Comparison of experimental and numerical sloshing loads in partially filled tanks

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1 Ships and Offshore Structures ISSN: (Print) X (Online) Journal homepage: Comparison of experimental and numerical sloshing loads in partially filled tanks S. Brizzolara, L. Savio, M. Viviani, Y. Chen, P. Temarel, N. Couty, S. Hoflack, L. Diebold, N. Moirod & A. Souto Iglesias To cite this article: S. Brizzolara, L. Savio, M. Viviani, Y. Chen, P. Temarel, N. Couty, S. Hoflack, L. Diebold, N. Moirod & A. Souto Iglesias (2011) Comparison of experimental and numerical sloshing loads in partially filled tanks, Ships and Offshore Structures, 6:1-2, 15-43, DOI: / To link to this article: Published online: 18 Feb Submit your article to this journal Article views: 499 View related articles Citing articles: 11 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 04 December 2017, At: 04:54

2 Ships and Offshore Structures Vol. 6, Nos. 1 2, 2011, Comparison of experimental and numerical sloshing loads in partially filled tanks S. Brizzolara a,l.savio a, M. Viviani a,y.chen b, P. Temarel b,n.couty c, S. Hoflack c, L. Diebold d,n.moirod d and A. Souto Iglesias e a Department of Naval Architecture and Marine Engineering, University of Genoa, Italy; b Ship Science, School of Engineering Sciences, University of Southampton, Southampton, UK; c Principia R.D., Nantes, France; d Bureau Veritas, Marine Division Research Department, Paris, France; e Naval Architecture Department (ETSIN), Technical University of Madrid (UPM), Madrid, Spain (Received 18 November 2009; final version received 10 August 2010) Sloshing describes the movement of liquids inside partially filled tanks, generating dynamic loads on the tank structure. The resulting impact pressures are of great importance in assessing structural strength, and their correct evaluation still represents a challenge for the designer due to the high level of nonlinearities involved, with complex free surface deformations, violent impact phenomena and influence of air trapping. In the present paper, a set of two-dimensional cases, for which experimental results are available, is considered to assess the merits and shortcomings of different numerical methods for sloshing evaluation, namely two commercial RANS solvers (FLOW-3D and LS-DYNA), and two academic software (Smoothed Particle Hydrodynamics and RANS). Impact pressures at various critical locations and global moment induced by water motion in a partially filled rectangular tank, subject to a simple harmonic rolling motion, are evaluated and predictions are compared with experimental measurements. Keywords: sloshing; CFD; RANS; SPH 1. Introduction Sloshing is a highly nonlinear movement of liquids inside partially filled tanks subject to oscillatory motions. This liquid movement generates dynamic loads on the tank structure, thus becoming a problem of importance in the design of marine structures in general and an especially important problem for some particular types of ships, such as LNG carriers (Tveitnes et al. 2004; Graczyk et al. 2007). In some cases, this water movement is used for damping ship motions (passive anti-roll tanks), especially for vessels with a low service speed (e.g. fishing vessels, supply vessels, and oceanographic and research ships) where active fin stabilisers would not produce a significant effect (Lloyd 1989). The sloshing problem has been investigated to a great extent in the last 50 years, with increasing levels of accuracy and computational efforts. First attempts were based on mechanical models of the phenomenon by adjusting the terms in the equation of motion (Graham and Rodriguez 1952; Lewison 1976). These types of techniques are used for cases where predictions need to be time-efficient, but not necessarily very accurate (Aliabadi et al. 2003). The second type of investigations solves a potential flow problem with a very sophisticated treatment of the free-surface boundary conditions (Faltinsen et al. 2005) that extends the classical linear wave theory by performing a multimodal analysis of the free-surface behaviour. This approach is very time-efficient and accurate for specific applications, but it does not allow to model overturning waves and may present problems when generic geometries and/or baffled tanks are considered. The third group of methods solves the nonlinear shallow-water equations (Stoker 1957) using different techniques (Verhagen and Van Wijngaarden 1965; Lee et al. 2002). The fourth group of methods used to deal with highly nonlinear free-surface problems is aimed at the numerical solution of the incompressible Navier Stokes equations. Frandsen (2004) solved the nonlinear potential flow problem with a finite difference method in a two-dimensional (2D) tank subjected to horizontal and vertical motions. He obtained very good results; however, this approach suffers from similar shortcomings as the aforementioned multimodal method. Celebi and Akyildiz (2002) solved the complete problem by using a finite difference scheme and a volume of fluid (VoF) formulation for tracking the free surface. Sames et al. (2002) presented results obtained from a commercial finite volume VOF method, applied to both rectangular and cylindrical tanks. Schellin et al. (2007) investigated the coupled ship/sloshing motions problem and obtained very promising results. In general, numerical techniques have significant problems when considering highly nonlinear waves and/or Corresponding author. viviani@dinav.unige.it ISSN: print / X online Copyright C 2011 Taylor & Francis DOI: /

3 16 S. Brizzolara et al. Figure 1. Tank geometry and position of the sensors. (This figure is available in colour online.) overturning waves, the effect of air cushions and fluid structure interactions. Considering the first problem, the smoothed particle hydrodynamics (SPH) meshless method appears to be a promising alternative to standard grid-based techniques, because of its intrinsic capability to capture surface deformations. There are relatively few publications on SPH applications to typical marine problems, with the work by Colagrossi and Landrini (2003) being one of the first applications in this field. In successive years, a number of applications were focussed on the assessment of slamming (Oger et al. 2006; Viviani et al. 2007a, b, 2009). Sloshing was considered by Souto Iglesias et al. (2006) and Delorme et al. (2009), through a comprehensive set of calculations, focusing on resultant global moment and its dependence on tank oscillating frequency. Nevertheless, problems related to the evaluation of local impact pressures are still considerable, with the presence of significant oscillations in the predictions. The activity described in the present paper was carried out in the framework of the EU-funded MARSTRUCT Network of Excellence. It encompasses three 2D (or infinite length) cases of a partially filled rectangular tank subject to an oscillatory rolling motion. Different motion periods and water levels were considered. Impact pressures and global moments were evaluated and measured. For the tests, experimental measurements were carried out by the Model Basin Research Group (CEHINAV) of the Naval Architecture Department (ETSIN) of the Technical University of Madrid (UPM), in the context of a comprehensive analysis of the sloshing phenomenon, as reported by Delorme and Souto Iglesias (2007, 2008). For the numerical evaluation, a range of numerical methods were applied to assess their merits and shortcomings. These are: a RANS (Reynolds averaged Navier Stokes equations) code developed by UoS (University of Southampton), hereafter referred to as RANSE-UoS; two commercial software for the solution of RANS equations, namely FLOW-3D applied by BV (Bureau Veritas) and LS-DYNA applied by PRI (Principia), hereafter referred to as RANSE-BV and RANSE-PRI, respectively; an SPH code developed by DINAV (University of Genoa), hereafter referred to as SPH. 2. Experimental set-up The experimental tests, used for benchmarking the various numerical methods, were performed by CEHINAV-ETSIN- UPM, as reported in Delorme and Souto Iglesias (2007, 2008). A rectangular tank having dimensions (in centimetres) as shown in Figure 1 was selected for the numerical investigations. The tank is cylindrical, and the dimension perpendicular to the plane shown in the figure is 62 mm. A sinusoidal rolling motion was imposed during the experiments, with an amplitude of 4 in all cases and different periods. The rolling axis was located 18.4 cm above the bottom line and the tank was fitted with a series of sensors, as indicated in Figure 1. The sensors are BTE6,000 Flush Mount, with a 500 mbar range. During the experiments, pressures corresponding to the two most critical positions were recorded. In parallel to pressure measurements, global torque measurements were also conducted. In particular, torque time history was measured during experiments for the tank partially filled and empty. Subsequently, the first harmonic Table 1. Filling levels and corresponding natural periods considered. Level/tank Natural Level symbol Level (cm) height (%) period T 0 (s) A B

