Sloshing in Arbitrary Shaped Tanks

Transcription

1 246 (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 hydraulic 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 particular design, where tank dimensions and structural arrangements are fixed, the following global parameters can vary to affect the load that is transmitted to any point on the tank structure : Cargo : varying density, viscosity Filling Level : anything below say 97% could be * Lloyd's Register of Shipping 71 Fenchurch Street, London, UK considered a partial filling Position of tank in ship : longitudinal, transverseand 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, principal 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 withstand 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 motions to the tank which result in realistic and representative short-term and long-term maximum responses. Section 3 details a transient two-dimensional finite difference program used to predict

2 Sloshing in Arbitrary Shaped Tanks 247 fluid response to imparted excitation. The computed pressures are at present used in a static structural analysis which is based on a plastic collapse theory. However, as the procedure produces pressure 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 irregular 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 comparatively 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 excitation 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 computer 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 excitation 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 rotational has its maximum at the ship's natural period and an exponential decay at higher and lower periods. 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 applications 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 translational 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 period, Ts, and the liquid natural period, Tn, are very close (synchronism). Conversely, as the separation 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 account 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 required, 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 compared with the harmonic or irregular forms which would necessitate numerous tests or a very long simulation to establish worst conditions.

3 248 Journal of The Society of Naval Architects of Japan, Vol The Mathematical Model The fluid motions and exerted pressures from a given excitation of a partially filled tank are evaluated using a mathematical model of the two-dimensional 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 generalisation 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 experiments showed qualitative agreement, while very good agreement was observed on comparisons of free surface motions. The adopted mathematical model provides a transient solution by progressing in small time increments. 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 adequately 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 viscous 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, Fig. 1 Typical sloshing excitation time histories of : variation of Period and of Roll, Sway and Heave amplitudes (Tolling Against Waves' condition) 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 internal 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 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 inclusion 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 unrealistic 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. Fig. 2 Free surface realisations at one eighth of a period intervals for a shallow filled rectangular tank in harmonic roll motion

4 Sloshing in Arbitrary Shaped Tanks 249 Fig. 5 Experimental and computed pressures in Fig. 3 Free surface and velocity field realisations for a rectangular tank with internal structure, in harmonic roll, sway and heave motions Pascals at transducer locations R1, R2 and free surface height in metres at position HR : Roll, h/d=0. 15, T = sec, ƒ³= O. 1 rads 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. 6 Experimental and computed pressures in 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. averaging process over the free surface 'step' accounts 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 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 rectangular and of constant width and height. From sensitivity 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- of more complex boundary tion conditions. Fugures 5, 6 and 7 show typical comparisons between computed and experimentally measured pressures, in Pascals, and free surface elevation, in metres, for a 1 : 40 model of a prismatic tank in

5 250 Journal of The Society of Naval Architects of Japan, Vol. 158 Fig. 7 Experimental and computed pressures in 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= sec, ƒ³=0.25 rads. 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 stepped one. Impact pressures, as differentiated from the dynamic ones, pose a more subtle problem when comparing theory with experiment and when they must be applied to consider the adequacy of a design. When the time step was halved in the computation then the impulsive pressure was seen to exactly double while the dynamic component remained 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 subsequently 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 incompressible and the other with the slightly compressible 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 (fraction 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 impacted 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 reasons, 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 desirable to proceed with computations employing such small time steps. The problem remained, however, what to do with the relative nature of impact 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 prismatic 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

6 Sloshing in Arbitrary Shaped Tanks 251 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. Furthermore, when the inspection of fluid velocities indicates 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 developments presently taking place on analysis of structures under impulsive loads. A limitation of LR. FLUIDS is that it can model only two-dimensional problems. In the experiments, from which Figures 5-7 are drawn, an attempt 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 natural 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 motion 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 distinctly 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 magnitude 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 likelihood 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 computer eigenvalue analysis has been purposely developed 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 manner the maximum pressures obtained at the same excitation period, in which case if roll and pitch resonances are well separated then the three-dimensional 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 girder, for the evaluation of dynamic loads on supports of independent tanks, and, when the sloshing induced dynamic loads are of sufficiently large magnitude, 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.

7 252 Journal of The Society of Naval Architects of Japan, Vol. 158 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) 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 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 vertical 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 moment on a rectangular tank and on the prismatic tank of figure 43). These comparisons covered ranges of the following variables : period and amplitude of rotational excitation, fill height and position 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 sloshing 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.

