Sloshing in Arbitrary Shaped Tanks

246

(Read at the Autumn Meeting of the Society of Naval Architects of Japan, November 1985)

Sloshing in Arbitrary Shaped Tanks

by N. E. Mikelis*, Member D. W. Robinson*

Summary

The paper describes a calculation procedure developed at Lloyd's Register of Shipping for

design purposes, based on a two-dimensional numerical finite-difference method which predicts

the sloshing behaviour of fluids in arbitrary shaped tanks when excited by ship motions in a

seaway.

The method overcomes some problems associated with model experiments. Furthermore,

tank shape, internal structure and excitation can be readily modelled and the calculation

provides realistic free surface behaviour and pressure time histories, including impacts. A standard excitation method is suggested which takes into account the important interactions

of fluid and ship natural periods and amplitudes of ship motions.

Other modes of excitation are also examined including one based on records of irregular

ships' motions and another based on a numerical model of the coupled phenomenon of sloshing

and ship motions.

1 Introduction

Sloshing of fluids in tanks is a phenomenon which

has concerned designers and Classification Societies

for a considerable time. Consequently efforts have

been devoted to produce procedures that ensure

structural adequacy by withstanding the predicted

sloshing pressures.

When liquid cargoes or ballast are carried in tanks

which are not completely full, ship motions can

create violent internal waves which impart dynamic

and impact pressures to the tank boundaries and

corresponding forces to the tank supports.

Although fill height is an important parameter

affecting the type of developed wave forms, which

include standing waves, travelling waves and hy-

draulic bores, it is now recognised that even very

high fillings in smooth tanks can cause problems").

Operational factors dictate that partial fillings are

desirable and therefore it is necessary to predict

the likely sloshing pressures at the design stage

for particular tanks and their structure. It is

hardly necessary to state that this is a difficult

problem given the non-linear nature of the phenomenon and the number of variables. For a partic-

ular design, where tank dimensions and structural

arrangements are fixed, the following global para-

meters can vary to affect the load that is transmit-

ted to any point on the tank structure :

Cargo : varying density, viscosity

Filling Level : anything below say 97% could be

considered a partial filling Position of tank in ship : longitudinal, transverse-

and vertical location of otherwise identical tanks in a ship result in different excitation and hence sloshing characteristics

Loading Condition of ship : resulting in different natural periods of motions

Ship Operation : speed, heading Ship Motions : in specific short-term sea states

and long-term wave climate From a Classification Society's viewpoint, the

complexity of the problem is enlarged still further to include different tank designs, ship types, prin-cipal dimensions and, in some cases operating areas. Notwithstanding the almost infinite variability, it is essential that the Classification Society has an independent, and cost effective means of ensuring that the structure of a tank is adequate to with-stand the loads -induced by sloshing. The scale of the problem dictates the need for mathematical modelling to represent the three components of an assessment procedure, namely, excitation, fluid behaviour and structural analysis. In stressing the complexity of the problem, the non-absolute nature of the three components becomes obvious, and naturally, to justify the application of any procedure, this must be calibrated by service experience.

Section 2 of this paper describes an excitation

procedure, the purpose of which is to impart mo-tions to the tank which result in realistic and re-

presentative short-term and long-term maximum responses. Section 3 details a transient two-dimen-sional finite difference program used to predict

* Lloyd's Register of Shipping 71 Fenchurch

Street, London, UK

Sloshing in Arbitrary Shaped Tanks 247

fluid response to imparted excitation. The comput-ed pressures are at present used in a static struc-tural analysis which is based on a plastic collapse theory. However, as the procedure produces pres-sure time histories, work is underway at Lloyd's Register to match these with a dynamic structural capability. In Section 4, two alternative methods of excitation are examined, namely, forced irregu-lar motions, which can be derived from full scale measurements or from calculations, and one based on a numerical model of the coupled phenomenon of sloshing and ship motions.

2 Excitation Procedure

In addition to the global variables mentioned in section 1, other factors such as wall flexibility,

gas cushioning, bubble content and the transient nature of dynamic signals complicate the treatment of the sloshing problem. Usually these phenomena are either not correctly scaled in experiments or not included in numerical solutions and therefore, the derived loads (measured or computed) cannot be treated as absolute but have to be applied com-

paratively on the basis of a procedure that also accounts for previous experience.