4 Ships and Offshore Structures 17 Table 2. Description of test cases. Table 3. Oscillating moment first harmonic values. Case Level/tank height (%) Period (s) Period/T 0 ( ) 1 A (18.3) B (43.7) B (43.7) Case M 0 (Nm) M 0 (Nm/m) φ(deg) of the moment response of the liquid with respect to the tank rolling centre was obtained for every case through post processing the data. The two filling levels considered in the present analysis, together with the corresponding natural period of oscillation T 0, are shown in Table 1. For both filling levels, experiments were carried out with the tank rolling at the resonance period (cases 1 and 3 for levels A and B, respectively) in order to evaluate the capability of the prediction methods used to capture large free surface deformations. Moreover, for case B a lower oscillating period (90% of the resonance period) was also considered (case 2), in which a marked beating phenomenon was observed. The main characteristics of the cases analysed are briefly summarised in Table 2. For case 1, pressure sensors are located at positions 1 and 6, and impacts were recorded on the sides for sensor 1, as shown in Figure 2. For cases 2 and 3, pressure sensors are located at positions 3 and 6. Case 2, as anticipated, is interesting due to the presence of beating type kinematics and pressures with peak values oscillating, as shown in Figure 3 for sensor 6. Regarding case 3, pressure peaks were recorded for both locations as shown in Figure 4, with impact events at the tank top (sensor 3) and pressure rises due to the incoming wave and water fall after impact at sensor 6. The resulting first harmonic of the liquid moment is reported in Table 3 for all cases, where M 0 is the first harmonic amplitude and φ is the phase lag with respect to oscillating motion. Equation (1) represents the time history, considering motion of period T : ( ) 2π M = M 0 sin T t φ. (1) Note that M 0, the amplitude of the first harmonic, is reported both for the real experimental case (tank with 62 mm third dimension) and for the equivalent 2D configuration, for which moment per unit length is given. The rolling motion during the experiments was a pure sinusoidal motion, except at the very beginning of the experiment to avoid infinite accelerations. A proper uncertainty analysis has not been conducted yet since it is hard to find a consistent approach that is Figure 2. Test case 1: level A, T = T 0, pressure at sensor 1.

5 18 S. Brizzolara et al. Figure 3. Test case 2: level B, T = 0.9 T 0, pressure at sensor 6. suitable for such cases. This is due to the strongly chaotic character of the pressure peaks, as discussed by Delorme et al. (2009), where the initial steps to a possible uncertainty analysis are discussed. In the present work, a statistical analysis has been performed, evaluating mean value, standard deviation and maximum deviations from the mean values, for cases 1 and 3. This analysis has been carried out in order to provide an insight into the variation of pressure Figure 4. Test case 3: level B, T = T 0, pressure at sensors 3 6. (This figure is available in colour online.)

6 Ships and Offshore Structures 19 Table 4. Details of method and idealisation: test case 1. Participant Method Idealisation details UoS RANSE (RANSE-UoS) grid DINAV SPH 37,138 real particles 1876 virtual particles PRI LS-DYNA (RANSE-PRI) grid BV FLOW 3D (RANSE-BV) grid peak values, for both numerical methods and experimental measurements. Finally, tank walls are made of 10 mm thick Plexiglass (E = 2.5 GPa). The lowest natural frequencies of the tank (2 khz) are above the frequencies found in the pressure registers and thus phenomena connected to hydroelasticity should not affect them, as reported for example in Lugni et al. (2006). 3. Description of methods Particulars of methods and idealisations used for modelling the sloshing phenomenon are summarised in Tables 4 and 5 for test case 1 and test cases 2 and 3, respectively. Calculations by BV are three-dimensional; hence, the experimental set-up was effectively reproduced. All other methods analysed the 2D problem. Only UoS considered air (and pressure variations within air), while in all other cases only water was considered (in VoF methods, air is assumed as void); none of the methods simulate air compressibility, and this is discussed later in this section. Real particles are used to represent the liquid (water in this case) for SPH, while virtual particles are used to represent the moving tank. A particle diameter of 1.5 mm has been adopted for either situation. Effective time histories were applied by both BV (case 1 only) and PRI, while pure harmonic oscillations were applied by BV (cases 2 and 3), UoS and DINAV. Calculations showed that this has not a significant impact once the results are stabilised after the initial transient. Finally, tank walls are considered rigid in all numerical calculations. This is in line with the experimental set-up, in which tank walls, as noted in the previous section, are sufficiently stiff to be considered rigid from a practical point of view. Table 5. Details of the method and idealisation: test cases 2 and 3. Participant Method Idealisation details UoS RANS E (RANSE-UoS) grid DINAV SPH 88,652 real particles 1876 virtual particles PRI LS-DYNA (RANSE-PRI) grid BV FLOW 3D (RANSE-BV) grid Descriptions of the methods used are provided in Sections More details are provided for the RANS method of UoS and the SPH method of DINAV as they are in-house developed. The description of the other two approaches is limited to the essentials because commercial codes are adopted RANSE approach developed by UoS (RANSE-UoS) The level-set formulation in a generalised curvilinear coordinates system, as presented by Price and Chen (2006), is used to simulate the free-surface waves generated by liquid sloshing in a container. The RANS equations are modified to account for variable density and viscosity for the twofluid flows (i.e. water air). By computing the flow fields in both the water and air regions, the location and transport of free surface inside a tank is implicitly captured. A brief description of the numerical method is given below. More details can be found in Chen et al. (2009a and 2009b). To construct an effective scheme to solve the incompressible two-fluid flow system, an artificial compressibility technique developed by Chorin (1967) is introduced by adding the pressure derivative term, with respect to the pseudo-time τ, to the continuity equation. The governing equations of the incompressible, immiscible two-fluid system in the Cartesian coordinate system are as follows: p β τ + u j x j = 0, (2) ρ u i t + ρ (u iu j + δ ij p) µ x j Re x j ( ui + u ) j + ρg i x j x i Fn + ρf 2 i = 0, (3) where a conventional Cartesian tensor notation is used to sum repeated indices. The spatial coordinates x i, velocity components u i and projection components of the gravitational acceleration in the axis directions g i, respectively, have been non-dimensionalised for each specific problem in terms of a characteristic length L, a characteristic velocity U 0 and gravitational acceleration g. The fluid density ρ and viscosity µ are made non-dimensional by the equivalent water values ρ w and µ w, respectively; the time t and the pressure p by L/U 0 and ρ w U0 2, respectively. In Equation (3), Re and Fn represent Reynolds and Froude numbers, respectively, which are defined as Re = LU 0 /ν w,fn= U 0 / Lg. (4) Except for the gravitational force, the external forces include the translational and rotational inertia forces; hence,

