in which both experimental and numericalfree surfaces are drawn at the same instant (fig. 7). The numerical one is on theright, the one whose free surface extents outside the tank due to the numericaltechnique used to compute the displacements.The tank filling is 60% of the tank height. The frequency of the displacement isnot at resonance. Two simulations are presented, the first one is a randomdisplacement with 3 cm amplitude, the second is harmonic with a period of0.72 s and the same amplitude. Two types of free surface deformations areobserved: in the first experiment, there are travelling waves and, in the second,they are rather standing waves.For the numerical simulation, the cell is 1 cm x 1 cm so the computing domainis made of 80 x 40 = 3200 cells. The initial time step is 0.05 s. For 10 s of sucha simulation, the CPU time is equal to about half an hour, if the sloshingphenomenon is not too much reducing the time step.

5. Conclusions

The general free surface deformation is fairly well simulated since, after manyseconds, the shape looks the same as the experimental one. For the local impactson the wall, the simulation is unable to reproduce very rapid displacements ofsmall quantities of fluid which often lead to drops of water. The viscosity of thesimulated fluid is not as low as the real one and the wave amplitudes are smaller.This is mainly due to the cell size; a difficult compromise has to be realisedbetween the "numerical viscosity" and the computing time in choosing the cellsize.The tests performed with very high water levels inside the tank demonstratedgreater discrepancies than the one presented here. In fact, the air captured anddisplaced by the water has a great influence on the water surface due to veryimportant air-flow velocities. These velocities are completely neglected in thenumerical simulation since, above the fluid, there is nothing.

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204 Marine, Offshore and Ice Technology

Experiments: Computations:

t = 10.4 s

f = 10.9 s

Figure 7: Comparison between experiments and numerical simulations

Improvements should be made to accelerate the computations so that turbulencecould be introduced inside the simulation procedure. Any tank shape and internalstructures can be studied. A circular tank is now under test.The simulation of the air above the fluid is also possible and will be tried in anear future.

Transactions on the Built Environment vol 5, © 1994 WIT Press, www.witpress.com, ISSN 1743-3509