Sloshing of Partially Filled LNG carriers.

R.H.M.Huijsmans, G.Tritschler1, G.Gaillarde R.P.D.Dallinga

Maritime Research Institute Netherlands, Wageningen,Netherlands

1 DCN, Brest, France.

Abstract

The motion of the LNG fluid inside gas carriers is normally restrictedby the loading condition of the vessel, ie either the vessel is oper-ated at near empty condition or at fully loaded condition. In thisway resonance or sloshing effects of the fluid on the ship’s hull arelimited. However nowadays the LNG carriers are considered to beoperating at intermediate loading conditions.LNG tanks are also foreseen to be used for anchored FPSO’s, withtheir filling ratio varying during the production.Subsequently LNG carrier will be sailing with partially filled LNGtanks and FPSO-LNG will be sometimes in conditions with partiallyfilled tanks. In this condition the LNG fluid is more likely to beinduced into resonance due to wave action and roll motions. Thisresonance or sloshing behaviour of the LNG fluid will lead to highimpact pressures on the thermal insulation. Due to the differentphysical properties of the LNG fluid with respect to water in termsof density and viscosity little is known on the behaviour of the LNGfluid in resonance condition. In this paper we will describe a studyof model test experiments and numerical procedures based on VoFtechniques for the behaviour of LNG in resonance condition.In the model test experiments special care is taken into the effect ofair bubbles and viscosity on the pressure impacts. Using High speedvideo the wave front formed by the bore of the LNG in resonanceis observed and the impact to the tank hull is measured.

Key words: Non–Linear Hydrodynamics, Sloshing, Im-

pacts, Volume of Fluid technique

Introduction

The problem of sloshing in the offshore and shipping industry isalready an old one. Sloshing can occur in storage tanks. Especiallyin LNG tanks the effects of sloshing can be quite detrimental.

Abramson et al [1] presented a study where all kind of damagesdue to sloshing are described. The sloshing in a tank at resonancefrequencies with a low filling grade gives rise to the forming of abore with a height similar to the liquid height.

This was already studied by e.g. Vugts and Bosch [3] and Verhagenand van Wijngaarden [2] in the sixties.

Next to experimental studies also a large number of numerical stud-ies have been reported. For small amplitude motions linear pertur-bation theory can be employed (see Faltinsen [6]).

For large motions the numerical description vary from semi-analyticapproaches based on the evolution of shallow water waves, non-linear BEM techniques, FEM techniques to VoF approaches. Ex-amples of these approaches are given in the papers by Faltinsen etal [7, 4, 5, 10, 11, 13] , Ferrant and Touze’ [29], Wu et al [9] andvan Daalen et al [8].

Many application are given to the 2-D sloshing problem whereasnowadays 3–D geometries can be modelled as well. The violentsloshing problem is determined through a highly non–linear freesurface motion. In these gravity driven flows viscosity effects gen-erally play a minor role. However in the case of liquid LNG, the fluidproperties are not so clear. The top layer of the LNG fluid consistsof liquid LNG with a rich content of gas bubbles. So the densityviscosity and vapour pressure may play a vital role. The presenceof bubbles makes potential flow type of modelling inadequate. Toovercome these kinds of deficiencies of potential flow solvers a Vol-ume of Fluid Solver is used. The LNG tank in this study is an opentank top without a roof. In this paper the main results presented,are on the validation of the impact pressures and wave heights dueto a rolling motion of the tank from experiments at model scale andcomputations. Details of the numerical approaches can be foundin Huijsmans et al [24, 25]

Mathematical Formulation

The resonant flow of the LNG liquid inside a storage tank is acomplex problem. In the high velocity region of the flow, the vis-cosity may play an important role. Furthermore, it can be imaginedthat the fluid sloshing against tank walls, is highly rotational. TheNavier-Stokes equations are the most complete mathematical de-scription for fluid flows, it can handle rotational flows and viscosity.For the LNG liquid the incompressible Navier-Stokes equations areused. The Navier-Stokes equations are found in conservation formby applying conservation of mass and conservation of momentumto a volume V :