7 20 S. Brizzolara et al. f i takes the following form: and vectors q, Q, F j,f v j and S are expressed as dω j f i = a i + ε ij k x k + ε ij k ε klm ω j ω l x m + 2ε ij k ω j u k, dt (5) where a i represents the translational acceleration components and ω i represents the rotational angular velocity components; ε ij k denotes the Levi-Civita symbol with repeating subscripts indicating summation. The free surface is defined as the zero-level set of a level-set function φ initialised as a signed distance function from the interface. In air, φ is set to a positive value and in water to a negative value as defined by φ(x i,t) > 0in air, φ(x i,t) = 0 on surface, (6) φ(x i,t) < 0 in water. Differentiating φ = 0 with respect to t, a transport equation is derived to describe the free-surface motion in the form φ t + u i φ x i = 0, (7) where u i is the local fluid velocity and, at any time, moving the interface is equivalent to updating φ by solving Equation (7). Because of the sharp variations of fluid properties at the free surface, there is a need to introduce a region of some finite thickness 2ε, over which a smooth but rapid change of density and viscosity occurs across the interface. First, a smoothed Heaviside function H ε (x) is defined, satisfying 1 if x>ε, H ε (x) = 0 if x< ε, 0.5(x + ε)/ε + 0.5sin(πx/ε)/π otherwise, (8) where ε is half the finite thickness of the interface in which the density and viscosity change. Using the above function, the corresponding smoothed density function ρ and smoothed viscosity function µ can be defined as ρ(φ) = 1 + (ρ a /ρ w 1)H ε (φ), (9) µ(φ) = 1 + (µ a /µ w 1)H ε (φ). (10) In terms of generalised coordinates, Equations (2) and (3) can be rewritten in a vector form as q Ɣ τ τ + Ɣ Q t + F j t ξ j + Fν j ξ j = S. (11) Here Ɣ τ and Ɣ t represent the two diagonal matrices, Ɣ τ = diag[1/β, 0, 0, 0], Ɣ t = diag[0, 1, 1, 1], q =J 1 (p, u, v, w) T, Q=J 1 (p/β,ρu,ρv,ρw) T, ρu j ρu 1 U j + ξj p x 1 F j = J 1 ρu 2 U j + ξj p, x 2 ρu 3 U j + ξj p x 3 F ν j = J 1 µ ξ j Re x m + ξ k y ( 0, ξ k u + ξ k u m x m ξ k x u m, ξ k w + ξ k ξ k x m ξ k z ) u T m, ξ k S = J 1 ρ ( 0,f 1 + g 1 /F n 2,f 2 + g 2 /F n 2, f 3 + g 3 /F n 2) T. ξ k, ξ k x m v ξ k (12) Here J = (ξ,η,ζ) / (x,y,z) is the Jacobian of the transformation, and the contra-variant velocity component U j is defined as U j = ξ j x m u m. (13) By approximating the pseudo-time derivative by an implicit Euler back forward difference formula and the time derivative by a second-order, three-point, backwarddifference implicit formula in Equation (11), one obtains Ɣ τ q m+1,n+1 q m,n+1 τ + Ɣ t 1.5Q m+1,n+1 2Q n + 0.5Q n 1 t + δ ξ F m+1,n+1 = 0. (14) Here, the superscript n denotes the nth physical time level, the superscript m denotes the level of the sub-iteration and δ ξ represents a spatial difference. For the sake of simplicity, only the convective flux derivative in one direction is presented, and the viscous and source terms in Equation (11) are omitted. For example, F denotes the convective flux F 1 in the ξ-direction. After linearising terms at the (m + 1)th time level and involving some simple algebraic manipulation, Equation (14) becomes ( 1 τ Ɣ τ ) t Ɣ t + A q δ ξ q + δ ξ F m,n+1 = r m,n+1, (15)

8 Ships and Offshore Structures 21 where A q = F q, rm,n+1 = 1.5Qm,n+1 2Q n + 0.5Q n 1 t and q = q m+1,n+1 q m,n+1. In this study, Roe s approximate Riemann solver, introduced by Roe (1981), is employed to calculate the numerical fluxes. Let us define Ɣ d = 1 τ Ɣ τ t Ɣ t. The flux Jacobian matrix and the Roe numerical flux expression in terms of vector q are, respectively, given by a q = Ɣ 1 d A q, F j+ 1 2 = 1 2 (F L + F R ) 1 2 ˆƔ d âq q. (16) Here F j+(1/2) represents the numerical flux at the cell interface and the tilde over each term implies they are evaluated using the Roe-averaged variables. Even if φ is initialised as a signed distance from a wave front, the level-set function no longer remains a distance function at later times. An iterative procedure, called re-initialisation, is used at each time step by solving a Hamilton Jacobi equation to ensure φ satisfies φ = 1, similar to Sussman et al. (1994). During the re-initialisation exercise, a process presented by Sussman et al. (1998) is introduced to preserve the fluid volume in each cell and improve the accuracy of solution to Equation (7). In the present investigation, no special treatment is required for the free surface as a two-fluid approach is used to solve the RANS equations in both water and air regions in a unified manner and the interface is only treated as a shift in fluid properties. The solid wall boundary condition is imposed by vanishing the normal velocity to the wall or setting the components of velocity on the wall to zero. The wall pressure is obtained by projecting the momentum equation along the normal to the wall. In addition to data provided in Tables 4 and 5, other computational conditions are listed in Table 6, indicating the values of half the finite thickness of the interface in which density and viscosity change ( z represents the vertical distance between two grid nodes and ε represents the interface half thickness) and time-step increment used in this study. In addition to data provided in the previous paragraph, other computational conditions include the half thickness of the interface ε, over which the density and viscosity change smoothly, and time-step increment t. In present work, these two parameters were set at ε = 1.5 z (where z represents the vertical distance between two grid nodes) and t = s for all three test cases. Considering calculation set-up (grid resolution, time step size and the interface thickness), the dependency of numerical solution on these parameters was systemically investigated, allowing to consider the one presented as the optimal choice. The detailed discussions can be found in Chen et al. (2009a). Finally, torque calculations for this method are provided for case 1 only LS-DYNA approach adopted by PRI (RANSE-PRI) LS-DYNA is an explicit finite element code solving the mechanics equations (here corresponding more precisely to the RANSE with Reynolds stresses neglected). The fluid domain is modelled using a multi-materials Eulerian formulation. For the advection of the variables at the integration points, a Van Leer scheme (second-order precision) is used and a half-index shift (HIS) method for the nodal variables. Each material is characterised by a volume fraction inside each element. It interacts in the calculation of the composite pressure for the partially filled cells within a VOF method. The dispersion of the volume fraction for these particular cells is limited with an interface reconstruction method (boundary between materials modelled as a plane). Air is modelled as void. The problem is addressed in two dimensions, although under-integrated 8-noded elements (one layer in the direction normal to the plane) are used to model the fluid domain in practice. The size of the fluid cells is of 10 mm edge length, chosen close to the sensor size. The mesh, using solid elements, can be seen in Figure 5 for case 1. Cases 2 and 3 are idealised in a similar manner, except for different level of water in accordance with the Table 6. Summary of results for case 1: level A, T = T 0, sensor 1. Experiments SPH RANSE-BV RANSE-UoS RANSE-PRI Mean (Pa) Diff (%) RMS (%) max (%) min (%)