For the procedure adopted at Lloyd's Register of Shipping, use is made of numerical simulation of liquid sloshing3). The objective of any scheme of excitation, which incidentally applies equally to

physical or mathematical modelling, is to ensure that one has considered a set of conditions that would give rise to a design maximum pressure envelope on the tank boundaries which can then be set against structural criteria to prevent damage.

There are normally two distinct forms of excita-tion adopted for sloshing analyses, namely, pure harmonic, where the response is examined at fixed

periods and amplitudes, and irregular, where a random motion is generated from selected spectra. At Lloyd's Register both these forms of excitation have been used but a type of 'Sloshing Excitation Spectrum' has finally been adopted, which employs a continuously and smoothly varying period and amplitude of motion. In this way a single com-

puter run covers excitation ranging from high amplitudes at periods close to the ship's natural

period, to lower amplitudes at other periods, so that the maximum design pressures are exposed at some combination of amplitude and period which is dependant on ship and tank conditions.

The adoption of a two-dimensional numerical solution necessitates the separate treatment of exci-tation in the longitudinal and transverse directions. Thus different runs simulate the roll/sway/heave and the pitch/heave motions. The effect of surge during excitation in the longitudinal direction has been ignored as it is considered to be negligible. All above degrees of freedom are excited at periods

which vary with time and at amplitudes which for

the translational motions are fixed, while the rota-

tional has its maximum at the ship's natural period

and an exponential decay at higher and lower peri-

ods. A narrower exponential function is used for

roll compared to pitch. Motion amplitudes and

variations with period are described by parametric

expressions which are based on numerous applica-

tions of the strip theory/sea spectra/wave climate

approach for a range of ship forms and short-term

wave spectra4•`6). To represent realistic minimum

values a lower limit is imposed to the exponential

decay of the rotational amplitudes (6•K in roll and

3•K in pitch). It should be pointed out that at high

periods of excitation the effect of introducing trans-

lational motions on the rotational one is small,

even if the amplitudes of the former motions are

large. Conversely at low periods of excitation the

combination of translations and rotation can result

in considerable sloshing when the relative phase

between the motions is chosen to model the 'rolling

against the wave' situation. Because at sea this

form of rolling is encountered at periods below the

ship's natural periody, the translational excitation

in the sloshing procedure was chosen to model the

`rolling against the wave' condition at all times

and at all periods. It is recognised that significant

sloshing would occur when the ship's natural per-

iod, Ts, and the liquid natural period, Tn, are very

close (synchronism). Conversely, as the separa-

tion between the two periods increases so sloshing

becomes less of a problem. In evaluating the liquid

period it is of course necessary to take into account

the effects of any internal stiffening and/or tank

chamfers.

The adopted excitation procedure takes into ac-

count the likelihood of synchronism and achieves

computational economy by restricting the range

of period variation to a convenient figure of 4

seconds. Based on numerical experiments which

used the 'Sloshing Excitation Spectrum' with long

period ranges and also from comparisons with the

Society's experience with sloshing problems, the

period range is centred so that it always includes

the ship's natural period but obtains a bias towards

the liquid period when the separation of periods

increases. In some cases this period separation

indicates where comprehensive analysis is not re-

quired, in which case a quasistatic approach is

adopted. Figure 1 provides an example of the

`Sloshing Excitation Spectrum'.

The adopted excitation procedure therefore

achieves a consistent and economic solution compar-

ed with the harmonic or irregular forms which

would necessitate numerous tests or a very long

simulation to establish worst conditions.

248 Journal of The Society of Naval Architects of Japan, Vol.158

3 The Mathematical Model

The fluid motions and exerted pressures from a

given excitation of a partially filled tank are evaluat-

ed using a mathematical model of the two-dimen-

sional problem. For this purpose the computer

program LR. FLUIDS3) has been developed, based

on the SOLA SURF code8) which in turn was based

on the MAC method9). The LR. FLUIDS program

may be considered as an extension and generalisa-

tion of the work by Navickas et at10. Navickas

used SOLA-SURF to model a two-dimensional

prismatic tank with a ceiling and extended the

code to model liquid compressibility during impacts

on the ceiling assuming small changes of density.