8 Sloshing in Arbitrary Shaped Tanks Coupled Sloshing and Ship Motions In this mode of excitation the simulation proceeds in time by a parallel and coupled set of computations of ship motion equations and of the sloshing analysis. As the liquid cargo moves, it transmits a force and a moment on the tank and consequently onto the ship. These liquid induced loads are computed for every time step by an integration of the pressures around the tank boundary, 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 dependent ship hydrodynamic data, from say a strip theory analysis. The definition of excitation is then simply made by the specification of the incident wave's height and period. It is also worth noting here that the additional computational effort 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 subsequently extended to a model of three and to a Fig. 11 Experimental and computed non-dimensional 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, incorporating 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) 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 correction for the presence of the free surface is not applied in the three degree coupled analysis, because the underlining phenomenon is implicitly accounted by the communication of liquid induced loads to the ship motions equations. In the particular 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 excitation provide a simple and direct verification of the coupling method when compared with the traditional 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 effect. Figures 13 and 15 illustrate the coupled sloshing

9 254 Journal of The Society of Naval Architects of Japan, Vol.158 and three degrees of freedom ship motions analysis. The wave amplitude is comparable in these two cases, but in figure 13 the wave period corresponds to the ship's natural period while in figure 15 it corresponds to the liquid cargo's natural period. 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). Fig.13 Coupled analysis of sloshing and three degrees of freedom ship motions for a products carrier ship (T=20, 5 sec, Ċ=2. 3 m) Fig.16 Computed time histories of ship motions from a coupled analysis for the case shown on figure 15 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 represents a consistent and efficient method for the determination 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 Fig. 14 Computed time histories of ship motions from a coupled analysis for the case shown on figure CPU seconds per simulated period, in the Society's mainframe computer. The exact figure depends 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. Fig.15 Coupled analysis of sloshing and three degrees of freedom ship motions for a products carrier ship (T =7. 6 sec, Ċ=2. 5 m) The paper finally discusses different schemes of excitation which can be used for a variety of studies.

10 Sloshing in Arbitrary Shaped Tanks 255 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 contribution in fundamental areas of this work. B: Breadth of tank Notation C : Sloshing induced moment on tank D: Depth of tank K: = (2 Ĕ)-1 ln(ė(t)/ė(t+ts)), roll decrement L: Length of tank OXY : Inertial frame of reference, Y positive upwards 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 ment and roll displacement, negative for lagging moment 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, ) Yoshimura, N., Tanaka, T., Endo, S., Jibiki, Y. and Umekawa, N.: 'The Estimation of maximum Sloshing Pressure on Newly Designed Membrane LNG Tanks' Tsu Research Laboratories Report, Nippon Kokan K. K., Japan, ) Mikelis, N. E., Miller, J. K. and Taylor, K. V.: 'Sloshing in Partially Filled Liquid Tanks and its Effect on Ship Motions : Numerical Simulations and Experimental Verification' The Naval Architect, RINA October, ) Blixell, C. A.: 'Calculation of Ship Responses in Regular Waves by Strip Theory' RATAS Report No. 5116, Lloyd's Register of Shipping, ) Robinson, D. W.: 'Calculation of Non-Dimensional Variance of Wave Induced Responses' Development Unit Report No. 225, Lloyd's Register of Shipping, ) Robinson, D. W.: `Long-Term Predictions of Wave-Induced Ship Responses. Rev. 2' Development Unit Report No. 28. Lloyd's Register of Shipping, ) Price, W. G. and Bishop R. E. D.: 'Probabilistic Theory of Ship Dynamics' Publ. Chapman and Hall, London, ) Hirt, C. W., Nichols, B. D. and Romero, N. C. : `SOLA -A Numerical Solution Algorithm for Transient Fluid Flows' Los Alamos Scientific Laboratory, Report LA-5852, ) Welch, J. E., Harlow, F. H., Shannon, J. P. and Daly, B. J.: 'The MAC Method, a Computing Technique for Solving Viscous, Incompressible, Transient Fluid-Flow Problems Involving Free Surfaces' Los Alamos Scientific Laboratory, Report LA-3425, ) Navickas, J., Peck, J. C., Bass III, R. L., Bowles, E. B., Yoshimura, N. and Endo, S.: 'Sloshing of Fluids at High-Fill Levels in ClosedTanks' ASME Winter Meetings, Washington D. C., pp. 191,198, ) Bass, R. L., Bowles, E. B., Trudell, R. W., Navickas, J., Peck, J. C., Yoshimura, N., Endo, S. and Pots, B. F. M.: 'Modelling Criteria for Scaled LNG Sloshing Experiments' ASME, June, ) Nichols, B. D. and Hirt, C. W.: 'Numerical Simulation of Hydrodynamic Impact Loads on Cylinders' Nuclear Science and Engineering, Vol. 68, No. 143, pp , ) Arai, M.: 'Experimental and Numerical Study of Sloshing Pressure in Liquid Cargo Tanks' Journees Vibrations Chocs, 1984 ` 1985, Lyon. 14) Mikelis, N. E. and Journee, J. M. J.: 'Experimental and Numerical Simulations of Sloshing Behaviour in Liquid Cargo Tanks and of its Effect on Ship Motions' Intern. Conference on Numerical Methods for Transient and Coupled Problems, Pine Ridge Press, Venice, 1984.