∮

Γ

~u · ~ndΓ = 0 (1)

∫

V

∂~u

∂tdV +

∮

Γ

~u~uT· ~ndΓ

= −1

ρ

∮

Γ

(p − µ∇~u) · ~ndΓ +

∫

V

~FdV (2)

with ~u the velocity vector and p the pressure, ρ and µ are re-spectively the density and the dynamic viscosity, ∇ is the gradientoperator and the vector ~F contains external body forces like gravity.Further, Γ is the boundary of volume V and ~n is the normal of thisboundary.To solve these equations, boundary conditions are required at thesolid boundaries, the free surface and at artificial boundaries (thein- and outflow boundaries). At the solid boundaries a no-slip con-dition is applied: ~u = 0, or when neglecting the shear stress atthe boundary a free-slip condition is implemented: un = 0 and∂ut

∂~n= 0. Where, un = ~u · ~n and ut = ~u · ~t denote the normal

and tangential velocity at the boundary respectively. The boundaryconditions at the free surface state that no tangential stress existsbetween the fluid and the air above it, and demand continuity ofnormal stresses:

µ

(

∂un

∂~t+

∂ut

∂~n

)

= 0 (3)

−p + 2µ∂un

∂~n= −p0 + 2γH (4)

Where p0 is the atmospheric pressure, γ the surface tension and2H denotes the total curvature of the boundary.Also an equation for the displacement of the free surface is required.Suppose the position of the free surface is given by s(~x, t) = 0, thenthe motion of the free surface is given by:

Ds

Dt=

∂s

∂t+ ~u · ∇s = 0 (5)

Numerical Model

Geometry and free surface capturing

For the discretisation of the mathematical model, the computa-tional domain is covered with a Cartesian grid. The collocationpoints of the velocities are in the centre of the cell faces, and thecollocation points of the pressure in the centre of the cell volume,see Fig. 1. To capture the position of the fluid inside the compu-tational domain, the volume of fluid in each cell is determined and

uu

w

w

p

p

p

p p

Figure 1: Location of thevelocity components andpressure

Fb=0.8

Fs=0.3

Ax=0.2

Az=0.5

Figure 2: Apertures of acell, fluid light grey, geom-etry dark grey.

stored in the fluid aperture Fs. Because a non-boundary-fitted gridis used, geometries and boundaries can cut through the grid cells,partly occupying these grid cells. To account for these cut-cell, ge-ometry volume and edge apertures are introduced, see Fig. 2. Thegeometry volume apertures Fb gives the fraction of the cell whichis available for fluid, the edge apertures, Ax, Ay and Az give thefraction of the cell faces which is available for the fluid flow. Ob-vious, the VOF function Fs and the geometry aperture Fb have tosatisfy: 0 ≤ Fs ≤ Fb ≤ 1.

F BBFF

S BFFS

E FFSE

E FSEE

Figure 3: Example of the cell labelling. The geometry isdark grey, the fluid is light grey and the air is white.

The volume apertures are used to label the cells, see Fig. 3. WhenFb > 0 the cell is labelled F-cell or fluid cell. The other cells haveFb = 0 and are labelled B-cell or boundary cell. The F-cells areunlabelled using the VOF function, to describe the position of thefree surface. A F-cell with Fs = 0 is labelled E-cell or empty cell.F-Cells adjacent to an E-cell are labelled S-cell or surface cells, theremaining F-cells are again labelled F -cells or fluid cells.

Discretisation of the Navier-Stokes Equations

A finite volume discretisation is used to discretise the Navier-Stokesequations in the conservation form. The discretisation is performedon a staggered grid; the conservation of mass is discretised arounda pressure collocation point, while the conservation of momentumis discretised around the velocity collocation points. Forward Euleris used for the temporal discretisation of the momentum equation.