It was reportedm that comparisons of compressible

and incompressible types of pressure with experi-

ments showed qualitative agreement, while very

good agreement was observed on comparisons of

free surface motions.

The adopted mathematical model provides a tran-

sient solution by progressing in small time incre-

ments. For each of these time steps an iterative

Finite Difference scheme updates the velocity and

pressure fields so that these satisfy the conservation

of momentum (Navier-Stokes) and conservation

of mass (Continuity) equations and also satisfy all

the boundary conditions which describe the tank

and its motion. The relevant equations and their

Finite Difference expressions are described adequate-

ly in references8•`10) while most of the LR. FLUIDS

developments are discussed in 3). For economy of

space only a brief description and comments are

considered necessary here.

The Navier-Stokes equations provide a non-linear

description of the problem which is necessary for

realistic modelling in view of the violent liquid

motions at resonance. The equations include vis-

cous effects with the laminar viscosity term. It

must be pointed out, however, that numerical

experiments have shown that viscosity does not

affect the liquid's sloshing response. Furthermore,

as discussed in11), this is confirmed by experiments

using liquids of different viscosities.

The incompressible continuity equation has been

modified to a slightly compressible one by the in-

clusion of a term describing small density changes

(eg. seem10)). The effect of this term is discussed

later in this section. In order to avoid the unreali-

stic pressure jumps due to the discontinuous nature

of time stepping the buffering scheme devised in

reference12) has been adopted. In this scheme, and

in its variant adopted by Arai13), the algorithm

progressively increases the pressure in cells where

the liquid is about to impact the tank's ceiling.

This pressure increase has the effect of slowing

the upward speed of the free surface until at the

moment of impact this speed is zero.

The code has been extended to include boundary

conditions which allow for any excitation composed

of two translational and one rotational motions.

The rotation is defined about any specified origin

and the relative phase angles are chosen at will.

Figure 1 provides one example of motion imparted

to a tank. In addition to the SOLA capabilities

LR. FLUIDS has boundary conditions which allow

the modelling of two-dimensional tanks with inter-

nal structure, chamfers or of U-shaped tanks.

Also the free surface boundary condition has been

improved to enable the treatment of steep free

surfaces and of free surfaces in the vicinity of

vertical internal structure. For this purpose an

Fig. 1 Typical sloshing excitation time histories

of : variation of Period and of Roll, Sway and Heave amplitudes (Tolling Against Waves' condition)

Fig. 2 Free surface realisations at one eighth of a

period intervals for a shallow filled rectan-

gular tank in harmonic roll motion


averaging process over the free surface 'step' ac-

counts for the fluid velocity and free surface slope

in the equation used to update the surface height

h, ie.

where v is the vertical fluid velocity, ii is the

averaged horizontal fluid velocity over the step and

dh/dx is the'effective, free surface slope. Figures

2,3 and 4 provide examples relevant to the above

discussion.

Typical Finite Difference grids employed by LR.

FLUIDS are composed of 300 to 400 cells, although

this number may be increased to account for details

of the internal structure. The cells are rectan-

gular and of constant width and height. From sen-sitivity studies it has been established that the

grid density used is adequate. The only area where

grid refinement improves results is in chamfers, which of necessity are modelled here as stepped

lines. This could be refined further by the adop-

tion of more complex boundary conditions.

Fugures 5, 6 and 7 show typical comparisons be-

tween computed and experimentally measured pres-

sures, in Pascals, and free surface elevation, in

metres, for a 1 : 40 model of a prismatic tank in

Fig. 3 Free surface and velocity field realisations

for a rectangular tank with internal struc-

ture, in harmonic roll, sway and heave

motions

Fig. 4 Free surface and velocity field realisations

at quarter period intervals, for a prismatic

tank in harmonic roll motion. Also shown

are the locations of pressure transducers and

of free surface height recorder for the tank

model of figures 5-7

Fig. 5 Experimental and computed pressures in

Pascals at transducer locations R1, R2 and

free surface height in metres at position

HR : Roll, h/D=0. 15, T =1. 532 sec, ƒ³=

O. 1 rads


Pascals at transducer locations R1, R2, R3 and

free surface height in metres at position

HR : Roll, h/D=0. 46, T =1.207 sec, ƒ³=0. 1

rads.