9 22 S. Brizzolara et al. Figure 5. Eulerian mesh for case 1 (water in dark, void in grey). (This figure is available in colour online.) test set-up. The behaviour of water is modelled with a polynomial equation of state, and no viscosity is defined. The fluid domain is surrounded with one layer of rigid elements, shown in grey in Figure 5, modelling the wall and preventing normal flow. The motion, as given by the roll time histories, is imposed to this rigid part. Subsequently, the rigid movement of the Eulerian grid (following the walls) is achieved by forcing the grid to follow the movement of three nodes of the rigid part (Aquelet et al. 2003). The time scheme is based on the finite difference method, which is conditionally stable. Thus, the time step is linked to the shortest duration for an acoustic wave to cross any element of the model (fluid or solid), close to 6 µs here. Pressure histories are directly post-treated in solid elements where sensors are located Smoothed particle hydrodynamics method, developed by DINAV (SPH) The SPH method used for this study has been developed by the authors following the methods presented by Monaghan (1994) and Liu and Liu (2003), which showed good ability in simulating the kinematics of hydrodynamic flows. For the derivation of fluid pressure on rigid boundaries, an approach similar to that presented by Oger et al. (2006) has been adopted. The present method has already been applied, with satisfactory outcomes, to slamming-related problems, as presented by Viviani et al. (2009). The foundation of the SPH method is the integral representation of a field function (e.g. speed, pressure or density) in the continuum domain, through its Kernel approximation: f (x) = f (x ) W(x x,h) dx. (17) The kernel function W(x x,h), which ideally should be a delta function, is approximated by a proper analytical function in a subdomain whose dimensions depend on the function itself adopted. Among possible formulations for the kernel functions, Viviani et al. (2009) tested both the standard Gaussian function (given by Equation (18)) and the piecewise cubic spline function (given by Equation (19)), as defined by Monaghan (1992) and Monaghan (1994), for validation with no significant differences in the results. W(r, h) = α d e r2, r = x x, (18) 2/ 3 r / 2 r 3 r 1 W(r, h) = α d 1/ 6 (2 r) 3 1 r 2. (19) 0 r>2 The final choice of the Gaussian formulation, adopted for the remaining the calculations in this paper, is because of its high numerical stability, being the function particularly smooth. To solve the problem numerically, the continuum is discretised in a number of particles, shown in Figure 6, each one representing a finite volume of fluid, so that integrals (17) are substituted by summations: f (x i ) = N f (x j ) W(x i x j,h ij ) V j, (20) j=1

10 Ships and Offshore Structures 23 where the strain rate tensor ε is defined as ( v ε lm m ) = + vl 2 x l x m 3 ( v) δlm (26) Figure 6. particles. kh Discrete representation of the continuum by smoothed where V j = m j /ρ j is the volume of the fluid represented by the particle j. In the framework of this discrete representation, the SPH approximation results in a linear and commutative operator. Exploiting these important properties, the gradient of a field function can be expressed as f (x i ) = 1 N [ m j f (xj ) f (x i ) ] i W ij, ρ i j=1 in which the gradient of the kernel function is i W ij = x i x j r ij W ij r ij = x ij r ij (21) W ij r ij. (22) Using these mathematical tools, it is possible to derive the SPH approximation of the main equations which govern the fluid motion, adopting a Lagrangian approach to follow in its motion each particle under internal forces between nearby interacting particles and external mass forces or boundary forces. Internal forces derived from the momentum and mass continuity equations (i.e. Navier Stokes equations): Dρ Dt Dv Dt = ρ v, (23) = g + 1 σ. (24) ρ In the mass continuity Equation (23), the fluid is considered slightly compressible in the whole domain of interest. This requires verifying that the value assumed by the modulus of the speed vector v is well below the limit of Mach < 0.1. In the momentum equation, σ is the generalised normal and stress tensor, defined as σ = p δ + τ = p δ + µ ε, (25) and δ is the Kronecker delta function (δ lm = 1, l = m; δ lm = 0 elsewhere). The smoothed particles discrete approximation of the conservation of mass can take several forms. The form adopted in this study is defined with reference to the particle i, which is imposed, and the generic particle j lying in its support domain, namely Dρ i Dt = N m j ν ij W ij, v ij = (v i v j ). j=1 (27) The Navier Stokes equations are approximated in the following way: Dv i Dt = N j=1 ( ) σ i + σ j m j i W ij, (28) ρ i ρ j where the generalised stress strain tensor is calculated as follows: σ i = p i δ + µ i ε i, (29) ε lm i = N j=1 2 3 m j v m W ij ij ρ j x l i N m j j=1 + N j=1 ρ j v ij i W ij m j v l W ij ij ρ j x m i δ lm. (30) In the present case, the viscous term is neglected; hence, Euler equations have been adopted, as already used successfully for slamming calculations by Viviani et al. (2009) and for sloshing calculations by Souto-Iglesias et al. (2006). To close the problem, the equation of state of the weakly compressible fluid is defined according to the following relation (i.e. Equation (31)), and it is solved to find the pressure on any particle whose density was calculated by means of Equation (27), namely p = c2 0 ρ 0 γ [( ρ ρ 0 ) γ 1], (31) where ρ 0 is reference density (1000 kg/m 3 ) and ρ is density of each particle. In Equation (31), the exponent constant γ is taken equal to 7 in accordance with Batchelor (1967), providing satisfactory results for various SPH applications. The value of the sound speed c 0 cannot, in general, be set to its effective value for practical reasons (i.e. time steps become too small). Thus, it is usually set in order to limit