FF FF F F F F

FF FS F F F F

SS* SE F F F F

EE EE S* S S F

EE EE E E E S*

Figure 4: Close-up of the cell labelling at the wave surface.

An extensive description of the discretisation based on apertures isgiven in [22].

Boundary conditions

The conservation of mass is applied at all pressure collocation pointsin F -cells and the conservation of momentum is applied aroundall velocity collocation points between two F -cells, an F -cell andan S-cell and between two S-cells, the so called FF -, FS- andSS-velocities. For the discretisation of Eq. 1 all velocities on thecell faces have to be known. Therefore, the FB-velocities needto be prescribed. For the discretisation of Eq. 2, not only theFB-velocities, but also the SB-velocities, BB-velocities, the SE-velocities and the EE-velocities adjacent to a SS-velocity need tobe prescribed. Further, the pressures on either side of the velocityaround which the momentum equation is discretised, is needed.This means that to discretise the momentum equation around FS-and SS-velocities the pressure in the S-cells needs to be prescribed.The FB-, SB- and the BB-velocities are prescribe using the no-slipor free-slip conditions on solid boundaries. When the boundary actsas an artificial boundary it should also prescribe the FB-, SB- andthe BB-velocities, this is discussed in the next section.The SE-velocities and the EE-velocities adjacent to a SS-velocityand the pressure in S-cells is prescribed using the free surface con-ditions, Eq. 3 and Eq. 4. Eq. 4 is used to prescribe the pressure.Neglecting the surface tension and the ∂un

∂~nterm, the pressure is

prescribed such that the pressure at the free surface equals the at-mospheric pressure using linear interpolation between the pressurecollocation points. The EE-velocities are prescribed using Eq. 3.In 2D this equation is simplified to:

µ

(

∂ux

∂z+

∂ux

∂z

)

= 0 (6)

assuming a either horizontal or vertical surface. The SE-velocitiesare in general prescribed by applying conservation of mass in theS-cell. When two SE-velocities occur around one S-cell, conser-vation of mass is required in each direction separately. During theaccuracy tests for wave simulation this was found to be introducinginaccuracies. A little sloping surface like the surface of an oceanwave, causes a staircase–like S-cell pattern, see Fig. 4. Besides ver-tical SE-velocities, this also causes some horizontal SE-velocities.Here there are two SE-velocities in around one S-cell (marked with

Designation Sym [] LNG TANKWidth B m 15.0LNG level H m 3.0Center of Rotation above keel CoR m 3.0

Table 1: LNG tank specifics

* in Fig. 4), thus in this case where no cut-cells are present, bothSE-velocities are set equal to the opposite velocity. This meansboth ∂ux

∂x= 0 and ∂ux

∂z= 0. However in the steepest part of

the wave where this is likely to happen, ∂ux

∂xis not equal to zero

but at it’s maximum. This error is circumvented by prescribing thehorizontal SE-velocities in wave simulations the same way the EE-velocities are prescribed. This implies the assumption of a more orless horizontal surface.

Free surface displacement

After the equations are discretised and the boundary condition areapplied, the discretised system of equations can be solved. Thesystem of equations is solved using Successive Over Relaxation orSOR, for more details about the solution method we refer to [22].After solving the system of equations the pressure and the velocityfield is obtained for the new time level. The new velocity field isused to update the free surface. The free surface is displaced usingthe original donor-acceptor method described by Hirt and Nicholsin [23]. In this method the free-surface is reconstructed using apiecewise constant reconstruction. Where the fluid is aligned withone of the coordinate axes. To avoid small bits of fluid to separatefrom the fluid, so called ’flotsam’ and ’jetsam’, the fluid is clippedto the surrounding fluid by using a local height function, detailsabout this height function can be found in [22]. After the fluidis displaced, the cell-labelling can be adjusted and the system ofequations rebuild.