250 Journal of The Society of Naval Architects of Japan, Vol. 158

pure roll and in a diagonal rotation. Other test conditions3), which also included pure pitching,

produced the same level of agreement to that shown in figures 5-7 here. The positions of the pressure transducers and free surface height recorder are

given in figure 4. The transducers which recorded roll response(R) were placed along the centreplane of the tank, while the transducers used for the 'diagonal' excitation(D) were on one of the two

edges which did not intersect with the axis of rotation. The signals from computation and from experiment are intentionally offset in the time scale as shown on figure 5, to ease visual comparison. The results shown in figures 5-7 are for resonant conditions for three different fill heights at one of two different amplitudes. The agreement is very good for the free surface and the dynamic pressure time histories. It is noted that the computation has also depicted the secondary peaks in the signals, resulting probably from smaller travelling waves. In other comparisons3)which included transducers on the chamfer of the tank, the computation was seen to overestimate the duration and the magnitude of the pressure pulse on the chamfer by a moderate amount, as it might reasonably be expected from the approximation of a sloping surface by a stepp-ed one.

Impact pressures, as differentiated from the dyna-mic ones, pose a more subtle problem when com-

paring theory with experiment and when they must be applied to consider the adequacy of a design. When the time step was halved in the com-

putation then the impulsive pressure was seen to exactly double while the dynamic component re-mained virtually unaffected. On reflection this is a consistent result, since it shows that the impulse

(force-time integral) is time-step independent. Numerical experiments were conducted with the LR. FLUIDS program, whereby a horizontal free surface was made to rise uniformly and subsequent-ly to impact on a horizontal ceiling which did not extend to the full length of the free surface. This allowed the fluid to escape after impact around both ends of the ceiling. A wide range of time steps was employed, ensuring that at the upper end of the range the free surface, and at the lower end of the range the sound, would travel only through a given fraction of a cell. Two versions of the LR. FLUIDS code were tested, one with the incom-

pressible and the other with the slightly compress-ible continuity equations. When the time step, was of the order that is normally adopted in sloshing calculations, the two versions gave identical results and both showed the impact pressure doubl ing when the time step was halved. Furthermore when the time step was reduced to the order which allows acoustic pressure waves to propagate (frac-tion of millisecond) then the incompressible version kept on exactly doubling the pressure with no limit in sight. The compressible version however reached a constant impact pressure with no effect from further reductions in time step size. The value of this pressure at the centre of the impacted area was precisely equal to the product of the fluid's density times the speed of sound in the fluid times

the velocity of the impacting free surface. The

pressure reduced away from the centre of the im-pacted surface, towards the open ends. Also fol lowing the impact a steep pressure wave propagated through the finite difference grid with speed equal

to the speed of sound in the fluid. Numerical diffusion however reduced the steepness of this wave as it progressed and as it reflected on different boundaries. All this behaviour was confirmed for various configurations and physical constants of the fluid.

LR. FLUIDS thus proved to be a promising tool for the study of impacts. For a number of rea-sons, however, an obvious one being economy in computation and another one being associated with

present inadequacies in implementing acoustic type of pressures in design, it was not considered desir-

able to proceed with computations employing such small time steps. The problem remained, how-ever, what to do with the relative nature of im-

pact pressure magnitudes. This was tackled by the development of an automatic selection of time step

which was calibrated using experimental data of

pressures on the chamfer and ceiling of the pris-matic tank discussed earlier. It should be pointed out that this development was completed after

producing the comparison for transducer R 7 in figure 7. The automatic selection of time step was


Pascals at transducer locations R3, D3, R4, D4,

R7, D7 and free surface height in metres at

position HR : Roll and Diagonal, h/D=0. 75,

T=1. 056 sec, ƒ³=0.25 rads.


designed to ensure that following an update of velocities no part of the fluid would have travelled more than a certain fraction of a cell. If this condition is violated the simulation steps back in time and repeats the computation with the time step set to half of the previous value. Further-more, when the inspection of fluid velocities indi-cates that a larger time step can be used, this is implemented at the next cycle. The numerical value of the cell fraction condition was chosen as one eighth(1/8) so as : (i) to provide a general agreement between measurement and computation and (ii) to produce impact pressures which when applied to the structural analysis -procedure adopted. at Lloyd's, fits well with the Society's long experience. In the future however a re-examination of this problem is likely to tie in with further deve-lopments presently taking place on analysis of struc-tures under impulsive loads.