11 24 S. Brizzolara et al. the Mach number to a value below 0.1 (Monaghan 1994) and, consequently, density variations in the incompressible flow to acceptable values. In particular, in accordance with previous works (e.g. Viviani et al. 2009), sound speed is kept to rather low values, but contemporarily limiting density variations to values below 1%. This, on the basis of previous experiences, allows to obtain a good compromise between computational times and pressure peak evaluation. In particular, lower values of sound speed would result in lower pressure peaks, and thus they must be avoided. On the contrary, higher values would not modify considerably pressure peak levels, introducing only a lower time step and thus a longer computational time. Density variation limitation can be easily achieved performing a preliminary calculation with a sound speed set on the basis of past experience, whose value is subsequently refined in order to fulfil conditions for the problem investigated. For all cases considered, a value of 50 m/s was sufficient to achieve this condition. Accordingly, a time step of 10 5 s was adopted, which is about half that required by the simple application of the Courant condition. Since SPH can be affected by lack of stability, various authors developed different methods which can help in reducing this problem, such as artificial viscosity and XSPH (Monaghan 1992). In the present calculations, on the basis of past experience (Viviani et al. 2007a,b; Viviani et al. 2009), it was decided not to opt for XSPH, and to consider an artificial viscosity term as shown in Equations (32): αc ij φ ij + βφij 2 ij = ρ ij 0 φ ij = h ij v ij x ij x ij +(εh ij ), 2 v ij x ij < 0, v ij x ij 0, (32) where c, ρ, h are sound speed, density, smoothing length and kernel function, respectively, and the subscript ij represents a mean value. A preliminary analysis was carried out to obtain the best set-up of parameters α, β and ε. Subsequently, they were all set to the low value of 0.01, thus reducing considerably artificial viscosity contribution. With reference to boundary treatment, repulsive forces are used as suggested by Monaghan (1994), adopting a force which is dependent on the inverse of the distance between fluid and boundary particles, according to the formulation of Lennard-Jones for molecular force, as described by Equation (33): (( f (r) = D r r0 ) p1 ( r0 ) p2 ) } r when r<r 2 0, r r (33) f (r) = 0 elsewhere, where p 1 = 12 and p 2 = 6, r 0 is the cutting-off distance, approximately equal to the smoothing length, and r is distance between real and boundary particles. It should be noted that in previous work on slamming (Viviani et al. 2009), a modified version of this approach was adopted, considering normal forces to the boundary instead of repulsive forces. This allowed to smooth pressure and density oscillations in the particles close to the boundary, deriving from the variation of the tangential force component. In the present paper, after some preliminary comparisons, the normal forces approach was not adopted, since it does not provide significant improvements in terms of pressure fluctuations for this sloshing problem, whilst requiring longer calculation times. Pressures on the boundary are calculated by means of an approach similar to that presented by Oger et al. (2006). The approach adopted is somehow simplified, since it evaluates the mean of pressure values on real particles in proximity of each boundary particle (namely in a rectangular region with width and height parallel and perpendicular to boundary surface, respectively). Nevertheless, the results show that it is quite effective. The size of the region adopted in the present paper, for averaging the pressure, extends 4 10 smoothing lengths in width and height, respectively. The time integration required by Equation (28), providing the velocity of each particle at each time step, is performed by a leapfrog scheme, schematically represented in the flow chart of Figure 7. At each time step, the particle position is updated (including that of the virtual particles in the moving boundary), then new density for each particle is calculated by means of Equation (27). At this point, by using the equation of state (31), it is possible to derive the pressure of each particle. These, together with Equation (30), are used to define the components of the generalised tensor σ in the Navier Stokes equations (28). To complete the forces acting on each particle, the external forces must be evaluated, namely the gravity and body forces, the repulsive forces from virtual particles and hence the artificial viscosity terms from Equation (32). At this stage, the righthand side of Equation (28) is known and can be integrated. The calculation procedure, repeated at each time step, is terminated with the evaluation of the new position of the fluid and virtual particles. According to the leapfrog scheme of integration, the density and speed of internal particles are updated at each half time step during each integration loop. As it is well known, one of the main shortcomings of SPH is the presence of different calibration/correction parameters, which can significantly affect calculation results. One of the aims of this paper, as regards SPH, was to obtain a satisfactory set-up procedure, whose applicability may be considered general and not specific for a single case. With this aim, a preliminary activity was performed considering three parameters which can affect SPH results, i.e. time step, artificial viscosity and particle size. As regards artificial viscosity parameters, a series of calculations with different settings has been carried out, covering the range of α. and β. from 0.3 to 0.01; in particular,

12 Ships and Offshore Structures 25 Figure 7. High-level flow chart of the time marching solution scheme. the values of the two parameters have always been kept equal to each other, and it was decided to modify the value of ε by the same percentage, starting from a value equal to 0.1 in correspondence with the maximum value of α and β. A similar analysis was carried out previously for slamming cases (see Viviani et al. 2009). In this case of sloshing calculations, satisfactory results were obtained in correspondence to values of α and β lower than 0.1, while for higher values calculations resulted clearly overdamped, without capturing any impact phenomenon. In particular, it was found that best results were obtained with values of α and β set to 0.05 and 0.01, and it was chosen to adopt the latter, which corresponds to a low artificial viscosity. As regards time step, the previous analysis was carried out adopting a value of 10 5 s. This is lower than the required time step for satisfaction of Courant condition, and in line with previous experiences on slamming calculations (Viviani et al. 2009). A further analysis was carried out, exploring the effect of a lower time step, but this resulted in an overdamping of Figure 8. Test case 1: level A, T = T 0, sensor 1 pressure time history up to t = 10 s. (This figure is available in colour online.)

13 26 S. Brizzolara et al. Figure 9. Test case 1: level A, T = T 0, sensor 1 pressure time history (t = s). (This figure is available in colour online.) the phenomenon and in an unsuccessful capturing of pressures and kinematics. As an example, this did not allow to predict sloshing on the tank top in correspondence with case 3. On the contrary, the application of higher time steps resulted in calculations divergence due to too high spurious density (and thus pressure) variations, probably due to boundary treatment adopted. It was therefore decided to keep 10 5 s as time step. Figure 10. Test case 1: level A, T = T 0, sensor 6 pressure time history up to t = 30 s. (This figure is available in colour online.)