General Presentation of Model test experiments

The model of the LNG tank (1:15.625) is tested on an oscillatingplatform at the premises of MARIN. The measurements included:global loads, wave heights near the tank wall, by means of arrays ofwave gauges, the pressure profiles using an array of small pressuregauges

Model of LNG Tank

The particulars of the LNG tanker are displayed in the table (1).

Instrumentation

Global loads

The 2–D section is kept fixed to a 3 degrees of freedom force trans-ducer attached to the oscillator.This system is constructed to measure forces (Fy and Fz) and theover turning roll moments (Mx),Wave Height probes:

8 wave probes are attached to the section as indicated in figure(5).Pressure gauges

10203040

50

100

10

960

480

Figure 5: 2–D section of LNG Tank with wave probes, mea-sures at model scale

1115

1014

9

8

13

12

7

6

5

4

3

2

1

Figure 6: Position of pressure transducers, center pres-sures are 3. cm apart

An array of 15 pressure gauges is placed at the side wall as indicatedin figure(6).

Experimental Arrangement

Study Physical parameters

The aim of this section is to find through numerical calculations, theconfiguration where the impact is the most significant. After vary-ing the water depth, roll oscillation period and motion amplitude inthe Volume of Fluid calculations we concluded that the conditionof the most significant impact would amount to water depth of 3.0m motion period of 6.5 seconds, roll oscillation of maximum 10degrees and the centre of rotation at 3.0m.

Description of the experiments

A series of experiments has been carried out to study the magni-tude of impact loads due to sloshing. The model consisted of asquare box with pressure sensors on the side face. Inside the modelten vertical wave probes were installed. The sampling frequency is5000 Hertz at the model scale. During the calibration, all pressuresensors are set at zero at the still water. The numerical parame-ters for theses condition were: vertical and horizontal resolution ofthe VoF mesh was 0.20m and the maximum update frequency usedamounted to 1000 Hrz.

Comparison between the experiments and the numeri-

cal simulations

The wave height

If we look at the Figure (7), we can remark that the accuracy ofthe Vof calculations satisfactorily predict the wave height, exceptat the peaks of the wave height.

Figure 7: Wave height measurements and computations atprobe 6

The pressures without the impact

If we compare the curves for the pressure between the experimentand the simulation, we can remark that computations give goodresults, apart from the impact conditions(see Figure (8)).

Figure 8: Pressure time traces at non–critical conditions

Maximal value of the impact for the tests with watertest A (deg) T(s) h (m) Max.(kPa) Nr sensor22 5 7.12 1.91 319 1321 5 6.78 1.91 327 320 5 6.47 1.91 323 1223 5 7.12 2.50 792 424 5 6.78 2.50 770 1425 5 6.47 2.50 663 1426 5 6.19 2.50 803 527 5 7.12 3.09 139 528 5 6.78 3.09 114 729 5 6.47 3.09 41 430 5 9.62 0.94 80 231 7.5 7.12 1.91 358 232 7.5 6.84 1.91 322 1333 7.5 7.91 2.50 230 234 7.5 7.49 2.50 575 835 7.5 7.12 2.50 722 1336 7.5 6.78 2.50 464 1037 10 7.91 1.91 188 138 5 6.19 2.50 803 5

Table 2: Overview of Maximum impact pressures

The prediction of the impact

The following table gives the values of the maximal impact thatwe obtain for the different tests. In these tests also different fluidproperties were investigated. By adding soap the amount of bubblesdrastically reduced. Also an oily substance was used instead ofwater. The oily substance was sunflower oil with a density 0.91kg/l and a viscosity of 70.10−3 [m2s−1].