A limitation of LR. FLUIDS is that it can model only two-dimensional problems. In the experi-ments, from which Figures 5-7 are drawn, an at-tempt was made to quantify the three-dimensional effect by imparting a diagonal excitation to the tank. In this situation, it was observed that when the forcing period was away from either the nat-ural period of the liquid in the roll plane, or of that in the pitch plane, there was little liquid motion. Also when the forcing period was at or

near either of the natural periods, the liquid mo-tion was confined in the plane of the motion whose

period was excited. This clear separation of responses is attributed to the fact that the tank's breadth to length ratio (=1. 86) resulted in distinct-ly different natural periods in the two principal

planes and thus each of these motions was excited separately at corresponding periods in the diagonal excitation experiments. This phenomenon is also reflected on the pressure recordings made on the edge of the tank, as seen for example in figure 7

where experimental measurements of pressure from diagonal tests follow the time history of those from

pure roll tests conducted at the same period of excitation. The wave height recorder was not functioning during the diagonal excitation tests and thus no data are available. In all cases tested

the pressures on the tank's edge obtained from diagonal excitation were never of dissimilar magni-tude from those recorded in the pure roll or pure

pitch tests at corresponding periods. Naturally, it is expected that if a tank's breadth to length

ratio tends to unity, then the roll and pitch natural

periods of the liquid would approach each other and the funnelling effect would magnify the loads on the tank's edges. In considering however the like-lihood or unlikelihood of diagonal flow, the effect of internal structure must not be ignored, since this

increases the natural period in the plane normal to

the direction the internal structure runs. A com-

puter eigenvalue analysis has been purposely devel-oped at Lloyd's for the calculation of the natural

period and of higher harmonics of liquid contained in two-dimensional tanks of arbitrary geometry.

To account for the three-dimensional effect of

flow on pressure, various authors have adopted the

square root of the sum of squares of pressures

obtained from separate roll and pitch experiments.

It may well be a sensible generalisation to adopt

this sum of squares in all cases, that is regardless

of the tank's breadth to length ratio. However it

would only be appropriate to combine in this man-

ner the maximum pressures obtained at the same

excitation period, in which case if roll and pitch

resonances are well separated then the three-dimen-

sional effect should again be predicted as being

negligible.

There are applications which require a knowledge

of the forces and moment exerted by the fluid on

a part of a tank's structure, such as an internal

member or a bulkhead, or in the complete tank.

These loads may be used for example for the esti-

mation of tripping moments on a stiffener or gir-

der, for the evaluation of dynamic loads on supports

of independent tanks, and, when the sloshing in-

duced dynamic loads are of sufficiently large magni-

tude, in the description of the coupled phenomenon

of sloshing and ship motions, as discussed in section

4. The fluid induced forces and moment are ob-

Fig.8 Time histories of forced rolling motion and

of computed sloshing induced horizontal and

vertical forces and moment on the tank of

figure 2.

252 Journal of The Society of Naval Architects of Japan, Vol. 158

tamed in LR. FLUIDS by integrating the pressure around the part of the tank's structure in question, or around all the 'wet' boundaries when the total loads are required. The integration is repeated every time step of the simulation. A technique for extrapolating pressure from the cell's centre, where it is computed, to the tank's boundary, where it is integrated, had to be devised3). Figures 8 and 9 provide examples of computed time histories of sloshing induced loads. Figure 8 shows the harmonic roll forced excitation and the fluid in duced forces and moment on the shallow filled rectangular tank of figure 2. For interest, the non-linear nature of the induced loads and the 10% increase of 'effective weight' of the fluid from the

given excitation are noted. Also as a check it is noted that following a Fourier analysis of the ver-tical force component signal, the constant term of the series exactly equals the static weight of the fluid. Figure 9 shows the time histories of forced harmonic excitation in roll, sway and heave and the liquid induced horizontal force on the bulkhead of the tank of figure 3. The considerable increase of the force, from the static value of about O. 3 MN to the dynamic value of 1. 3 MN, should be noted.