14 Ships and Offshore Structures 27 Figure 11. Test case 1: level A, T = T 0, global torque time history up to t = 25 s. (This figure is available in colour online.) Finally, particle size was also analysed; this parameter is very important, since it is obviously linked to total particle number and, in its turn, to computational time. In particular, three rather different particle sizes have been considered (i.e. 1.5, 3 and 6 mm). The increase in the particle size, as expected, resulted in a progressively worse capturing of kinematics, due to the lower resolution. In particular, in correspondence with the largest size, it prevented completely to capture impacts on the tank top. The intermediate set-up allowed to capture impacts at tank top; however, it provided a worse evaluation of spray after impact and of free-surface shape. Moreover, considering pressure results, adoption of the intermediate particle size resulted in anomalous highfrequency oscillations. This problem is probably due to the lower resolution of this set-up, which tends to amplify pressure waves. As a consequence, the lower particle size was adopted in the present work. The set-up chosen was considered satisfactory and applied to all cases, demonstrating to have a broader validity, not limited to the case used for parameters calibration Flow-3D approach, adopted by BV (RANSE-BV) FLOW-3D solves the transient Navier Stokes equations (without neglecting viscosity) by a finite volume/finite difference method in a fixed Eulerian rectangular grid. One of its distinctive features is the Fractional Area Volume Obstacle Representation (FAVOR) technique, which allows for the definition of solid boundaries within the Eulerian grid. Using such a technique, definition of boundaries and obstacles is carried out independent of the grid generation. The one-phase flow option was used for this calculation, considering air as void. Boundary condition imposed at tank boundaries is the no-slip condition. Regarding time variations, the real time roll series was used as input motion for the first case shown in Table 2. On the contrary, for cases 2 and 3 harmonic excitations were adopted for the sake of simplicity, since it was found that differences were not important. Finally, the x-direction was also considered (thus having a 3D case), with a depth equal to 6.2 cm, as utilised in the experiments Compressibility and air mixing As it is well known, the effect of dispersed bubbles or entrapped air in a two-fluid sloshing flow is important when the excited surface wave impacts on the sides or roof of a container. The presence of air as a trapped bubble or dispersed air reduces the density of water and dramatically changes acoustic properties in the water. The local pressures are then influenced by air compressibility. As anticipated, none of the methods adopted in the present study allows to consider these effects (considering air as void in most of the cases or incompressible in the case of RANSE-UoS).

15 28 S. Brizzolara et al. Figure 12. Kinematics capturing with SPH: (a) overturning wave and (b) flow rise after impact. (This figure is available in colour online.)

16 Ships and Offshore Structures 29 Figure 13. Kinematics capturing with LS-DYNA (RANSE-PRI): (a) overturning wave and (b) flow rise after impact. (This figure is available in colour online.) If the two cases in which violent phenomena occur (cases 1 and 3) are considered, the following considerations may arise. As regards case 1, from the analysis of experimental results, it can be seen that in correspondence with some impact events (e.g. first impact), aerated impacts may have occurred during experiments. This could have a relevant effect on the maximum pressure peak (see e.g. Godderidge et al. 2009). Nonetheless, for the type of comparison that has been performed, covering some consecutive impact events without focusing on the detailed physics of a specific impact, taking into account that factor was not considered necessary. Considering numerical results, mean values of pressure peaks are captured correctly (with rather small deviations) by almost all methods, as reported in Section 4.1, thus evidencing a low effect of air cushions. Considering case 3, experiments evidenced that impacts presented no gas entrapment at the tank top; thus, the nature of peak pressure is purely impulsive and compressibility is not an issue at sensor 3; this observation is partially confirmed by the numerical calculations, which present peak values in the range of the experimental calculations (even if with a larger scatter). On the contrary, computed values in correspondence with sensor 6 at water level at rest are all in rather good agreement, but tend to overestimate experimental pressure peak value, evidencing a possible cushion effect, even if this cannot be identified clearly from experiments. In test case 2, impacts were not recorded and pressure at the sensor presented mainly a hydrostatic character; thus, compressibility is not important. Figure 14. Kinematics capturing of the overturning wave with flow-3d (RANSE-BV). (This figure is available in colour online.) Figure 15. Kinematics capturing of the overturning wave with RANSE-UoS. (This figure is available in colour online.)

17 30 S. Brizzolara et al. Figure 16. Kinematics of the overturning wave at two successive periods during experiments. (This figure is available in colour online.) Some research work has been done by authors to address compressibility issues (Chen et al. 2009b) adopting RANSE-UoS and considering test case 3. Results obtained show a good agreement between calculated and measured impact pressures at sensor 3, when the water impacts the roof of the tank. For this filling level, the calculated pressure signal does not show any oscillations, which confirms that cavities are not formed at the upper corner of the tank during impact events. The first and second peaks of the impact pressure at sensor 6 are qualitatively captured and a better agreement is found than that obtained by the incompressible two-fluid flow model used in this investigation; nevertheless, considering peak values, they are still considerably overestimated. Reasons for this will be investigated in future work. Regarding water/air mixing, this can occur when in violent sloshing flow the wave front is in the process of breaking before it hits the tank wall. The unique method Figure 17. Test case 2: level B, T = 0.9 T 0, sensor 6 pressure time history up to t = 16 s. (This figure is available in colour online.)

18 Ships and Offshore Structures 31 Figure 18. Test case 2: level B, T = 0.9 T 0, sensor 6 pressure time history (t = s). (This figure is available in colour online.) considering a bi-fluid flow adopted in this investigation (RANSE-UoS) assumes air and liquid phase immiscible. In general, it is not practical to use a mesh with its size as small as tiny bubbles. As an alternative, a compressible two-phase flow model assuming a homogeneous mixture of liquid and gas may be needed to analyse this problem (Bredmose et al. 2009; Dias et al. 2010). Nevertheless, this was beyond the scope of this work, in which a more general comparison of different approaches is carried out, considering different cases without focusing on a specific problem. 4. Results and discussion 4.1. Test case 1: water level A, T = T 0 In general, this is the case for which the most comprehensive analysis was carried out. In particular, as discussed above, Figure 19. Test case 2: level B, T = 0.9 T 0, global torque time history up to t = 20 s. (This figure is available in colour online.)

19 32 S. Brizzolara et al. Figure 20. Kinematics capturing with SPH: (a) maximum peak and (b) minimum peak. (This figure is available in colour online.)