Considering the measured and the calculated impact pressure, thefollowing observations can be made. Sometimes the VoF compu-tations give good results for the prediction of the impact (test 29,Figure (10) and (test 20, Figure (11)) and sometimes the resultsof computations and experiments are completely different (test 26,Figure (9), in detail Figure (12))

We can remark that the position of the most important impact onthe wall correspond with the top of the bore. The photos taken

Figure 9: Comparison VoF simulation and Experiments

35 40 45 50 55 60 65

0

20

40

60

80

100test29−impact pressure from experiment and VoF Computations

time[s]

pres

sure

[kP

a]

Figure 10: Location of Wave probes and pressure transduc-ers

from the test 26 as displayed in figures (14) to (19), confirm this(time taken at model scale).

Influence of the nature of fluid

Some tests were done with other fluids: water with soap and oil.The aim of soap is to influence the content of the bubbles in thewater.

The following table on next page presents the different test with thevalue of the impact and the number of the pressure sensor wherethe impact is the most significant.

Summary of the Tests

From the time series of the pressure profiles a spatial representationof the pressure can be determined. In the figures (20) to (22) thepressure distribution at the side wall around the time of impact aredisplayed. From the shown free surface elevation near the side wallwe see virtually no movement, indicating that at time of impact theflow is oriented in lateral direction increasing its added mass effectin that direction.

Figure 11: Location of Wave probes and pressure transduc-ers

Figure 12: Selected part of time trace with one impact

Discussion and Conclusions

From the forced oscillation roll tests we observe a very sharp pres-sure peak, lasting approximately 0.01 second. This corresponds toa near flat wedge impact in water. This was observed from thehigh speed video pictures, which clearly indicate a formation of anearly vertical bore hitting the side tank wall. Obviously this alsohas to be modelled in the VoF simulations. The resolution in termsof grid size and temporal resolution did not allow for such shorttime impacts. The VoF simulations are based on Cartesian grids,making it hard to generate enough resolution near the tank wall atthe time of impact, this of course within reasonable computationtimes. From the wave height computations and measurements wesee a difference of around 20% in height and a small shift in theperiod of the bore. This can be explained that due to the factthat for the smaller computed wave height the wave travels fasterthan the larger measured wave height, due to non-linear dispersioneffects.

From the experiments one may also conclude that the effect of re-ducing the bubble content in the liquid did not produce significantly

Figure 13: Location of Wave probes and pressure transduc-ers

Maximal value of the impact for the tests with waterFluid A Tφ (s) h (m) Max.(kPa) sensorwater 5 6.78 1.91 327 3water 5 6.47 1.91 323 12water + soap 5 6.78 1.91 252 2oil 5 6.78 1.91 148 12oil 5 6.47 1.91 195 12water 5 6.19 2.50 803 5water + soap 5 6.19 2.50 802 10oil 5 6.19 2.50 516

different results. However, when applying a more oily substance,this leads to less violent fluid motions in the tank, which in turn didlead to lower impact pressures. The viscosity of this oily substancehowever will probably not reflect the viscosity properties of the LNGliquid. The viscosity of liquid LNG with an appreciable amount ofbubbles is not well known.The effect of the vapour pressure within the bubbles was not mod-elled. Especially for fluid with a lot of gaseous content hydro-elasticeffects may play a more dominant role. In future model test the airpressure within the bubbles should be reduced according to Thoma’slaw as used frequently in cavitation research for ship propellers.

Figure 14: snap shot at t=t0

Figure 15: snap shot at t=t0+0.008s

Figure 16: snap shot at t=t0+0.016s

Figure 17: snap shot at t=t0+0.024s

Figure 18: snap shot at t=t0+0.032s

Figure 19: snap shot at t=t0+0.040s

-2 -1 0 1 2 3 4 5-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4step 40: t = 34.131 s

Figure 20: Pressure profile 0.001 sec before the time ofmaximum pressure

-2 -1 0 1 2 3 4 5-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4step 39: t = 34.1302 s

Figure 21: Pressure profile at the time of maximum pressure

-2 -1 0 1 2 3 4 5-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4step 38: t = 34.1294 s

Figure 22: Pressure profile 0.001 sec after the time of max-imum pressure

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