To establish confidence in the use of the computed sloshing induced loads, comparisons were conducted with experimental measurements of induced mo-ment on a rectangular tank and on the prismatic tank of figure 43). These comparisons covered ran-

ges of the following variables : period and ampli-tude of rotational excitation, fill height and posi-tion of the centre of rotation. Figure 10 shows for the prismatic tank such a comparison for the magnitude of the first harmonic component of the induced moment signal and for the phase angle between this harmonic and the forced oscillation. Results for a range of dimensionless frequencies are presented. In all tested cases the agreement between computation and experiment has been shown3) to be very satisfactory.

4 Other Forms of Excitation

A particular strength of the LR. FLUIDS slosh-ing simulation computer program is the generality of excitation it allows. In addition to the excitation procedure described in Section 2 of this paper, the following alternatives are available :

4.1 Harmonic Forced Excitation In this case the user decides if the tank is to

be excited in 1, 2 or 3 degrees of freedom. The harmonic forced motion is then defined by a period, an amplitude and a phase angle for each of the invoked degrees of freedom. The phase angles can be specified such that a realistic representation of the ship's behaviour is simulated. For example, the ship may be forced to roll 'with' or 'against' the wave, and this would depend on the relation between the period of excitation and the ship's natural period.

4.2 Irregular Forced Excitation The tank's motion can also be specified as records

of irregular signals of displacements, velocities and accelerations. These signals may be recordings of real ship motions or can be generated numerically from sea spectra.

Fig. 9 Time histories of forced motions and of

computed sloshing induced horizontal force

per unit length on the end wall (bulkhead) of the tank of figure 3

Fig. 10 Experimental and computed dimensionless

amplitude and phase angle of sloshing

induced moment on the tank of figure 4

(pure roll, h/D=0. 45, ƒ³=0. 1 rads)


4.3 Coupled Sloshing and Ship Motions In this mode of excitation the simulation pro-

ceeds in time by a parallel and coupled set of com-

putations of ship motion equations and of the slosh-ing analysis. As the liquid cargo moves, it trans-mits a force and a moment on the tank and con-sequently onto the ship. These liquid induced loads are computed for every time step by an in-tegration of the pressures around the tank bound-ary, as discussed earlier, and are introduced in equations which model ship motions. In turn these equations are solved, thus providing values of displacements, velocities and accelerations which are used to excite the sloshing simulation in the subsequent time step.

This process, which is repeated for as long as is required, relies on the input of frequency depen-dent ship hydrodynamic data, from say a strip theory analysis. The definition of excitation is then simply made by the specification of the inci-dent wave's height and period. It is also worth noting here that the additional computational ef-fort required for the coupled solution is negligible.

This method of analysis has been verified3,14) by comparisons with model scale measurements on a

products carrier in beam waves of various heights and periods. The ship model has tanks built in, which carry either solid or liquid cargo. The comparisons were initially conducted using a one degree of freedom (roll) coupling. This was sub-sequently extended to a model of three and to a

pseudo-five degrees of freedom coupling. For the sake of completeness the results from

these comparisons are shown on figures 11 and 12

for free and for forced rolling respectively. Apart

from the good agreement demonstrated for forced

rolling on figure 12, excellent agreement was also

found in the free rolling comparisons where the

measured and computed natural periods of the ship

were practically identical for both solid and liquid

cargo conditions. The metacentric height correc-

tion for the presence of the free surface is not

applied in the three degree coupled analysis, be-

cause the underlining phenomenon is implicitly ac-

counted by the communication of liquid induced

loads to the ship motions equations. In the parti-

cular case of ship, tank and filling level shown

here, the natural periods of ship and of liquid

cargo happen to be well separated. As expected,

during the slow large motions resulting from waves

with periods near the ship's natural period, the

computed free surface remains almost horizontal.

Therefore the results of computations at such ex-

citation provide a simple and direct verification of

the coupling method when compared with the tra-

ditional free surface correction to the metacentric

height. When however the liquid cargo and ship

natural periods are closer, then the free surface

correction used for assessing stability would clearly

be inadequate since the liquid would not remain

horizontal and the assumed wedge shape of mass

transfer would underestimate the free surface ef-

fect.