20 Ships and Offshore Structures 33 Figure 21. Kinematics capturing with LS-DYNA (RANSE-PRI): (a) maximum peak and (b) minimum peak. (This figure is available in colour online.) a relatively large number of computations was performed using SPH to analyse the effect of some parameters, and in particular of artificial viscosity. Results of this analysis are not included in this paper, and only the final set-up results are presented (artificial viscosity parameters set to 0.01). As mentioned earlier, the influence of artificial viscosity is very pronounced, and in particular higher values result in an overdamping of all phenomena with lower pressure peaks (with impact pressures almost vanishing when values of α and β in artificial viscosity are set equal to 0.3) and without overturning waves. Predicted and measured pressures at sensor 1 are shown in Figures 8 and 9, the first 10 s of the simulation/experiment in Figure 8 and the remainder up to 30 s in Figure 9. In general, order of magnitude of pressure peaks is captured by all methods applied. In all predictions, the first peak is captured with sufficient accuracy (considering mean values in the whole time history). The second peak is also captured reasonably well, even though the accuracy of the amplitude predicted is not as good as that of the first peak. SPH calculations provide a higher fourth peak, similar to the magnitudes obtained by RANSE-PRI later on in the simulation (see Figure 9). RANSE-UoS results provide a peak which has the correct order of magnitude but is narrower in form. In general, predictions using RANSE-UoS and RANSE-BV appear to be more stable, whilst RANSE-PRI and SPH results contain more oscillations. The extended simulation, shown in Figure 9, contains predictions using the RANSE-BV and RANSE-PRI methods. Both methods, as can be seen, still provide accurate results. Nevertheless, RANSE-BV calculations appear to be more robust (with less oscillations) and closer to experimental measurements. Calculations with SPH were not extended above 10 s due to the long calculation times involved. Instead, attention was paid to calibrating of various parameters, as explained above. Once a satisfactory set-up was obtained (in terms of artificial viscosity parameters, time step and other important parameters), this was applied to other cases to verify its general nature. In order to have a more detailed insight into these results and the possible uncertainties involved, a basic statistical analysis was carried out, as mentioned in Section 2. Results are shown in Table 6. In particular, mean values of pressure peaks from experiments and calculations are reported, together with the corresponding discrepancies. The error is rather low, as can be seen, with the maximum deviation of 15% for RANSE-BV results. Maximum, minimum and RMS values were also obtained and they are shown in Table 6 as a percentage of the mean, in order to provide a general impression of the oscillating nature of predictions and measurements. These basic statistics are in line with the comments made earlier on the higher instability of results from SPH and RANSE-PRI compared with those obtained from RANSE-BV and RANSE-UoS, which show behaviour more similar to that observed in the experiments. Pressure results obtained for sensor 6 are shown in Figure 10. In this case, pressure values are lower by an order of magnitude compared with those for sensor 1, representing more sensor wetting, hence a less critical point, rather than real impact phenomena. Keeping this in mind, only RANSE-BV and RANSE-UoS methods are able to predict values close to the experimental pressure peaks. SPH overestimates the pressure peaks and, in some cases, results in anomalous negative values, which are also present for RANSE-UoS. The pressure time history, predicted by all methods, is different from the experimental method, with a faster pressure decay than measured. Predictions by

21 34 S. Brizzolara et al. Figure 22. Test case 3: level B, T = T 0, sensor 6 pressure time history (a) first few oscillations and (b) t = 4 12 s. (This figure is available in colour online.) RANSE-PRI appear to be only capable of signalling the sensor wetting occurrences after the first 10 s of calculations, with the presence of a repetitive phenomenon having the same period as the experimental peaks. However, pressure peaks are, by and large, overestimated and anomalous negative pressure values are predicted. The global torque time history is shown in Figure 11. Only results after the first period are reported, cutting the initial transient phase and focusing on the stabilised values. This allows a better comparison with experimental results which, as already mentioned, are referred only to the first harmonic of the oscillating torque time history.

22 Ships and Offshore Structures 35 Figure 23. Test case 3: level B, T = T 0, sensor 6 pressure time history for t = s. (This figure is available in colour online.) Predictions using SPH correspond, as mentioned before, to the best set-up obtained from the preliminary analysis. It should be noted, however, that once the global torque is considered, instead of local pressures, differences between various set-ups are reduced, resulting in similar predictions for a wide range of settings of artificial viscosity terms. This is due to the fact that pressure integration needed to obtain the resultant torque acts as a low-pass filter, smoothing pressure oscillations and cutting the effect of localised pressure peaks. There are still some differences with Figure 24. Test case 3: level B, T = T 0, sensor 3 pressure time history up to 30 s. (This figure is available in colour online.)

23 36 S. Brizzolara et al. Figure 25. Test case 2: level B, T = T 0, global torque time history up to t = 15 s. (This figure is available in colour online.) reference to torque peaks. For example, the peak torque values obtained using the lowest α and β values in artificial viscosity are approximately 10 15% higher than those obtained with high α and β. The following comments summarise the comparison between predictions and measurements. SPH tends to underestimate by about 10% the effective oscillating torque amplitude. RANSE-PRI results are more consistent with experimental measurements even though, in general, they fluctuate more than other predictions, and, in the central part of the simulation, they appear to suffer from an anomalous behaviour which could be associated with a sort of beating phenomenon, not recorded in the experiments. RANSE- UoS and RANSE-BV results provide the best agreement with experiments, with a lower mean error and a higher stability. Finally, examples of kinematic capturing of the sloshing phenomenon are presented: in Figure 12 for SPH, in Figure 13 for RANSE-PRI, in Figure 14 for RANSE-BV, in Figure 15 for RANSE-UoS and in Figure 16 for the experiments. These figures show good qualitative capturing of sloshing impact phenomena and overturning waves Test case 2: water level B, T = 0.9 T 0 Predicted and measured pressure time histories for sensor 6 are shown in Figures 17 (up to 16 s) and 18 (10 40 s). This case, as mentioned earlier, is considerably different from the other cases, since strong impacts are not expected, just flow oscillations. The interest in this case is due to the presence of a beating-type phenomenon, with pressure peaks of oscillating amplitude. All methods applied capture the beating phenomenon qualitatively, even though most of the numerical codes do not allow for simulating the almost complete quiescence of fluid with pressures near to zero. Only RANSE-PRI predictions allow for obtaining the low pressures corresponding to the beating, although they appear to contain more instabilities than the other methods. Regarding pressure peaks, RANSE-UoS calculations overestimate them in the whole time history, whilst the other methods seem to capture them with a higher accuracy. For example, predictions using SPH and LS-DYNA (RANSE-PRI) capture correctly maximum pressures of the first beating, whilst Flow-3D (RANSE-BV) overestimate them. SPH and RANSE-BV capture best the shape of the pressure oscillations, with a sort of secondary peak during the excitation phase and a more bell-shaped curve in the damping phase. Two anomalous pressure peaks were encountered during calculations when using SPH, showing a degree of instability. RANSE-BV calculations appear to be more robust. Examining the extended simulations, shown in Figure 18, RANSE-PRI calculations continue to capture the beating amplitude (though with higher peaks in the excitation phase and lower peaks in the damping phase ), whilst

24 Ships and Offshore Structures 37 Figure 26. Kinematics capturing with SPH : impact on the tank top. (This figure is available in colour online.)