Figures 13 and 15 illustrate the coupled sloshing

Fig. 11 Experimental and computed non-dimen-

sional logarithmic decrement and computed

angular displacement of a products carrier

ship model in free rolling from 10•K, with

solid and with liquid cargo in three tanks

filled at h/D=0. 45 (tank shown on figure

4)

Fig.12 Experimental and computed roll response

of a products carrier ship model, incorpora-ting three tanks (shown on figure 4) filled at h/D=0. 45 with solid and with liquid cargo in beam waves at zero forward speed

(values appropriate to full scale)

254 Journal of The Society of Naval Architects of Japan, Vol.158

and three degrees of freedom ship motions analy-

sis. The wave amplitude is comparable in these

two cases, but in figure 13 the wave period corres-

ponds to the ship's natural period while in figure

15 it corresponds to the liquid cargo's natural pe-

riod. In the former the resulting roll amplitude

is relatively large but does not excite sloshing (10•K

in the steady state, as is shown on figure 14 which

illustrates the computed time history of motion).

However, while the shorter wave of figure 15 does

not cause any appreciable roll (2. 5•K in the steady

state, as shown on figure 16) considerable sloshing

is induced.

5 Concluding Remarks

A procedure based on a numerical solution for

assessing the effect of sloshing in partially filled

tanks has been discussed. This procedure repre-

sents a consistent and efficient method for the de-

termination of design loads.

Using the LR. FLUIDS program, together with

specially developed computer animated graphics

output, a designer can 'see' the effect of varying

tank configurations and filling levels and can obtain

pressure time histories in a fraction of the time

and cost of equivalent model tests.

The computer resources required by LR. FLUIDS

are not excessive, typically being between 20 and

40 CPU seconds per simulated period, in the Soci-

ety's mainframe computer. The exact figure de-

pends on the severity of motion, amount of detail

in the modelled tank, fill height etc.

The two-dimensional analysis is applicable to

most tank designs in view of the usually large

separation between the natural periods of the liquid

cargo in the longitudinal and in the transverse

directions of the tank. For some of the cases

where three-dimensional effects may be important,

the two-dimensional approach may need calibration

in the lines originally proposed in section 3.

The paper finally discusses different schemes of

excitation which can be used for a variety of stud-

ies.

Fig.13 Coupled analysis of sloshing and three

degrees of freedom ship motions for a pro-

ducts carrier ship (T=20, 5 sec, ƒÄ=2. 3 m)

Fig. 14 Computed time histories of ship motions

from a coupled analysis for the case shown

on figure 13

Fig.15 Coupled analysis of sloshing and three

degrees of freedom ship motions for a pro-

ducts carrier ship (T =7 . 6 sec, ƒÄ=2. 5 m)

Fig.16 Computed time histories of ship motions

from a coupled analysis for the case shown

on figure 15


Acknowledgements

The authors express their gratitude to Lloyd's

Register of Shipping for permission to publish this

paper and also acknowledge Mr J. K Miller's con-

tribution in fundamental areas of this work.

Notation

B: Breadth of tank

C : Sloshing induced moment on tank

D: Depth of tank

K: = (2 ƒÎ)-1 ln(ƒÓ(t)/ƒÓ(t+Ts)), roll decrement

L: Length of tank

OXY : Inertial frame of reference, Y positive up-

wards

T : Period of forced motion

Tn: Natural period of liquid

Ts: Natural rolling period of ship

g: Gravitational constant

h: Liquid height in tank

t: Time

εt : Phase angle between sloshing induced mo.

ment and roll displacement, negative for lag-

ging moment

ζ : Wave height amplitude

μα : =C/(ρgB3L), non-dimensional amplitude of

sloshing induced moment

ρ : Density

φ : Amplitude of roll motion

φ : rotational displacement in roll

φ : = (φ(t)+φ(t+Ts))/2, average roll displace-

ment in free rolling tests

ω : Frequency of excitation

References

1) Bass, R. L., Bowles, E. B. and Cox, P. A.:

'Liquid Dynamic Loads in LNG Cargo Tanks'

Trans. SNAME, Vol. 88, pp. 103•`126, 1980.

2) Yoshimura, N., Tanaka, T., Endo, S., Jibiki, Y.

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Sloshing in Arbitrary Shaped Tanks

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