25 38 S. Brizzolara et al. Figure 27. Kinematics capturing with RANSE-PRI: impact on the tank top. (This figure is available in colour online.) other methods tend to obtain a smoother phenomenon with lower oscillations due to a possible overdamping and, in general, with overestimated pressures. Timing of the beating phenomenon is not correctly captured by RANSE-PRI and RANSE-UoS, showing an apparent shift with increasing calculation time. RANSE-BV captures it correctly. SPH calculations have not been extended due to long computational times. The predicted and measured global torque time histories, evaluated using SPH and RANSE-BV, are shown in Figure 19. In this case, a comparison with experimental results is more difficult than in case 1. This is due to the fact that experimental data are provided in terms of the first harmonic of torque oscillations, which in this case involving the beating phenomenon does not describe satisfactorily the complete time history. It has already been seen that all methods are able to capture the beating phenomenon mentioned in the pressure time histories. Torque values predicted by SPH show the same qualitative trends for the beating in the first period, although, as already noted when discussing the pressure results, the torque does not vanish during the quiescent part. For the second beating, the SPH predictions appear to be more damped, consistent with predicted pressure results. This increase in damping may be related to numerical integration problems. Predictions by RANSE-BV show higher values of global torque, again consistent with predicted pressure time histories, and a lower damping in second beating. Figure 28. Kinematics capturing with RANSE-BV: impact on the tank top. (This figure is available in colour online.)

26 Ships and Offshore Structures 39 Figure 29. Kinematics capturing with RANSE-UoS: impact on the tank top. (This figure is available in colour online.) Finally, examples of kinematics capturing are shown in Figure 20, using SPH, and Figure 21 using RANSE-PRI. The maximum and minimum fluid oscillation peaks during the pseudo-period are of particular interest in these figures. Maximum oscillations are similar for both SPH and RANSE-PRI. On the other hand, the minimum oscillations are not completely captured by SPH, and RANSE-PRI captures an almost horizontal surface, consistent with predicted pressures Test case 3: water level B, T = T 0 While in the previous test case (case 2), very violent impact phenomena were not recorded during experiments, in this case impacts were recorded by sensor 3, at the top of the tank. Pressure peaks were also recorded by sensor 6 situated at the calm water level, even though smoother flow motions were observed at this location. Predicted and measured pressures at sensor 6 are shown in Figure 22 (up to 12 s) and Figure 23 (between 10 and 30 s). The pressures for sensor 3 are shown in Figure 24. In general, pressure peak magnitudes for sensor 6 have been captured by all methods applied. On the other hand, there are problems associated with the predictions for sensor 3, similar to that observed for sensor 6 in case 1. The mean experimental pressure peak value for sensor 6 is approximately 1300 Pa. All methods tend to overestimate it, in particular predictions by Flow-3D (RANSE-BV), SPH and RANSE-UoS are approximately 25 30% higher. LS-DYNA (RANSE-PRI) predictions show a more oscillatory behaviour, with maximum values approximately 90% higher and minimum values 30% lower than measurements, though the trend is captured satisfactorily. Examining pressure time history, SPH and RANSE-BV results appear to better reproduce the secondary pressure peak, whilst predictions by RANSE-UoS have a larger second peak, and RANSE-PRI results lie in between. It should be noted that RANSE-PRI and RASNE-BV simulations were run for an extended time, showing relative good stability considering the high surface deformations and nonlinearities involved in this case. As regards sensor 3, different considerations have to be made. All methods are capable of capturing the sloshing impact on the tank top. The experimental pressures range between 500 and 1000 Pa, excluding the two initial higher peaks. All numerical methods have a rather variable behaviour with reference to predicted pressures. RANSE-BV predictions contain oscillating values with higher peaks up to 1500 Pa and lower values with almost no impact. Predictions by RANSE-PRI and SPH show a similar trend and even higher peaks in some cases (2000 and 2500 Pa, respectively). RANSE-UoS predictions also follow a similar trend, but show a decreasing tendency with increasing time. Furthermore, both RANSE-PRI and RANSE-BV simulations have a quiet period between t = 15 and 20 s, during which pressure peaks are very low. Experimental pressure peaks are effectively lower in the time range mentioned earlier, though not as low as those obtained from calculations. SPH results are not available above 12 s. RANSE-UoS predictions appear to capture the pressure decrease, even though data are not available to confirm the possible pressure rises after t = 17 s. It is worth emphasising that experimental results in this case should be interpreted only as a measure of impact occurrence, as the absolute values can be affected by some problem due to the very high frequency of impacts. The statistical analysis of predicted and measured pressures for sensors 6 and 3 is shown in Tables 7 and 8, Table 7. Summary of results for case 3: level B, T = T 0,sensor6. Experiments SPH RANSE-BV RANSE-UoS RANSE-PRI Mean (Pa) Diff (%) RMS (%) max (%) min (%)

27 40 S. Brizzolara et al. Figure 30. Kinematics during experiments: sloshing impacts at the two sides. (This figure is available in colour online.) respectively. The data contained in these tables tend to confirm the above comments. In particular, it can be seen that pressure peaks for sensor 6 tend to be overestimated, with higher oscillations for RANSE-PRI and lower oscillations for SPH, RANSE-BV and RANSE-UoS. The mean pressure value for sensor 3 is captured well by RANSE-UoS and RANSE-PRI. In this case, as already mentioned, high oscillations are present in all calculations, and, to a lesser extent, the experimental data. This is underlined by the high RMS value and by the large difference between the maximum and minimum values, shown in Table 8. RANSE-UoS results appear to be more stable, with RMS values similar to those from the experiment, though as already pointed out, the peaks have a decreasing tendency, as if progressive damping were occurring. Predicted and measured global torque time histories, using SPH, RANSE-BV and RANSE-PRI, are shown in Figure 25. Global torque calculation using either SPH or RANSE-BV are in good agreement with experimental results, with a similar harmonic behaviour, but higher superimposed peaks (approximately 35% higher). On the other hand, predictions using RANSE-PRI show a general overestimation of peaks, together with higher oscillations. Examples of kinematics capturing are shown in Figure 26, using SPH, Figure 27, using RANSE-PRI, Figure 28, using RANSE-BV, Figure 29, using RANSE- UoS, and Figure 30, obtained from the experiments. These figures demonstrate the capability of each code to capture the impact with the tank top, showing rather good agreement between all prediction methods used. 5. Conclusions An evaluation of different numerical methods for the simulation of sloshing was presented in this paper, through comparisons with experimental measurements provided by CEHINAV-ETSIN-UPM for three significantly different test cases. In particular, efforts focussed on pressures predicted using SPH, an in-house RANS code (RANSE-UoS) and commercial software LS-DYNA (RANSE-PRI) and FLOW-3D (RANSE-BV). In addition, comparisons were made to assess global torque calculation and kinematics capturing capability of the codes used. This work, together with a second work on three further cases with sway motion instead of rolling motion (Viviani et al. 2010), allowed to assess the capabilities of the different methods investigated, considering both Lagrangian and Eulerian approaches, commercial and own developed codes. Results of this analysis are considered a significant achievement, taking into account that not many papers with this particular approach exist (see for example Cariou and Casella 1999 and more recently Kim et al and other papers presented at ISOPE 2009 sloshing minisymposium benchmark). Particular attention has been devoted to assess the possibility of obtaining accurate enough results on rather Table 8. Summary of results for case 3: level B, T = T 0,sensor3. Experiments SPH RANSE-BV RANSE-UoS RANSE-PRI Mean (Pa) Diff (%) RMS (%) max (%) min (%)